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+ 
+RECOMMED: A Comprehensive Pharmaceutical 
+Recommendation System 
+ 
+Mariam Zomorodi1,*, Ismail Ghodsollahee2, Pawel Plawiak1,3, U. Rajendra Acharya4, 5, 6 
+1 Department of Computer Science, Faculty of Computer Science and Telecommunications, Cracow University of 
+Technology, Krakow, Poland 
+2 Department of Computer Engineering, Ferdowsi University of Mashhad 
+3 Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Gliwice, Poland 
+4 Department of ECE, Ngee Ann Polytechnic, 535 Clementi Road, Singapore 599 489, Singapore 
+5 Department of Biomedical Engineering, School of Science and Technology, SUSS University, Singapore 
+6 Department of Biomedical Informatics and Medical Engineering, Asia University, Taichung, Taiwan 
+ 
+Objectives: To extract datasets containing useful information from two drug databases and 
+recommend a list of drugs to physicians and patients with high accuracy while considering 
+the most important features of patients and drugs. The history and review of the target 
+patient and similar patients, and drug information, are used as a reference to recommend 
+drugs. 
+Methods: A comprehensive pharmaceutical recommendation system was designed based 
+on the patients’ and drugs’ features extracted from Drugs.com and Druglib.com. First, data 
+from these databases were combined, and a dataset of patients and drug information was 
+built. Secondly, the patients and drugs were clustered, and then the recommendation was 
+performed using different ratings provided by patients, and importantly by the knowledge 
+obtained from patients and drug specifications, and considering drug interactions. To the 
+best of our knowledge, we are the first group to consider patients’ conditions and history 
+in the proposed approach for selecting a specific medicine appropriate for that particular 
+user. Our approach applies artificial intelligence (AI) models for the implementation. 
+Sentiment analysis using natural language processing approaches is employed in pre-
+
+processing along with neural network-based methods and recommender system algorithms 
+for modeling the system. In our work, patients’ conditions and drugs’ features are used for 
+making two models based on matrix factorization. Then we used drug interaction to filter 
+drugs with severe or mild interactions with other drugs. 
+We developed a deep learning model for recommending drugs by using data from 2304 
+patients as a training set, and then we used data from 660 patients as our validation set. 
+After that, we used knowledge from critical information about drugs and combined the 
+outcome of the model into a knowledge-based system with the rules obtained from 
+constraints on taking medicine.  
+Results: The results show that our recommendation system is able to recommend the best 
+combination of medicines according to the existing real-life prescriptions available. It also 
+has the best accuracy and other metrics in recommending a specific drug compared to other 
+existing approaches, which are generally based only on patient ratings or comments. Our 
+proposed model improves the accuracy, sensitivity, and hit rate by 26%, 34%, and 40%, 
+respectively, compared with conventional matrix factorization. In addition, it improves the 
+accuracy, sensitivity, and hit rate by an average of 31%, 29%, and 28% compared to other 
+machine learning methods. We have also open-sourced our implementation in Python. 
+Conclusions: Our proposed RECOMMED system extracts all vital information from the 
+drug,  patient databases and considers  all necessary factors for recommending accurate 
+medicine which can be trusted more by doctors and patients. We have shown the efficacy 
+of our proposed model in real test cases.   
+Keywords: recommendation system; drug recommendation system; drug information 
+extraction; hybrid recommendation method 
+ 
+1- INTRODUCTION 
+Recommendation systems (RS) are knowledge extraction systems that use information retrieval 
+approaches to help people make better decisions and discover items through a complex information 
+space [1], [2]. They have been around for many years, and with the advancement in machine 
+
+learning approaches, their use has been widened, and it helps people to make more appropriate 
+decisions in using different products. The popularity of using RS in different fields has increased 
+since the announcement of the Netflix Prize competition that aimed to predict movie rates [3]. The 
+application of recommender systems is very extensive: from entertainment to e-commerce, the 
+tourism industry, and medical recommender systems. Also, with rapid progress in artificial 
+intelligence, there has been a greater acceleration in the application of recommender systems and 
+their development. 
+Medical recommender systems are a particular type of recommender system, and they have some 
+distinct features that make them special: They have to be used very carefully, and because they 
+affect people’s health, there are many concerns about using RS for them. On the other hand, many 
+people die every year because of medication errors. It has been reported as the third leading cause 
+of death in the world [4]. This makes the use of intelligent systems in medical science valuable 
+and necessary. Drug prescription is also vital for physicians, and it involves considering different 
+aspects. Patient history of using drugs, the specification of drugs for diseases related to the 
+recommendation in question, and the drug's effectiveness for that specific case are among such 
+concerns.  
+Having as many different medicines as 24000 [5] in just one database, they can benefit from a 
+recommendation system to perform a set of suggestions for a particular patient with a specific 
+disease to help physicians in prescribing the most appropriate medicines and also help patients to 
+have a better choice in using drugs. 
+Recommender systems can be distinguished by the degree of risk imposed when a user accepts a 
+recommendation [6]. In this regard, the medical domain can be seen as high risk, mostly due to the 
+recommendation given to the user. 
+On the other hand, while having a comprehensive drug recommender system is important, 
+designing a complete system requires a dataset of drugs with patient ratings, reviews, and also 
+information about drugs. 
+We gathered this information from two different and well-known databases Drugs.com [5] and 
+Druglib.com [7] and we built three datasets which train the system and construct the final model. 
+Finally, putting all of them together, we proposed a novel drug recommender system called 
+
+RECOMMED that learns the patient and drug features and their previous drugs taken, and also the 
+user reviews for different drugs to recommend a new drug to a patient.  
+The novelty of this work lies in the following parts: 
+1- Propose a pharmaceutical recommender system by considering the features of patients and 
+drugs, including patients’ conditions, age, gender, drug side effects, and drug categories. 
+2- Performing pre-processing steps on databases Druglib.com and Drugs.com websites to 
+gather the appropriate data for our recommendation system, leading to comprehensive 
+datasets for drug information. 
+3- Our system considers sentiment analysis of reviews, the prescriptions of doctors, and 
+different similarity measures for recommending a medicine and its dose and other 
+recommendations, including side effects and warnings for their usage. 
+4- Our system consists of a knowledge-based component to exclude drugs with serious side 
+effects for a specific patient. 
+5- We proposed a model to predict the efficiency of medicine for patients. 
+ 
+In the next section, we provide some background in the general and general recommender systems 
+field; later in Section 3 we particularly introduce the drug recommender systems and their 
+challenges. Then in Section 4 we provide the current state-of-the-art of recommender systems 
+methods, specifically drug recommender systems. Section 5 is an elaborate explanation of our 
+proposed comprehensive drug recommender system in detail. Section 6 provides the results of this 
+work, plus a discussion about them. Finally, in Section 7 we conclude the paper and present the 
+future directions of this research. 
+ 
+2- RECOMMENDATION SYSTEMS BACKGROUND 
+Recommender systems are decision-making systems that extract information from different kinds 
+of knowledge. For many years, many recommender systems have been in various domains with 
+different purposes [8]. In this section, we review the basic concepts of various types of 
+recommender systems and the way they are categorized. 
+
+According to the type of data that recommender systems use to make decisions, the algorithms 
+utilized in recommender systems have two major categories: - collaborative filtering (CF) and -
+content-based filtering (CB). CF approaches are further divided into user-based and item-based 
+approaches. CF and CB approaches have shown acceptable results when they are used in 
+recommending different kinds of products like movies, books, and music. 
+Also, the recommendation system can combine these two major techniques, usually called hybrid 
+recommendations. In addition, there are some specific types of recommender systems that have 
+their strength in various domains. One of these types is a knowledge-based recommender system.  
+Here, we briefly introduce the major recommender system techniques considered in this work. 
+2-1 Collaborative recommender systems 
+The idea behind this group of recommendation methods is to use a measure of similarity between 
+users or items to recommend something to a given user. It states that if two users share some 
+interest in the past, they will likely have similar interests in the future. A collaborative approach is 
+based on the rating a user gives to items; in its basic form, it doesn’t need any other information 
+about users and items. CF approaches can be divided into two basic types, neighborhood methods 
+(also known as memory-based) and latent factor models. Neighborhood methods are divided into 
+one of the following two basic methods: 
+2-1-1 User-based neighborhood recommender system 
+This approach aims at suggesting the products based on the similarity between users. In this regard, 
+several similarity measures can be used.  
+We denote 𝒰 as the set of users, ℐ as the set of items, and 𝑅 as the set of existing ratings, 
+Pearson Correlation (PC) is one of the popular ones, which is computed as equation (1) for users 
+𝑢 and 𝑣 [9]: 
+𝑃𝑒𝑎𝑟𝑠𝑜𝑛_𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛(𝑢, 𝑣) =
+∑
+(𝑟𝑢𝑖−𝑟̅𝑢)(𝑟𝑣𝑖−𝑟̅𝑣)
+𝑖∈ℐ𝑢𝑣
+√∑
+(𝑟𝑢𝑖−𝑟̅𝑢)2
+𝑖∈ℐ𝑢𝑣
+√∑
+(𝑟𝑣𝑖−𝑟̅𝑣)2
+𝑖∈ℐ𝑢𝑣
+ 
+(1) 
+In this equation, ℐ𝑢 is the set of items rated by user 𝑢 and ℐ𝑢𝑣 is the items rated by both 𝑢 and 𝑣.  
+Also, 𝑟𝑢𝑖 is the rating of the user 𝑢 for a new item 𝑖 and 𝑟̅𝑢 is the average of the ratings given by 𝑢. 
+
+And the prediction for the rating of user 𝑢 for item 𝑖 is calculated as equation (2): 
+𝑝𝑟𝑒𝑑(𝑢, 𝑖) =
+∑
+𝑠𝑖𝑚(𝑢,𝑗)∗(𝑟𝑖,𝑗−𝑟̅𝑗)
+𝑗 ∈𝑢𝑠𝑒𝑟𝑠
+∑
+𝑠𝑖𝑚(𝑢,𝑗)
+𝑗 ∈𝑢𝑠𝑒𝑟𝑠
++ 𝑟̅𝑢 
+(2) 
+Where 𝑠𝑖𝑚 is the measure of similarity between user 𝑢 and item 𝑖 and 𝑢𝑠𝑒𝑟𝑠 is the set of users 
+most similar to user 𝑢. Therefore, the ratings are weighted by the similarity measure in this 
+prediction. 
+2-1-2 Item-based neighborhood recommender system 
+In contrast to user-based recommender systems, item-based recommender systems use item 
+similarity to suggest a product to a specific user. Similar to the user-based approach, here the 
+prediction for user 𝑢 for item 𝑖 is also calculated as equation (3): 
+𝑝𝑟𝑒𝑑(𝑢, 𝑖) =
+∑
+𝑠𝑖𝑚(𝑖,𝑗)∗(𝑟𝑢,𝑗−𝑟̅𝑗)
+𝑗 ∈𝑁
+∑
+𝑠𝑖𝑚(𝑖,𝑗)
+𝑗 ∈𝑁
++ 𝑟̅𝑖 
+(3) 
+Where 𝑠𝑖𝑚 is the measure of similarity between items 𝑖 and 𝑗 and 𝑁 is the set of items similar to 
+item 𝑖 rated by 𝑢. 
+2-1-3 Matrix factorization 
+One of the biggest challenges for standard methods in CF is the sparsity of the rating matrix (or 
+user-item matrix); model-based CF can help overcome this challenge. There are many ways to 
+build models based on which we can make recommendations. Matrix factorization is one the 
+popular methods with the idea of decomposition of a matrix into the product of two or maybe three 
+matrices.  
+Having a dataset of the ratings of various users for different items, this model transforms the rating 
+matrix to the individual user and item matrices. 
+So the model is defined as follows: 
+Assume there is a set of users 𝑈 and items 𝐷, with rating matrix 𝑅(𝑀 × 𝑁), which is the ratings 
+given by users on items. 𝑀 and 𝑁 are the total numbers of users and items, respectively. Matrix 
+factorization in recommender systems aims to find 𝑘 total latent features/factors by decomposing 
+𝑅 according to equation (4) to user matrix 𝑈 and item matrix 𝐼. 
+
+𝑅 ≈ 𝑈 × 𝐼𝑇 = 𝑅̂ 
+(4) 
+U is a 𝑀 × 𝑘 embedding matrix and, 
+I is a 𝑁 × 𝑘 embedding matrix. 
+2-2 Content-based recommender systems 
+Content-based recommendation uses the attributes of the users or user profile and the attributes of 
+items to recommend an item to a user [10]. Providing this information requires extra work and 
+effort to represent items properly and build a user profile appropriate for the recommendation 
+process [10]. 
+This kind of recommender system learns the user preferences and tries to recommend items similar 
+to the user's preferences. 
+Having 𝐷 rated items by user 𝑈, content-based RS aims to find the rating for item 𝑖, which is not 
+seen by user 𝑈. 
+In this method, items’ features are extracted, then used to find similarities between items. Then, in 
+a simple nearest neighbor approach top-𝑛 nearest neighbors of item 𝑖 in 𝐷 are selected. This 
+selection is based on a similarity measure like cosine similarity which is calculated as equation 
+(5): 
+cos(𝜃) =
+𝑿 .𝒀
+‖𝑿‖×‖𝒀‖ =
+∑
+𝑋𝑖𝑌𝑖
+𝑛
+𝑖=1
+√∑
+𝑋𝑖
+2
+𝑛
+𝑖=1
+√∑
+𝑌𝑖
+2
+𝑛
+𝑖=1
+    
+(5) 
+The ratings of these 𝑛 items are used to predict the rating for item 𝑖 by user 𝑈.  
+In most content-based recommender systems, item features are textual descriptions and don’t have 
+well-defined values. So, natural language processing approaches like TF-IDF or the bag-of-words 
+are used to assign numerical values to the textual features. 
+2-3 Hybrid recommender systems 
+Hybrid approaches combine different recommender system algorithms to make a more accurate 
+system that considers the benefits of different approaches for recommending an item to the users. 
+
+A combination of content-based and collaborating filtering is the most common type of 
+hybridization method [11]. 
+2-4 Knowledge-based recommender systems 
+This RS aims to produce recommendations based on existing rules that satisfy a user’s needs. In 
+the context of drug recommendation, this knowledge involves many different conditions. For 
+example, death reports for a specific drug and drug interactions are two important information that 
+a drug recommender system has to consider before recommending a list of medicine to a patient. 
+2-5 AI-based recommendation systems 
+Over time many different artificial intelligent approaches have been applied to recommendation 
+systems. However, the tendency to use AI methods in recommender systems is mostly because of 
+the big data availability and diversity of recommendation systems approaches, which can benefit 
+from AI, particularly machine learning algorithms. 
+Deep learning as a subfield of machine learning has attracted many researchers from a broad 
+variety of disciplines due to its learning capabilities from data. Recently there have been many 
+researches on deep learning-based recommendation systems [12], and Multilayer Perceptron 
+(MLP), Autoencoder (AE), Convolutional Neural Networks (CNN), Recurrent Neural Networks 
+(RNN), are among the mostly used deep learning models in RS [13]. Many of these deep learning-
+based approaches have contributed to the works on CB, CF, and other types of RS [14]. Also, some 
+works utilize hybrid deep networks, like the combination of RNN and CNN [15]. Moreover, to 
+integrate the advantages of memorization and generalization for recommender systems, a wide & 
+deep neural network has been used [16], and the model shows better results with increased 
+acquisitions on the Google Play app.  
+2-6 Other types of recommendation systems 
+Although these techniques are the basic and mostly used recommender system approaches, several 
+more types of recommender systems are suggested in the literature, and authors in [17] give a 
+detailed classification. 
+ 
+
+Medical and drug recommendation is one of the important applications of recommender systems 
+which uses techniques in recommender systems to recommend medicine, predict the usefulness of 
+drugs, etc. In the following sections, the position of recommender systems in medical science, 
+particularly in the pharmaceutical sector and the state-of-the-art in this field, is discussed. 
+ 
+3- RECOMMENDER SYSTEMS IN MEDICAL SCIENCE 
+ 
+One of the attractive and important applications of recommendation systems is medical 
+recommendations and drug products. 
+Here are the major differences between medical recommender systems and other recommender 
+systems: 
+- 
+Medical recommender systems care more about the health of patients than to make a profit. 
+- 
+Security is the primary goal in drug recommender systems.  
+- 
+Many existing recommender system techniques cannot be used, and others must be used 
+with caution because of safety issues. 
+- 
+In the long term, time is considered an important factor in recommending a drug. There are 
+many situations where some drugs' negative effects are discovered over time. One example 
+is the drug zimeldine [18]. So, a comprehensive medical recommender system should 
+consider ratings in different time stamps. 
+In the drug recommender system, the domain is medicine, and the exact contents to be 
+recommended are one or more of the following lists: 
+1. A list of drugs, at least one. 
+2. The dose, is the amount of drug taken at one time. 
+3. The frequency at which the drug doses are taken over time. 
+4. Duration, which is how long the drug is taken. 
+Numbers 2 to 4 in the above list are referred to as the dosage. Therefore, we can define a drug 
+recommender system as a smart system that is able to recommend a list of drugs plus their dosages 
+
+with high accuracy in terms of a real prescription of a physician and also to have a positive effect 
+on a patient, which available data can partially verify. 
+It should be noted that, of course, there is no medicine recommender system that we can trust 
+thoroughly, and like other artificial intelligence systems applied in healthcare, their use and ethical 
+issues must be addressed appropriately [19], [20]. 
+ 
+4- LITERATURE REVIEW 
+ 
+Medical recommender systems have been around for many years, even before the emergence of 
+recommender systems as a new field in computer science. According to [21], medicine 
+recommender systems fall into two broad categories named “ontology and rule-based 
+approaches” and “data mining and machine learning-based” approaches. Ontology-based 
+recommender systems use the hierarchical organization of users and items to improve the 
+recommendation [22].  
+Data mining and machine learning algorithms in the medical field are used to predict and 
+recommend things like drug usefulness, having a disease [23], [24], the condition of the user, or 
+ratings [25], [26]. For example, SVM, backpropagation neural network, and ID3 decision tree have 
+been used in [27] for recommending drugs. The performance of these approaches has been 
+compared in the above work, and the authors have shown that SVM has better accuracy and 
+efficiency compared to the other algorithms. Their data set contains patients’ features age, sex, 
+blood pressure, cholesterol, Na and K levels, and drug.  
+Some other researchers, while described as medical or medicine recommender systems, consider 
+a detection and classification task where the dataset which is trained has some patient attributes, 
+and based on that, the objective of the work is the detection or prediction of a disease and then for 
+each disease a set of medicines is recommended [27]. 
+Sentiment analysis of drug reviews is one of the basic approaches for drug recommendations [28], 
+[29], [30], [31], [32]. The sentiment analysis in these works mainly aims to recommend a drug or 
+extract useful information like adverse drug reactions. 
+
+In [31], different deep learning approaches, such as CNN, LSTM, and BERT, have been 
+investigated for sentiment analysis of patients’ drug reviews. In another work, the combination of 
+CNN-RNN has been applied  
+In addition to recommendation systems, sentiment analysis and opinion mining of drug reviews is 
+an active research area in drug review processing [33]. This analysis can be used for automatic 
+opinion mining and recommending drugs. 
+A hybrid knowledge-based medical prescription approach has been presented in [34]. The authors 
+use historical medical prescriptions to recommend a list of medicines to physicians. The approach uses the 
+similarity between cases where a case is medical information like demography, treatment, age, sex, 
+symptoms, and diagnosis. Based on the degree of similarity, a drug list is produced. The list is 
+complemented by Bayesian reasoning, where a model of the conditional probability of drugs is built. This 
+approach has been applied in Humphrey & Partners Medical Services Limited medical center. 
+Some works in medical recommendation have focused on particular drugs like diabetes [35]. Their 
+model is based on the ontology of medical knowledge and a decision decision-making approach 
+for multiple criteria and computes the medication. Then by using the entropy, the information 
+about patient’ history has been computed, and finally, the most appropriate medications have been 
+recommended to the physicians. 
+Many recommender system approaches have not been well considered in the medical and 
+pharmaceutical recommendations. However, using polarity in sentiment analysis of user 
+comments is one of the important parts of using NLP in recommendation systems. It can be viewed 
+as determining whether a word or phrase in the document or even a whole document is positive, 
+negative, or neutral in general.  
+Figure 1 shows the broad classification of different recommendation system approaches in 
+pharmaceutical research. We can see a growing tendency to use machine learning approaches in 
+this field. 
+ 
+
+ 
+Figure 1: Broad classification of recommendation system techniques. 
+ 
+5- Material and methods 
+In this section, we formulate the medicine recommender system problem and present our approach 
+for the general medicine recommender system. 
+Many recommendation systems, like collaborative filtering and content-based approaches, mostly 
+rely on past information to make decisions for the current situation. It is not always the case in the 
+domain of drug recommendation. The patient condition is different compared to the other patients 
+and compared to the same patient over time. So, in addition to the history information like general 
+rates, reviews, and the effective rate of the drug, it is necessary to use the patient's current condition 
+to make a more accurate decision. We also cannot rely on diversity-based recommendations as it 
+is used in some recommender systems, like the one used in Netflix, even if the drug is not rated 
+high, it can be suitable for some patients.  
+On the other hand, many recommender systems rely on knowledge from users; when there is a 
+lack of users’ knowledge, we cannot personalize them. While we have an adequate dataset for our 
+recommendation task, the problem emerges when new inputs enter the system. In our medicine 
+Drug 
+Recommendation 
+Systems
+Collaborative Filtering
+Memory Based
+User-Based Filtering
+WaveLet [36]
+Item-Based Filtering
+Model Based
+Matrix Factorization
+SVD
+SVD++
+Content Based
+Machine Learning
+T-Recs [37]
+Knowledge-Based
+Machine Learning
+Data mining Framework 
+[38]
+Ontology-based
+MCDM & Entropy [35]
+SWRL [39] , [40], [41]
+GalenOWL [42]
+Panacea [43]
+SemMed [44]
+Hybrid
+CB & Knowledge Based
+Machine Learning
+LOD cloud mining [45]
+User CF & Knowledge 
+Based
+Machine Learning
+DiaTrack [46]
+Personalized Clinical 
+Prescription [47]
+CADRE [48]
+Item Based CF & 
+Knowledge Based
+Machine Learning
+Data Driven[49]
+
+recommender system, these inputs can be new patients or new drugs. Cold start problem is a term 
+used for this problem, and it is a challenging issue in designing any recommender system. We 
+reduced this effect by applying a clustering-based approach. Because drugs are clustered into a 
+specific category, we can put a new drug in the category which belongs to it, so we use the same 
+rating for the new drugs as those in that category. This is effective in solving the cold start problem 
+in our recommender system.  
+Proposed method 
+Since every recommendation technique has its own benefits, a universal recommender system 
+should be able to take advantage of all of these techniques to improve the outcome of a 
+recommender system. The drug recommendation system in our work has the benefits of different 
+recommendation categories and combines their advantages by using several steps. First, natural 
+language processing and machine learning algorithms are applied in the context of basic 
+recommender system techniques.  
+This section discusses all phases of our model for building a comprehensive drug recommender 
+system. 
+This paper presents a novel hybrid drug recommender system (RS) with features of several 
+recommender systems. It uses natural language processing (NLP) and other machine learning 
+techniques to implement the system. The proposed RS approach is a new recommendation system 
+method for pharmaceutical recommendation, which can be considered a hybrid of CB, CF model-
+based, knowledge-based, and AI-based methods. Here in this section, we elaborate on each step 
+toward the final drug recommendation for each patient. After a very intensive web crawling 
+through two well-known pharmaceutical websites, Drugs.com and Druglib.com, and building 
+three different datasets, feature extraction and modeling are performed. Then in the next step, 
+recommendations for proper drugs are performed. At the final stage, the list of drugs is refined 
+based on defined rules in addition to the ratings and drug features which is an important aspect of 
+our medicine recommender system. 
+ 
+
+ 
+Figure 2: Components of RECOMMED drug recommendation system in the training stage. 
+Figure 2 presents the whole RECOMMED model in the training stage of our work, consists of 
+four components, and we elaborate on each phase of our approach in more detail in the following 
+parts: 
+5-1 Dataset extraction 
+In this work, any recommendation for drugs and their dosage is based on the patients’ features like 
+age, gender, previous illness, and other drugs they consume, and drugs’ features like drug 
+classification, side effects, and drug interactions. So, in this phase, the extraction of user features, 
+drug features, and drug interaction datasets from Drugs.com  and Druglib.com databases is 
+accomplished. In the second step of this phase, the dataset is prepared for clustering and modelling 
+the recommendation system. The review field in the drug recommendation database contains users' 
+and caregivers opinions about drugs' effectiveness. According to our knowledge, none of the 
+existing datasets have complete and comprehensive patient and drug information. 
+ 
+  
+Start
+ 
+ 
+Remove HTML Frames, 
+ 
+Tags and advertisement
+ 
+End
+ 
+  
+Crawl Review Webpages
+ 
+Dataset 
+Extraction 
+Modeling 
+User and Drug 
+Feature Set 
+  
+Keyword Search
+ 
+Drug 
+Interation Set 
+Pre-
+processing 
+  
+Feature Extraction 
+ 
+Normalized 
+User and Drug 
+Feature Set 
+  
+Combining User Comment
+ 
+Rates And Effectivness
+ 
+  
+Normalize Feature Sets
+ 
+User Clusters 
+Feature Set 
+User Rating 
+Matrix 
+Drug Clusters 
+Feature Set 
+Drug Rating 
+Matrix 
+  
+Clustering Users
+ 
+  
+Clustering Drugs
+ 
+  
+Generating 
+ 
+User Rating
+ 
+  
+Generating 
+ 
+Drug Rating
+ 
+Knowledge-
+based 
+User Weights  
+& Biases Sets 
+Drug Weights 
+& Biases Sets 
+  
+Initialize Weights & Biases
+ 
+  
+Forward Propagation
+ 
+  
+Backward Propagation
+ 
+  
+Update Weights & Biases
+ 
+Error < Threshold
+ 
+  
+New User Registration
+ 
+  
+Compute 10 High Rated 
+Drug for Recommendation
+ 
+Drug[i]
+ 
+ Interaction?
+ 
+  i=0
+ 
+i<10
+ 
+  i=i+1
+ 
+User Features  
+Filter Set 
+Drug[i] 
+ 
+Allowed?
+ 
+  
+ Recommendation
+ 
+append(Drug[i])
+ 
+
+We built three different datasets named users, drugs, and interactions. 
+5.1.1 Drugs and users datasets 
+In this work, Druglib.com and Drugs.com were employed to extract information about patients 
+and drugs and build two datasets named drugs and patients. We should mention that there are also 
+other databases for drug information and recommendations, like SIDER [50], for drug side effects. 
+We will include them in future works to build a complete dataset for drugs. Three features 
+consisting of side effects, benefits, and membership in a given drug category were considered for 
+drugs. 
+First, different drug categories and side effects were extracted in tables 1 and 2. There are 150 
+different drug categories, and 128 different side effects were extracted from the Druglib.com 
+database. 
+TABLE   1 - DRUG CATEGORIES LIST 
+Category 
+Index 
+Acetylcholine-Agonists 
+1 
+Adrenergic-Alpha-Agonists 
+2 
+ 
+… 
+Vasodilators 
+150 
+ 
+TABLE   2 -DRUGS SIDE EFFECTS 
+Side Effects 
+Index 
+Completed suicide 
+1 
+Confusional state 
+2 
+ 
+… 
+Wrong drug administered 
+128 
+ 
+Then drug benefits were also extracted and combined with the information in the above tables, and 
+finally, the drugs dataset was prepared, as is partially shown in Table 3. 
+TABLE 3- DRUGS DATASET 
+Benefits 
+Side Effects 
+Drug Category 
+Drug Name 
+Index 
+88 
+… 
+2 
+1 
+128 
+… 
+2 
+1 
+150 
+… 
+2  
+1 
+0 
+… 
+1 
+1 
+0 
+… 
+0 
+0 
+0 
+… 
+0 
+1 
+Hytrin Terazosin 
+1 
+0 
+… 
+0 
+0 
+0 
+… 
+1 
+1 
+0 
+… 
+0 
+1 
+Mirtazapine 
+2 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+0 
+… 
+0 
+0 
+0 
+… 
+0 
+0 
+1 
+… 
+0 
+0 
+Proscar Finasteride 
+480 
+
+We also extracted the users dataset of patient features and comments on different drugs. Six 
+features are considered for users datasets: age, gender, current disease (condition), other 
+conditions, other drugs are taken, and user level, which is patient or caregiver.  
+Table 4 represents the structure of this dataset. 
+TABLE3- USERS DATASET 
+Comment 
+Side 
+Effects 
+Effective
+ness 
+Overall 
+Rating 
+Drug 
+Name 
+Other 
+Drug 
+Other 
+Conditio
+n 
+Conditio
+n 
+Genus 
+Age 
+Level 
+index 
+… 
+Severe Side 
+Effects 
+Ineffective
+ 
+1 
+Mirtazapine
+ 
+None
+ 
+Sleeplessness
+ 
+Depression
+ 
+Male 
+22 
+Patient
+ 
+1 
+… 
+Moderate 
+Side Effects 
+Ineffective
+ 
+2 
+Mirtazapine
+ 
+None
+ 
+None
+ 
+Depression
+ 
+Male 
+38 
+Patient
+ 
+2 
+ ... 
+ ... 
+ ... 
+ ... 
+ ... 
+ ... 
+ ... 
+ ... 
+ ... 
+ ... 
+ ... 
+… 
+… 
+Mild Side 
+Effect 
+Moderately 
+Effective
+ 
+4 
+Proscar 
+Finasteride 
+ 
+None
+ 
+None
+ 
+Hair loss
+ 
+Male
+ 
+ 
+28 
+Patient 
+3294 
+ 
+5.1.2 Interactions dataset 
+The last dataset prepared in this work is the interactions dataset. This information is important for 
+recommending the appropriate medicine list to the patients. We extracted drug interaction 
+information from Drugs.com, and after mapping drugs’ names with their counterparts in 
+Druglib.com, the interaction dataset, partially presented in Table 5, was created with 180 drug 
+interaction information. 
+TABLE   4 -DRUG INTERACTION DATASET 
+ 
+Abilifish
+ 
+…
+ 
+Cimbaita
+ 
+…
+ 
+Syntroid
+ 
+…
+ 
+zyban
+ 
+Abilifish
+ 
+- 
+- 
+Moderate
+ 
+…
+ 
+- 
+…
+ 
+- 
+Accupril
+ 
+- 
+- 
+- 
+…
+ 
+Moderate
+ 
+…
+ 
+- 
+Aciphex
+ 
+- 
+- 
+Major
+ 
+…
+ 
+- 
+…
+ 
+- 
+…
+ 
+…
+ 
+…
+ 
+…
+ 
+…
+ 
+…
+ 
+…
+ 
+…
+ 
+Zyban
+ 
+- 
+- 
+- 
+…
+ 
+- 
+…
+ 
+- 
+
+5.2 Dataset preparation 
+ 
+In this phase, our dataset is prepared for creating the recommendation model in the next step. First, 
+using Natural Language Processing (NLP) techniques, user and drug features are extracted, and 
+then normalization and clustering are accomplished to prepare the datasets for modeling the 
+recommendation system. Here, we elaborate on each of these steps: 
+ 
+5.2.1 Feature extraction 
+The first pre-processing step is feature extraction from user feature and drug features datasets. 
+Bag-Of-Words (BOW) method is used for this purpose. 
+NLP for extracting drug and user features 
+The feature extraction was mainly performed using natural language processing (NLP) techniques. 
+Two well-known methods to extract text features by NLP are Bag-of-Words (BOW) and term 
+frequency-inverse term frequency (TF-IDF).  
+Our proposed pharmaceutical recommendation system uses the BOW feature extraction method 
+to perform feature extraction from database texts. This method consists of four steps: 
+• Text-pre-processing pre-processing  
+• Vocabulary creation 
+• Building feature matrix 
+• Polarity of user comments 
+Here, every part of this process has been described: 
+Text-pre-processing pre-processing  
+In the text-  pre-processing step, all punctuations and symbols are removed, and abbreviations are 
+converted into their full names or phrases. Some of these conversions are presented in Table 6. 
+Moreover, spelling mistakes were corrected using the TexBlob library of Python, and stop words 
+were removed using a predefined list of stop words. 
+ 
+
+TABLE 6- EXAMPLES OF ABBREVIATIONS TO FULL NAME CONVERSIONS 
+Original Form
+ 
+Abbreviations
+ 
+high blood pressure
+ 
+HBP
+ 
+chronic obstructive pulmonary disease
+ 
+COPD
+ 
+premenstrual syndrome
+ 
+PMS
+ 
+obsessive-compulsive disorder
+ 
+OCD
+ 
+ 
+Vocabulary creation 
+Using NLP techniques, a vocabulary of words is created in the second step of feature extraction. 
+For this purpose, an array of words is created by checking all registered words in the dataset. This 
+array is constructed from unique words of the dataset and their frequency. To deal with the random 
+filling of the feature matrix, words are rearranged according to their frequency. Moreover, to deal 
+with the sparseness of the feature matrix, words with low frequency are removed. Some of the 
+most frequent words extracted from the datasets created and discussed in the previous section can 
+be seen in Table 7. 
+ 
+TABLE 7. EXAMPLES OF MOST FREQUENT WORDS IN DATASETS. 
+Frequency
+ 
+Term
+ 
+33
+ 
+Pain
+ 
+22
+ 
+Infection
+ 
+15
+ 
+Surgery
+ 
+13
+ 
+Chronic
+ 
+ 
+Building feature matrix 
+The feature matrix is created in the third step of extracting the features. For this purpose, a unique 
+word is assigned to each matrix column, and a new row is considered for each user review. Each 
+cell of this matrix represents the existence of the word in the user’s review, which is essentially 
+zero or one. 
+ 
+Polarity of user comments (PUC) 
+We used NLP and opinion mining to extract PUC. This approach aims at extracting the opinion of 
+users as a positive or negative comment. The output of this component is used in the users’ rating 
+matrix. 
+
+ 
+ALGORITHM1. COMMENT POLARITY ACQUISITION 
+Input:𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠, 𝑆𝑡𝑜𝑝𝑊𝑜𝑟𝑑𝑠 
+Output: 𝑃𝑜𝑙𝑎𝑟𝑖𝑡𝑦𝑂𝑓𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠 
+1. 𝑅𝑒𝑚𝑜𝑣𝑖𝑛𝑔 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 𝐿𝑒𝑡𝑡𝑒𝑟𝑠 𝑎𝑛𝑑 𝐸𝑚𝑜𝑗𝑖𝑠 from 𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠  
+2. 𝑅𝑒𝑚𝑜𝑣𝑖𝑛𝑔 𝑆𝑡𝑜𝑝𝑊𝑜𝑟𝑑𝑠 𝑓𝑟𝑜𝑚 𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠 
+3. 𝑊𝑜𝑟𝑑 𝑇𝑜𝑘𝑒𝑛𝑖𝑧𝑒 (𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) 
+4. 𝑊𝑜𝑟𝑑𝐿𝑒𝑚𝑎𝑡𝑖𝑎𝑡𝑖𝑜𝑛(𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) 
+5. 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑊𝑜𝑟𝑑𝑠(𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) 
+6. 𝑇𝑒𝑥𝑡𝐵𝑙𝑜𝑏(𝑈𝑠𝑒𝑟𝐶𝑜𝑚𝑚𝑒𝑛𝑡𝑠) 
+ 
+Combined User Rating Acquisition 
+To have a more accurate rating for drugs, we considered the combined user comments and ratings 
+from different sources. This overall rating is called Combined User Rating Acquisition (CUR) 
+parameter and is obtained from analyzing user comments and ratings as follows: 
+1. 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑅𝑎𝑡𝑖𝑛𝑔 ∈ 𝑍, 0 ≤ 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑅𝑎𝑡𝑖𝑛𝑔 ≤ 10 
+2. 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑛𝑒𝑠𝑠 ∈ 𝐸, 𝐸 = {Ineffective, Marginally Effective, Moderately Effective,
+Considerably Effective, Highly Effective} 
+3. 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡 ∈ 𝑆, 𝑆 = {𝑁𝑜 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝑀𝑖𝑙𝑑 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝑀𝑜𝑑𝑒𝑟𝑎𝑡𝑒 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡,
+𝑆𝑒𝑣𝑒𝑟𝑒 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡, 𝐸𝑥𝑡𝑒𝑟𝑒𝑚𝑙 𝑆𝑒𝑣𝑒𝑟𝑒 𝑆𝑖𝑑𝑒 𝐸𝑓𝑓𝑒𝑐𝑡} 
+4. User Comment 
+  
+CUR parameter is calculated as equation (6), and the above parameters are replaced by CUR in 
+the user feature matrix: 
+CUR =
+(𝑂𝑣𝑒𝑟𝑎𝑙𝑙𝑅𝑎𝑡𝑖𝑛𝑔
+10
++𝐷𝑂𝐸
+4
+)
+2
+− 𝐷𝑂𝑆
+4 +𝑃𝑈𝐶
+2
+      
+(6) 
+In equation (6), DOE (Degree of Effectiveness) represents the degree of drug effectiveness. The 
+user selects the effectiveness of a drug from a list of five different options: Ineffective, Marginally 
+Effective, Moderately Effective, Considerably Effective, and Highly Effective, and it takes a number 
+in the range [0-4]. Similarly, DOS (Degree of Side Effects) is the degree that a drug has a side 
+effect (range [0-4]), and the numbers applied in the denominator are for normalization purpose. 
+PUC (Polarity of User Comments) is calculated using Natural Language Processing (NLP), and 
+opinion mining techniques and the nltk library in Python are used in this regard. Algorithm 1 shows 
+the steps of the work for calculating PUC. 
+
+Normalization- After extracting features from drug and user datasets, these features should also 
+be normalized to perform better in training the model.  
+ 
+ 
+Combined User 
+Rating (CUR) 
+Drug Name 
+Other Drug 
+Other Condition 
+Condition 
+Genus 
+Age 
+Level 
+index 
+ 
+Comment 
+Side Effects (DOS) 
+Effectiveness (DOE) 
+Overall Rating 
+Drug Name 
+Other Drug 
+Other Condition 
+Condition 
+Genus 
+Age 
+Level 
+Index 
+0.05 
+Mirtazapine 
+None 
+Sleeplesness 
+depression 
+male 
+22 
+patient 
+1 
+ 
+0.8 
+(Obtained 
+from 
+PUC) 
+0.75 
+0 
+0.1 
+Mirtazapine 
+None 
+Sleeplesness 
+depression 
+male 
+22 
+patient 
+1 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+ 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+… 
+0.1 
+Proscar 
+None 
+None 
+Hair loss 
+male 
+28 
+patient 
+3294 
+ 
+0.2 
+0.25 
+0.5 
+0.4 
+Proscar 
+None 
+None 
+Hair loss 
+male 
+28 
+patient 
+3294 
+ 
+Figure 3- Combination of different user ratings for a given drug 
+ 
+Figure 3 is the final user rating dataset after applying the combined user rating acquisition stage. 
+This stage converts the dataset on the left side into the right side dataset. Each column in both 
+datasets has a given user’s features along with the drug name they rate. In the left side dataset, we 
+can see different ratings of the user, and then in the right side dataset, these ratings are combined 
+into CUR using equation (6). 
+ 
+5.3 Clustering 
+Clustering is considered one of the main steps in a recommender system for improving the 
+diversity, consistency, and reliability [51], which has been considered in many works in 
+recommender systems, particularly for reducing the sparsity of data [52], [53]. Due to the 
+sparseness of the rating matrix, we consider a clustering-based approach, and patients are clustered 
+
+before performing the matrix factorization, which is elaborated in the next part. This clustering is 
+mostly required because users usually review only one drug corresponding to a specific disease, 
+so the rating matrix is highly sparse. Clustering can help group the users and drugs with similar 
+features and significantly resolve the sparsity problem. Users are clustered based on their gender, 
+age, comments, and being patient or caregiver. It is clear that after clustering, each class of users 
+reviews several drugs, which can improve the matrix factorization process. 
+We used a modified K-means algorithm in [54] to perform this clustering. While the original K-
+means algorithm is unsupervised, which is used for clustering, the number of clusters is pre-
+determined, and so it couldn't be utilized in the same way in our proposed drug recommendation 
+system. Therefore, in this paper, we employed the U-Kmeans method [54]. This method performs 
+the unsupervised K-means and determines the best cluster numbers that lead to better classification 
+performance.  
+If each row of the dataset and the center of each cluster are represented by F= {𝑓1, … , 𝑓𝑛}  and A= 
+{𝑎1, … , 𝑎𝑘} respectively, the K-means objective function is defined as (7). 
+      
+𝐽(𝑀, 𝐴) = ∑
+∑
+𝑀𝑖𝑗‖𝑓𝑖 − 𝑎𝑗‖
+𝑘
+𝑗=1
+𝑛
+𝑖=1
+    
+(7) 
+ 
+ Where in (7), 𝑘 is the number of clusters, 𝑛 is the number of dataset features, and 𝑀𝑖𝑗 indicates 
+the membership of 𝐹𝑖 to the 𝑗𝑡ℎ cluster. In the K-means algorithm, this objective function must be 
+minimized. In [55], an entropy-based method is proposed to improve K-means. In this method, to 
+determine the centers of the clusters, Equation (8) is added to the objective function. 
+𝐵𝑛 ∑
+𝑎𝑗
+𝑘
+𝑗=1
+ln 𝑎𝑗   
+(8) 
+       In (8), the effect of the cluster imbalance is added to the objective function. As can be seen in 
+(9), when the  𝐵𝑛 coefficient of the improved objective function is zero. The following K-means 
+objective function is obtained. 
+ 
+𝐽(𝑀, 𝐴) = ∑
+∑
+𝑀𝑖𝑗‖𝑥𝑖 − 𝑎𝑗‖
+𝑘
+𝑗=1
+𝑛
+𝑖=1
+− 𝐵 ∑
+𝜂𝑗
+𝑘
+𝑗=1
+ln 𝜂𝑗 
+ 
+(9) 
+ 
+
+       Where in this equation, 𝜂𝑗 represents the number of members of a cluster, which is determined 
+by (10). 
+𝜂𝑗 =
+∑
+𝑀𝑖𝑗
+𝑛
+𝑖=1
+𝑥𝑖
+∑
+𝑀𝑖𝑗
+𝑛
+𝑖=1
+   
+(10) 
+ 
+       In [54], equation (11) is considered to determine the optimized number of clusters. By adding 
+this term to equation (10), the final objective function is obtained as (12). 
+L ∑
+∑
+𝑀𝑖𝑗
+𝑘
+𝑗=1
+ln 𝑎𝑗
+𝑛
+𝑖=1
+     
+(11) 
+𝐽(𝑀, 𝐴, 𝑎) = ∑
+∑
+𝑀𝑖𝑗‖𝑥𝑖 − 𝑎𝑗‖
+𝑘
+𝑗=1
+𝑛
+𝑖=1
+− 𝐵 ∑
+𝑎𝑗
+𝑘
+𝑗=1
+ln 𝑎𝑗 − L ∑
+∑
+𝑀𝑖𝑗
+𝑘
+𝑗=1
+ln 𝑎𝑗
+𝑛
+𝑖=1
+    
+(12) 
+      The pseudocode of the U-K-means classification method based on the approach in [54] is 
+presented in Algorithm 2. 
+Algorithm   .
+2  Our modified Pseudo code of U-Kmeans based on [54]. 
+1. 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑐(0) = 𝑛, 𝛼𝑘
+(0) =
+1
+𝑛 , 𝑎𝑘
+(0) = 𝑥𝑖  
+2. 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑙𝑒𝑎𝑟𝑛𝑖𝑛𝑔 𝑟𝑎𝑡𝑒𝑠 𝐿(0) = 𝐵(0) = 1  
+3. 𝑆𝑒𝑡 𝑡 = 0 , 𝜀 > 0 
+4. 𝑤ℎ𝑖𝑙𝑒 𝑚𝑎𝑥‖𝑎𝑘
+𝑡+1 − 𝑎𝑘
+𝑡 ‖ < 𝜀 
+5.       If ‖𝑥𝑖 − 𝛼𝑘‖2 − 𝐿𝑙𝑛𝛼𝑘 = min
+1≤𝑘≤𝑐‖𝑥𝑖 − 𝑎𝑘‖2 − 𝐿𝑙𝑛𝛼𝑘 
+6.             𝑀𝑖𝑘
+(𝑡+1) = 1 
+7.       Else 
+8.             𝑀𝑖𝑘
+(𝑡+1) = 0 
+9.       𝐿(𝑡+1) = 𝑒−𝑐(𝑡+1)/250 
+10.       𝛼𝑘
+(𝑡+1) = ∑
+𝑀𝑖𝑘
+𝑛 + (
+𝐵
+𝐿) 𝛼𝑘
+(𝑡) ln 𝑎𝑘
+𝑡 − ∑
+𝛼𝑠
+𝑡
+𝑐
+𝑠=1
+𝑛
+𝑖=1
+ln 𝑎𝑠
+𝑡 
+11.       𝐵𝑡+1 = 𝑚𝑖𝑛 (
+∑
+exp (−𝜂𝑛|𝑎𝑘
+𝑡+1−𝑎𝑘
+𝑡 |)
+𝑐
+𝑘=1
+𝑐
+,
+1− max
+1≤𝑘≤𝑐(1
+𝑛 ∑
+𝑀𝑖𝑘
+𝑛
+𝑖=1
+)
+− max
+1≤𝑘≤𝑐 𝑎𝑘
+𝑡 ∑
+ln 𝑎𝑘
+𝑡
+𝑐
+𝑘=1
+) 
+12.       𝑢𝑝𝑑𝑎𝑡𝑒 𝐶𝑡 𝑡𝑜 𝐶𝑡+1 𝑏𝑦 𝑑𝑖𝑠𝑐𝑎𝑟𝑑 𝑡ℎ𝑜𝑠𝑒 𝑐𝑙𝑢𝑠𝑡𝑒𝑟 𝑤𝑖𝑡ℎ 𝑎𝑘
+𝑡+1 ≤
+1
+𝑛 
+13.       𝑎𝑘
+∗ =
+𝑎𝑘
+∗
+∑
+𝑎𝑠∗
+𝑐(𝑡+1)
+𝑠=1
+ 
+14.       𝑀𝑖𝑘
+∗ =
+𝑀𝑖𝑘
+∗
+∑
+𝑀𝑖𝑠
+∗
+𝑐(𝑡+1)
+𝑠=1
+ 
+15.       𝑎𝑘 =
+∑
+𝑀𝑖𝑘𝑥𝑖𝑗
+𝑛
+𝑖=1
+∑
+𝑀𝑖𝑘
+𝑛
+𝑖=1
+ 
+16.       𝑖𝑓 𝑡 ≥ 60 𝑎𝑛𝑑 𝑐(𝑡−60) − 𝑐𝑡 = 0 
+17.             𝐵(𝑡+1) = 0 
+18.       t=t+1 
+ 
+ 
+ 
+
+TABLE 8: RATE MATRIX WITHOUT CLASSIFICATION 
+619
+ 
+618
+ 
+617
+ 
+616
+ 
+615
+ 
+…
+ 
+6
+ 
+5
+ 
+4
+ 
+3
+ 
+2
+ 
+1
+ 
+Drugs
+ 
+ 
+Users
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+1
+ 
+1
+ 
+3
+ 
+ 
+ 
+ 
+3
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+2
+ 
+ 
+ 
+ 
+5
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+3
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+…
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+1
+ 
+ 
+979
+ 
+ 
+3
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+980
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+3
+ 
+ 
+ 
+ 
+ 
+ 
+981
+ 
+ 
+TABLE  9: RATE MATRIX AFTER CLASSIFICATION 
+619
+ 
+618
+ 
+617
+ 
+616
+ 
+615
+ 
+…
+ 
+6
+ 
+5
+ 
+4
+ 
+3
+ 
+2
+ 
+1
+ 
+Drugs
+ 
+ 
+Users
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+1
+ 
+1
+ 
+3
+ 
+ 
+ 
+ 
+3
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+2
+ 
+ 
+ 
+ 
+5
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+3
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+…
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+1
+ 
+ 
+38
+ 
+ 
+3
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+39
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+3
+ 
+ 
+ 
+ 
+ 
+ 
+40
+ 
+ 
+ 
+5.4 Modeling 
+In the next step, the clustering outcome is used to build a recommender system model able to 
+recommend the best drugs. Later, we filter the model's output with a knowledge-based component 
+for safety reasons. 
+ 
+Neural Network-based Matrix Factorization 
+Matrix factorization is a popular method for recommender systems aiming at finding two 
+rectangular matrices called user and item matrices with smaller sizes than the rating matrix [56]. 
+The dot product between these two matrices results in the rating matrix.  
+
+To reduce the computational overhead, copeTo reduces the computational overhead, cope with the 
+sparsity of the ratings, and increase accuracy. We proposed a neural network-based matrix 
+factorization technique. The first two matrices, Rating and Effectiveness, are constructed by 
+extracting information from Druglib.com.  
+In our model, the rating matrix 𝑅𝑎𝑡𝑖𝑛𝑔 ∈ R𝑛∗𝑚 is estimated as the multiplication of two matrices 
+𝐶𝑙𝑢𝑠𝑡𝑒𝑟𝑠𝑛∗𝑘 and 𝐷𝑟𝑢𝑔𝑠𝑘∗𝑚 as (13): 
+𝑅𝑎𝑖𝑛𝑔 ≈ Clusters. 𝐷𝑟𝑢𝑔𝑠𝑇    (13) 
+ 
+This model applies a neural network algorithm to estimate the users’ comments for each medicine. 
+Clustered users and drugs and users’ and drugs’ features are used in building this new model, as 
+illustrated in Figure 4. 
+Figure 4- Our proposed customized matrix factorization method. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+𝑏𝑚 
+ 
+𝑏2 
+𝑏1 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Drug 
+Cluster k
+ 
+…
+ 
+Drug 
+Cluster 2
+ 
+Drug 
+Cluster 1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+Drug m
+ 
+…
+ 
+Drug 2
+ 
+Drug 1
+ 
+ 
+ 
+   
+ 
+𝑐1,𝑚 
+ 
+𝑐1,2 
+𝑐1,1 
+Drug 
+Feature 
+1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+𝑑1,𝑚 
+ 
+𝑑1,2 
+𝑑1,1 
+Latent 
+Factor 1
+ 
+ 
+ 
+𝑐2,𝑚 
+ 
+𝑐2,2 
+𝑐2,1 
+Drug 
+Feature 
+2
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+𝑑2,𝑚 
+ 
+𝑑2,2 
+𝑑2,1 
+Latent 
+Factor 2
+ 
+ 
+ 
+ 
+ 
+…
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+…
+ 
+𝑐𝑘,𝑚 
+ 
+𝑐𝑘,2 
+𝑐𝑘,1 
+Drug 
+Feature 
+k
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+𝑑𝑘,𝑚 
+ 
+𝑑𝑘,2 
+𝑑𝑘,1 
+Latent 
+Factor k
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+User 
+Feature k
+ 
+…
+ 
+ 
+User 
+Feature 2
+ 
+User 
+Feature 1
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+User 
+Feature k
+ 
+…
+ 
+User 
+Feature 2
+ 
+User 
+Feature 1
+ 
+ 
+𝑟1,𝑚 
+ 
+𝑟1,2 
+𝑟1,1 
+ 
+𝑢1,𝑘 
+ 
+ 
+𝑢1,2 
+𝑢1,1 
+User 
+1
+ 
+ 
+𝑏1 
+ 
+𝑟1,𝑚 
+ 
+𝑟1,2 
+𝑟1,1 
+ 
+𝑐1,𝑘 
+ 
+𝑐1,2 
+𝑐1,1 
+User 
+Cluster 
+1
+ 
+𝑟1,𝑚 
+ 
+𝑟2,2 
+𝑟2,1 
+ 
+𝑢2,𝑘 
+ 
+ 
+𝑢2,2 
+𝑢2,1 
+User 
+2
+ 
+ 
+𝑏1 
+ 
+𝑟1,𝑚 
+ 
+𝑟2,2 
+𝑟2,1 
+𝑐2,𝑘 
+ 
+𝑐2,2 
+𝑐2,1 
+User 
+Cluster 
+2
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+…
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+…
+ 
+𝑟𝑛,𝑚 
+ 
+𝑟𝑛,2 
+𝑟𝑛,1 
+ 
+𝑢𝑛,𝑘 
+ 
+ 
+𝑢𝑛,2 
+𝑢𝑛,1 
+User 
+n
+ 
+ 
+𝑏𝑛 
+ 
+𝑟𝑛,𝑚 
+ 
+𝑟𝑛,2 
+𝑟𝑛,1 
+𝑐𝑛,𝑘 
+ 
+𝑐𝑛,2 
+𝑐𝑛,1 
+User 
+Cluster 
+n
+ 
+User Ratings
+ 
+ 
+ 
+ 
+ 
+ 
+Drug Ratings
+ 
+ 
+ 
+User embedding 
+(User Weights) 
+Drug Cluster Features 
+sEmbedding 
+Drug Embedding 
+(Drug Weights) 
+ 
+ Drug Bias 
+ 
+User Cluster Features 
+ User Bias 
+
+ 
+The input to the neural network is user and drug-clustered features. Drug Embedding and User 
+Embedding matrices are the input to this network, and drug and user are the network's outputs. 
+With sparse rating matrices, the forward and backward pass calculations are accomplished just for 
+non-zero ratings to reduce the computation load. The neural network layer output is calculated as: 
+��𝑧+1 = 𝑓𝑧+1(∑
+𝑤𝑖
+𝑧+1. 𝜓𝑧+1(𝑛, 𝑚). 𝑎𝑖
+𝑧 + 𝑏𝑗
+𝑧+1
+𝐾
+𝑖=1
+)      𝑖 ∈ (1, 𝐾), 𝑗 ∈ (1, 𝑀𝑁), 𝑧 ∈ (0, 𝑍 − 1), 𝑛 ∈
+(0, 𝑁), 𝑚 ∈ (0, 𝑀) 
+(14) 
+In this equation, 𝑓𝑧+1 is the activation function, 𝑤𝑖
+𝑧+1 are the weights, 𝑏𝑗
+𝑧+1 are the biases, 𝑍 is the 
+number of layers, 𝑀 is the number of drugs, 𝑁 is the number of users, 𝑎𝑍 is the output of the 
+network, 𝐾 is the number of the features for drugs or users, and 𝑀𝑁 represents the number of 
+drugs in the user network and represents the number of users in the drugs network. And finally 
+𝜓𝑧+1 is the rating existence function defined as: 
+{𝜓𝑧+1(𝑛, 𝑚) = 1      𝑖𝑓(𝑅𝑎𝑡𝑖𝑛𝑔 𝑛, 𝑚  𝑒𝑥𝑖𝑡 𝑜𝑟 𝑧 < 𝑍 − 1)
+𝜓𝑧+1(𝑛, 𝑚) = 0      𝑖𝑓(𝑅𝑎𝑡𝑖𝑛𝑔 𝑛, 𝑚 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡)                 
+(15) 
+Also, the backward pass calculations are as equations (16) to (19) for the output and hidden layers 
+respectively: 
+For the output: 
+∆𝑜𝑢𝑡 = (𝑅(𝑛,𝑚) − 𝑎𝑍). 𝜓𝑍(𝑛, 𝑚). 𝑓𝑧+1′(𝑎𝑍)    𝑛 ∈ (0, 𝑁), 𝑚 ∈ (0, 𝑀) 
+(16) 
+∆𝑊𝑍 = ∆𝑜𝑢𝑡. 𝑎𝑍. 𝛾𝑍 
+(17) 
+For the hidden layers: 
+∆𝐻𝑖𝑑𝑑𝑒𝑛𝑧 = 𝑓′(𝑎𝑧). ∑ ∆𝑜𝑢𝑡𝑖
+𝑖
+. 𝑤𝑖
+𝑧         𝑖 ∈ (1, 𝐾), 𝑧 ∈ (0, 𝑍 − 1) 
+(18) 
+∆𝑤𝑧 = ∆𝐻𝑖𝑑𝑑𝑒𝑛𝑧. 𝑎𝑧. 𝛾𝑧       𝑧 ∈ (0, 𝑍 − 1) 
+(19) 
+In these equations, 𝑓𝑧+1′ is the gradient of the activation function, 𝑅(𝑛,𝑚) is the rating 
+corresponding to the users or drugs, ∆𝑊𝑍 is the error correction for the output layer, ∆𝑤𝑧 are the 
+error corrections for the hidden layers, and 𝛾𝑧 is the learning rate. The weight updates are also 
+according to equation (20): 
+𝑤𝑛𝑒𝑤
+𝑧
+= 𝑤𝑜𝑙𝑑
+𝑧
++ ∆𝑤𝑧 𝑧 ∈ (0, 𝑍) (20) 
+
+ 
+5.5 Knowledge-based component 
+After modeling the recommendation system, several constraints on the model output are applied. 
+The final stage in the recommendation process is based on the knowledge-based technique. The 
+knowledge-based recommendation is a specific recommender system that can be used in 
+combination with other algorithms or alone.  
+The aim of using this module is its huge impact in increasing the safety of the recommendations.  
+We extracted and gathered rules in the drug recommendation domain as queries. These rules are 
+based on Drug Interactions and Adverse Events. Using these rules, we can prevent recommending 
+drugs that lead to events like death, hospitalization, disability, and life-threatening events. The 
+flowchart of this component has been extracted from Figure 2 and redrawn in Figure 5. 
+The set of these rules which our knowledge-based component considers falls into these two 
+categories: 
+- 
+Based on patients’ features: 
+o Gender is allowed to recommend a drug. 
+o The age is allowed for recommending a drug. 
+- 
+Based on drug interactions: 
+o The recommended drug has no interaction with other drugs taken by the user. 
+
+ 
+Figure 5- Knowledge-based component of our proposed approach based on the bottom left of 
+Figure 2. 
+ 
+Table 10 presents knowledge-based rules based on patient’s features that have been considered in 
+this work. For example, according to this table, a drug can only be recommended if the patient's 
+age is in the allowed range and the gender is allowed for recommending the drug. 
+TABLE 10- USER FEATURE-BASED RULES 
+Zometa 
+ 
+Actemra 
+Abilify 
+Drug Name 
+31 
+… 
+29 
+24 
+Minimum 
+Not allowed age 
+ranges 
+67 
+… 
+64 
+45 
+Maximum 
+0 
+… 
+1 
+0 
+None 
+Allowed gender 
+0 
+… 
+0 
+0 
+Female 
+0 
+… 
+0 
+0 
+Male 
+1 
+… 
+0 
+1 
+Both 
+ 
+ Start
+ 
+  
+New User Registration
+ 
+User Weights  
+& Biases Sets 
+Drug Weights 
+& Biases Sets 
+  
+Compute 10 High Rated 
+Drug Recommendation 
+Drug[i] Interaction?
+ 
+  I=0
+ 
+I<10
+ 
+  I=i+1
+ 
+Drug 
+Interation Set 
+ End
+ 
+ 
+ Recommendation.append(Drug[i]
+ 
+Knowledge-
+based 
+Recommendations 
+based on the model 
+  
+User Features  
+Filter Set 
+Drug[i] Allowed?
+ 
+Drug Recommendation
+ 
+
+For our proposed knowledge-based component, another adverse events dataset is generated from 
+Druglib.com. The structure of this dataset is presented in Table 11. Features in this dataset include 
+age, gender, the name of the drug taken by a given patient, its adverse event, reaction, and other 
+drugs used by the patient. 
+TABLE11-ADVERSE EVENTS DATASET 
+Other Drug 
+Adverse Event 
+Reaction 
+Genus 
+Age 
+Drug Name 
+Index 
+- 
+Death 
+ 
+male 
+63 
+Ability 
+(Airipiprazole) 
+1 
+... 
+… 
+… 
+… 
+… 
+... 
+… 
+Insulin 
+Death; 
+Hospitalization 
+ 
+female 
+47 
+Acterma 
+12 
+… 
+… 
+... 
+… 
+… 
+… 
+… 
+Fluticasone propiate; 
+Salmeterol; Carbemazepine 
+Hospitalization 
+ 
+male 
+50 
+Zyprexa 
+2486 
+ 
+We used Gaussian and Poisson distribution for patients' age and gender from the above dataset for 
+the adverse events of using a specific drug. These adverse events can be death, hospitalization, 
+disability, or other life-threatening events. 
+Since, in this case, we require the average and standard deviation, by using Poisson and Gaussian 
+distribution, it is possible to compute the allowed gender for recommending a drug to a patient 
+using much less memory than machine learning for this specific task. 
+Assume that on average, by recommending a drug 𝛾 for 𝜂 times to patients with 𝑔𝑒𝑛𝑑𝑒𝑟 =
+ 𝑓𝑒𝑚𝑎𝑙𝑒 they experience one of the adverse events mentioned in Table 11, then the probability 
+that by recommending drug 𝛾 to a female patient she experiences one of the adverse events is 
+calculated as equation (21): 
+𝑃𝑛(𝑥) = 𝑒−𝜆𝐹𝑒𝑚𝑎𝑙𝑒
+𝛶
+𝜆𝐹𝑒𝑚𝑎𝑙𝑒
+𝛶
+     𝜆𝐹𝑒𝑚𝑎𝑙𝑒
+𝛶
+=
+𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐴𝑑𝑣𝑒𝑟𝑠𝑒 𝐸𝑣𝑒𝑛𝑡
+𝜂
+  
+(21) 
+And similarly, for a male patient, this probability is calculated as equation (22): 
+𝑃𝑛(𝑥) = 𝑒−𝜆𝑀𝑎𝑙𝑒
+𝛶
+𝜆𝑀𝑎𝑙𝑒
+𝛶
+     𝜆𝑀𝑎𝑙𝑒
+𝛶
+=
+𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐴𝑑𝑣𝑒𝑟𝑠𝑒 𝐸𝑣𝑒𝑛𝑡
+𝜂
+ 
+(22) 
+ 
+
+Using the above calculations, if the probability of an adverse event for each gender and each 
+medicine is more than a given threshold value, the medicine is removed from the list and is not 
+recommended to the patient. In this paper, we set the 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 =  50%. 
+Also, normal distribution was used for setting the rules related to the patients ages. Suppose the 
+average and standard deviation of a patient’s age who have taken medicine 𝛾 and has an adverse 
+event is represented by 𝜇 and 𝜎, respectively. In that case, the normal distribution function related 
+to age is as equation (23): 
+𝑓(𝑥) =
+1
+√2𝜋σ𝛶 𝑒
+−1
+2(𝑥−μ𝛶
+σ𝛶 )
+2
+ 
+(23) 
+and so for patients who are taking medicine 𝛾, equation (17) for age range has to be met to 
+minimize the adverse event (24): 
+𝑋𝜖 (𝜇𝛶 − 1.96 (
+𝜎𝛶
+√𝑛) , 𝜇𝛶 + 1.96 (
+𝜎𝛶
+√𝑛)) , 1 − 𝑎 = 95%, 𝑍0.975 = 1.96 
+(24) 
+ 
+In this research, we used the rules related to the users’ features and the medicine rules and drug 
+interactions we are also considering. In this regard, the drug interactions dataset was used to 
+exclude recommendations for drugs having high interactions with other drugs. 
+ 
+6- RESULTS AND DISCUSSION 
+This section discusses our proposed drug recommendation system implementation and the newly 
+generated datasets. First, we explain the extracted and newly generated datasets and then we will 
+demonstrate the results of our implemented system. 
+ 
+The dataset 
+As discussed in the proposed method, we used the information from two databases of drugs 
+Druglib.com [7] and Drugs.com [5]. The first database Druglib.com is a comprehensive resource 
+for drug information. For each drug, a variety of information such as description, side-effects, drug 
+ratings & reviews by patients, and clinical pharmacology has been provided. Also, Drugs.com is 
+
+another database for drug information, and many recommendation systems have been suggested 
+that use this database to build their models. Both the original and the revised version of Drugs.com 
+have been used in RS to evaluate the performance of the approaches. 
+We crawled these pharmaceutical websites to construct our intended datasets with the required 
+features in a structured way. As a result, we gathered much useful information about drugs and 
+patients’ conditions and collected them into three datasets as follows:  
+- 
+The first extracted dataset is the Rating dataset consists of patients’ features and their 
+ratings on drugs consisting of 3294 samples. 
+- 
+The second dataset consists of Drug features containing drug categories, side effects, and 
+benefits. 
+- 
+The last dataset is the Interaction dataset containing interactions between drugs. 
+To evaluate the performance of our system, we used the most popular existing machine learning 
+evaluation metrics. Accuracy, sensitivity (recall), specificity, and precision were the basic metrics 
+that we applied to our model. 
+We used 70 percent of the samples (2304 samples) in the dataset for training our model, 20 percent 
+(660 samples) for evaluation, and 10 percent (330 samples) for the test. 
+After obtaining the values for true positive (TP), false positive (FP), true negative (TN), and false 
+negative (FN), different metrics can be calculated.  
+We compared our results with the existing approaches in [27], [48], [57], [58], and [59]. We 
+implemented the algorithms in these papers with the datasets they have applied. 
+In [27], SVM and recurrent neural network (RNN) ve been used to recommend a drug to a patient. 
+In [48] the authors first considered the clustering of drugs according to the drug information, like 
+the algorithm proposed in this paper. Then collaborative filtering is used to recommend a drug. 
+But unlike our work, they haven’t considered the classification of users and their features. Finally, 
+in [57], an improved matrix factorization has been used, filters the results using NSGA-III to 
+improve the accuracy, diversity, novelty, and recall.  
+Table 12 represents the comparison results between our work and other drug recommendation 
+systems in terms of important machine learning metrics. 
+
+TABLE 12: COMPARISON RESULT OF OUR PROPOSED RECOMMENDATION SYSTEM 
+WITH OTHER STATE-OF-THE-ART APPROACHES 
+F1-Measure
+ 
+Precision
+ 
+Specificity
+ 
+Sensitivity
+ 
+Accuracy
+ 
+ 
+0.07
+ 
+0.04
+ 
+0.33
+ 
+0.75
+ 
+0.34
+ 
+SVM[26]
+ 
+0.18
+ 
+0.31
+ 
+0.86
+ 
+0.13
+ 
+0.31
+ 
+Neural Network [26]
+ 
+0.41
+ 
+0.32
+ 
+0.54
+ 
+0.61
+ 
+0.55
+ 
+Kmeans User CF [48]
+ 
+0.39
+ 
+0.41
+ 
+0.66
+ 
+0.39
+ 
+0.63
+ 
+NSGA III [57]
+ 
+0.38
+ 
+0.33
+ 
+0.49
+ 
+0.45
+ 
+0.48
+ 
+Conventional MF [58]
+ 
+0.45
+ 
+0.36
+ 
+0.38
+ 
+0.60
+ 
+0.45
+ 
+MLP [59]
+ 
+0.65
+ 
+0.62
+ 
+0.64
+ 
+0.69
+ 
+0.65
+ 
+Proposed Method
+ 
+ 
+Comparison results consist of the F2 measure, ROC, and confusion matrix of different approaches 
+depicted in Figures 6 to 8. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 6- Comparison result of F2 measure metric 
+ 
+0
+0.2
+0.4
+0.6
+F2 Measure
+SVM [26]
+Backpropagation [26]
+Kmeans CF [48]
+NSGA II [57]
+ConvMF [58]
+MLP [59]
+Proposed Method
+
+ 
+Figure 7- comparison result of ROC. 
+ 
+ 
+Figure 8- comparison result of the confusion matrix. 
+
+ROC
+10
+True Positive Rate
+0.8
+0.6
+0.4
+ProposedMethod
+Backpropagation
+0.2
+KmeansUserCF
+Conventional MF
+0.0
+RandomClassifier
+0.0
+0.2
+0.4
+0.6
+0.8
+10
+FalsePositiveRateProposed Method
+110
+100
+S
+proper
+108
+47
+ActualLabels
+90
+80
+70
+not proper
+113
+60
+ 50
+PredictedLabelsConventional MF
+100
+I Labels
+proper
+53
+90
+Actual
+80
+not proper
+105
+108
+70
+60
+Predicted LabelsMLP
+120
+110
+Labels
+proper
+73
+47
+100
+06
+Actual
+80
+not_proper
+128
+82
+70
+60
+F 50
+PredictedLabelsSVM
+200
+175
+Labels
+proper
+10
+5
+150
+125
+Actual
+100
+75
+not_proper
+208
+107
+50
+25
+PredictedLabels 
+Figure 9 -Confusion matrix obtained for the  proposed method. 
+ 
+The construction of a confusion matrix for different ratings is also shown in Figure 9. The 
+predictions are compared with actual ratings of users, and drugs for the case when they are 
+considered separately and combined according to our proposed approach. 
+One of the important components of our recommender system is the final knowledge-based 
+approach. This component prevents death, hospitalization, and disability by considering drug 
+interactions and the user’s age. The adverse Events Dataset is used in this regard to our system's 
+performance for recognizing such cases and recommending the appropriate drugs. This dataset 
+contains 2486 samples, where 80% of them are used for rule extraction, and the remaining 20% 
+are for the test. 
+
+JustUserRatings
+160
+140
+Labels
+proper
+37
+19
+120
+100
+Actual
+80
+not proper
+112
+162
+ 60
+40
+20
+PredictedLabelsustDrugRatings
+180
+160
+S
+proper
+30
+14
+140
+Label
+120
+Actual
+100
+80
+not proper
+98
+188
+60
+40
+F 20
+PredictedLabelsUsers&DrugRatings
+110
+100
+S
+proper
+108
+47
+Label
+90
+Actual
+80
+70
+not proper
+113
+60
+ 50
+PredictedLabelsThe following parameters are considered for the evaluation: 
+𝐷𝑒𝑎𝑡ℎ 𝑅𝑎𝑡𝑖𝑜 =
+𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐷𝑒𝑎𝑡ℎ
+𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑅𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛 
+𝐷𝑖𝑠𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑅𝑎𝑡𝑖𝑜 =
+𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐷𝑖𝑠𝑎𝑏𝑖𝑙𝑖𝑡𝑦 
+𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑅𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛 
+𝐻𝑜𝑠𝑝𝑖𝑡𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑖𝑜 =
+𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐻𝑜𝑠𝑝𝑖𝑡𝑎𝑙𝑖𝑧𝑎𝑡𝑖𝑜𝑛 
+𝑇𝑜𝑡𝑎𝑙 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑅𝑒𝑐𝑜𝑚𝑚𝑒𝑛𝑑𝑎𝑡𝑖𝑜𝑛 
+For comparison, the system's performance for different adverse events was calculated one time 
+without a knowledge-based component and the second time using this component. 
+Table 13 and Figure 10 represent the results of this comparison. 
+TABLE 13- COMPARISON RESULTS OF KNOWLEDGE-BASED COMPONENT 
+Adverse event 
+Without knowledge-based 
+component 
+With knowledge-based 
+component 
+Death rate 
+44% 
+6% 
+Hospitalization 
+15% 
+2% 
+Disability 
+4% 
+0.7% 
+  
+Knowledge-based component is an essential part of a drug recommendation system in reducing 
+adverse events and improving the quality of recommendations. 
+ 
+Figure 10- Comparison result of adding knowledge-based in the recommendation system. 
+We also considered one more important metric for recommender systems evaluations: hit rate. 
+0
+10
+20
+30
+40
+50
+Death Rate
+Disability Rate
+Hospitalization Rate
+With Knowledge Base
+Without Knowledge base
+
+The data set's testing samples (330) are utilized in hit-rate evaluation. The hit-rate in evaluation is 
+calculated by the ratio of the total hits in the top 10 recommended drugs returned for all users and 
+the total testing samples. So if 𝜂 is the number of relevant predicted drugs for all users, and 𝑁 is 
+the total number of testing samples, according to [60], the hit-rate is calculated as equation (25): 
+ℎ𝑖𝑡_𝑟𝑎𝑡𝑒 =
+𝜂
+𝑁  (25) 
+The result of hit rate evaluation is represented in Figure 11. As it can be seen from this figure, our 
+proposed approach has the ℎ𝑖𝑡 𝑟𝑎𝑡𝑒 =  0.49, which is better than all other approaches. 
+ 
+ 
+Figure 11- Top-10 hit-rate recommendation systems. 
+ 
+The next evaluation metric is cumulative hit-rate, which represents the number of hits with ratings 
+above a given threshold and ignores the predicted ratings lower than the threshold. The result of 
+the cumulative hit-rate with the threshold set to 4 is shown in Figure 12. The cumulative hit-rate 
+is calculated as (26): 
+𝐶𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝐻𝑖𝑡 − 𝑅𝑎𝑡𝑒 =
+𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ℎ𝑖𝑡𝑠 𝑤𝑖𝑡ℎ 𝑟𝑎𝑡𝑖𝑛𝑔 𝑎𝑏𝑜𝑣𝑒 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑
+𝑁
+ 
+(26) 
+The utilization of this threshold makes a better match with the user’s interest in the recommended 
+drug. 
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Top-10 Hit Rate
+SVM [26]
+NeuralNetwork [26]
+Kmeans Used CF [48]
+NSGA III [57]
+Conventional MF [58]
+MLP [59]
+Proposed Method
+
+ 
+Figure 12- Top-10 cumulative hit-rate of recommendation systems. 
+ 
+Our results are encouraging in the field of drug recommendations. It has combined the benefits of 
+basic recommender approaches with less computational overhead through a novel modeling 
+approach and using statistical methods. It also classifies drugs and users in terms of their features, 
+leading to high accuracy compared to state-of-the-art algorithms. However, better results can be 
+achieved by considering the characteristics of diseases and recommending drugs based on disease 
+features in addition to the features of patients and drugs. 
+7- CONCLUSION 
+In this paper, we proposed a comprehensive drug recommender system that takes advantage of all 
+basic recommender system techniques and applies natural language processing, neural network-
+based matrix factorization, and, more importantly, employing knowledge-based recommendations 
+to recommend the most accurate drugs to patients.  Compared with conventional matrix 
+factorization, our proposed method improves the accuracy, sensitivity, and hit rate by 26%, 34%, 
+and 40%, respectively. In comparison with other machine learning approaches, we obtained an 
+accuracy, sensitivity, and hit rate by an average of 31%, 29%, and 28%, respectively. Our approach 
+can be used as an adjunct tool torecommend drugs to patients and improve the quality of 
+prescriptions and reduce the errors caused by medical practitioners. 
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+Top-10 Cumulative Hit Rate
+SVM [26]
+NeuralNetwork [26]
+Kmeans Used CF [48]
+NSGA III [57]
+Conventional MF [68]
+MLP [59]
+Proposed Method
+
+In the future, we will extend the knowledge and information extraction from drug databases and 
+include all existing patient features in the user features. Also, we are going to consider the features 
+of the disease in the recommendation. These features can be captured by general practitioners and 
+help improve the proposed drug recommender system performance and make more accurate 
+recommendations by having more relevant features. In the final output of the recommendation, we 
+also include the dosage and effectiveness of a drug in addition to the list of drugs. Also, in our 
+future work, we will extract the information from other drug resources like the SIDER database 
+for drug side effects. 
+At the end, it should be noted that a physician should approve the recommended medicines for 
+safety reasons. 
+ 
+Data availability 
+The 
+datasets 
+generated 
+during 
+the 
+current 
+study 
+are 
+available 
+in 
+the 
+https://github.com/DatasetsLibrary/RECOMMED repositories. In addition, preprocessed datasets and 
+source code of this study are also available at https://github.com/DatasetsLibrary/RECOMMEDTool. 
+ 
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+

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+Distributionally Robust Multi-objective Bayesian Optimization under
+Uncertain Environments
+Yu Inatsu1,∗
+Ichiro Takeuchi2,3
+1 Department of Computer Science, Nagoya Institute of Technology
+2 Department of Mechanical Systems Engineering, Nagoya University
+3 RIKEN Center for Advanced Intelligence Project
+∗ E-mail: inatsu.yu@nitech.ac.jp
+ABSTRACT
+In this study, we address the problem of optimizing multi-output black-box functions under uncertain
+environments.
+We formulate this problem as the estimation of the uncertain Pareto-frontier (PF) of a
+multi-output Bayesian surrogate model with two types of variables: design variables and environmental
+variables.
+We consider this problem within the context of Bayesian optimization (BO) under uncertain
+environments, where the design variables are controllable, whereas the environmental variables are assumed
+to be random and not controllable. The challenge of this problem is to robustly estimate the PF when the
+distribution of the environmental variables is unknown, that is, to estimate the PF when the environmental
+variables are generated from the worst possible distribution.
+We propose a method for solving the BO
+problem by appropriately incorporating the uncertainties of the environmental variables and their probability
+distribution.
+We demonstrate that the proposed method can find an arbitrarily accurate PF with high
+probability in a finite number of iterations.
+We also evaluate the performance of the proposed method
+through numerical experiments.
+1. Introduction
+In many industrial applications, we encounter the problem of optimizing the multi-output black-box function
+under uncertain environments. For example, in the problem of optimizing growing conditions for crops, we want
+to optimize several conditions such as fertilizer levels to maximize crop quality and yield under an uncertain
+environment such as weather conditions.
+To formulate this problem, let f (1)(x, w) and f (2)(x, w) be a pair of outputs of a black-box function that
+we want to simultaneously maximize, where x ∈ X and w ∈ Ω are the design variables (such as fertilizer levels)
+and environmental variables (such as weather conditions) defined in domains X and Ω, respectively, where the
+former is controllable and the latter is not. To characterize the uncertainty of the environmental variables w,
+we assume that it is sampled from an unknown probability distribution, P †. Because we do not know P †, we
+consider the case where we know only A, which is a family of candidate distributions for w.
+This study aims to identify a distributionally robust Pareto-frontier (DR-PF) in the above setting, which is
+formulated as a PF of the following two functions:
+F (1)(x) =
+inf
+p(w)∈A
+�
+Ω
+f (1)(x, w)p(w)dw,
+F (2)(x) =
+inf
+p(w)∈A
+�
+Ω
+f (2)(x, w)p(w)dw.
+Figure 1 shows an example of the problem setup.
+To identify a DR-PF, it is necessary to predict it and quantify its uncertainty. In this study, under the
+assumption that f (1)(x, w) and f (2)(x, w) follow a Gaussian process (GP), we developed a Bayesian optimization
+(BO) method to find a lower bound of the DR-PF by considering the uncertainty of environmental variables w
+and the uncertainty of the probability distribution for w. Specifically, we propose an acquisition function (AF)
+that enables us to sequentially select the controllable design variable x in a sample-efficient manner to obtain
+the DR-PF.
+To this end, various technical challenges need to be resolved. One difficulty is that, even when f (1)(x, w)
+and f (2)(x, w) are GPs, F (1)(x) and F (2)(x) are not GPs anymore. Therefore, we derive a non-trivial credible
+intervals of F (1)(x) and F (2)(x) considering that they are defined as the infima of integrated GPs. Furthermore,
+although a naive formulation of multi-objective BOs is computationally expensive, the proposed AF has the
+advantage that it can be evaluated efficiently. We also conducted a theoretical analysis of the proposed BO
+method to prove that the proposed BO method can find an arbitrarily accurate DR-PF with a high probability
+in a finite number of iterations under mild conditions.
+1.1. Related Work
+For black-box function optimization problems, BOs have been popularly used [Settles, 2009, Shahriari et al., 2016]
+in which GP [Williams and Rasmussen, 2006] is often employed as a surrogate model. An optimization problem
+1
+arXiv:2301.11588v1  [stat.ML]  27 Jan 2023
+
+Multi-objective function
+� �, � = � � �, � , � � �, �
+(a)
+Objective function 1
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+(b)
+Candidate distributions of 
+-4
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+0.0
+0.1
+0.2
+0.3
+0.4
+Candidate distribution of w
+w
+Density
+-4
+-2
+0
+2
+4
+0.0
+0.1
+0.2
+0.3
+0.4
+Candidate distribution of w
+w
+Density
+-4
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+0.1
+0.2
+0.3
+0.4
+Candidate distribution of w
+w
+Density
+・
+・
+・
+・
+・
+・
+・��・
+・
+・
+・・・
+・
+・
+・・・
+・
+(c)
+Expected value of � � (�, �)
+Expected value of � � (�, �)
+Candidate Pareto-frontiers
+Distributionally robust 
+Pareto-frontier
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+Pareto-frontier
+Figure
+1: Conceptional diagram of a DR-PF. The upper and lower color maps in (a) represent objective
+functions f (1)(x, w) and f (2)(x, w), respectively, where x and w are the scalar design and environmental variable,
+respectively. The plots within the frame in (b) represent multiple candidate distributions for w. A dashed line
+in (c) is the expected PF for each candidate distributions of w. A red solid line is the DR-PF, which is defined
+by the worst-case expectation of the candidate distributions, provides a lower bound of uncertain PF. The
+objective of this study is to efficiently identify the DR-PF.
+of multiple black-box functions is typically formulated as a Pareto optimization problem, and BO methods for
+such problems have been also studied [Zuluaga et al., 2016, Suzuki et al., 2020].
+Many studies have been conducted on BOs under uncertain environments. For example, [Bogunovic et al., 2018]
+proposed a BO method to maximize the worst-case function value with respect to a shift in the input. In several
+studies [Beland and Nair, 2017, Toscano-Palmerin and Frazier, 2018, Oliveira et al., 2019, Fr¨ohlich et al., 2020,
+Gessner et al., 2020], a BO problem to maximize the expected value of a black-box function with respect to the
+input distribution was considered. Furthermore, several studies considered the simultaneous optimization of
+multiple black-box functions under the assumption that the distribution of environmental variables is known.
+For example, [Iwazaki et al., 2021] dealt with constrained optimization and Pareto optimization problems for the
+mean and variance of a black-box function with respect to environmental variables, [Qing et al., 2022] considered
+Pareto optimization problems for the expected values of multiple objective functions, and [Amri et al., 2021]
+dealt with chance-constrained optimization problems that is an extension of constrained optimization problems
+to the input uncertainty setting.
+Distributionally robust optimization (DRO) was first introduced by [Scarf, 1958]. Because DRO is an im-
+portant topic in the context of robust optimization and has been the subject of numerous studies, we refer an
+exhaustive survey of DRO to [Rahimian and Mehrotra, 2019]. In recent years, several studies on DRO in the
+context of BOs (DRO-BOs) have been conducted. [Kirschner et al., 2020] and [Nguyen et al., 2020] proposed
+BO methods to efficiently find the design variable that maximizes F (1)(x). DRO-BOs have also been stud-
+ied under multiple black-box functions. [Inatsu et al., 2022] proposed a BO method for distributionally robust
+chance-constrained problems, which is an extension of the chance-constrained problem to DRO. However, to the
+best of our knowledge, there are no prior studies on BOs for Pareto optimization under the DRO framework.
+1.2. Contributions
+The contributions of this study are as follows:
+• We develop a BO method for identifying DR-PFs called DR-PF BO method. Specifically, we propose a
+novel AF for the DR-PF BO, which is computationally inexpensive and has theoretical guarantees.
+• Under mild conditions, we prove that the DR-PF BO method can find an arbitrarily accurate PF with a
+high probability in a finite number of iterations.
+• Additionally, we prove that with a specification of an appropriate family of candidate distributions, even
+if the true distribution is unknown, the DR-PF BO method can find an arbitrarily accurate expected PF
+on the true distribution.
+2
+
+• We confirm the performance of the proposed method through numerical experiments with benchmark
+functions and simulator-based functions.
+2. Preliminaries
+Let f (1) : X × Ω → R and f (2) : X × Ω → R be the expensive-to-evaluate black-box functions1, where X
+and Ω are the finite sets. For each input (x, w) ∈ X × Ω, the values of f (1)(x, w) and f (2)(x, w) are observed
+with observation noise as y(1) = f (1)(x, w) + ε(1) and y(2) = f (2)(x, w) + ε(2), where ε(1) and ε(2) are random
+samples from independent normal distributions with the mean zero and variances σ(1)2
+noise and σ(2)2
+noise, respectively.
+In this study, the environmental variable w ∈ Ω was assumed to be a discrete random variable that follows an
+unknown probability distribution P †. The two distinct phases called the development phase and use phase exist
+in the literature of BOs under uncertainty. In the development phase, environmental variables are completely
+controllable as design variables, whereas they are stochastic and uncontrollable in the use phase. In this study,
+we consider the development phase, and the use phase is described in Appendix A. Furthermore, we denote the
+family of candidate distributions of w as A and consider the following class of distributions:
+A = {p(w) | d(p(w), p∗(w)) ≤ ξ},
+where p∗(w) is a user-specified reference distribution, d(·, ·) is a given distance metric function between dis-
+tributions, and ξ ≥ 0. This means that we consider a set of candidate distributions whose distance from the
+reference distribution is not larger than a certain threshold. Subsequently, the distributionally robust expecta-
+tions F (1)(x) and F (2)(x) for each design variable x ∈ X are defined as follows:
+F (1)(x) =
+inf
+p(w)∈A
+�
+w∈Ω
+f (1)(x, w)p(w),
+F (2)(x) =
+inf
+p(w)∈A
+�
+w∈Ω
+f (2)(x, w)p(w).
+The objective of this study is to efficiently identify the PF determined by F (1)(x) and F (2)(x). Let F (x) =
+(F (1)(x), F (2)(x)) for each x ∈ X, and let F (E) = {F (x) | x ∈ E} for a set E ⊂ X. Furthermore, for a set
+B ⊂ R2, we denote the domain dominated by B and the PF of B, respectively, by
+Dom(B) = {y ∈ R2 |∃ y′ ∈ B s.t. y ⪯ y′},
+Par(B) = ∂(Dom(B)).
+Here, for a point a = (a1, . . . , am) and b = (b1, . . . , bm), a ⪯ b implies ai ≤ bi for any i ∈ {1, . . . , m}. For a set
+C, ∂(C) denotes the boundary of C. The PF determined by F (1)(x) and F (2)(x) can then be written as
+Z∗ = Par(F (X)).
+2.1. Gaussian Process
+In this study, we use GP surrogate models for black-box functions f (1) and f (2). First, we assume that
+f (1) and f (2) follow GP priors GP(0, k(1)((x, w), (x′, w′))) and GP(0, k(2)((x, w), (x′, w′))), respectively, where
+k(1)((x, w), (x′, w′)) and k(2)((x, w), (x′, w′)) are the positive-definite kernels. For l ∈ {1, 2}, given a dataset
+{(xi, wi, y(l)
+i )}t
+i=1, where t is the number of queried instances, the posterior distribution of f (l) is a GP, and its
+posterior mean µ(l)
+t (x, w) and posterior variance σ(l)2
+t
+(x, w) are given by
+µ(l)
+t (x, w) = k(l)
+t (x, w)⊤(K(l)
+t
++ σ(l)2
+noiseIt)−1y(l)
+t ,
+σ(l)2
+t
+(x, w) = k(l)((x, w), (x, w)) − k(l)
+t (x, w)⊤(K(l)
+t
++ σ(l)2
+noiseIt)−1k(l)
+t (x, w),
+where k(l)
+t (x, w) is the t-dimensional vector, whose jth element is k(l)((x, w), (xj, wj)), y(l)
+t
+= (y(l)
+1 , . . . , y(l)
+t )⊤,
+It is the t × t identity matrix, K(l)
+t
+is the t × t matrix whose (j, k) element is k(l)((xj, wj), (xk, wk)), with a
+superscript ⊤ indicating the transpose of vectors or matrices.
+3. Proposed Method
+Here, we propose a BO method to efficiently identify Z∗. Because the functions f (1)(x, w) and f (2)(x, w)
+are random variables following GPs, F (1)(x) and F (2)(x) are also random variables. Therefore, a reasonable
+approach is to construct credible intervals for F (1)(x) and F (2)(x), and use them to estimate the PF. However,
+1Note that the method proposed in this study can be straightforwardly extended to the case where there are more than three
+objective functions f(1), f(2), f(3), . . ..
+3
+
+unlike f (1)(x, w) and f (2)(x, w), F (1)(x) and F (2)(x) do not follow GPs. Therefore, we cannot directly obtain
+credible intervals using the properties of GPs. In Section 3.1, we construct credible intervals for F (1)(x) and
+F (2)(x) using the method proposed by [Kirschner et al., 2020] and provide a method for estimating the PF
+based on the constructed credible intervals.
+3.1. Credible Intervals and PF Estimation
+For each input (x, w) ∈ X × Ω and time t, the credible interval of f (1)(x, w) is denoted by Q(f (1))
+t
+(x, w) =
+[l(f (1))
+t
+(x, w), u(f (1))
+t
+(x, w)]. Here, the lower value l(f (1))
+t
+(x, w) and the upper value u(f (1))
+t
+(x, w) are given as
+l(f (1))
+t
+(x, w) = µ(1)
+t (x, w) − β1/2
+1,t σ(1)
+t
+(x, w),
+u(f (1))
+t
+(x, w) = µ(1)
+t (x, w) + β1/2
+1,t σ(1)
+t
+(x, w),
+where β1/2
+1,t
+≥ 0 is a user-specified tradeoff parameter.
+We then define the credible interval Q(F (1))
+t
+(x) ≡
+[l(F (1))
+t
+(x), u(F (1))
+t
+(x)] of F (1)(x). Here, the lower and upper values are respectively given by
+l(F (1))
+t
+(x) =
+inf
+p(w)∈A
+�
+w∈Ω
+l(f (1))
+t
+(x, w)p(w),
+u(F (1))
+t
+(x) =
+inf
+p(w)∈A
+�
+w∈Ω
+u(f (1))
+t
+(x, w)p(w).
+(3.1)
+Notably, if the L1- (or L2-) norm is used as the distance d(·, ·) between the distributions, the problem of
+obtaining the lower and upper values in (3.1) is reduced to a linear programming problem (or a second-order
+cone programming problem). In either case, the existence of a fast solver of these problems enabled us to obtain
+Q(F (1))
+t
+(x). Similarly, we define credible intervals Q(f (2))
+t
+(x, w) = [l(f (2))
+t
+(x, w), u(f (2))
+t
+(x, w)] for f (2)(x, w) and
+Q(F (2))
+t
+(x) ≡ [l(F (2))
+t
+(x), u(F (2))
+t
+(x)] for F (2)(x). Next, for any input x ∈ X and any subset E ⊂ X, we define
+LCBt(x), UCBt(x) and LCBt(E) as follows:
+LCBt(x) = (l(F (1))
+t
+(x), l(F (2))
+t
+(x)),
+UCBt(x) = (u(F (1))
+t
+(x), u(F (2))
+t
+(x)),
+LCBt(E) = {LCBt(x) | x ∈ E}.
+The estimated PF solution set ˆΠt ⊂ X for the design variables is then defined as follows:
+ˆΠt = {x ∈ X | LCBt(x) ∈ Par(LCBt(X))}.
+Figure 2 (a) shows a conceptual diagram of LCBt(x) and UCBt(x), and (b) shows a conceptual diagram of
+Par(LCBt(X)) and ˆΠt.
+3.2. Acquisition Function
+Here, we propose an AF for determining the next evaluation point. First, for each point a ∈ Rm and subset
+B ⊂ Rm, we denote the closeness between them as
+dist(a, B) = inf
+b∈B d∞(a, b),
+where d∞(a, b) denotes the metric function given by d∞(a, b) = max{|a1 − b1|, . . . , |am − bm|}. Using this, we
+define AF at(x) for x ∈ X as
+at(x) = dist(UCBt(x), Dom(LCBt(ˆΠt))).
+We then select the following evaluation points, as described in the following definition:
+Definition 3.1. The next design variable, xt+1, to be evaluated is selected as follows:
+xt+1 = argmax
+x∈X
+at(x),
+and the next environmental variable, wt+1, to be evaluated is selected as
+wt+1 = argmax
+w∈Ω
+{σ(1)2
+t
+(xt+1, w) + σ(2)2
+t
+(xt+1, w)}.
+4
+
+𝐹(2)(𝒙)
+𝐹(1)(𝒙)
+𝐹(2)(𝒙)
+𝐹(1)(𝒙)
+𝐹(2)(𝒙)
+𝐹(1)(𝒙)
+(a)
+(b)
+(c)
+𝒙1
+𝒙2
+𝒙3
+𝒙4
+𝒙5
+𝒙6
+𝒙7
+Figure 2: Conceptual diagrams of LCBt(x), UCBt(x), Par(LCBt(X)), ˆΠt and AFs for seven input points
+x1, . . . , x7. At each point x in the left figure (a), LCBt(x) and UCBt(x) indicate the lower left point and the
+upper right point of the dashed rectangular region, respectively. In (b), the PF (red line) computed using each
+LCBt(x) is Par(LCBt(X)), and because it is constructed by LCBt(x1), LCBt(x2), LCBt(x3), LCBt(x7), ˆΠt
+is given by ˆΠt = {x1, x2, x3, x7}. In (c), the light red region indicates Dom(LCBt(ˆΠt)), the region dominated
+by the red points (LCBt(ˆΠt)), and at(x) is the closeness between the light red region and UCBt(x) (purple
+point). The furthest point is represented by the purple triangle, UCBt(x4). Thus, the next design variable to
+be evaluated is x4.
+Figure 2 (c) shows a conceptual diagram of the AF at(x). Here, AF at(x) can be computed analytically
+using the following lemma:
+Lemma 3.1. Let UCBt(x) = (u1, u2) and LCBt(ˆΠt) = {(l(i)
+1 , l(i)
+2 ) | 1 ≤ i ≤ k}. Then, at(x) can be computed
+as follows:
+˜at(x) = min
+1≤i≤k max{u1 − l(i)
+1 , u2 − l(i)
+2 },
+at(x) = max{˜at(x), 0}.
+Notably, when the number of objective functions is m ≥ 3, at(x) is easily extended as follows:
+˜at(x) = min
+1≤i≤k′ max{u1 − l(i)
+1 , . . . , um − l(i)
+m },
+at(x) = max{˜at(x), 0},
+(3.2)
+where UCBt(x) = (u1, . . . , um) and LCBt(ˆΠt) = {(l(i)
+1 , . . . , l(i)
+m ) | 1 ≤ i ≤ k′}. The proofs of Lemma 3.1 and
+(3.2) are presented in Appendix B. From Lemma 3.1, once LCBt(ˆΠt) is computed, the maximum value of at(x)
+can be analytically obtained by performing 2|X| times inf calculations and computing u(F (1))
+t
+(x) and u(F (2))
+t
+(x)
+for all x ∈ X. On the other hand, an AF based on exact posterior distributions of target functions such as the
+expected hypervolume improvement [Emmerich, 2005] for ordinary Pareto optimization, requires approximation
+by sampling from f (1)(x, w) and f (2)(x, w) under this problem setting. However, in each posterior sample, the
+inf calculation must be performed again to calculate F (1)(x) and F (2)(x) for all design variables. Therefore, if
+the number of Monte Carlo samples is M, M times more inf calculations are required compared to the proposed
+AF. The comparison of computational times is given in Section 5.
+3.3. Stopping Condition
+Here, we describe the stopping conditions of the proposed algorithm. From Fig. 2 (c), AF at(x) represents
+the closeness of the pessimistic PF and the optimistic predictive value of F (x).
+That is, if this value is
+sufficiently small, there is little room for improvement in the PF; therefore, it is reasonable to use it as the
+stopping condition. Let ϵ > 0 be a user-specified parameter. Then the algorithm is terminated if at(x) ≤ ϵ is
+satisfied. The pseudocode for the proposed algorithm is given in Algorithm 1.
+4. Theoretical Analysis
+Here, we provide the theorems for the accuracy and convergence of the proposed algorithm. The details of
+the proofs are presented in Appendix B. First, to provide theoretical results, we assume that f (1) and f (2) follow
+5
+
+Algorithm 1 BO for identifying DR-PF
+Input: GP priors GP(0, k(1)), GP(0, k(2)), parameter ξ ≥ 0, tradeoff parameters {β1,t}t≥0, {β2,t}t≥0, stopping
+parameter ϵ > 0
+t ← 1
+while at(x) > ϵ do
+Compute Q(F (1))
+t
+(x) and Q(F (2))
+t
+(x) for each x ∈ X
+Select the next evaluation point (xt, wt)
+Observe y(1)
+t
+= f (1)(xt, wt) + ε(1)
+t
+and y(2)
+t
+= f (2)(xt, wt) + ε(2)
+t
+at the point (xt, wt)
+Update GPs by adding observed points
+t ← t + 1
+end while
+Output: Return ˆΠt as the estimated set of design variables comprising the DR-PF
+GPs GP(0, k(1)((x, w), (x′, w′))) and GP(0, k(2)((x, w), (x′, w′))), respectively. Moreover, we assume that the
+prior variances k(1)((x, w), (x, w)) ≡ σ(1)2
+0
+(x, w) and k(2)((x, w), (x, w)) ≡ σ(2)2
+0
+(x, w) satisfy
+max
+(x,w)∈X×Ω σ(1)2
+0
+(x, w) ≤ 1,
+max
+(x,w)∈X×Ω σ(2)2
+0
+(x, w) ≤ 1.
+Furthermore, let κ(1)
+T
+and κ(2)
+T
+be the maximum information gains of f (1) and f (2) at time T, respectively.
+Notably, the maximum information gain is a measure often used in theoretical analyses of GP-based BO (see,
+e.g., [Srinivas et al., 2010]). Here, for each j ∈ {1, 2}, using the mutual information I(y(j); f (j)) between y(j)
+and f (j), κ(j)
+T
+can be expressed as
+κ(j)
+T
+=
+max
+A⊂X×Ω I(y(j)
+A ; f (j)).
+Next, to quantify the goodness of the predicted ˆΠt, we define an ϵ-accurate Pareto region Zϵ. With user-specified
+positive numbers ϵ and ϵ = (ϵ, ϵ), we define Zϵ as
+Zϵ = {y ∈ R2 |∃ y′ ∈ Z∗ s.t. y ⪯ y′ and ∃y′′ ∈ Z∗ s.t. y′′ ⪯ y + ϵ}.
+That is, Zϵ is the set of points that lie inside Z∗ and within ϵ in the sense of d∞(·, ·)-distance. The concept of
+Zϵ was also used in [Zuluaga et al., 2016]. Using Zϵ, we define the accuracy of ˆΠt in terms of the following two
+aspects:
+Definition 4.1 (Accuracy for ˆΠt). Let ϵ be a positive value. We then define ˆΠt as an ϵ-accurate estimated
+solution set if the following holds:
+F (ˆΠt) ⊂ Zϵ.
+(4.1)
+Moreover, we define ˆΠt as an ϵ-accurate estimated Pareto solution set if the following holds:
+Par(F (ˆΠt)) ⊂ Zϵ.
+(4.2)
+It is easy to obtain a set that satisfies either (4.1) or (4.2). Generally, by ignoring F (2)(x) and focusing
+only on the maximization of F (1)(x), the maximization point x∗ can be estimated using methods such as
+[Kirschner et al., 2020]. Subsequently, by letting ˆΠt = {x∗}, (4.1) is satisfied with high probability. Similarly,
+if we predict that all points constitute the PF, that is, ˆΠt = X, then (4.2) is satisfied because Par(F (ˆΠt)) =
+Par(F (X)) = Z∗. The following theorem guarantees that the proposed algorithm satisfies both (4.1) and (4.2)
+with a high probability.
+Theorem 4.1. Let t ≥ 1 and δ ∈ (0, 1) and define β1,t = β2,t = 2 log(2|X × Ω|π2t2/(6δ)) ≡ βt. In addition, let
+ϵ > 0 be a user-specified stopping parameter. Then, when Algorithm 1 terminates, ˆΠt satisfies (4.1) and (4.2)
+with a probability of at least 1 − δ.
+Theorem 4.1 does not indicate whether the algorithm terminates or not. The following theorem guarantees
+the convergence of Algorithm 1.
+Theorem 4.2. Under the same conditions as those in Theorem 4.1, let T be the smallest positive integer that
+satisfies the following inequality:
+βT (C1κ(1)
+T
++ C2κ(2)
+T )
+T
+≤ ϵ2
+4 ,
+where C1 = 2/ log(1+σ(1)−2
+noise ) and C2 = 2/ log(1+σ(2)−2
+noise ). Then, Algorithm 1 terminates after at most T trials.
+6
+
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+Simulator setting
+iteration
+R1
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+Random
+UCB_F1
+UCB_F2
+MVA
+EHI
+Proposed
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+Simulator setting
+iteration
+R2
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+Random
+UCB_F1
+UCB_F2
+MVA
+EHI
+Proposed
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+Uncontrollable setting
+iteration
+R1
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+0
+100
+200
+300
+400
+500
+0
+5
+10
+15
+20
+25
+30
+35
+Random
+UCB_F1
+UCB_F2
+MVA
+EHI
+Proposed
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+Uncontrollable setting
+iteration
+R2
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+0
+100
+200
+300
+400
+500
+0
+20
+40
+60
+80
+Random
+UCB_F1
+UCB_F2
+MVA
+EHI
+Proposed
+Figure 3: Average values of R1 and R2 for each method in Simulator and Uncontrollable settings. The length
+of each error bar represents twice the standard error.
+0
+20
+40
+60
+80
+100
+0
+5
+10
+15
+20
+25
+30
+35
+SIR (case1)
+iteration
+R1
+0
+20
+40
+60
+80
+100
+0
+5
+10
+15
+20
+25
+30
+35
+0
+20
+40
+60
+80
+100
+0
+5
+10
+15
+20
+25
+30
+35
+0
+20
+40
+60
+80
+100
+0
+5
+10
+15
+20
+25
+30
+35
+0
+20
+40
+60
+80
+100
+0
+5
+10
+15
+20
+25
+30
+35
+0
+20
+40
+60
+80
+100
+0
+5
+10
+15
+20
+25
+30
+35
+Random
+UCB_F1
+UCB_F2
+MVA
+EHI
+Proposed
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+SIR (case1)
+iteration
+R2
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+0
+20
+40
+60
+80
+100
+0
+20
+40
+60
+80
+Random
+UCB_F1
+UCB_F2
+MVA
+EHI
+Proposed
+0
+20
+40
+60
+80
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+SIR (case2)
+iteration
+R1
+0
+20
+40
+60
+80
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+20
+40
+60
+80
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+20
+40
+60
+80
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+20
+40
+60
+80
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+20
+40
+60
+80
+100
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Random
+UCB_F1
+UCB_F2
+MVA
+EHI
+Proposed
+0
+20
+40
+60
+80
+100
+0
+10
+20
+30
+40
+SIR (case2)
+iteration
+R2
+0
+20
+40
+60
+80
+100
+0
+10
+20
+30
+40
+0
+20
+40
+60
+80
+100
+0
+10
+20
+30
+40
+0
+20
+40
+60
+80
+100
+0
+10
+20
+30
+40
+0
+20
+40
+60
+80
+100
+0
+10
+20
+30
+40
+0
+20
+40
+60
+80
+100
+0
+10
+20
+30
+40
+Random
+UCB_F1
+UCB_F2
+MVA
+EHI
+Proposed
+Figure 4: Average values of R1 and R2 for each method in Case1 and Case2 in SIR model experiments. The
+length of each error bar represents twice the standard error.
+Table 1: Computational time (second) and the ratios of the computational time to that of the proposed method
+Random
+UCB F1
+UCB F2
+MVA
+EHI
+Proposed
+Computational time
+0.000
+0.068
+0.067
+0.135
+16.21
+0.139
+(Standard error)
+(0.000)
+(0.000)
+(0.000)
+(0.001)
+(0.011)
+(0.001)
+Computational time ratio
+0.000
+0.496
+0.488
+0.985
+118.68
+1
+(Standard error)
+(0.000)
+(0.004)
+(0.004)
+(0.006)
+(0.548)
+(0)
+Here, because the maximum information gains κ(1)
+T
+and κ(2)
+T
+are known to be sublinear with respect to T
+under mild assumptions [Srinivas et al., 2010], and the order of β1,T = β2,T is O(log T), the positive integer T
+satisfying the inequality in Theorem 4.2 exists.
+We emphasize that Theorem 4.1 also holds in the use phase setting and a similar theorem holds for Theorem
+4.2 under mild additional conditions. Moreover, by appropriately designing the family of candidate distributions
+using the empirical distribution as the reference distribution, the proposed method provides an arbitrarily
+accurate solution for the expected PF based on the true distribution, even when the true distribution is unknown.
+Details are provided in Appendix A.
+5. Numerical Experiments
+Here, we confirm the performance of the proposed method using synthetic functions and real-world simulation
+models. In this experiment, we used a one-dimensional design variable x and environmental variable w and the
+following Gaussian kernels:
+k(1)((x, w), (x′, w′)) = σ2
+f,1 exp(−∥ν − ν′∥2
+2/L1),
+k(2)((x, w), (x′, w′)) = σ2
+f,2 exp(−∥ν − ν′∥2
+2/L2),
+where ν = (x, w) and ν′ = (x′, w′). Moreover, we used the L1-norm as the distance between distributions. In
+all experiments except computational time experiments, we used the following two indicators R1 and R2 based
+on (4.1) and (4.2) to evaluate the goodness of ˆΠt estimated by each method:
+R1 = inf{a ∈ R | F (ˆΠt) ⊂ Za},
+R2 = inf{a ∈ R | Par(F (ˆΠt)) ⊂ Za}.
+Experimental details and additional experiments, which are not included in the main body, are described in
+Appendix C.
+7
+
+5.1. Synthetic Function
+We evaluate the performance in our proposed method through synthetic functions. First, the input space
+X × Ω was a set of 50 × 50 grid points equally spaced in [−10, 10] × [−10, 10]. In this experiment, we used the
+following (scaled) Himmelblau’s function f (1)(x, w), which is commonly used as a benchmark function in BO
+studies [Andrei, 2008], and sinusoidal function f (2)(x, w) as black-box functions:
+f (1)(x, w) = (x2 + w − 11)2
+150
++ (x + w2 − 7)2
+150
+− C,
+f (2)(x, w) = (80 sin(1.5x) − 50 cos(2w))/1.5,
+where C = 3321.291/150 is a constant to set the mean to zero.
+Under this setting, we compared the following six methods:
+Random: Determine the next evaluation point xt+1 at random.
+UCB F1: Select the next evaluation point by xt+1 = argmaxx∈X u(F (1))
+t
+(x).
+UCB F2: Select the next evaluation point by xt+1 = argmaxx∈X u(F (2))
+t
+(x).
+MVA: Select the next evaluation point by xt+1 = argmaxx∈Mt∪ˆΠt λt(x), where Mt and λt(x) are given in
+Section 3.2 of [Iwazaki et al., 2021].
+EHI: Select the next evaluation point xt+1 by maximizing an expected hypervolume improvement for the
+DR-PF calculated based on posterior means.
+Proposed: Select the next evaluation point xt+1 by Definition 3.1.
+Here, UCB F1 (or UCB F2) focuses on the maximization of F (1)(x) (or F (2)(x)) and does not consider the
+identification of the DR-PF. In contrast, MVA focuses on reducing the uncertainty of a potential optimal set,
+which is a set of input points that may constitute the DR-PF. The EHI method is the strategy that extends the
+expected hypervolume improvement strategy, which is commonly used in BO for ordinary Pareto optimization
+problems, to the DR-PF identification problem. Because the expected hypervolume improvement for the DR-PF
+cannot be calculated analytically, we approximated it using Monte Carlo sampling with a sample size of 100.
+Experiments were conducted under the following two settings for the observation of w:
+Simulator setting: At each time t, arbitrary w can be selected.
+Uncontrollable setting: At each time t, w cannot be selected and is observed as a random sample from the
+uniform distribution on Ω.
+The Simulator setting and Uncontrollable setting correspond to the development phase and use phase,
+respectively. In Simulator setting, we used p∗(w) = 1/50, and the next environmental variable to be evaluate
+for each method except Random was selected by
+wt+1 = argmax
+w∈Ω
+(σ(1)2
+t
+(xt+1, w) + σ(2)2
+t
+(xt+1, w)).
+In Random, wt+1 was selected as a random sample from the uniform distribution on Ω. In Uncontrollable
+setting, we allowed the use of a different reference distribution p∗
+t (w) for each iteration t and used the empirical
+distribution of w as p∗
+t (w).
+Under this setup, one initial point was taken at random and the algorithm was run until the number of
+iterations reached 500. This simulation was repeated 100 times and the average values of R1 and R2 at each
+iteration were calculated. From Fig. 3, it can be confirmed that R1 and R2 in Random are not zero even after
+500 iterations for both settings. In UCB F1 and UCB F2, the value of R1 is good but the value of R2 is
+not good because they focus on one of the black-box functions. For MVA, EHI, and Proposed, which focus
+on improving the DR-PF, R1 and R2 tend to be zero in both settings, but Proposed converges to zero more
+quickly.
+5.2. Computational Time Experiments
+We confirmed the computational time required to select xt+1 and wt+1 using each method. We performed the
+same experiment as in Simulator setting in the previous section to evaluate the computational time. Under
+this setup, one initial point was taken at random and the algorithm was run until the number of iterations
+reached 500. We computed the average computational time over 500 iterations. We also computed the ratio
+of the computational time of each method to that of the proposed method. From Table 1, it can be confirmed
+that Random, which does not require inf calculations, is faster than the proposed method, and UCB F1 (or
+8
+
+UCB F2), which uses only u(F (1))
+t
+(x) (or u(F (2))
+t
+(x)) required inf calculations, is about half the computational
+time of the proposed method.
+The proposed method and MVA using both u(F (1))
+t
+(x) and u(F (2))
+t
+(x) have
+comparable computational speed. On the other hand, EHI, which performs the same number of inf calculations
+for each Monte Carlo sample as the proposed method requires, takes about 100 times longer than the proposed
+method because the number of Monte Carlo samples is 100.
+5.3. Infection Simulation
+We applied the proposed method to the Pareto optimization problem using a simulation model of a real-
+world infectious disease. We used the SIR model [Kermack and McKendrick, 1927], which is commonly used
+as the infection simulation model. In this experiment, we used the SIR model which has the infection rate
+β ∈ [0, 1] and the recovery γ ∈ [0, 1]. The input space X × Ω was defined as the set of grid points when the
+region [0.01, 0.5] × [0.01, 0.5] was equally divided into 50 × 50 grid points. Using the SIR model, we defined the
+following two risk functions which can be interpreted as economic risks:
+r1(β, γ) = n(β, γ) − 450β + 800γ − C1,
+r2(β, γ) = n(β, γ) − C2,
+where n(β, γ) is the maximum number of infected individuals during a given period, calculated using the SIR
+model, and C1 and C2 are constants. Notably, r1(β, γ) and r2(β, γ) were also used in [Inatsu et al., 2022]. In
+this experiment, we used the same parameter setting as them. To adapt it to our problem setup, we multiplied
+them by minus one because risk functions should be minimized. Because β and γ can be interpreted as both
+design variables and environmental variables, we considered the following two cases:
+Case1: f (1)(x, w) = −r1(x, w) and f (2)(x, w) = −r2(x, w), where x and w are the infection rate and recovery
+rate, respectively.
+Case2: f (1)(x, w) = −r1(x, w) and f (2)(x, w) = −r2(x, w), where x and w are the recovery rate and infection
+rate, respectively.
+In this experiment, we considered Simulator setting as in Section 5.1.
+Under this setup, one initial point was taken at random and the algorithm was run until the number of
+iterations reached 100. This simulation was repeated 100 times and the average values of R1 and R2 at each
+iteration were calculated. From Fig. 4, it can be confirmed that the proposed method achieves equal or better
+performance in all settings.
+6. Conclusion
+In this study, we proposed an efficient BO method for identifying the DR-PF. We proved that the proposed
+method has theoretical guarantees on accuracy and convergence. Furthermore, through numerical experiments,
+we confirmed that the proposed method outperforms other comparative methods. Future work includes extend-
+ing the method to the case where w is a continuous random variable.
+Acknowledgement
+This work was partially supported by MEXT KAKENHI (20H00601), JST CREST (JPMJCR21D3, JP-
+MJCR21D3), JST Moonshot R&D (JPMJMS2033-05), JST AIP Acceleration Research (JPMJCR21U2), NEDO
+(JPNP18002, JPNP20006) and RIKEN Center for Advanced Intelligence Project.
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+10
+
+Appendix
+A. Generalization of Problem Setup
+Here, we consider a more general problem setting, including the setting introduced in the main text. Specif-
+ically, we consider the following three settings:
+• The case with three or more objective functions.
+• Environment variables cannot be controlled even during optimization.
+• The setting where reference distributions, control parameter ξ, and candidate distribution family A set
+differently at each time.
+Furthermore, by combining (ii) and (iii), we show that under appropriate assumptions, the solution to the
+distributionally robust Pareto optimization problem is also a good solution to the Pareto optimization problem
+defined by the true expectation.
+A.1. Extended Problem Setting
+Let f (1), . . . , f (m) : X × Ω → R be expensive-to-evaluate black-box functions, where 2 ≤ m. Also let X and
+Ω be finite sets. Here, for each (x, w) ∈ X ×Ω and j ∈ {1, . . . , m} ≡ [m], the value of f (j)(x, w) is observed with
+Gaussian noise ε(j) as y(j) = f (j)(x, w) + ε(j), where ε(1), . . . , ε(m) are mutually independent, and ε(j) follows
+Normal distribution with mean zero and variance σ(j)2
+noise, that is, ε(j) ∼ N(0, σ(j)2
+noise). In this section, we assume
+that w ∈ Ω is a discrete random variable and follows some unknown distribution P †. Here, for environmental
+variables, we consider either settings in the development phase or settings in the use phase. That is, in the
+former, environmental variables are completely controllable as design variables; in the latter, environmental
+variables are uncontrollable and observed as realizations from the true distribution. Furthermore, let At denote
+the candidate distribution family of P † at each time t, and consider the following At:
+At = {p(w) | d(p(w), p∗
+t (w)) ≤ ξt},
+where p∗
+t (w) is a user-specified reference distribution, d(·, ·) is a given distance function between distribu-
+tions, and ξt ≥ 0. Then, for each design variable x ∈ X and time t, the distributionally robust expectations
+F (1)
+t
+(x), . . . , F (m)
+t
+(x) are defined as follows:
+F (j)
+t
+(x) =
+inf
+p(w)∈At
+�
+w∈Ω
+f (j)(x, w)p(w), j ∈ [m].
+Hereafter, we aim to efficiently identify the PF Z∗
+t determined by F (1)
+t
+(x), . . . , F (m)
+t
+(x). For each x ∈ X, subset
+E ⊂ X and time t, let Ft(x) = (F (1)
+t
+(x), . . . , F (m)
+t
+(x)) and Ft(E) = {Ft(x) | x ∈ E}. Moreover, for a set
+B ⊂ Rm, the domain Dom(B) dominated by B and the Pareto-frontier Par(B) of B are given by
+Dom(B) = {y ∈ Rm |∃ y′ ∈ B s.t. y ⪯ y′},
+Par(B) = ∂(Dom(B)).
+Then, the PF Z∗
+t defined by F (1)
+t
+(x), . . . , F (m)
+t
+(x) can be expressed as follows:
+Z∗
+t = Par(Ft(X)).
+A.1.1. Gaussian Process
+Next, we construct predictive models for the black-box functions.
+As in the main body, GPs are used to
+model the black-box functions f (1), . . . , f (m). First, for each j ∈ [m], assume that f (j) follows a GP prior
+GP(0, k(j)((x, w), (x′, w′))), where k(j)((x, w), (x′, w′)) is a positive-definite kernel.
+Then, under the given
+dataset {(xi, wi, y(j)
+i )}t
+i=1, the posterior distribution of f (j) is again a GP, and its posterior mean µ(j)
+t (x, w)
+and posterior variance σ(j)2
+t
+(x, w) are given by
+µ(j)
+t (x, w) = k(j)
+t (x, w)⊤(K(j)
+t
++ σ(j)2
+noiseIt)−1y(j)
+t ,
+σ(j)2
+t
+(x, w) = k(j)((x, w), (x, w)) − k(j)
+t (x, w)⊤(K(j)
+t
++ σ(j)2
+noiseIt)−1k(j)
+t (x, w),
+where k(j)
+t (x, w) is the t-dimensional vector whose kth element is k(j)((x, w), (xk, wk)), y(j)
+t
+= (y(j)
+1 , . . . , y(j)
+t )⊤,
+K(j)
+t
+is the t × t matrix whose (k, l)th element is k(j)((xk, wk), (xl, wl)) and It is the t × t identity matrix.
+11
+
+A.2. Proposed Method in the Generalized Setting
+Here, we propose a BO method for efficiently identifying Z∗
+t . Using the same argument as the method used in
+the main body, we construct credible intervals interval for F (1)
+t
+(x), . . . , F (m)
+t
+(x) using the method proposed by
+[Kirschner et al., 2020], and give an estimation method for the PF based on the constructed credible intervals.
+A.2.1. Composition of Credible Intervals and Pareto-frontier Estimation
+For each (x, w) ∈ X ×Ω, j ∈ [m] and time t, let Q(f (j))
+t
+(x, w) = [l(f (j))
+t
+(x, w), u(f (j))
+t
+(x, w)] be a credible interval
+of f (j)(x, w). Here, l(f (j))
+t
+(x, w) and u(f (j))
+t
+(x, w) are given by
+l(f (j))
+t
+(x, w) = µ(j)
+t (x, w) − β1/2
+j,t σ(j)
+t (x, w),
+u(f (j))
+t
+(x, w) = µ(j)
+t (x, w) + β1/2
+j,t σ(j)
+t (x, w),
+where β1/2
+j,t ≥ 0. Then, the credible interval for F (j)
+t
+(x) is denoted as Q(F (j)
+t
+)
+t
+(x) ≡ [l(F (j)
+t
+)
+t
+(x), u(F (j)
+t
+)
+t
+(x)], where
+its lower and upper are given by
+l
+(F (j)
+j
+)
+t
+(x) =
+inf
+p(w)∈At
+�
+w∈Ω
+l(f (j))
+t
+(x, w)p(w),
+u
+(F (j)
+j
+)
+t
+(x) =
+inf
+p(w)∈At
+�
+w∈Ω
+u(f (j))
+t
+(x, w)p(w).
+Next, for each x ∈ X, subset E ⊂ X and time t, we define LCB(m)
+t
+(x), UCB(m)
+t
+(x) and LCB(m)
+t
+(E) as
+LCB(m)
+t
+(x) = (l(F (1)
+t
+)
+t
+(x), . . . , l(F (m)
+t
+)
+t
+(x)), UCB(m)
+t
+(x) = (u(F (1)
+t
+)
+t
+(x), . . . , u(F (m)
+t
+)
+t
+(x)),
+LCB(m)
+t
+(E) = {LCB(m)
+t
+(x) | x ∈ E}.
+Using this, we define the estimated Pareto solutions set ˆΠ(m)
+t
+⊂ X for design variables as
+ˆΠ(m)
+t
+= {x ∈ X | LCB(m)
+t
+(x) ∈ Par(LCB(m)
+t
+(X))}.
+Hereafter, for simplicity, we denote LCB(m)
+t
+(x), UCB(m)
+t
+(x), LCB(m)
+t
+(E) and ˆΠ(m)
+t
+as LCBt(x), UCBt(x),
+LCBt(E) and ˆΠt, respectively.
+A.2.2. Acquisition Function
+Here, we propose an AF to determine the next evaluation point. Similar to the main body, we define the AF
+at(x) for x ∈ X as
+at(x) = dist(UCBt(x), Dom(LCBt(ˆΠt))).
+Then, the next evaluation point is selected as follows:
+Definition A.1 (For the setting in the development phase). The next design variable xt+1 to be evaluated is
+selected by
+xt+1 = argmax
+x∈X
+at(x).
+Similarly, the next environmental variable wt+1 to be evaluated is selected by
+wt+1 = argmax
+w∈Ω
+{σ(1)2
+t
+(xt+1, w) + · · · + σ(m)2
+t
+(xt+1, w)}.
+Notably, since wt+1 cannot be selected in the setting at the use phase, wt+1 is the realized value from P †
+at time t + 1. Here, at(x) can be computed analytically by the following lemma:
+Lemma A.1. Let UCBt(x) = (u1, , . . . , um) and LCBt(ˆΠt) = {(l(i)
+1 , . . . , l(i)
+m ) | 1 ≤ i ≤ k}. Then, at(x) can
+be computed as follows:
+at(x) = max{˜at(x), 0},
+˜at(x) = min
+1≤i≤k max{u1 − l(i)
+1 , . . . , um − l(i)
+m }.
+A.2.3. Stopping Condition
+Here, we give a stopping condition of our proposed algorithm. As in the main body, let ϵ > 0 be a user-
+specified stopping parameter. Then, algorithms terminate if at(x) ≤ ϵ is satisfied. Finally, the pseudo-codes
+of the proposed algorithm in the development phase and use phase settings are given in Algorithm 2 and 3,
+respectively.
+12
+
+Algorithm 2 BO for identifying DR-PF in the development phase setting
+Input: GP prior GP(0, k(j)), candidate distribution family At, tradeoff parameter {βj,t}t≥0, stopping param-
+eter ϵ > 0, j ∈ [m]
+t ← 1
+while at(x) > ϵ do
+Compute Q(F (j)
+t
+)
+t
+(x) for each x ∈ X and j ∈ [m]
+Select the next evaluation point (xt, wt)
+Observe y(j)
+t
+= f (j)(xt, wt) + ε(j)
+t
+for each j ∈ [m]
+Update GPs by adding observed points
+t ← t + 1
+end while
+Output: Return ˆΠt as the estimated set of design variables comprising the DR-PF
+Algorithm 3 BO for identifying DR-PF in the use phase setting
+Input: GP prior GP(0, k(j)), candidate distribution family At, tradeoff parameter {βj,t}t≥0, stopping param-
+eter ϵ > 0, j ∈ [m]
+t ← 1
+while at(x) > ϵ do
+Compute Q(F (j)
+t
+)
+t
+(x) for each x ∈ X and j ∈ [m]
+Select the next evaluation point xt
+Generate wt from P †
+Observe y(j)
+t
+= f (j)(xt, wt) + ε(j)
+t
+for each j ∈ [m]
+Update GPs by adding observed points
+t ← t + 1
+end while
+Output: Return ˆΠt as the estimated set of design variables comprising the DR-PF
+A.3. Theoretical Analysis
+Here, we give theorems on the accuracy and convergence of the proposed algorithms. First, to give theoretical
+guarantees, we assume that for each j ∈ [m], f (j) follows GP GP(0, k(j)((x, w), (x′, w′))). In addition, we assume
+that each prior variance k(j)((x, w), (x, w)) ≡ σ(j)2
+0
+(x, w) satisfies
+max
+(x,w)∈X×Ω σ(j)2
+0
+(x, w) ≤ 1,
+∀j ∈ [m].
+Furthermore, let κ(j)
+T
+be a maximum information gain for f (j) at time T. Moreover, as in the main body,
+we define an ϵ-accurate Pareto region Zϵ,t to quantify the goodness of the predicted ˆΠt as input points that
+constitute the PF. For a positive number ϵ and the m-dimensional vector ϵ = (ϵ, . . . , ϵ), we define Zϵ,t as
+Zϵ,t = {y ∈ Rm |∃ y′ ∈ Z∗
+t s.t. y ⪯ y′ and ∃y′′ ∈ Z∗
+t s.t. y′′ ⪯ y + ϵ}.
+Using Zϵ,t, we define the accuracy of ˆΠt as follows:
+Definition A.2 (Accuracy for ˆΠt). Let ϵ be a positive number.
+Then, we define ˆΠt to be an ϵ-accurate
+estimated solution set if the following holds:
+Ft(ˆΠt) ⊂ Zϵ,t.
+(A.1)
+In addition, we define ˆΠt to be an ϵ-accurate estimated Pareto solution set if the following holds:
+Par(Ft(ˆΠt)) ⊂ Zϵ,t.
+(A.2)
+Then, the following theorem guarantees that the proposed algorithms satisfy both (A.1) and (A.2) with a
+high probability.
+Theorem A.1. Let t ≥ 1 and δ ∈ (0, 1), and define β1,t = · · · = βm,t = 2 log(m|X × Ω|π2t2/(6δ)) ≡ βt.
+Moreover, let ϵ > 0 be a user-specified stopping parameter.
+Then, when Algorithm 2 terminates, with a
+probability of at least 1 − δ, ˆΠt satisfies both (A.1) and (A.2) for any At.
+Next, we give a theorem on convergence in the development phase setting. The following theorem gives
+convergence guarantees for Algorithm 2:
+13
+
+Theorem A.2. Under the same condition as in Theorem A.1, let T be the smallest positive integer satisfying
+the following inequality:
+βT (C1κ(1)
+T
++ · · · + Cmκ(m)
+T
+)
+T
+≤ ϵ2
+4 ,
+where Cj = 2/ log(1 + σ(j)−2
+noise ) and j ∈ [m]. Then, Algorithm 2 terminates after at most T iterations.
+Next, we give the theorem on convergence under the setting in the use phase. Unlike the setting in the
+development phase, we cannot control w in the use phase. Therefore, to make a reasonable inference, the
+uncertainty at all points must be able to be reduced stochastically. However, if the value of the true probability
+function p†(w) at some w ∈ Ω is zero, the uncertainty of f (j)(x, w) containing this point cannot be sufficiently
+small. To avoid this problem, we make the following assumption on the true probability function:
+min
+w∈Ω p†(w) ≡ pmin > 0.
+Then, the following theorem holds:
+Theorem A.3. Under the same condition as in Theorem A.1, assume that pmin > 0. Let T be the smallest
+positive integer satisfying the following inequality:
+βT ( ˜C1κ(1)
+T
++ · · · + ˜Cmκ(m)
+T
++ ˜C)
+T
+≤ ϵ2
+4 ,
+where ˜Cj = (4p−1
+min)/ log(1 + σ(j)−2
+noise ), ˜C = 8mp−1
+min log(8m/δ) and j ∈ [m]. Then, with a probability of at least
+1 − δ, Algorithm 3 terminates after at most T trials.
+Finally, we show that under appropriate assumptions, the solution to the distributionally robust Pareto
+optimization problem is also a good solution to the Pareto optimization problem defined by the true expectation
+in the use phase setting. Here, the expectation function ˜F (j)(x) determined by the true probability function
+p†(w) is given by
+˜F (j)(x) =
+�
+w∈Ω
+f (j)(x, w)p†(w), j ∈ [m].
+In addition, for each x ∈ X and E ⊂ X, let ˜F (x) = ( ˜F (1)(x), . . . , ˜F (m)(x)) and ˜F (E) = { ˜F (x) | x ∈ E}.
+Then, the PF ˜Z∗ defined by ˜F (1)(x), . . . , ˜F (m)(x) can be expressed as follows:
+˜Z∗ = Par( ˜F (X)).
+Furthermore, as in the case of Z∗
+t , we define an ϵ-accurate Pareto region ˜Zϵ for ˜Z∗. For a positive number ϵ
+and an m-dimensional vector ϵ = (ϵ, . . . , ϵ), we define ˜Zϵ as
+˜Zϵ = {y ∈ Rm |∃ y′ ∈ ˜Z∗ s.t. y ⪯ y′ and ∃y′′ ∈ ˜Z∗ s.t. y′′ ⪯ y + ϵ}.
+Using ˜Zϵ, we define the accuracy of ˆΠt for ˜Z∗.
+Definition A.3 (Accuracy of ˆΠt for ˜Z∗). Let ϵ be a positive number. Then, we define ˆΠt to be an ϵ-accurate
+estimated solution set for ˜Z∗ if the following holds:
+Ft(ˆΠt) ⊂ ˜Zϵ.
+Moreover, we define ˆΠt to be an ϵ-accurate estimated Pareto solution set for ˜Z∗ if the following holds:
+Par(Ft(ˆΠt)) ⊂ ˜Zϵ.
+Then, the following theorem holds:
+Theorem A.4. Under the use phase setting, let t ≥ 1 and δ ∈ (0, 1), and define β1,t = · · · = βm,t =
+2 log(m|X × Ω|π2t2/(6δ)) ≡ βt. Moreover, let ϵ > 0 be a user-specified stopping parameter. Furthermore, let
+the reference distribution p∗
+t (w) be the empirical distribution function for w, and define ξt and the distance
+between distributions d(·, ·) as
+ξt = |Ω|
+�
+1
+2t log
+�|Ω|π2t2
+3δ
+�
+,
+d(p1(w), p2(w)) =
+�
+w∈Ω
+|p1(w) − p2(w)|.
+Then, if Algorithm 3 terminates at t ≥ T after time T, with a probability of at least 1−2δ, ˆΠt is the 2ϵ-accurate
+estimated set and estimated Pareto set, that is, the following holds:
+Ft(ˆΠt) ⊂ ˜Z2ϵ,
+Par(Ft(ˆΠt)) ⊂ ˜Z2ϵ.
+Here, T is the smallest positive integer satisfying the following inequality:
+∀n ≥ T, 2β1/2
+1
+ξn ≤ ϵ.
+(A.3)
+14
+
+B. Proofs
+Here, we give proofs of Lemma A.1 and Theorem A.1–A.4. Notably, Lemma 3.1 and Theorem 4.1–4.2 in the
+main body are special cases of Lemma A.1 and Theorem A.1–A.2, respectively. First, we prove Lemma A.1.
+Proof. Let UCBt(x) = (u1, . . . , um) ≡ u and LCBt(ˆΠt) = {(l(i)
+1 , . . . , l(i)
+m ) | 1 ≤ i ≤ k} ≡ L.
+Here, if
+u ∈ Dom(L), then the following holds from the definition of dist(a, B):
+at(x) = dist(u, Dom(L)) =
+inf
+b∈Dom(L) d∞(u, b) = d∞(u, u) = 0.
+In addition, since u ∈ Dom(L), there exists (l(i)
+1 , . . . , l(i)
+m ) such that uj ≤ l(i)
+j
+for any j ∈ [m]. Thus, we have
+max{u1 − l(i)
+1 , . . . , um − l(i)
+m } ≤ 0. This implies that
+˜at(x) = min
+1≤i≤k max{u1 − l(i)
+1 , . . . , um − l(i)
+m } ≤ 0
+and max{˜at(x), 0} = 0. Therefore, we get at(x) = max{˜at(x), 0}. Next, we consider the case where u /∈
+Dom(L). Let at(x) = η. Then, noting that u /∈ Dom(L), for any i ∈ {1, . . . , k}, there exists j ∈ [m] such that
+uj > l(i)
+j . This implies that
+˜at(x) = min
+1≤i≤k max{u1 − l(i)
+1 , . . . , um − l(i)
+m } ≡ ˜η > 0
+and max{˜at(x), 0} = ˜at(x) = ˜η. For this ˜η, there exists i such that
+uj − l(i)
+j
+≤ ˜η
+∀j ∈ [m].
+Hence, we have ˜u ≡ (u1 − ˜η, . . . , um − ˜η) ∈ Dom(L) because uj − ˜η ≤ l(i)
+j
+for any j ∈ [m]. Thus, from the
+definition of at(x), the following holds:
+η = at(x) = dist(u, Dom(L)) =
+inf
+b∈Dom(L) d∞(u, b) ≤ d∞(u, ˜u) = ˜η.
+Here, we assume η < ˜η. Then, noting that Dom(L) is the closed set, there exists ˜l = (˜l1, . . . , ˜lm) ∈ Dom(L)
+such that d∞(u, ˜l) = η. Therefore, ˜l can be expressed as ˜l = (u1 − s1, . . . , um − sm), where 0 ≤ |sj| ≤ η and at
+least one of s1, . . . , sm is η. Thus, since (u1 − η, . . . , um − η) ⪯ ˜l, noting that (u1 − η, . . . , um − η) ∈ Dom(L)
+there exists i such that
+uj − η ≤ l(i)
+j
+∀j ∈ [m].
+This implies that max{u1 − l(i)
+1 , . . . , um − l(i)
+m } ≤ η. Hence, it follows that
+˜η = min
+1≤i≤k max{u1 − l(i)
+1 , . . . , um − l(i)
+m } ≤ η.
+However, this is a contradiction with η < ˜η. Consequently, we obtain at(x) = max{˜at(x), 0}.
+Next, we prove Theorem A.1.
+Proof. First, we prove Ft(ˆΠt) ⊂ Zϵ,t. Since ˆΠt ⊂ X, for any y ∈ Ft(ˆΠt), there exists y′ ∈ Z∗
+t such that
+y ⪯ y′. Thus, it is sufficient to show that there exists y′′ ∈ Z∗
+t such that y′′ ⪯ y +ϵ. Here, under the theorem’s
+assumption, with a probability of at least 1 − δ, the following holds for any (x, w) ∈ X × Ω, j ∈ [m] and time
+t ≥ 1:
+f (j)(x, w) ∈ Q(f (j))
+t
+(x, w),
+where this relation can be derived by using Lemma 5.1 of [Srinivas et al., 2010]. Hence, we have F (j)
+t
+(x) ∈
+Q(F (j)
+t
+)
+t
+(x). In addition, since Z∗
+t is the closed set, for any x ∈ ˆΠt, there exist a ≥ 0 and Ft(x) + (a, . . . , a) ≡
+y′′ ∈ Z∗
+t such that
+dist(Ft(x), Z∗
+t ) = d∞(Ft(x), y′′) ≤ d∞((l(F (1)
+t
+)
+t
+(x), . . . , l(F (m)
+t
+)
+t
+(x)), y′′′),
+where y′′′ ∈ Z∗
+t can be given by using s ≥ a ≥ 0 as y′′′ = (l(F (1)
+t
+)
+t
+(x), . . . , l(F (m)
+t
+)
+t
+(x)) + (s, . . . , s). Here, for some
+ˆx ∈ X satisfying y′′′ ⪯ Ft(ˆx), the following holds:
+d∞((l(F (1)
+t
+)
+t
+(x), . . . , l(F (m)
+t
+)
+t
+(x)), y′′′) ≤ dist(Ft(ˆx), Dom(LCBt(ˆΠt))).
+15
+
+Moreover, the right hand side is bounded from above as
+dist(Ft(ˆx), Dom(LCBt(ˆΠt)))
+≤ dist(UCBt(ˆx), Dom(LCBt(ˆΠt)))
+≤ max
+x†∈X dist(UCBt(x†), Dom(LCBt(ˆΠt))) = at(xt+1).
+Therefore, if at(xt+1) ≤ ϵ, then d∞(Ft(x), y′′) ≤ ϵ. It follows that y′′ ⪯ Ft(x) + ϵ. Since x is an arbitrary
+element of ˆΠt, we have Ft(ˆΠt) ⊂ Zϵ,t. Next, we prove Par(Ft(ˆΠt)) ⊂ Zϵ,t. As before, noting that ˆΠt ⊂ X, for
+any y ∈ Par(Ft(ˆΠt)), there exists y′ ∈ Z∗
+t such that y ⪯ y′. In addition, since Z∗
+t is the closed set, for any
+y ∈ Par(Ft(ˆΠt)), there exist a ≥ 0 and y + (a, . . . , a) ≡ y′′ ∈ Z∗
+t such that
+dist(y, Z∗
+t ) = d∞(y, y′′).
+Here, for some ˆx ∈ X satisfying y′′ ⪯ Ft(ˆx), the following holds:
+d∞(y, y′′) ≤ d∞(y′′′, Ft(ˆx)) ≤ dist(Ft(ˆx), Dom(LCBt(ˆΠt))),
+where y′′′ ∈ Par(Ft(ˆΠt)) can be given by using s′ ≥ a ≥ 0 as y′′′ = Ft(ˆx) − (s′, . . . , s′). Then, we obtain
+dist(Ft(ˆx), Dom(LCBt(ˆΠt)))
+≤ dist(UCBt(ˆx), Dom(LCBt(ˆΠt)))
+≤ max
+x†∈X dist(UCBt(x†), Dom(LCBt(ˆΠt))) = at(xt+1).
+Thus, if at(xt+1) ≤ ϵ, then d∞(y, y′′) ≤ ϵ. This implies that y′′ ⪯ y + ϵ. Consequently, since y is an arbitrary
+element of Par(Ft(ˆΠt)), we have Par(Ft(ˆΠt)) ⊂ Zϵ,t.
+Next, we prove Theorem A.2.
+Proof. Let xt = argmaxx∈X at−1(x). Here, since LCB t−1(xt) ∈ Dom(LCB t−1(ˆΠt−1)), the following in-
+equality holds:
+at−1(xt) = dist(UCBt−1(xt), Dom(LCB t−1(ˆΠt−1)))
+≤ d∞(UCB t−1(xt), LCB t−1(xt))
+= max
+1≤j≤m{u
+(F (j)
+t−1)
+t−1
+(xt) − l
+(F (j)
+t−1)
+t−1
+(xt)}.
+Therefore, from the definition of u
+(F (j)
+t−1)
+t−1
+(xt) and l
+(F (j)
+t−1)
+t−1
+(xt), we get
+u
+(F (j)
+t−1)
+t−1
+(xt) − l
+(F (j)
+t−1)
+t−1
+(xt) ≤ 2β1/2
+t
+max
+w∈Ω σ(j)
+t−1(xt, w).
+Hence, the following inequality holds:
+a2
+t−1(xt) ≤ 4βt max
+1≤j≤m max
+w∈Ω σ(j)2
+t−1 (xt, w).
+Furthermore, since wt is selected by
+wt = argmax
+w∈Ω
+(σ(1)2
+t−1 (xt, w) + · · · + σ(m)2
+t−1 (xt, w)),
+the following holds:
+a2
+t−1(xt) ≤ 4βt max
+1≤j≤m max
+w∈Ω σ(j)2
+t−1 (xt, w)
+≤ 4βt(σ(1)2
+t−1 (xt, wt) + · · · + σ(m)2
+t−1 (xt, wt)).
+In addition, let T be the number given by Theorem A.2. Then, we get
+T min
+1≤t≤T a2
+t−1(xt) ≤
+T
+�
+t=1
+a2
+t−1(xt)
+≤ 4βT
+m
+�
+j=1
+T
+�
+t=1
+σ(j)2
+t−1 (xt, wt)
+≤ 4βT (C1κ(1)
+T
++ · · · + Cmκ(m)
+T
+),
+16
+
+where the last inequality can be derived by using Lemma 5.3 and 5.4 of [Srinivas et al., 2010].
+Therefore,
+dividing both sides by T, we obtain
+min
+1≤t≤T a2
+t−1(xt) ≤ 4βT
+C1κ(1)
+T
++ · · · + Cmκ(m)
+T
+T
+≤ ϵ2.
+Hence, we get min1≤t≤T at−1(xt) ≤ ϵ. This implies that there exists t′ ∈ {1, . . . , T} such that at′−1(xt′) ≤ ϵ.
+Next, we prove Theorem A.3.
+Proof. Using the same argument as in the proof of Theorem A.2, we have
+a2
+t−1(xt) ≤ 4βt max
+1≤j≤m max
+w∈Ω σ(j)2
+t−1 (xt, w).
+Furthermore, since pmin > 0, the following inequality holds:
+max
+w∈Ω σ(j)2
+t−1 (xt, w) ≤
+�
+wΩ
+σ(j)2
+t−1 (xt, w)
+=
+�
+wΩ
+(p†(w))−1p†(w)σ(j)2
+t−1 (xt, w)
+≤ p−1
+min
+�
+wΩ
+p†(w)σ(j)2
+t−1 (xt, w)
+≡ p−1
+minEw[σ(j)2
+t−1 (xt, w)].
+Using this, we get
+max
+1≤j≤m max
+w∈Ω σ(j)2
+t−1 (xt, w) ≤
+m
+�
+j=1
+max
+w∈Ω σ(j)2
+t−1 (xt, w)
+≤ p−1
+min
+m
+�
+j=1
+Ew[σ(j)2
+t−1 (xt, w)]
+= p−1
+minEw
+�
+�
+m
+�
+j=1
+σ(j)2
+t−1 (xt, w)
+�
+� .
+Here, let T be the number given by Theorem A.3. Then, we obtain
+T min
+1≤t≤T a2
+t−1(xt) ≤
+T
+�
+t=1
+a2
+t−1(xt)
+≤ 4βT p−1
+min
+T
+�
+t=1
+Ew
+�
+�
+m
+�
+j=1
+σ(j)2
+t−1 (xt, w)
+�
+� .
+(B.1)
+Noting that �m
+j=1 σ(j)2
+t−1 (xt, w) is a non-negative random variable and �m
+j=1 σ(j)2
+t−1 (xt, w) ≤ m, from Lemma 3
+of [Kirschner and Krause, 2018], the following holds with a probability of at least 1 − δ:
+T
+�
+t=1
+Ew
+�
+�
+m
+�
+j=1
+σ(j)2
+t−1 (xt, w)
+�
+� ≤ 2
+m
+�
+j=1
+T
+�
+t=1
+σ(j)2
+t−1 (xt, w) + 4m log 1
+δ + 8m log 4m + 1.
+Moreover, since 4m log 1
+δ ≤ 8m log 1
+δ and 1 ≤ 8m log 2, we have
+4m log 1
+δ + 8m log 4m + 1 ≤ 8m log 1
+δ + 8m log 4m + 8m log 2
+= 8m log 8m
+δ
+and
+T
+�
+t=1
+Ew
+�
+�
+m
+�
+j=1
+σ(j)2
+t−1 (xt, w)
+�
+� ≤ 2
+m
+�
+j=1
+T
+�
+t=1
+σ(j)2
+t−1 (xt, w) + 8m log 8m
+δ .
+(B.2)
+17
+
+Hence, by substituting (B.2) into (B.1), from Lemma 5.3 and 5.4 of [Srinivas et al., 2010], we get
+T min
+1≤t≤T a2
+t−1(xt) ≤ 4βT ( ˜C1κ(1)
+T
++ · · · + ˜Cmκ(m)
+T
++ ˜C).
+Therefore, from the definition of T, dividing both sides by T, we obtain
+min
+1≤t≤T a2
+t−1(xt) ≤ 4βT
+˜C1κ(1)
+T
++ · · · + ˜Cmκ(m)
+T
++ ˜C
+T
+≤ ϵ2.
+Thus, we get min1≤t≤T at−1(xt) ≤ ϵ. This implies that there exists t′ ∈ {1, . . . , T} such that at′−1(xt′) ≤ ϵ.
+Finally, we prove Theorem A.4.
+Proof. Suppose that p∗
+t (w) is the empirical distribution function of w. Then, from Hoeffding’s inequality, the
+following inequality holds for any w:
+P(|p∗
+t (w) − p†(w)| ≥ λ) ≤ 2 exp(−2tλ2).
+Here, let
+λ =
+�
+1
+2t log
+�|Ω|π2t2
+3δ
+�
+.
+Then, with a probability of at least 1 − δ, the following holds for any t ≥ 1 and w ∈ Ω:
+|p∗
+t (w) − p†(w)| ≤ λ.
+In addition, from the definition of d(·, ·), we get
+d(p∗
+t (w), p†(w)) =
+�
+w∈Ω
+|p∗
+t (w) − p†(w)| ≤ |Ω|λ = ξt.
+This implies that p†(w) ∈ At. Furthermore, from the definition of ˜F (j)(x) and F (j)
+t
+(x), the following inequality
+holds for any t ≥ 1, j ∈ [m] and x ∈ X:
+F (j)
+t
+(x) ≤ ˜F (j)(x).
+Therefore, it follows that
+Ft(x) ⪯ ˜F (x)
+∀x ∈ X,∀ t ≥ 1.
+(B.3)
+Here, for any x ∈ X, t ≥ 1 and j ∈ [m], let ¯p(j)
+t,x(w) ∈ At be the probability function satisfying
+F (j)
+t
+(x) =
+�
+w∈Ω
+f (j)(x, w)¯p(j)
+t,x(w).
+Then, the following inequality holds:
+| ˜F (j)(x) − F (j)
+t
+(x)| ≤
+�
+w∈Ω
+|f (j)(x, w)||p†(w) − ¯p(j)
+t,x(w)|.
+Moreover, from Lemma 5.1 of [Srinivas et al., 2010], with a probability of at least 1 − δ, the following holds for
+any x ∈ X, w ∈ Ω and j ∈ [m]:
+|f (j)(x, w)| ≤ β1/2
+1
+σ(j)
+0 (x, w) ≤ β1/2
+1
+.
+Hence, we have
+˜F (j)(x) − F (j)
+t
+(x) ≤ | ˜F (j)(x) − F (j)
+t
+(x)|
+≤ β1/2
+1
+�
+w∈Ω
+|p†(w) − ¯p(j)
+t,x(w)|
+= β1/2
+1
+d(p†(w), ¯p(j)
+t,x(w))
+≤ β1/2
+1
+(d(p†(w), p∗
+t (w)) + d(p∗
+t (w), ¯p(j)
+t,x(w))) ≤ 2β1/2
+1
+ξt.
+In addition, let T be the smallest positive integer satisfying (A.3). Then, for any t ≥ T, the following inequality
+holds:
+˜F (j)(x) ≤ F (j)
+t
+(x) + ϵ.
+18
+
+Thus, we obtain
+˜F (x) ⪯ Ft(x) + ϵ
+∀x ∈ X,∀ t ≥ T.
+(B.4)
+Therefore, by combining (B.3) and (B.4), we have
+Par(Ft(X)) ⊂ ˜Zϵ
+∀t ≥ T.
+(B.5)
+Finally, from Theorem A.1, the following holds at t′, the time at which the algorithm terminates:
+Ft′(ˆΠt′) ⊂ Zϵ,t′, Par(Ft′(ˆΠt′)) ⊂ Zϵ,t′.
+(B.6)
+Consequently, if t′ ≥ T, using (B.5) and (B.6) we get
+Ft′(ˆΠt′) ⊂ ˜Z2ϵ, Par(Ft′(ˆΠt′)) ⊂ ˜Z2ϵ.
+C. Experimental Details and Additional Experiments
+Here, we give experimental details and additional experiments in Section 5.
+Experimental Setup
+The experimental parameters used in each experiment are described in Table 2.
+Table 2: Experimental parameters for each setting
+Parameters
+Simulator setting
+σ2
+f,1 = 1000, L1 = 2, σ(1)2
+noise = 10−4, β1/2
+1,t = 3, σ2
+f,2 = 1000, L2 = 2, σ(2)2
+noise = 10−4, β1/2
+2,t = 3, ξ = 0.05
+Uncontrollable setting
+σ2
+f,1 = 1000, L1 = 2, σ(1)2
+noise = 10−4, β1/2
+1,t = 3, σ2
+f,2 = 1000, L2 = 2, σ(2)2
+noise = 10−4, β1/2
+2,t = 3, ξ = 0.05
+SIR (Case1)
+σ2
+f,1 = 5000, L1 = 0.1, σ(1)2
+noise = 10−8, β1/2
+1,t = 3, σ2
+f,2 = 105, L2 = 0.01, σ(2)2
+noise = 10−4, β1/2
+2,t = 2, ξ = 0.15
+SIR (Case2)
+σ2
+f,1 = 104, L1 = 0.1, σ(1)2
+noise = 10−3, β1/2
+1,t = 2, σ2
+f,2 = 105, L2 = 0.1, σ(2)2
+noise = 10−3, β1/2
+2,t = 3, ξ = 0.15
+MVA
+The MVA method is based on reducing the uncertainty in the potential optimal set Mt and the
+estimated PF solution set ˆΠt. Using ˆΠt, Mt is defined as follows:
+Mt = {x ∈ X \ ˆΠt |∀ x′ ∈ ˆΠt, u(F (1))
+t
+(x) > l(F (1))
+t
+(x′) or u(F (2))
+t
+(x) > l(F (2))
+t
+(x′)}.
+The uncertainty λt(x) is given by
+λt(x) =
+�
+(u(F (1))
+t
+(x) − l(F (1))
+t
+(x))2 + (u(F (2))
+t
+(x) − l(F (2))
+t
+(x))2.
+EHI
+The EHI method is based on the expected hypervolume improvement for a bounded region defined by
+PFs and reference points. Let B ⊂ R2 be a set, and let r = (r1, r2) ∈ R2 be a reference point satisfying r ⪯ B.
+Then, let us denote the bounded region dominated by B and r, by
+Dom(B; r) = Dom(B) ∩ [r1, ∞) × [r2, ∞).
+In EHI, for each j ∈ {1, 2} we calculated the estimated value of F (j)(x) by using posterior means as
+µ(F (j))
+t
+(x) =
+inf
+p(w)∈A
+�
+w∈Ω
+µ(j)
+t (x, w)p(w).
+For a point x ∈ X a subset E ⊂ X, we define µ(F )
+t
+(x) and µ(F )
+t
+(E) as
+µ(F )
+t
+(x) = (µ(F (1))
+t
+(x), µ(F (2))
+t
+(x)),
+µ(F )
+t
+(E) = {µ(F )
+t
+(x) | x ∈ E}.
+In our experiments, we defined the reference point rt = (rt,1, rt,2) for each iteration t as
+rt,1 = min
+x∈X µ(F (1))
+t
+(x),
+rt,2 = min
+x∈X µ(F (2))
+t
+(x).
+Then, the expected hypervolume improvement for x ∈ X is given by
+EHIt(x) = EF (1)(x),F (2)(x)[Vol(Dom(µ(F )
+t
+(X) ∪ {(F (1)(x), F (2)(x))}; rt) \ Dom(µ(F )
+t
+(X); rt))].
+19
+
+Table
+3: Computational time (second) and computational time ratio for each setting when Nx = 50 and
+Nw = 100
+Random
+UCB F1
+UCB F2
+MVA
+EHI
+Proposed
+Computational time
+0.000
+0.181
+0.185
+0.375
+45.77
+0.364
+(Standard error)
+(0.000)
+(0.001)
+(0.001)
+(0.001)
+(0.025)
+(0.001)
+Computational time ratio
+0.000
+0.500
+0.510
+1.031
+126.15
+1
+(Standard error)
+(0.000)
+(0.002)
+(0.002)
+(0.003)
+(0.327)
+(0)
+Table
+4: Computational time (second) and computational time ratio for each setting when Nx = 100 and
+Nw = 50
+Random
+UCB F1
+UCB F2
+MVA
+EHI
+Proposed
+Computational time
+0.000
+0.134
+0.135
+0.272
+31.97
+0.268
+(Standard error)
+(0.000)
+(0.001)
+(0.001)
+(0.001)
+(0.015)
+(0.001)
+Computational time ratio
+0.000
+0.503
+0.507
+1.019
+119.92
+1
+(Standard error)
+(0.000)
+(0.002)
+(0.002)
+(0.004)
+(0.346)
+(0)
+Table
+5: Computational time (second) and computational time ratio for each setting when Nx = 100 and
+Nw = 100
+Random
+UCB F1
+UCB F2
+MVA
+EHI
+Proposed
+Computational time
+0.000
+0.360
+0.362
+0.719
+91.82
+0.726
+(Standard error)
+(0.000)
+(0.001)
+(0.001)
+(0.002)
+(0.099)
+(0.002)
+Computational time ratio
+0.000
+0.496
+0.500
+0.991
+127.02
+1
+(Standard error)
+(0.000)
+(0.001)
+(0.001)
+(0.002)
+(0.412)
+(0)
+Here, for a bounded set A, Vol(A) represents the hypervolume of A.
+Because F (1)(x) and F (2)(x) do not
+follow GPs, we cannot calculate EHIt(x) analytically. Thus, we approximate it by using samples from posterior
+distributions.
+Let M be a number of Monte Carlo sampling, and let (f (j)
+t,(l)(x, w1), . . . , f (j)
+t,(l)(x, w|Ω|)) be an
+lth sample from the posterior distribution of (f (j)(x, w1), . . . , f (j)(x, w|Ω|)) at iteration t, where 1 ≤ l ≤ M,
+j ∈ {1, 2} and x ∈ X. Then, for each t, we calculate F (1)
+t,(l)(x) and F (2)
+t,(l)(x) as
+F (1)
+t,(l)(x) =
+inf
+p(w)∈A
+�
+w∈Ω
+f (1)
+t,(l)(x, w)p(w),
+F (2)
+t,(l)(x) =
+inf
+p(w)∈A
+�
+w∈Ω
+f (2)
+t,(l)(x, w)p(w).
+Using this, we approximate EHIt(x) as
+1
+M
+M
+�
+l=1
+Vol(Dom(µ(F )
+t
+(X) ∪ {(F (1)
+t,(l)(x), F (2)
+t,(l)(x))}; rt) \ Dom(µ(F )
+t
+(X); rt)).
+Additional Computational Time Experiments
+We also compared the computational time of each method
+by changing input space settings conducted in Section 5.2. In this experiment, the input space X × Ω was a set
+of grid points divided into [−10, 10] × [−10, 10] equally spaced at Nx × Nw. We compared computational times
+using (50, 100), (100, 50) and (100, 100) as (Nx, Nw). From Table 3–5, it can be confirmed that the results are
+similar to the experimental results conducted in Section 5.2.
+20
+

8tE1T4oBgHgl3EQfUAOX/content/tmp_files/2301.03085v1.pdf.txt

@@ -0,0 +1,726 @@
+Granger causality test for heteroskedastic and
+structural-break time series using generalized least squares
+Hugo J. Bello ∗
+January 10, 2023
+Abstract
+This paper proposes a novel method (GLS Granger test) to determine causal relation-
+ships between time series based on the estimation of the autocovariance matrix and gener-
+alized least squares. We show the effectiveness of proposed autocovariance matrix estimator
+(the sliding autocovariance matrix) and we compare the proposed method with the classical
+Granger F-test with via a synthetic dataset and a real dataset composed by cryptocurren-
+cies. The simulations show that the proposed GLS Granger test captures causality more
+accurately than Granger F-tests in the cases of heteroskedastic or structural-break residu-
+als. Finally, we use the proposed method to unravel unknown causal relationships between
+cryptocurrencies.
+Keywords:
+Granger Test, Causality, Generalized Least Squares, Wald Test.
+1
+Introduction
+Granger causality is a statistical concept that helps us determine whether the information in
+one time series is useful in predicting another. It is widely used in numerous fields, including
+economics, finance, geology, genetics and neuroscience, to understand the relationships between
+variables and to identify possible causal connections.
+The main tool for studying time series causality is the Granger causality F-test ([Gra69],
+[Gra80]) which is based on ordinary least squares (OLS) estimation and therefore is tied to the
+assumptions of OLS (among them homocedasticity of the residuals)
+In many real-world applications, we are confronted with time series suffering from a variety
+of problems such as heteroscedasticity (non-equal variance amongst time observations) or struc-
+tural breaks (the appearance of two or more stationary periods with different means). In these
+situations OLS is not recommended and Granger F-test can be inadequate.
+The aim of this paper is to present a method for determining Granger causality based on
+generalized least squares based on the estimation of the autocovariance matrix of the residuals.
+1.1
+Preliminaries
+In this section we will present introducing notation and definitions. We also review the notion
+of Granger causality.
+∗hugojose.bello@uva.es Department of Applied Mathematics, University of Valladolid (Campus Soria)
+1
+arXiv:2301.03085v1  [stat.ME]  8 Jan 2023
+
+1.1.1
+Granger causality F-test
+The Granger causality ([Gra69], [Gra80]) test is a statistical hypothesis test for determining
+whether one time series x = (xt)N
+t=0 is useful in forecasting another y = (yt)N
+t=0. In particular
+Granger causality focuses on the possibility of x and y to predict future values of y significatively
+better than those of y alone. In this case it is said that x Granger causes y
+The idea behind this is that a cause should be helpful in predicting the future effects, beyond
+what can be predicted solely based on their own past values.
+The test null hypothesis states that the linear regression model
+yt = β0 +
+p
+�
+k=1
+βkyt−k + εt
+(1)
+approximates y significatively better than the model
+yt = β0 +
+p
+�
+k=1
+βkyt−k +
+p
+�
+k=1
+β′
+kxt−k + εt
+(2)
+If the null hypothesis is correct, this will imply that the lagged values of x add explanatory
+power to the prediction of y and therefore the process behind x causes y.
+To test if model (2) is significatively more accurate than (1) the following statistic is used
+(SSRRM − SSRUM)/N
+SSEUM/(N − 2)(p − 1)
+(3)
+where SSRRM and SSRUM are the sum of residuals for the restricted and unrestricted models
+respectively
+SSRRM = Σt(yt − β0 −
+p
+�
+k=1
+βkyt−k)2
+SSRUM = Σt(yt − β0 −
+p
+�
+k=1
+βkyt−k −
+p
+�
+k=1
+β′
+kxt−k)2
+If the necessary assumptions for ordinary least of squares are satisfied, (3) follows a F(p, N −
+2p − 1) distribution under the null hypothesis.
+Observation 1. Notice that 1 can be understood as writing y as a linear combination of the
+lagged time series Bky (using the backward operatior notation), that is:
+y = β0 +
+L
+�
+k=1
+βkBky + ε
+Similarly 1
+y = β0 +
+L
+�
+k=1
+βkBky +
+L
+�
+k=1
+βkBkx + ε
+Therefore Granger causality can be understood as the use of a F-test to compare two multi-
+linear regression models.
+2
+
+1.1.2
+Generalized Least Squares
+For a linear regression model of the form
+yi = β1 xi1 + β2 xi2 + · · · + βp xip + εi,
+(4)
+The following assumptions must be satisfied for ordinary least squares (OLS) to have the
+desired asymptotic properties:
+(P1) Correct specification.
+Te underlying process generating the data must be in esence
+linear.
+(P2) Strict exogeneity. The errors in the regression must have conditional mean zero: E[ ε | X ] = 0..
+(P3) No linear dependence. The regressors must all be linearly independent.
+(P4) Homoscedasticity E[ε2
+i |X] = σ2 ∀i. The error term has the same variance σ2 in each
+observation.
+(P5) No autocorrelation E[εiεj|X] = 0 ∀i ̸= j. The errors are uncorrelated between observa-
+tions.
+(P6) Normality. It is sometimes additionally assumed that the errors have normal distribution
+conditional on the regressors
+If homocedasticity (P4) or non-autocorrelation (P5) assumptions are not satisfied we can use
+Generalized Least Squares (GLS) to better approximate the parameters of (4).
+Using the standard matricial notation (4) can be written as
+y = Xβ + ε
+where y = (yi), X = (xT
+1 . . . xT
+p ) is the design matrix and ε is the error term. if the error
+term satisfies that E[ε|X] = 0 and denoting cov[ε|X] = Ω the non-singular covariance matrix of
+the residuals. the GLS estimate for β is
+�βGLS = argminβ(y − Xβ)T Ω(y − Xβ) =
+�
+XTΩ−1X
+�−1 XTΩ−1y
+(5)
+(see [Gre03, §9.3]) Notice that for Ω = σ2I (where I is the identity matrix) we are under the
+assumptions of OLS and the resulting estimator is
+�βOLS =
+�
+XTX
+�−1 XTy
+(6)
+It is known that
+E[�βGLS] = βGLS
+(7)
+cov[�βGLS] = V = (XT Ω−1X)−1
+(8)
+In fact
+√
+N(�βGLS − βGLS) −→D N(0, V )
+1.1.3
+Wald Test
+The Wald test is a statistical hypothesis test that assesses constrains on statistical parameters
+for regression models based on the weighted distance between an unrestricted estimate and its
+hypothesized value under the null hypothesis (see [Gre03, §5.3]).
+Let �β the sample estimate for the regression model (GLS or OLS) model with covariance V
+as described before. If Q hypothesis on the p parameters are expressed in the form of a Q × p
+matrix R:
+3
+
+H0 : Rβ = r
+H1 : Rβ ̸= r
+The wald test statistic is
+(R�β − r)T ·
+�
+R�V RT · 1
+n
+�−1
+· (R�β − r)
+(9)
+Under the null hypothesis, the Wald statistic (9) follows a F(Q, N − p) distribution.
+Granger F-test as a Wald Test
+The Granger F-Test described in (3) is in fact a particular case of the Wald test. If we consider
+�βOLS, then we can stablish the unrestricted regression model
+y = β0 + β1By + . . . + βpBpy + β′
+1Bx + . . . + β′
+pBpx
+(10)
+Where B is the backshift operator Bkx = (xt−k)t. So for y to be caused by x the coefficients
+β′
+1, . . . β′
+p must be zero, therefore we need to test the hypothesis
+H0 :β′
+k = 0 for all k ≤ p
+(11)
+H1 :β′
+k ̸= 0 for some k ≤ p
+Defining the matrices
+R =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+...
+0
+1
+...
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+; β =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+β1
+...
+βp
+β′
+1
+...
+β′
+p
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+; r =
+�
+�
+�
+0
+...
+0
+�
+�
+�
+We can codify the test 11 as
+H0 :Rβ = 0
+(12)
+H1 :Rβ ̸= 0
+It can be shown [Gre03, §5.4], that with this notation the corresponding wald statistic (9) in
+fact coincides with the F-test statistic (3)
+2
+Methodology
+The Granger F-test (3) assumes that conditions (4) are satisfied. We aim to present a version
+of the Granger test based on generalized least squares, for that we need �Ω, an estimate for the
+covariance matrix cov[ε|X] = Ω.
+4
+
+In most cases Ω is not known and a reasonable approach is to use the βOLS to obtain the
+residuals
+rt(β) = yt − β0 −
+p
+�
+k=1
+yt−kβk −
+p
+�
+k=1
+xt−kβ′
+k
+And attempt to estimate Ω using the covariance matrix of rt. This procedure is often called
+feasible least squares.
+The difficulty lies in the fact that the covariance matrix of a time series (which is often called
+autocovariance matrix) is not known in general. To overcome this problem in many cases rt is
+assumed to follow a known model such as AR(1), whose theoretical autocovariance matrix is
+known and can be obtained from the model parameters. This approach is very restrictive since
+in general rt can take many forms, for this reason we will first tackle the following problem:
+Problem 2. Given a time series x = (xt) how can we estimate the covariance matrix Ω =
+(cov(xt, xt′))t,t′?
+Since in general the previous problem can be really difficult to tackle we will impose certain
+assumptions.
+We will focus on the following realm of very general time series: the locally
+jointly-stationary which as we see admit a convenient estimation of their covariance (which
+we will call the the sliding autocovariance matrix)
+2.1
+Locally jointly-stationarity and the sliding autocovariance matrix
+Definition 3. A time series x = (xt) is locally jointly-stationary if there exists an increas-
+ing sequence of time instances 0 < t1 < t2 < . . . < tn (called time breaks) such that each
+subsequence
+x(k) = xtk:tk+1 = {xt : t ∈ [tk, tk+1]}
+is stationary and jointly-stationary with respect to the rest of the subsequences.
+Recall that two time series x, y are jointly-stationary if they satisfy cov(xt, yt) = cov(xt+h, yt+h)
+Example 4. Stationary time series are locally jointly-stationary. This is very easy to verify
+since every subseries of a stationary time series will be cross stationary with any other subseries.
+One can consider any instance T and the initial subseries x0:T is trivially cross stationary with
+the rest of the series by definition.
+Example 5. A time series (xt) is called stationary with structural breaks if it satisfies
+xt = α + δDt + εt
+where
+Dt =
+�
+1
+if t ≥ TB + 1
+0
+otherwise
+for α, δ ∈ R and εt stationary. These time series were introduced by Perron (see [Per89] and
+[LS01]).
+Property 6. Stationary time series with structural breaks are locally jointly-stationary.
+5
+
+Proof. Consider the subsampled time series at = α + εt defined for values of t between 0 and Tb,
+and bt = α + δ + εt. Since εt is stationary, this two time series are jointly-stationary:
+cov(at1, bt2) = cov(α + εt1, α + δ + εt1)
+= cov(εt1, εt2) = cov(εt1+h, εt2+h)
+= cov(α + εt1+h, α + δ + εt1+h)
+= cov(at1+h, bt2+h)
+Therefore x satisfies definition 3, taking the partition 0 ≤ t ≤ Tb and Tb < t.
+Definition 7. Given a time series x = (xt), we define the window of length τ at t0 as
+wτ(x, t0) = xt0:t0−τ = {xt : t ∈ [t0 − τ, t0]}
+Definition 8. Let x = (xt) be a time series, and two time instants t1, t2. Consider the windows
+of length τ
+w1 = wτ(x, t1)
+w2 = wτ(x, t2)
+We will call the windowed sample autocovariance of length τ at t1, t2 to
+�γτ(t, t′) = �γw1w2(t, t′)
+(13)
+= 1
+τ
+τ
+�
+k=0
+(w1
+k − w1)(w2
+k − w2)
+(14)
+= 1
+τ
+τ
+�
+k=0
+(xt−k − w1)(xt′−k − w2)
+(15)
+which coincides with the sample cross-covariance 1 between the time series wτ(x, t) and
+wτ(x, t′)
+Property 9. Let x = (xt) be a locally jointly-stationary time series with time breaks 0 < t1 <
+t2 < . . . < tn. Given t, t′, taking
+τ = argmin
+1≤m≤n
+T =t,t′
+|T − tm|
+the windowed sample autocovariance of length τ is an estimator for cov(xt, xt′)
+Proof. Suppose that the time break immediately lower that t is tm and that the one immediately
+lower than t′ is tm′. We will consider first the case that tm and t′
+m are different.
+Notice that (following the notation in 3), by the choice of τ
+wτ(x, t, τ) = wτ(x(tm), t, τ)
+wτ(x, t′, τ) = wτ(x(tm′), t′, τ)
+1The sample cross-covariance of two (jointly-stationary) time series x and y is defined as �γxy(h) = �(xt+h −
+x)(yt+h − y). See example 1.23 [SSS00]
+6
+
+Therefore, �γτ(t, t′), the windowed sample autocovariance of length τ coincides with the sample
+cross covariance of the previous two subseries windows wτ(x(tm), t, τ), wτ(x(tm′), t′, τ).
+Since the subsequences x(tm) and x(tm′) are jointly-stationary, �γτ(t, t′) estimates the covari-
+ance
+cov(wτ(x(tm), t, τ), wτ(x(tm′), t′, τ))
+which must coincide with cov(x(tm)
+t
+, x(tm′)
+t′
+) = cov(xt, xt′).
+If tm = t′
+m then in the previous argument x(tm) = x(tm′) and the same consequence follows
+using the stationarity of x(tm).
+Observation 10. Notice that since the sample cross-covariance for jointly-stationary time se-
+ries is a biased estimator, the windowed sample autocovariance �γτ(t, t′) is a biased estimator.
+Nevertheless, in the case that the expected value E[x] is known, replacing the average by the
+expected value in the formula the cross-covariance becomes unbiased and therefore the same
+holds for �γτ(t, t′) in view of the previous proof.
+Definition 11. The sliding autocovariance matrix of length τ is the N × N matrix Ωτ
+defined thewindowed sample autocovariance �γτ(t, t′) for every pair of time instances, that is
+Ωτ = (�γτ(t, t′))t,t′
+(16)
+By prop. 9 Ωτ is an estimator for the covariance matrix Ω in the case that the series x is
+locally jointly-stationary. By prop. 6 if x is stationary with structural breaks, Ωτ estimates Ω.
+Observation 12. Notice that in (16) for low values of t, t′ the estimation �γτ(t, t′))t,t′ becomes
+imprecise due to the small number of values before. One way to fix this problem in certain
+situations is to complete the time series with values before 0 using x−t = xt.
+Example 13. Consider time series xt = φ1xt−1 + εt with φ1 = 0.9. Since the time series follows
+an AR(1) model, we know that the autocovariance
+cov(xt, xt+h) = φh
+1 · var(xt)
+(17)
+Figure 1 shows a simulation of this time seres at the top, the theoretical covariance matrix
+using the previous formula 17 is shown on the left side and the sliding covariance matrix estimated
+using (16) is shown on the left. For the imprecision around low values we used the procedure
+described in obs. 12. To calculate the autocovariance matrix we used the value τ = N/3 where
+N is the sample size for the time series, lower values of τ produce similar estimations.
+7
+
+Figure 1: Autocovariance matrix and estimated sliding autocovariance matrix.
+2.2
+Generalized least squares Granger causality test
+Going back to Granger causality test, in this section we present a novel Granger causality test
+based on wald tests and the estimation of the covariance matrix of the residuals via the sliding
+autocovariance matrix.
+Method 14 (GLS Granger Causality test). Given two time series x, y, in order to assess
+whether x causes y with lag L, we follow the following procedure, which is a variation of the
+classical Granger causality test:
+1. Use OLS to obtain an estimate (βOLS, β′
+OLS) for the model
+yt = β0 +
+p
+�
+k=1
+βkyt−k +
+p
+�
+k=1
+β′
+kxt−k + εt
+(18)
+2. Using the residuals of the previous model
+rt = yt − β0 −
+p
+�
+k=1
+βkyt−k −
+p
+�
+k=1
+β′
+kxt−k
+estimate their covariance matrix using the sliding autocovariance matrix Ωτ as in (16)
+8
+
+at=0.9at-1+Et
+0.06
+0.04
+0.02
+0.00
+0.02
+0.04
+0.06
+0
+200
+400
+600
+800
+1000
+0
+0
+0.0004
+200
+200
+0.0002
+400
+400
+0.0000
+600
+600
+-0.0002
+800
+800
+-0.0004
+0
+200
+400
+600
+800
+0
+200
+400
+600
+800
+Theorethical autocovariancematrix
+Estimatedautocovariancematrix3. Use GLS and the previous covariance matrix to estimate again the parameters (βGLS, β′
+GLS)
+in the model (18) as in (5).
+4. Use a Wald test with null hypothesis
+H0 : β′
+GLS k = 0 for all k ≤ p
+If the null hypothesis is rejected conclude that x causes y otherwise conclude the opposite.
+3
+Results
+We proceed now to assess the efficacy of the proposed GLS Granger causality test (14) in com-
+parison with the classical Granger F-Test (sec. 1.1.1).
+3.1
+Simulated dataset
+Given a time series x we applied the following procedure to consistently generate a caused series
+y. The procedure consists of defining y in the following way
+yt =
+L
+�
+k=1
+xt−k · βk + εt
+(19)
+where βk are generated randomly and ε is a time series that can be constructed in several
+ways depending in the type of causality that we want to simulate. We can consider:
+(M1) ε stationary time series, for instance a white noise εt ∼ N(0, σ2).
+(M2) ε is stationary with structural breaks, for instance considering εt =∼ N(µt, σ2) with µt = 0
+for t ≤ tb and µt = µ for t > tb.
+(M3) ε non-stationary time series with changing variance for instance εt ∼ N(0, (t · σ)2).
+On the other hand, to simulate non causality, we will simply simulate two time series x
+and y by using auto-regressive processes with different parameters
+xt = xt−1φ + εt
+(AR1)
+yt = yt−1φ′ + εt
+Observation 15. Notice that a regression model applied to predict y from the lagged time
+series Bky and Bky (using the backward operator as in obs. 1). The regression parameters will
+approximate β1, . . . βL residuals of the regression will approximate ε.
+With this in mind, (M1) will produce time series in which classical Granger F-test will be
+very effective. In contrast (M2) will give us stationary with structural breaks residuals and (M3)
+will produce heteroskedastic residuals, therefore the classical Granger F-test will be less effective
+with these time series.
+For this reason the introduced dataset (simulated using M1, M2 and M3) will be suitable for
+comparing the proposed GLS Granger causality test with the classical Granger F-Test.
+9
+
+Example 16. In the following figure 2 we show three examples of the generated dataset using
+(M1), (M2) and (M3).
+Notice that in the graphs of figure 2 we can observe the causality, in the sense that changes
+x (shown in blue) cause changes in y after a number of lags.
+In the second graph the figure we see the structural break introduced in the residual of y
+(M2).
+Finally, in the last graph of the figure we appreciate the residual with growing variance
+introduced by (M3) in y.
+Figure 2: Three pairs of time series generated using the previous method (for L = 15 and random
+β1, . . . , βL). In blue, the original time series x generated using an AR(1) process. In red the
+caused time series y generated using the three methods (M1), (M2) and (M3) respectively.
+Experiment results
+We perform four experiments, each with 150 pairs of time series each with 600 points. The lag
+used to simulate the causality is L = 15.
+The first three experiments consist on testing the performance of Classical Granger against
+the proposed GLS Granger by simulating causal relationships using methods (M1), (M2) and
+(M3). For the sake of this comparison, we record the percentage of correct predictions by each
+method.
+The last experiment attempts to search for false positives. In this experiment we generate
+non-caused time series using the method (AR1) described before.
+Simulated causal
+relationships
+simulation
+procedure
+% of correct classical
+Granger F-test
+% of correct
+GLS-Granger
+y caused by x
+(M1) stationary residual
+75.0%
+96.6%
+y caused by x
+(M2) structural breaks residual
+57.3%
+85.5%
+y caused by x
+(M3) heteroskedastic residual
+32.6%
+42.6%
+y not caused by x
+(AR1)
+94.0%
+94.7%
+Table 1: Experiment results table
+The window length τ used for the sliding autocovariance matrix estimation was τ = N/5
+where N is the number of observations. This value was obtained using cross-validation, but
+greater values of τ produced very similar results.
+In view of table 1, the proposed method gets more accurate results than the Granger F-test
+in every one of the datasets simulated.
+10
+
+4 -
+10
+3 -
+2 -
+-5
+-10
+-1
+-15 
+-20
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0
+25
+50
+100
+125
+150
+175
+200
+25
+50
+75
+100
+125
+150
+175
+2003.1.1
+Real dataset
+Cryptocoins are known for their volatility and interdependence. Granger causality is a known tool
+to study the interdependence of cryptocoins, for instance [KCP21] found a strong relationship
+between Bitcoin (BTN) and Ethereum (ETH) and [Yav22] points out complex interdependence
+between the main cryptocoins.
+We will use a dataset composed by the values of the main 10 cryptocurrencies (Bitcoin,Ethereum,
+Aave, BinanceCoin, Cardano, ChainLink, Cosmos, CryptocomCoin, Dogecoin, EOS, Iota, Lite-
+coin, Monero) from July 2020 to July 2021.
+The trend component of these time series was removed applying first order differentiation.
+Even after differentiation a progressive change in variance is observed in the time series (see
+figure 3). This suggests that even though in some cases the series pass a Augmented Dickey-
+Fuller stationarity test, the OLS estimation performed in every Granger F-test will be imprecise
+or problematic. This situation has many similarities to the simulation preformed in (M3), for
+this reason our proposed method is more suitable to deal with the heteroskedastic behavior of
+the residuals. (see sec. 3.1 and fig. 2).
+Figure 3: Differentated data of the cryptocoin Ethereum from July 2020 to July 2021
+We applied a Granger F-test and our proposed GLS Granger test on each pair of cryptocoins
+considered composing causals graphs, i.e. a graph that has as nodes all the time series and as
+edges the causal relationships (if x causes y we draw x → y)
+We used the lag L = 1, the optimal lag was obtained using Akaike Information Criterion
+(AIC).
+The result is shown in Figure 4. The left graph of 4 shows the Granger F-test causal graph,
+whereas the right graph show the resulting GLS Granger Graph. We obtained a very connected
+causal network as it is to expect from the behavior of cryptocurrencies. Interestingly, the pro-
+posed method was able to capture more causal relationships, showing an even more connected
+network. It also noteworthy that the GLS Granger graph shows many causal relationship that
+connect the two leading cryptocoins Bitcoin and Ethereum with the rest of them. For instance
+the proposed method finds the relations Ethereum → cardano, Bitcoin → EOS, Bitcoin →
+ChainLink, Iota → Bitcoin.
+11
+
+600
+400
+200
+-200
+-400
+-600
+-800Figure 4: Causal graphs
+4
+Conclusions and Future work
+In this paper, we propose a generalization of the Granger F-test to uncover the temporal causal
+structures from heteroskedastic and structural-breaks time series trough the estimation of the
+residual autocovariance matrix and GLS.
+We demonstrate its effectiveness on four simulation datasets and one real application dataset.
+For future work, we are interested in researching other uses of the sliding covariance matrix
+in the field of time series classification and machine learning.
+Code availability
+Datasets and scripts for this article are available at github: https://github.com/Granger-Causality-
+GLS
+References
+[Gra69]
+Clive WJ Granger, Investigating causal relations by econometric models and cross-
+spectral methods, Econometrica: journal of the Econometric Society (1969), 424–438.
+[Gra80]
+, Testing for causality: a personal viewpoint, Journal of Economic Dynamics
+and control 2 (1980), 329–352.
+[Gre03]
+William H Greene, Econometric analysis, Pearson Education India, 2003.
+[KCP21] Myeong Jun Kim, Nguyen Phuc Canh, and Sung Y Park, Causal relationship among
+cryptocurrencies: A conditional quantile approach, Finance Research Letters 42 (2021),
+101879.
+[LS01]
+Junsoo Lee and Mark Strazicich, Testing the null of stationarity in the presence of a
+structural break, Applied Economics Letters 8 (2001), no. 6, 377–382.
+[Per89]
+Pierre Perron, The great crash, the oil price shock, and the unit root hypothesis, Econo-
+metrica: journal of the Econometric Society (1989), 1361–1401.
+[SSS00]
+Robert H Shumway, David S Stoffer, and David S Stoffer, Time series analysis and its
+applications, vol. 3, Springer, 2000.
+12
+
+cosms
+binance
+cosmes
+binance
+opero
+oherc
+dogec
+dogec新
+chainL
+chainLink
+ave
+cryptocomcpin
+cryptocomcin
+etherum
+etherum
+cardao
+ec
+oin
+cal
+iota
+ota
+Titecotn
+Classical Granger F-test causal graph
+GLS Granger test causal graph[Yav22]
+G¨UL Yavuz, Causality and cointegration in cryptocurrency markets, Uluslararası
+˙Iktisadi ve ˙Idari ˙Incelemeler Dergisi (2022), no. 34, 129–142.
+13
+

8tE1T4oBgHgl3EQfUAOX/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf,len=251
+page_content='Granger causality test for heteroskedastic and  structural-break time series using generalized least squares  Hugo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Bello ∗  January 10, 2023  Abstract  This paper proposes a novel method (GLS Granger test) to determine causal relation-  ships between time series based on the estimation of the autocovariance matrix and gener-  alized least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' We show the effectiveness of proposed autocovariance matrix estimator  (the sliding autocovariance matrix) and we compare the proposed method with the classical  Granger F-test with via a synthetic dataset and a real dataset composed by cryptocurren-  cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The simulations show that the proposed GLS Granger test captures causality more  accurately than Granger F-tests in the cases of heteroskedastic or structural-break residu-  als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Finally, we use the proposed method to unravel unknown causal relationships between  cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Keywords:  Granger Test, Causality, Generalized Least Squares, Wald Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  1  Introduction  Granger causality is a statistical concept that helps us determine whether the information in  one time series is useful in predicting another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' It is widely used in numerous fields, including  economics, finance, geology, genetics and neuroscience, to understand the relationships between  variables and to identify possible causal connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  The main tool for studying time series causality is the Granger causality F-test ([Gra69],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  [Gra80]) which is based on ordinary least squares (OLS) estimation and therefore is tied to the  assumptions of OLS (among them homocedasticity of the residuals)  In many real-world applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' we are confronted with time series suffering from a variety  of problems such as heteroscedasticity (non-equal variance amongst time observations) or struc-  tural breaks (the appearance of two or more stationary periods with different means).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' In these  situations OLS is not recommended and Granger F-test can be inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  The aim of this paper is to present a method for determining Granger causality based on  generalized least squares based on the estimation of the autocovariance matrix of the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1  Preliminaries  In this section we will present introducing notation and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' We also review the notion  of Granger causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  ∗hugojose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='bello@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='es Department of Applied Mathematics, University of Valladolid (Campus Soria)  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='03085v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='ME]  8 Jan 2023  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1  Granger causality F-test  The Granger causality ([Gra69], [Gra80]) test is a statistical hypothesis test for determining  whether one time series x = (xt)N  t=0 is useful in forecasting another y = (yt)N  t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' In particular  Granger causality focuses on the possibility of x and y to predict future values of y significatively  better than those of y alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' In this case it is said that x Granger causes y  The idea behind this is that a cause should be helpful in predicting the future effects, beyond  what can be predicted solely based on their own past values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  The test null hypothesis states that the linear regression model  yt = β0 +  p  �  k=1  βkyt−k + εt  (1)  approximates y significatively better than the model  yt = β0 +  p  �  k=1  βkyt−k +  p  �  k=1  β′  kxt−k + εt  (2)  If the null hypothesis is correct, this will imply that the lagged values of x add explanatory  power to the prediction of y and therefore the process behind x causes y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  To test if model (2) is significatively more accurate than (1) the following statistic is used  (SSRRM − SSRUM)/N  SSEUM/(N − 2)(p − 1)  (3)  where SSRRM and SSRUM are the sum of residuals for the restricted and unrestricted models  respectively  SSRRM = Σt(yt − β0 −  p  �  k=1  βkyt−k)2  SSRUM = Σt(yt − β0 −  p  �  k=1  βkyt−k −  p  �  k=1  β′  kxt−k)2  If the necessary assumptions for ordinary least of squares are satisfied,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' (3) follows a F(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' N −  2p − 1) distribution under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Notice that 1 can be understood as writing y as a linear combination of the  lagged time series Bky (using the backward operatior notation), that is:  y = β0 +  L  �  k=1  βkBky + ε  Similarly 1  y = β0 +  L  �  k=1  βkBky +  L  �  k=1  βkBkx + ε  Therefore Granger causality can be understood as the use of a F-test to compare two multi-  linear regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='2  Generalized Least Squares  For a linear regression model of the form  yi = β1 xi1 + β2 xi2 + · · · + βp xip + εi,  (4)  The following assumptions must be satisfied for ordinary least squares (OLS) to have the  desired asymptotic properties:  (P1) Correct specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Te underlying process generating the data must be in esence  linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  (P2) Strict exogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The errors in the regression must have conditional mean zero: E[ ε | X ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='.  (P3) No linear dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The regressors must all be linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  (P4) Homoscedasticity E[ε2  i |X] = σ2 ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The error term has the same variance σ2 in each  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  (P5) No autocorrelation E[εiεj|X] = 0 ∀i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The errors are uncorrelated between observa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  (P6) Normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' It is sometimes additionally assumed that the errors have normal distribution  conditional on the regressors  If homocedasticity (P4) or non-autocorrelation (P5) assumptions are not satisfied we can use  Generalized Least Squares (GLS) to better approximate the parameters of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Using the standard matricial notation (4) can be written as  y = Xβ + ε  where y = (yi), X = (xT  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' xT  p ) is the design matrix and ε is the error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' if the error  term satisfies that E[ε|X] = 0 and denoting cov[ε|X] = Ω the non-singular covariance matrix of  the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' the GLS estimate for β is  �βGLS = argminβ(y − Xβ)T Ω(y − Xβ) =  �  XTΩ−1X  �−1 XTΩ−1y  (5)  (see [Gre03, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='3]) Notice that for Ω = σ2I (where I is the identity matrix) we are under the  assumptions of OLS and the resulting estimator is  �βOLS =  �  XTX  �−1 XTy  (6)  It is known that  E[�βGLS] = βGLS  (7)  cov[�βGLS] = V = (XT Ω−1X)−1  (8)  In fact  √  N(�βGLS − βGLS) −→D N(0, V )  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='3  Wald Test  The Wald test is a statistical hypothesis test that assesses constrains on statistical parameters  for regression models based on the weighted distance between an unrestricted estimate and its  hypothesized value under the null hypothesis (see [Gre03, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Let �β the sample estimate for the regression model (GLS or OLS) model with covariance V  as described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' If Q hypothesis on the p parameters are expressed in the form of a Q × p  matrix R:  3  H0 : Rβ = r  H1 : Rβ ̸= r  The wald test statistic is  (R�β − r)T ·  �  R�V RT · 1  n  �−1  (R�β − r)  (9)  Under the null hypothesis, the Wald statistic (9) follows a F(Q, N − p) distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Granger F-test as a Wald Test  The Granger F-Test described in (3) is in fact a particular case of the Wald test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' If we consider  �βOLS, then we can stablish the unrestricted regression model  y = β0 + β1By + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' + βpBpy + β′  1Bx + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' + β′  pBpx  (10)  Where B is the backshift operator Bkx = (xt−k)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' So for y to be caused by x the coefficients  β′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' β′  p must be zero, therefore we need to test the hypothesis  H0 :β′  k = 0 for all k ≤ p  (11)  H1 :β′  k ̸= 0 for some k ≤ p  Defining the matrices  R =  �  �  �  �  �  �  �  �  �  �  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  0  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  1  �  �  �  �  �  �  �  �  �  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' β =  �  �  �  �  �  �  �  �  �  �  β1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  βp  β′  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  β′  p  �  �  �  �  �  �  �  �  �  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' r =  �  �  �  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  0  �  �  �  We can codify the test 11 as  H0 :Rβ = 0  (12)  H1 :Rβ ̸= 0  It can be shown [Gre03, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='4], that with this notation the corresponding wald statistic (9) in  fact coincides with the F-test statistic (3)  2  Methodology  The Granger F-test (3) assumes that conditions (4) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' We aim to present a version  of the Granger test based on generalized least squares, for that we need �Ω, an estimate for the  covariance matrix cov[ε|X] = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  4  In most cases Ω is not known and a reasonable approach is to use the βOLS to obtain the  residuals  rt(β) = yt − β0 −  p  �  k=1  yt−kβk −  p  �  k=1  xt−kβ′  k  And attempt to estimate Ω using the covariance matrix of rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' This procedure is often called  feasible least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  The difficulty lies in the fact that the covariance matrix of a time series (which is often called  autocovariance matrix) is not known in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' To overcome this problem in many cases rt is  assumed to follow a known model such as AR(1), whose theoretical autocovariance matrix is  known and can be obtained from the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' This approach is very restrictive since  in general rt can take many forms, for this reason we will first tackle the following problem:  Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Given a time series x = (xt) how can we estimate the covariance matrix Ω =  (cov(xt, xt′))t,t′?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Since in general the previous problem can be really difficult to tackle we will impose certain  assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  We will focus on the following realm of very general time series: the locally  jointly-stationary which as we see admit a convenient estimation of their covariance (which  we will call the the sliding autocovariance matrix)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1  Locally jointly-stationarity and the sliding autocovariance matrix  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' A time series x = (xt) is locally jointly-stationary if there exists an increas-  ing sequence of time instances 0 < t1 < t2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' < tn (called time breaks) such that each  subsequence  x(k) = xtk:tk+1 = {xt : t ∈ [tk, tk+1]}  is stationary and jointly-stationary with respect to the rest of the subsequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Recall that two time series x, y are jointly-stationary if they satisfy cov(xt, yt) = cov(xt+h, yt+h)  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Stationary time series are locally jointly-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' This is very easy to verify  since every subseries of a stationary time series will be cross stationary with any other subseries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  One can consider any instance T and the initial subseries x0:T is trivially cross stationary with  the rest of the series by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' A time series (xt) is called stationary with structural breaks if it satisfies  xt = α + δDt + εt  where  Dt =  �  1  if t ≥ TB + 1  0  otherwise  for α, δ ∈ R and εt stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' These time series were introduced by Perron (see [Per89] and  [LS01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Property 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Stationary time series with structural breaks are locally jointly-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Consider the subsampled time series at = α + εt defined for values of t between 0 and Tb,  and bt = α + δ + εt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Since εt is stationary, this two time series are jointly-stationary:  cov(at1, bt2) = cov(α + εt1, α + δ + εt1)  = cov(εt1, εt2) = cov(εt1+h, εt2+h)  = cov(α + εt1+h, α + δ + εt1+h)  = cov(at1+h, bt2+h)  Therefore x satisfies definition 3, taking the partition 0 ≤ t ≤ Tb and Tb < t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Given a time series x = (xt), we define the window of length τ at t0 as  wτ(x, t0) = xt0:t0−τ = {xt : t ∈ [t0 − τ, t0]}  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Let x = (xt) be a time series, and two time instants t1, t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Consider the windows  of length τ  w1 = wτ(x, t1)  w2 = wτ(x, t2)  We will call the windowed sample autocovariance of length τ at t1, t2 to  �γτ(t, t′) = �γw1w2(t, t′)  (13)  = 1  τ  τ  �  k=0  (w1  k − w1)(w2  k − w2)  (14)  = 1  τ  τ  �  k=0  (xt−k − w1)(xt′−k − w2)  (15)  which coincides with the sample cross-covariance 1 between the time series wτ(x, t) and  wτ(x, t′)  Property 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Let x = (xt) be a locally jointly-stationary time series with time breaks 0 < t1 <  t2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' < tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Given t, t′, taking  τ = argmin  1≤m≤n  T =t,t′  |T − tm|  the windowed sample autocovariance of length τ is an estimator for cov(xt, xt′)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Suppose that the time break immediately lower that t is tm and that the one immediately  lower than t′ is tm′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' We will consider first the case that tm and t′  m are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Notice that (following the notation in 3), by the choice of τ  wτ(x, t, τ) = wτ(x(tm), t, τ)  wτ(x, t′, τ) = wτ(x(tm′), t′, τ)  1The sample cross-covariance of two (jointly-stationary) time series x and y is defined as �γxy(h) = �(xt+h −  x)(yt+h − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' See example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='23 [SSS00]  6  Therefore, �γτ(t, t′), the windowed sample autocovariance of length τ coincides with the sample  cross covariance of the previous two subseries windows wτ(x(tm), t, τ), wτ(x(tm′), t′, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Since the subsequences x(tm) and x(tm′) are jointly-stationary, �γτ(t, t′) estimates the covari-  ance  cov(wτ(x(tm), t, τ), wτ(x(tm′), t′, τ))  which must coincide with cov(x(tm)  t  , x(tm′)  t′  ) = cov(xt, xt′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  If tm = t′  m then in the previous argument x(tm) = x(tm′) and the same consequence follows  using the stationarity of x(tm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Observation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Notice that since the sample cross-covariance for jointly-stationary time se-  ries is a biased estimator, the windowed sample autocovariance �γτ(t, t′) is a biased estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Nevertheless, in the case that the expected value E[x] is known, replacing the average by the  expected value in the formula the cross-covariance becomes unbiased and therefore the same  holds for �γτ(t, t′) in view of the previous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The sliding autocovariance matrix of length τ is the N × N matrix Ωτ  defined thewindowed sample autocovariance �γτ(t, t′) for every pair of time instances, that is  Ωτ = (�γτ(t, t′))t,t′  (16)  By prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' 9 Ωτ is an estimator for the covariance matrix Ω in the case that the series x is  locally jointly-stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' By prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' 6 if x is stationary with structural breaks, Ωτ estimates Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Observation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Notice that in (16) for low values of t, t′ the estimation �γτ(t, t′))t,t′ becomes  imprecise due to the small number of values before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' One way to fix this problem in certain  situations is to complete the time series with values before 0 using x−t = xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Example 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Consider time series xt = φ1xt−1 + εt with φ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Since the time series follows  an AR(1) model, we know that the autocovariance  cov(xt, xt+h) = φh  1 · var(xt)  (17)  Figure 1 shows a simulation of this time seres at the top, the theoretical covariance matrix  using the previous formula 17 is shown on the left side and the sliding covariance matrix estimated  using (16) is shown on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' For the imprecision around low values we used the procedure  described in obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' To calculate the autocovariance matrix we used the value τ = N/3 where  N is the sample size for the time series, lower values of τ produce similar estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  7  Figure 1: Autocovariance matrix and estimated sliding autocovariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='2  Generalized least squares Granger causality test  Going back to Granger causality test, in this section we present a novel Granger causality test  based on wald tests and the estimation of the covariance matrix of the residuals via the sliding  autocovariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Method 14 (GLS Granger Causality test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Given two time series x, y, in order to assess  whether x causes y with lag L, we follow the following procedure, which is a variation of the  classical Granger causality test:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Use OLS to obtain an estimate (βOLS, β′  OLS) for the model  yt = β0 +  p  �  k=1  βkyt−k +  p  �  k=1  β′  kxt−k + εt  (18)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Using the residuals of the previous model  rt = yt − β0 −  p  �  k=1  βkyt−k −  p  �  k=1  β′  kxt−k  estimate their covariance matrix using the sliding autocovariance matrix Ωτ as in (16)  8  at=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='9at-1+Et  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='06  0  200  400  600  800  1000  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='0004  200  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='0002  400  400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='0000  600  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='0002  800  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='0004  0  200  400  600  800  0  200  400  600  800  Theorethical autocovariancematrix  Estimatedautocovariancematrix3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Use GLS and the previous covariance matrix to estimate again the parameters (βGLS, β′  GLS)  in the model (18) as in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Use a Wald test with null hypothesis  H0 : β′  GLS k = 0 for all k ≤ p  If the null hypothesis is rejected conclude that x causes y otherwise conclude the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  3  Results  We proceed now to assess the efficacy of the proposed GLS Granger causality test (14) in com-  parison with the classical Granger F-Test (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1  Simulated dataset  Given a time series x we applied the following procedure to consistently generate a caused series  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The procedure consists of defining y in the following way  yt =  L  �  k=1  xt−k · βk + εt  (19)  where βk are generated randomly and ε is a time series that can be constructed in several  ways depending in the type of causality that we want to simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' We can consider:  (M1) ε stationary time series, for instance a white noise εt ∼ N(0, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  (M2) ε is stationary with structural breaks, for instance considering εt =∼ N(µt, σ2) with µt = 0  for t ≤ tb and µt = µ for t > tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  (M3) ε non-stationary time series with changing variance for instance εt ∼ N(0, (t · σ)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  On the other hand, to simulate non causality, we will simply simulate two time series x  and y by using auto-regressive processes with different parameters  xt = xt−1φ + εt  (AR1)  yt = yt−1φ′ + εt  Observation 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Notice that a regression model applied to predict y from the lagged time  series Bky and Bky (using the backward operator as in obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The regression parameters will  approximate β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' βL residuals of the regression will approximate ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  With this in mind, (M1) will produce time series in which classical Granger F-test will be  very effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' In contrast (M2) will give us stationary with structural breaks residuals and (M3)  will produce heteroskedastic residuals, therefore the classical Granger F-test will be less effective  with these time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  For this reason the introduced dataset (simulated using M1, M2 and M3) will be suitable for  comparing the proposed GLS Granger causality test with the classical Granger F-Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  9  Example 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' In the following figure 2 we show three examples of the generated dataset using  (M1), (M2) and (M3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Notice that in the graphs of figure 2 we can observe the causality, in the sense that changes  x (shown in blue) cause changes in y after a number of lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  In the second graph the figure we see the structural break introduced in the residual of y  (M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Finally, in the last graph of the figure we appreciate the residual with growing variance  introduced by (M3) in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Figure 2: Three pairs of time series generated using the previous method (for L = 15 and random  β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' , βL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' In blue, the original time series x generated using an AR(1) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' In red the  caused time series y generated using the three methods (M1), (M2) and (M3) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Experiment results  We perform four experiments, each with 150 pairs of time series each with 600 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The lag  used to simulate the causality is L = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  The first three experiments consist on testing the performance of Classical Granger against  the proposed GLS Granger by simulating causal relationships using methods (M1), (M2) and  (M3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' For the sake of this comparison, we record the percentage of correct predictions by each  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  The last experiment attempts to search for false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' In this experiment we generate  non-caused time series using the method (AR1) described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Simulated causal  relationships  simulation  procedure  % of correct classical  Granger F-test  % of correct  GLS-Granger  y caused by x  (M1) stationary residual  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='0%  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='6%  y caused by x  (M2) structural breaks residual  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='3%  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='5%  y caused by x  (M3) heteroskedastic residual  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='6%  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='6%  y not caused by x  (AR1)  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='0%  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='7%  Table 1: Experiment results table  The window length τ used for the sliding autocovariance matrix estimation was τ = N/5  where N is the number of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' This value was obtained using cross-validation, but  greater values of τ produced very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  In view of table 1, the proposed method gets more accurate results than the Granger F-test  in every one of the datasets simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  10  4 -  10  3 -  2 -  5  10  1  15  20  0  25  50  75  100  125  150  175  200  0  25  50  100  125  150  175  200  25  50  75  100  125  150  175  2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1  Real dataset  Cryptocoins are known for their volatility and interdependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Granger causality is a known tool  to study the interdependence of cryptocoins, for instance [KCP21] found a strong relationship  between Bitcoin (BTN) and Ethereum (ETH) and [Yav22] points out complex interdependence  between the main cryptocoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  We will use a dataset composed by the values of the main 10 cryptocurrencies (Bitcoin,Ethereum,  Aave, BinanceCoin, Cardano, ChainLink, Cosmos, CryptocomCoin, Dogecoin, EOS, Iota, Lite-  coin, Monero) from July 2020 to July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  The trend component of these time series was removed applying first order differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Even after differentiation a progressive change in variance is observed in the time series (see  figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' This suggests that even though in some cases the series pass a Augmented Dickey-  Fuller stationarity test, the OLS estimation performed in every Granger F-test will be imprecise  or problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' This situation has many similarities to the simulation preformed in (M3), for  this reason our proposed method is more suitable to deal with the heteroskedastic behavior of  the residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' (see sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='1 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Figure 3: Differentated data of the cryptocoin Ethereum from July 2020 to July 2021  We applied a Granger F-test and our proposed GLS Granger test on each pair of cryptocoins  considered composing causals graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' a graph that has as nodes all the time series and as  edges the causal relationships (if x causes y we draw x → y)  We used the lag L = 1, the optimal lag was obtained using Akaike Information Criterion  (AIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  The result is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' The left graph of 4 shows the Granger F-test causal graph,  whereas the right graph show the resulting GLS Granger Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' We obtained a very connected  causal network as it is to expect from the behavior of cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' Interestingly, the pro-  posed method was able to capture more causal relationships, showing an even more connected  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' It also noteworthy that the GLS Granger graph shows many causal relationship that  connect the two leading cryptocoins Bitcoin and Ethereum with the rest of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content=' For instance  the proposed method finds the relations Ethereum → cardano, Bitcoin → EOS, Bitcoin →  ChainLink, Iota → Bitcoin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  11  600  400  200  200  400  600  800Figure 4: Causal graphs  4  Conclusions and Future work  In this paper, we propose a generalization of the Granger F-test to uncover the temporal causal  structures from heteroskedastic and structural-breaks time series trough the estimation of the  residual autocovariance matrix and GLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  We demonstrate its effectiveness on four simulation datasets and one real application dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  For future work, we are interested in researching other uses of the sliding covariance matrix  in the field of time series classification and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  Code availability  Datasets and scripts for this article are available at github: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='com/Granger-Causality-  GLS  References  [Gra69]  Clive WJ Granger, Investigating causal relations by econometric models and cross-  spectral methods, Econometrica: journal of the Econometric Society (1969), 424–438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
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+page_content=' 3, Springer, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
+page_content='  12  cosms  binance  cosmes  binance  opero  oherc  dogec  dogec新  chainL  chainLink  ave  cryptocomcpin  cryptocomcin  etherum  etherum  cardao  ec  oin  cal  iota  ota  Titecotn  Classical Granger F-test causal graph  GLS Granger test causal graph[Yav22]  G¨UL Yavuz, Causality and cointegration in cryptocurrency markets, Uluslararası  ˙Iktisadi ve ˙Idari ˙Incelemeler Dergisi (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}
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+page_content='  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE1T4oBgHgl3EQfUAOX/content/2301.03085v1.pdf'}

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+CENTRAL H-SPACES AND BANDED TYPES
+ULRIK BUCHHOLTZ, J. DANIEL CHRISTENSEN, JARL G. TAXER˚AS FLATEN, AND EGBERT RIJKE
+Abstract. We introduce and study central types, which are generalizations of Eilenberg–Mac Lane
+spaces. A type is central when it is equivalent to the component of the identity among its own self-
+equivalences. From centrality alone we construct an infinite delooping in terms of a tensor product
+of banded types, which are the appropriate notion of torsor for a central type. Our constructions are
+carried out in homotopy type theory, and therefore hold in any ∞-topos.
+Even when interpreted into the ∞-topos of spaces, our main results and constructions are new.
+In particular, we give a description of the moduli space of H-space structures on an H-space which
+generalizes a formula of Arkowitz–Curjel and Copeland which counts the number of path components
+of this moduli space.
+Contents
+1.
+Introduction
+1
+2.
+H-spaces and evaluation fibrations
+3
+2.1.
+H-space structures
+3
+2.2.
+Evaluation fibrations
+7
+2.3.
+Unique H-space structures
+9
+3.
+Central types
+10
+4.
+Bands and torsors
+13
+4.1.
+Types banded by a central type
+13
+4.2.
+Tensoring bands
+15
+4.3.
+Bands and torsors
+17
+5.
+Examples and non-examples
+18
+5.1.
+The H-space of G-torsors
+19
+5.2.
+Eilenberg–Mac Lane spaces
+20
+5.3.
+Products of Eilenberg–Mac Lane spaces
+21
+References
+21
+1. Introduction
+In this paper we study H-spaces and their deloopings. We work in homotopy type theory, so our
+results apply to any ∞-topos. Many of our results are new, even for the ∞-topos of spaces.
+A key concept is that of a central type. A pointed type A is central if the map (A → A)(id) → A
+sending a function f to f(pt) is an equivalence. Here (A → A)(id) denotes the identity component of
+the type of all self-maps of A, and pt denotes the base point of A. Every central type is a connected
+H-space, and a connected H-space is central precisely when the type A →∗ A of pointed self-maps is
+a set. We prove this and other characterizations of central types in Proposition 3.6. It follows, for
+example, that every Eilenberg–Mac Lane space K(G, n), with G abelian and n ≥ 1, is central. We
+show in Section 5.3 that some, but not all, products of Eilenberg–Mac Lane spaces are central. We
+don’t know whether every central type is a product of Eilenberg–Mac Lane spaces.
+Our first result is:
+Date: January 6, 2023.
+1
+arXiv:2301.02636v1  [math.AT]  6 Jan 2023
+
+2
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+Theorem 4.6. Let A be a central type. Then A has a unique delooping.
+The key ingredient of this result and much of the paper is that we have a concrete description of
+the delooping of A. It is given by the type BAut1(A) :≡ ΣX:U∥A = X∥0 of types banded by A, which
+is the 1-connected cover of BAut(A). As an example, since K(G, n) is central for G abelian and n ≥ 1,
+this gives an alternative way to define K(G, n + 1) in terms of K(G, n), as previously indicated by the
+first author [Buc19].
+We also show:
+Theorem 4.10. Let A be a central type. Then every pointed map f : A →∗ A is uniquely deloopable
+to a map Bf : BAut1(A) →∗ BAut1(A).
+It follows that the type of pointed self-maps of BAut1(A) is a set, since it is equivalent to A →∗ A.
+One of the motivations for studying BAut1(A) is that one can define a tensoring operation. Given
+two banded types X and Y in BAut1(A), the type X∗ = Y has a natural banding, where X∗ is a certain
+dual of X. We write X ⊗ Y for this banded type, and show in Theorem 4.19 that it makes BAut1(A)
+into an abelian H-space. Combined with Theorem 4.6, Theorem 4.10, and the characterization of
+central types mentioned earlier, we therefore deduce:
+Corollary 4.20. For a central type A, the type BAut1(A) is again central. Therefore, A is an infinite
+loop space, in a unique way. Moreover, every pointed map A →∗ A is infinitely deloopable, in a unique
+way.
+Our tensoring operation gives a new description of the H-space structure on K(G, n), which will
+be helpful for calculations of Euler classes in work in progress and is what originally motivated this
+work.
+We also give an alternate description of the delooping of a central type A as a certain type of
+A-torsors, and give an analogous description of K(G, 1) for any group G.
+To prove the above results, we first need to further develop the theory of H-spaces. One difference
+between our work and classical work in topology is that we emphasize the moduli space HSpace(A)
+of H-space structures on a pointed type A, rather than just the set of components. For example, we
+prove:
+Theorem 2.27. Let A be an H-space such that for all a : A, the map a · − is an equivalence. Then
+the type HSpace(A) of H-space structures on A is equivalent to the type A ∧ A →∗ A of pointed maps.
+This generalizes a classical formula of Arkowitz–Curjel and Copeland, which plays a key role in
+classical results on the number of H-space structures on various spaces. The classical formula only de-
+termines the path components of the type of H-space structures, while our formula gives an equivalence
+of types. From our formula it immediately follows, for example, that the type of H-space structures on
+the 3-sphere is Ω6S3. The proof of Theorem 2.27 uses evaluation fibrations, which generalize the map
+appearing in the definition of “central.” In fact, these evaluation fibrations play an important role in
+much of the paper. For example, we include results relating the existence of sections of an evaluation
+fibration to the vanishing of Whitehead products, and use this to show that no even spheres besides
+S0 admit H-space structures.
+In Proposition 3.3 we show that every central type has a unique H-space structure, in the strong
+sense that the type HSpace(A) is contractible. We prove several results about types with unique
+H-space structures. For example, we show that such H-space structures are associative and coherently
+abelian, and that every pointed self-map is an H-space map, a weaker version of the delooping above.
+We also give an example showing that not every type with a unique H-space structure is central.
+We note that these results rely on us defining “H-space” to include a coherence between the two
+unit laws (see Definition 2.1).
+Outline. In Section 2, we give results about H-spaces which do not depend on centrality, including a
+description of the moduli space of H-space structures, results about Whitehead products and H-space
+
+CENTRAL H-SPACES AND BANDED TYPES
+3
+structures on spheres, and results about unique H-space structures. In Section 3, we define central
+types, show that central types have a unique H-space structure, give a characterization of which H-
+spaces are central, and prove other results needed in the next section. Section 4 is the heart of the
+paper. It defines the type BAut1(A) of bands for a central type A, shows that it is a unique delooping
+of A, proves that it is an H-space under a tensoring operation, and shows that central types and their
+self-maps are uniquely infinitely deloopable. We also give the alternate description of the delooping
+in terms of A-torsors. Finally, Section 5 gives examples and non-examples of central types, mostly
+related to Eilenberg–Mac Lane spaces and their products.
+Notation and conventions. In general, we follow the notation used in [Uni13]. For example, we
+write path composition in diagrammatic order: given paths p : x = y and q : y = z, their composite
+is p � q. The reflexivity path is written refl.
+Given a type A and an element a : A, we write (A, a) for the type A pointed at a. If A is already a
+pointed type with unspecified base point, then we write pt for the base point. If A and B are pointed
+types, and f, g : A →∗ B are pointed maps, then f =∗ g is the type of pointed homotopies between f
+and g. If A is an H-space, then we write x · y for the product of two elements x, y : A (unless another
+notation for the multiplication is given). For a pointed type A, we write HSpace(A) for the type of
+H-space structures on A with the basepoint as the identity element (Definition 2.1). We write Sn for
+the n-sphere, and U for a fixed universe of types.
+Acknowledgements. We would like to thank David Jaz Myers for many lively discussions on the
+content of this paper, especially related to bands and torsors. We also thank David W¨arn for fruitful
+discussions and for sharing drafts of his forthcoming work.
+Egbert Rijke gratefully acknowledges the support by the Air Force Office of Scientific Research
+through grant FA9550-21-1-0024, and support by the Slovenian Research Agency research programme
+P1-0294. Dan Christensen and Jarl Flaten both acknowledge the support of the Natural Sciences and
+Engineering Research Council of Canada (NSERC), RGPIN-2022-04739.
+2. H-spaces and evaluation fibrations
+In Section 2.1, we begin by recalling the notion of a (coherent) H-space structure on a pointed type
+A. We discuss the type of pointed extensions of a map B ∨ C →∗ A to B × C, and show that the type
+of H-space structures on A is equivalent to the type of pointed extensions of the fold map. We relate
+the existence of extensions to the vanishing of Whitehead products, and use this to show that there
+are no H-space structures on even spheres except S0. In addition, we show that for any n-connected
+H-space A, the Freudenthal map π2n+1(A) → π2n+2(ΣA) is an isomorphism, not just a surjection.
+In Section 2.2, we study evaluation fibrations. We show that the type of H-space structures is
+equivalent to a type of sections of an evaluation fibration, and use this to show that the type of
+H-space structures on a left-invertible H-space A is equivalent to A ∧ A →∗ A, generalizing a classical
+formula of Arkowitz–Curjel and Copeland. It immediately follows, for example, that the type of H-
+space structures on the 3-sphere is Ω6S3. We end with a result relating the existence of sections of an
+evaluation fibration to the vanishing of Whitehead products.
+Section 2.3 is a short section which studies the case when the type of H-space structures is con-
+tractible. We stress that this is not the same as HSpace(A) having a single component, which is what
+is classically meant by “A has a unique H-space structure.” This situation is interesting in its own
+right. We show that such H-space structures are associative and coherently abelian, and we prove
+that all pointed self-maps of A are automatically H-space maps.
+2.1. H-space structures. We begin by giving the notion of H-space structure that we will consider
+in this paper.
+Definition 2.1. Let A be a pointed type.
+(1) A non-coherent H-space structure on A consists of a binary operation µ : A → A → A,
+along with two homotopies µl : µ(pt, −) = idA and µr : µ(−, pt) = idA;
+
+4
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+(2) A (coherent) H-space structure on A consists of a non-coherent H-space structure µ on
+A along with a coherence µlr : µl(pt) =µ(pt,pt)=pt µr(pt).
+(3) We write HSpace(A) for the type of (coherent) H-space structures on A.
+When the H-space structure is clear from the context we may write x · y :≡ µ(x, y). Any H-space
+structure yields a non-coherent H-space structure by forgetting the coherence.
+Suppose A has a
+(non)coherent H-space structure µ.
+(4) If µ(a, −) : A → A is an equivalence for all a : A, then µ is left-invertible, and we write
+x\y :≡ µ(x, −)−1(y). Right-invertible is defined dually, and we write x/y :≡ µ(−, y)−1(x).
+(5) The twist µT of µ is the natural (non)coherent H-space structure with operation
+µT (a0, a1) :≡ µ(a1, a0).
+When we say “H-space” we mean the coherent notion—we will only say “coherent” for emphasis.
+The notion of H-space structure considered in [Uni13, Def. 8.5.4] corresponds to our non-coherent H-
+space structures. While many constructions can be carried out for non-coherent H-spaces (such as the
+Hopf construction), the coherent case is more natural for our purposes. Moreover, any non-coherent
+H-space can be made coherent by simply changing one of the unit laws:
+Proposition 2.2. Any non-coherent H-space structure on a pointed type A gives rise to a coherent
+H-space structure with the same underlying binary operation.
+Proof. Let (A, µ, µl, µr) be a non-coherent H-space. We define a new homotopy µ′
+r : µ(−, pt) = id as
+the concatenation of paths
+µ(x, pt)
+µ(x, µ(pt, pt))
+µ(x, pt)
+x.
+apµ(x)(µr(pt))−1
+apµ(x)(µl(pt))
+µr(x)
+We claim that µl(pt) = µ′
+r(pt). To see this, it suffices to show that the square
+µ(pt, µ(pt, pt))
+µ(pt, pt)
+µ(pt, pt)
+pt
+apµ(pt)(µl(pt))
+apµ(pt)(µr(pt))
+µr(pt)
+µl(pt)
+commutes. We will show that the top path is equal to µl(µ(pt, pt)), and this turns the square into a
+naturality square for the homotopy µl, which always commutes. To see that
+apµ(pt)(µl(pt)) = µl(µ(pt, pt)),
+observe that µl is a homotopy µ(pt) = id, and for any homotopy H : f = id we have apf Hx = Hf(x)
+for all x.
+□
+The proposition implies that the types of non-coherent and coherent H-space structures on a pointed
+type are logically equivalent. However, they are not generally equivalent as types (see Remark 3.4).
+We’ll be interested in abelian and associative H-spaces later on.
+Definition 2.3. Let A be an H-space with multiplication µ.
+(1) If there is a homotopy h : Πa,bµ(a, b) = µ(b, a) then µ is abelian.
+(2) If µ = µT in HSpace(A) then µ is coherently abelian.
+(3) If there is a homotopy α : Πa,b,c:Aµ(µ(a, b), c) = µ(a, µ(b, c)) then µ is associative.
+The following lemma gives a convenient way of constructing abelian H-space structures, and will
+be used in Theorem 4.19.
+Lemma 2.4. Let A be a pointed type with a binary operation µ, a symmetry σa,b : µ(a, b) = µ(b, a)
+for every a, b : A such that σpt,pt = refl, and a left unit law µl : µ(pt, −) = idA. Then A becomes an
+abelian H-space with the right unit law induced by symmetry.
+
+CENTRAL H-SPACES AND BANDED TYPES
+5
+Proof. For b : A, the right unit law is given by the path σb,pt � µl(b) of type µ(b, pt) = b. For coherence
+we need to show that the following triangle commutes:
+µ(pt, pt)
+µ(pt, pt)
+pt .
+µl
+σpt,pt
+µl
+By our assumption that σpt,pt = refl, the triangle is filled reflµl.
+□
+For any right-invertible H-space A, for b : A one can define the two operations (−)/b and (−)·(pt/b)
+of type A → A. If A is associative, then these coincide:
+Lemma 2.5. Let A be an associative H-space. For any a, b : A, we have that a/b = a · (pt/b).
+Proof. For all a, b : A we have (a · (pt/b)) · b = a · ((pt/b) · b) = a · pt = a. Thus by dividing by b on
+the right, we deduce a · (pt/b) = a/b.
+□
+We collect a few basic facts about H-spaces. The following lemma generalizes a result of Evan
+Cavallo, who formalized the fact that unpointed homotopies between pointed maps into a homogeneous
+type A can be upgraded to pointed homotopies. Being a homogeneous type is logically equivalent to
+being a left-invertible H-space [Cav21]. Here we do not need to assume left-invertibility, and we factor
+this observation through a further generalization.
+Lemma 2.6. Let A be a pointed type, and consider the following three conditions:
+(1) A is an H-space.
+(2) The evaluation map (idA = idA) → (pt = pt) has a section.
+(3) For every pointed type B and pointed maps f, g : B →∗ A, there is a map (f = g) → (f =∗ g)
+which upgrades unpointed homotopies to pointed homotopies.
+Then (1) implies (2) and (2) implies (3).
+Proof. To show that (1) implies (2), suppose that A is an H-space, and let p : pt = pt. For any x : A
+we define the path px : x = x to be the concatenation
+x
+x · pt
+x · pt
+x .
+µ−1
+r
+apµ(x)(p)
+µr
+This defines a map s : (pt = pt) → (idA = idA). To see that this map is a section of the evaluation
+map, it suffices to show that the square
+pt · pt
+pt · pt
+pt
+pt
+apµ(pt)(p)
+µr
+µr
+p
+commutes. To see this, note that µr = µl. If we replace µr by µl in the above square, we obtain a
+naturality square of homotopies, which always commutes.
+We next show that (2) implies (3). Let f, g : B →∗ A be pointed maps and let H : f = g be an
+unpointed homotopy. By path induction on H, we can assume we have a single function f : B → A
+with two pointings, fpt and f ′
+pt : f(pt) = pt. Our goal is to define a homotopy K : f = f such that
+Kpt = r, where r :≡ fpt · f ′pt : f(pt) = f(pt). By path induction on fpt, we can assume that the
+basepoint of A is f(pt). By (2), we have s : (f(pt) = f(pt)) → (idA = idA) such that s(p, f(pt)) = p
+for all p : f(pt) = f(pt). For b : B, define Kb to be s(r, f(b)). Then Kpt = r, as required.
+□
+The following result is straightforward and has been formalized, so we do not include a proof.
+
+6
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+Proposition 2.7. Suppose A is a (left-invertible) H-space. For any pointed type B, the mapping
+type B →∗ A based at the constant map is naturally a (left-invertible) H-space under pointwise mul-
+tiplication.
+Similarly, for any type B, the mapping type B → A based at the constant map is a
+(left-invertible) H-space under pointwise multiplication.
+□
+In particular, if A is left-invertible then for any f : B →∗ A there is a self-equivalence of B →∗ A
+which sends the constant map to f—namely, the pointwise multiplication by f on the left.
+Our next goal is to rule out H-space structures on even spheres using Brunerie’s computation of
+Whitehead products. (See [Bru16, Section 3.3] for their definition.) To do so, we prove some results
+about Whitehead products from [Whi46] which relate to H-spaces.
+Definition 2.8. Let α : B →∗ A and β : C →∗ A be pointed maps. An (α, β)-extension is a
+pointed map f : B × C →∗ A equipped with a pointed homotopy filling the following diagram:
+B ∨ C
+A
+B × C .
+α∨β
+f
+Remark 2.9. It is equivalent to consider the type of unpointed (α, β)-extensions consisting of unpointed
+maps f : B × C → A and unpointed fillers. The additional data in a pointed extension is a path
+fpt : f(pt, pt) = pt and a 2-path that determines fpt in terms of the other data.
+These form a
+contractible pair.
+When α and β are maps between spheres, Whitehead instead says that f is “of type (α, β)” but we
+prefer to stress that we work with a structure and not a property, as the following lemma illustrates:
+Lemma 2.10. H-space structures on a pointed type A correspond to (idA, idA)-extensions.
+□
+The proof consists of straightforward reshuffling of data.
+Lemma 2.11. If A is an H-space, then there is an (α, β)-extension for every pair α : B →∗ A and
+β : C →∗ A of pointed maps.
+Proof. Using naturality of the left and right unit laws and coherence, one can show that the map
+(b, c) �→ α(b)·β(c) : B ×C → A is an (α, β)-extension. Alternatively, observe that the (α, β)-extension
+problem factors through the (idA, idA)-extension problem via the map α × β : B × C → A × A.
+□
+The lemmas explain the relation between H-space structures and (α, β)-extensions, which are in
+turn related to Whitehead products via the next two results.
+Proposition 2.12 ([Whi46, Corollary 3.5]). Let m, n > 0 be natural numbers and consider two
+pointed maps α : Sm →∗ A and β : Sn →∗ A. The type of (α, β)-extensions is equivalent to the type
+of witnesses that the map [α, β] : Sm+n−1 →∗ A is constant (as a pointed map).
+Proof. Consider the diagram of pointed maps below, where the composite of the top two maps is [α, β]
+and the left diamond is a pushout of pointed types:
+Sm ∨ Sn
+Sm+n−1
+Sm × Sn
+A .
+1
+α∨β
+f
+An (α, β)-extension is the same as a pointed map f along with a pointed homotopy filling the top-right
+triangle. Since the bottom-right triangle is filled by a unique pointed homotopy, an (α, β)-extension
+thus corresponds exactly to the data of a filler in the outer diagram, i.e., a homotopy witnessing that
+[α, β] is constant as a pointed map.
+□
+
+CENTRAL H-SPACES AND BANDED TYPES
+7
+With the notation of the previous proposition, we have the following:
+Corollary 2.13 ([Whi46, Corollary 3.6]). Suppose A is an H-space. Then [α, β] is constant.
+Proof. The follows from Lemma 2.11 and Proposition 2.12.
+□
+Using the above results, we can rule out H-space structures on even spheres in positive dimensions.
+Proposition 2.14. The n-sphere merely admits an H-space structure if and only if [ιn, ιn] = 0. In
+particular, there are no H-space structures on the n-sphere when n > 0 is even.
+Proof. The implication (→) is immediate by Corollary 2.13. Conversely, Proposition 2.12 implies that
+[ιn, ιn] = 0 if and only if an (idSn, idSn)-extension merely exists, which by Lemma 2.10 happens if and
+only if Sn merely admits an H-space structure.
+Finally, Brunerie showed that [ιn, ιn] = 2 in π2n−1(Sn) for even n > 0 [Bru16, Proposition 5.4.4],
+which by the above implies that Sn cannot admit an H-space structure.
+□
+We also record the following result and a corollary.
+Proposition 2.15. Let A be a left-invertible H-space. The unit η : A →∗ ΩΣA has a pointed retrac-
+tion, given by the connecting map δ : ΩΣA →∗ A associated to the Hopf fibration of A.
+Proof. Let δ : ΩΣA →∗ A be the connecting map associated to the Hopf fibration of A. Recall that
+for a loop p : N = N, we have δ(p) :≡ p∗(pt) where p∗ : A → A denotes transport and A is the fibre
+above N. By definition of the Hopf fibration, a path merid(a) : N =ΣA S sends an element x of the
+fibre A to a · x. Now define a homotopy δ ◦ η = id by
+δ(η(a)) ≡ δ(merid(a) � merid(pt)−1) = merid(pt)−1
+∗ (merid(a)∗(pt)) ≡ pt\(a · pt) = a.
+Finally, we promote this to a pointed homotopy using Lemma 2.6.
+□
+It follows that for any n-connected H-space A, the Freudenthal map π2n+1(A) → π2n+2(ΣA) is an
+isomorphism, not just a surjection. In particular, we have:
+Corollary 2.16. The natural map π5(S3) → π6(S4) is an isomorphism.
+□
+The fact that the unit η : A →∗ ΩΣA has a retraction when A is a left-invertible H-space also follows
+from James’ reduced product construction, as shown in [Jam55]. Using [Bru16], one can see that this
+goes through in homotopy type theory. However, the above argument is much more elementary. We
+don’t know if this argument had been observed before.
+2.2. Evaluation fibrations. We now begin our study of evaluation fibrations and their relation to
+H-space structures and (α, β)-extensions from the previous section. Given a pointed map f : B →∗ A,
+we will simply write ev : (B → A, f) →∗ A for the map which evaluates at pt : B. This map is
+pointed since f is. If no map f is specified, then we mean that f ≡ id.
+In a moment we will define evaluation fibrations to be the restriction of ev to a component, but
+first we make a useful observation.
+Definition 2.17. Let e : X →∗ A and g : B →∗ A be pointed maps. A pointed lift of g through
+e consists of a pointed map s : B →∗ X along with a pointed homotopy e ◦ s =∗ g. If g ≡ id, then s
+is more specifically a pointed section of e.
+Proposition 2.18. Let f : B →∗ A and g : C →∗ A be pointed maps. The type of (f, g)-extensions
+is equivalent to the type of pointed lifts of g through ev : (B → A, f) →∗ A.
+□
+We stress that the domain of ev is the type of unpointed maps B → A, pointed by (the underlying
+map of) f. The proof of the statement is a straightforward reshuffling of data. Diagrammatically, it
+gives a correspondence between the dashed arrows below, with pointed homotopies filling the triangles:
+B ∨ C
+A
+(B → A, f)
+B × C
+C
+A
+f∨g
+ev
+g
+
+8
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+Combining Lemma 2.10 with the previous proposition, we deduce:
+Corollary 2.19. Let A be a pointed type. The type of H-space structures on A is equivalent to the
+type of pointed sections of ev : (A → A, id) →∗ A.
+□
+Remark 2.20. Phrased another way, an H-space structure on a pointed type A is equivalent to a family
+µ : Π(a:A)(A, pt) →∗ (A, a).
+If A is a higher inductive type with a point pt, one can define µ(pt) :≡ id to simplify the task.
+Definition 2.21. Let A be a type and a : ∥A∥0. The path component of a in A is
+A(a) :≡ Σa′:A(|a′|0 = a).
+If a : A then we abuse notation and write A(a) for A(|a|0), and in this case A(a) is pointed at (a, refl).
+Definition 2.22. For any pointed map α : B →∗ A, the evaluation fibration (at α) is the pointed
+map evα : (B → A)(α) →∗ A induced by evaluating at the base point of B.
+Observe that the component (A → A)(id) is equivalent to (A ≃ A)(id), since being an equivalence is
+a property of a map. We permit ourselves to pass freely between the two.
+Since pointed maps out of connected types land in the component of the base point of the codomain,
+we have the following consequence of Corollary 2.19.
+Corollary 2.23. Let A be a pointed, connected type. The type of H-space structures on A is equivalent
+to the type of pointed sections of evid : (A ≃ A)(id) →∗ A.
+□
+For certain H-spaces, various evaluation fibrations become trivial:
+Proposition 2.24. Suppose A is a left-invertible H-space. We have a pointed equivalence over A
+(A → A)
+(A →∗ A) × A
+A ,
+ev
+∼
+pr2
+where the mapping spaces are both pointed at their identity maps. This pointed equivalence restricts
+to pointed equivalences (A ≃ A) ≃∗ (A ≃∗ A) × A over A, and (A → A)(id) ≃∗ (A →∗ A)(id) × A(pt)
+over A(pt).
+Proof. Define e : (A → A) → (A →∗ A) × A by e(f) :≡ (a �→ f(pt)\f(a), f(pt)) where the first
+component is a pointed map in the obvious way. Clearly e is a map over A, and moreover e is pointed.
+It is straightforward to check that the triangle above is filled by a pointed homotopy. (One could also
+apply Lemma 2.6, but a direct inspection suffices in this case.)
+Finally, it’s straightforward to check that e has an inverse given by
+(g, a) �→ (x �→ a · g(x)).
+Hence e is an equivalence, as desired. The restrictions to equivalences and path components follow by
+functoriality.
+□
+The hypotheses of the proposition are satisfied, for example, by connected H-spaces.
+Example 2.25. We obtain three pointed equivalences for any abelian group A and n ≥ 1:
+�
+K(A, n) → K(A, n)
+�
+≃∗ Ab(A, A) × K(A, n),
+�
+K(A, n) ≃ K(A, n)
+�
+≃∗ AutAb(A) × K(A, n), and
+�
+K(A, n) → K(A, n)
+�
+(id) ≃∗ K(A, n).
+
+CENTRAL H-SPACES AND BANDED TYPES
+9
+Example 2.26. Taking A :≡ S3 in the previous proposition, by virtue of the H-space structure on
+the 3-sphere constructed in [BR18], we get three pointed equivalences:
+(S3 → S3) ≃∗ Ω3S3 × S3,
+(S3 ≃ S3) ≃∗ Ω3
+±1S3 × S3,
+and
+(S3 ≃ S3)(id) ≃∗ (S3 ≃∗ S3)(id) × S3,
+where Ω3
+±1S3 :≡ (Ω3S3)(1)⊔(Ω3S3)(−1) and 1 and −1 refer to the corresponding elements of π3(S3) = Z.
+By combining our results thus far, we obtain the following equivalence which generalizes a classical
+formula of [Cop59, Theorem 5.5A], independently shown by [AC63], for counting homotopy classes of
+H-space structures on certain spaces.
+Theorem 2.27. Let A be a left-invertible H-space. The type HSpace(A) of H-space structures on A
+is equivalent to A ∧ A →∗ A.
+Proof. By Corollary 2.19, the type of H-space structures on A is equivalent to the type of pointed
+sections of ev : (A → A) → A. By Proposition 2.24, this type is equivalent to the type of pointed
+sections of pr2 : (A →∗ A) × A → A, which are simply pointed maps A →∗ (A →∗ A, id), where
+the codomain is pointed at the identity. The latter type is equivalent to A →∗ (A →∗ A), where
+the codomain is pointed at the constant map, by Proposition 2.7. Finally, this type is equivalent to
+A ∧ A →∗ A by the smash–hom adjunction for pointed types [vDoo18, Theorem 4.3.28].
+□
+Example 2.28. It follows from the proposition that HSpace(S1) ≃ 1 and HSpace(S3) ≃ Ω6S3.
+We record the following result which relates Whitehead products and evaluation fibrations.
+Proposition 2.29 ([Han74, Lemma 2.2]). Let n, m ≥ 2 and let α : πm(Sn).
+The evaluation
+fibration evα : (Sm → Sn)(α) → Sn merely has a section if and only if the Whitehead product
+[α, ιn] : πn+m−1(Sn) vanishes.
+Proof. As we are proving a proposition, we may pick a representative α : Sm →∗ Sn throughout.
+Using Proposition 2.18 and that Sn is connected, we see that [α, ιn] vanishes if and only if there
+merely exists a pointed section of evα. The fibre of the forgetful map from pointed sections of evα
+to unpointed sections of evα over some section (s, h) is equivalent to
+�
+k:s(pt,−)=α
+h(pt) =s(pt,pt)=pt k(pt) � αpt,
+where αpt : α(pt) = pt is the pointing of α. This fibre is (−1)-connected since s lands in the component
+of α and the inner part of the Σ-type is a double path space of Sn with n ≥ 2. In other words, this
+forgetful map is an epimorphism. A pointed section of evα therefore merely exists if and only if an
+unpointed section merely exists, completing the proof.
+□
+2.3. Unique H-space structures. We collect results about H-space structures which are unique, in
+the sense that the type of H-space structures is contractible. In particular, we give elementary proofs
+that such H-space structures are automatically coherently abelian and associative. Moreover, pointed
+self-maps of such are automatically H-space self-maps.
+Lemma 2.30. Let A be a pointed type and suppose HSpace(A) is contractible.
+Then the unique
+H-space structure µ on A is coherently abelian.
+Proof. Since HSpace(A) is contractible, there is an identification µ = µT of H-space structures. (Here,
+µT is the twist, defined in Definition 2.1.)
+□
+For the next result, we use the definition of the smash product from [vDoo18, Definition 4.3.6]
+(see also [CS20, Definition 2.29]) which avoids higher paths. For pointed types (X, x0) and (Y, y0),
+the smash product X ∧ Y is the higher inductive type with point constructors sm : X × Y →
+X ∧ Y and auxl, auxr : X ∧ Y , and path constructors gluel : �
+y:Y sm(x0, y) = auxl and gluer :
+�
+x:X sm(x, y0) = auxr.
+It is pointed by auxl.
+The smash product was shown to be associative
+in [vDoo18, Definition 4.3.33].
+
+10
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+Proposition 2.31. Suppose A is a pointed type with a unique H-space structure, and suppose moreover
+that this H-space structure is left-invertible. Then any pointed map f : A →∗ A is an H-space map,
+i.e., we have f(a · b) = f(a) · f(b) for all a, b : A.
+Proof. Let f : A →∗ A be a pointed map. We will define an associated map ν : A ∧ A →∗ A, which
+records how f deviates from being an H-space map. We define ν(sm(a, b)) :≡
+�
+f(a · b)/f(b)
+�
+/f(a),
+ν(auxl) :≡ pt, and ν(auxr) :≡ pt. For b : A, we have a path ν(sm(pt, b)) ≡
+�
+f(pt · b)/f(b)
+�
+/f(pt) =
+�
+f(b)/f(b)
+�
+/pt = pt/pt = pt, and similarly for the other path constructor. Since A admits a unique
+H-space structure, the type A∧A →∗ A is contractible by Theorem 2.27. Consequently, ν is constant,
+whence for all a, b : A we have
+�
+f(a · b)/f(b)
+�
+/f(a) = pt, and therefore
+f(a · b) = f(a) · f(b).
+□
+Remark 2.32. Note that when A and B are two pointed types, each with unique H-space structures,
+it is not necessarily the case that every pointed map f : A →∗ B is an H-space map. For example, the
+squaring operation gives a natural transformation H2(X; Z) → H4(X; Z) which is represented by a
+map K(Z, 2) →∗ K(Z, 4). But since squaring isn’t a homomorphism, this map isn’t an H-space map.
+Proposition 2.33. Suppose A is a pointed type with a unique H-space structure which is left-invertible.
+Then the H-space structure is necessarily associative.
+Proof. Let a : A. Define a map ν : A ∧ A →∗ A as follows. We let ν(sm(b, c)) :≡ ((a · b) · c)/(a · (b · c)),
+ν(auxl) :≡ pt, and ν(auxr) :≡ pt. For c : A, we have a path ν(sm(pt, c)) ≡ ((a · pt) · c)/(a · (pt · c)) =
+(a · c)/(a · c) = pt, and similarly for the other path constructor. Since A admits a unique H-space
+structure, the type A ∧ A →∗ A is contractible by Theorem 2.27. Consequently, for each a, ν is
+constant. It follows that for all a, b, c : A we have ((a · b) · c)/(a · (b · c)) = pt, and therefore
+(a · b) · c = a · (b · c).
+□
+Note that if A ∧ A →∗ A is contractible, then it follows from the smash-hom adjunction that
+A∧n →∗ A is contractible for each n ≥ 2, where A∧n denotes the smash power.
+3. Central types
+In this and the next section we focus on pointed types which we call central. Centrality is an
+elementary property with remarkable consequences. For example, in the next section we will see that
+every central type is an infinite loop space (Corollary 4.20). To show this, we require a certain amount
+of theory about central types. We first show that every central type has a unique H-space structure.
+When A is already known to be an H-space, we give several conditions which are equivalent to A
+being central. From this, it follows that every Eilenberg–Mac Lane space K(G, n), with G abelian and
+n ≥ 1, is central. We also prove several other results which we will need in the next section.
+Definition 3.1. Let A be a pointed type. The center of A is the type ZA :≡ (A → A)(id), which
+comes with a natural map evid : ZA →∗ A (see Definition 2.22). If the map evid is an equivalence,
+then A is central.
+Remark 3.2. The terminology “central” comes from higher group theory. Suppose A :≡ BG is the
+delooping of an ∞-group G. The center of G is the ∞-group ZG :≡ Πx:G(x = x) with delooping
+BZG :≡ (BG ≃ BG)(id), which is our ZA.
+Central types and H-spaces are connected through evaluation fibrations:
+Proposition 3.3. Suppose that A is central. Then A admits a unique H-space structure. In addition,
+A is connected, so this H-space structure is both left- and right-invertible.
+Proof. Since evid is an equivalence, it has a unique section. By Corollary 2.23, we deduce that A has
+a unique H-space structure µ. It follows from Lemma 2.30 that it is coherently abelian. Finally, the
+equivalence evid : (A → A)(id) ≃ A implies that A is connected. Then, since µ(pt, −) and µ(−, pt) are
+both equal to the identity, it follows that µ is left- and right-invertible.
+□
+
+CENTRAL H-SPACES AND BANDED TYPES
+11
+It follows from Proposition 2.33 and Lemma 2.30 that the unique H-space structure on a central
+type is associative and coherently abelian.
+Remark 3.4. In contrast, the type of non-coherent H-space structures on a central type A is rarely
+contractible. We’ll show here that it is equivalent to the loop space ΩA. First consider the type of
+binary operations µ : A → (A → A) which merely satisfy the left unit law. This is equivalent to the
+type of maps A → (A → A)(id), since A is connected. Such a map µ satisfies the right unit law if and
+only if the composite evid ◦µ : A → A is the identity map. In other words, µ must be a section of the
+equivalence evid, so there is a contractible type of such µ.
+The left unit law says that µ sends pt to id. After post-composing with evid, it therefore says that
+it sends pt to id(pt), which equals pt. So the type of left unit laws is pt = pt, i.e., the loop space ΩA.
+Note that we imposed the left unit law both merely and purely, but that doesn’t change the type. So
+it follows that the type of all non-coherent H-space structures on a central type A is ΩA.
+We give conditions for an H-space to be central, in which case the H-space structure is the unique
+one coming from centrality. For the next two results, write
+F :≡ Σf:A→∗A∥f = id∥
+for the fibre of evid : (A → A)(id) →∗ A over pt : A. Note that the equality f = id is in the type of
+unpointed maps A → A.
+Lemma 3.5. Suppose that A is a connected H-space. Then F ≃ (A →∗ A)(id).
+Proof. By our assumptions, Proposition 2.24 gives a trivialization of evid over A:
+t : (A → A)(id) ≃∗ (A →∗ A)(id) × A.
+Passing to the fibres of evid and pr2 over pt : A gives the desired equivalence.
+□
+The lemma can also be shown using Lemma 2.6.
+Proposition 3.6. Let A be a pointed type. Then the following are logically equivalent:
+(1) A is central;
+(2) A is a connected H-space and A →∗ A is a set;
+(3) A is a connected H-space and A ≃∗ A is a set;
+(4) A is a connected H-space and A →∗ ΩA is contractible;
+(5) A is a connected H-space and ΣA →∗ A is contractible.
+Proof. (1) =⇒ (2): Assume that A is central. Then Proposition 3.3 implies that A is a connected H-
+space. Since A is a left-invertible H-space, so is A →∗ A, by Proposition 2.7. Therefore all components
+of A →∗ A are equivalent to (A →∗ A)(id), and thus to F by Lemma 3.5. Now, F is contractible since
+evid is an equivalence, and consequently A →∗ A is a set since all of its components are contractible.
+(2) =⇒ (3): This follows from the fact that A ≃∗ A embeds into A →∗ A.
+(3) =⇒ (1): If (A ≃∗ A) is a set, then its component (A →∗ A)(id) is contractible. Therefore F is
+contractible, by Lemma 3.5. It follows that evid is an equivalence, since A is connected. Hence A is
+central.
+(3) ⇐⇒ (4): Since A is a left-invertible H-space, so is A →∗ A. The latter is therefore a set if and
+only if the component of the constant map is contractible, which is true if and only if the loop space
+Ω(A →∗ A) is contractible. Finally, the equivalence Ω(A →∗ A) ≃ (A →∗ ΩA) shows that this is true
+if and only if A →∗ ΩA is contractible.
+(4) ⇐⇒ (5): This follows from the equivalence (A →∗ ΩA) ≃ (ΣA →∗ A).
+□
+Example 3.7. Consider the Eilenberg–Mac Lane space K(G, n) for n ≥ 1 and G an abelian group.
+It is a pointed, connected type.
+Since K(G, n) ≃ Ω K(G, n + 1), it is an H-space.
+By [BvDR18,
+Theorem 5.1], K(G, n) ≃∗ K(G, n) is equivalent to the set of automorphisms of G. It therefore follows
+from Proposition 3.6 that K(G, n) is central. We will see in Proposition 5.9 a more self-contained
+proof of this result.
+
+12
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+Example 3.8. Brunerie showed that π4(S3) ≃ Z/2 [Bru16]. Therefore, S4 →∗ S3 is not contractible,
+and so S3 is not central, by Proposition 3.6(5). Since this is in the stable range, it follows that Sn is
+not central for n ≥ 3.
+Remark 3.9. For a pointed type A, we have seen that A being central is logically equivalent to A being
+a connected H-space such that A ≃∗ A is a set. It is natural to ask whether the reverse implication
+holds without the assumption that A is an H-space. However, this is not the case. Consider, for
+example, the pointed, connected type K(G, 1) for a non-abelian group G. Then K(G, 1) ≃∗ K(G, 1)
+is equivalent to the set of group automorphisms of G. If K(G, 1) were central, then G would be twice
+deloopable, which would contradict G being non-abelian.
+By the previous proposition, the type A →∗ A is a set whenever A is central. Presently we observe
+that it is in fact a ring.
+Corollary 3.10. For any central type A, the set A →∗ A is a ring under pointwise multiplication
+and function composition.
+Proof. It follows from A being a commutative and associative H-space that the set A →∗ A is an
+abelian group. The only nontrivial thing we need to show is that function composition is linear. Let
+f, g, φ : A →∗ A, and consider a : A. By Proposition 2.31, φ is an H-space map. Consequently,
+�
+φ ◦ (f · g)
+�
+(a) ≡ φ(f(a) · g(a)) = φ(f(a)) · φ(g(a)) ≡
+�
+(φ ◦ f) · (φ ◦ g)
+�
+(a).
+□
+The following remark gives some insight into the nature of the ring A →∗ A.
+Remark 3.11. If BG is an ∞-group and X is a pointed type, recall that a bundle over X is G-principal
+if it is classified by a map X →∗ BG (see e.g. [Sco20, Def. 2.23] for a formal definition which easily
+generalizes to arbitrary ∞-groups). In particular, it is not hard to see that the Hopf fibration of G
+(as the loop space of BG) is a G-principal bundle, i.e., classified by a map ΣG →∗ BG.
+In Proposition 4.4 we will see that any central type A has a delooping BAut1(A). This means we
+have equivalences
+(A →∗ A) ≃
+�
+A →∗ (A ≃ A)(id)
+�
+≃ (ΣA →∗ BAut1(A)).
+Thus we see that A →∗ A is the ring of principal A-bundles over ΣA. The equivalence above maps the
+identity id : A →∗ A to the Hopf fibration of A (as a principal A-bundle), meaning the Hopf fibration
+is the multiplicative unit from this perspective.
+In the remainder of this section we collect various results which are needed later on. The first result
+is that “all” of the evaluation fibrations of a central type A are equivalences:
+Proposition 3.12. Let A be a central type and let f : A →∗ A be a pointed map. The evaluation
+fibration evf : (A → A)(f) →∗ A is an equivalence, with inverse given by a �→ a · f(−).
+Proof. The type A → A is a left-invertible H-space via pointwise multiplication, by Proposition 2.7.
+So there is an equivalence (A → A)(id) → (A → A)(f) sending g to f · g. Since f is pointed, we have
+evf(f · g) ≡ (f · g)(pt) ≡ f(pt) · g(pt) = pt · g(pt) = g(pt) = evid(g).
+In other words, evf ◦(f · −) = evid, which shows that evf is an equivalence. Since f is pointed, the
+stated map is a section of evf, hence is an inverse.
+□
+Corollary 3.13. Let A be a central type, let f : A →∗ A, and let g : (A → A)(f). Then for all a : A,
+we have g(a) = g(pt) · f(a).
+□
+Any central type has an inversion map, which plays a key role in the next section.
+Definition 3.14. Suppose that A is central. The inversion map id∗ : A → A sends a to a∗ :≡ pt/a.
+
+CENTRAL H-SPACES AND BANDED TYPES
+13
+The defining property of a∗ is that a∗·a = pt. Since A is abelian, we also have a·a∗ = pt, so it would
+have been equivalent to define the inversion to be a �→ a\pt. From associativity of a central H-space
+it follows that pt∗ = pt and a∗∗ = a for all a, so the inversion map id∗ is a pointed self-equivalence of
+A and an involution.
+A curious property is that on the component of id∗, inversion of equivalences is homotopic to the
+identity. This comes up in the next section.
+Proposition 3.15. The map φ �→ φ−1 : (A ≃ A)(id∗) → (A ≃ A)(id∗) is homotopic to the identity.
+Proof. Let φ : (A ≃ A)(id∗). We need to show that φ = φ−1, or equivalently that φ(φ(pt)) = pt, since
+evid is an equivalence. (Note that φ ◦ φ : (A ≃ A)(id).) Using Corollary 3.13, we have that
+φ(φ(pt)) = φ(pt) · φ(pt)∗ = pt.
+□
+4. Bands and torsors
+We begin in Section 4.1 by defining and studying types banded by a central type A, also called
+A-bands. We show that the type BAut1(A) of banded types is a delooping of A, that A has a unique
+delooping, and that every pointed self-map A →∗ A has a unique delooping.
+In Section 4.2, we show that BAut1(A) is itself an H-space under a tensoring operation, from which
+it follows that it is again a central type. Thus we may iteratively consider banded types to obtain
+an infinite loop space structure on A, which is unique. As a special case, taking A to be K(G, n) for
+some abelian group G produces a novel description of the infinite loop space structure on K(G, n), as
+described in Section 5.2.
+In Section 4.3, we define the type of A-torsors, which we show is equivalent to the type of A-bands
+when A is central, thus providing an alternate description of the delooping of A. The type of A-torsors
+has been independently studied by David W¨arn, who has shown that it is a delooping of A under the
+weaker assumption that A has a unique H-space structure.
+4.1. Types banded by a central type. We now study types banded by a central type A. On this
+type we will construct an H-space structure, which will be seen to be central.
+Definition 4.1. For a type A, let BAut1(A) :≡ ΣX:U∥A = X∥0. The elements of BAut1(A) are types
+which are banded by A or A-bands, for short. We denote A-bands by Xp, where p : ∥A = X∥0 is
+the band. The type BAut1(A) is pointed by A|refl|0.
+Given a band p : ∥A = X∥0, we will write ˜p : ∥X ≃ A∥0 for the associated equivalence.
+Remark 4.2. It’s not hard to see that BAut1(A) is a connected, locally small type—hence essentially
+small, by the join construction [Rij17].
+The characterization of paths in Σ-types tells us what paths between banded types are.
+Lemma 4.3. Consider two A-bands Xp and Yq. A path Xp = Yq of A-bands corresponds to a path
+e : X = Y between the underlying types making the following triangle of truncated paths commute:
+A
+X
+Y .
+p
+q
+|e|0
+In other words, there is an equivalence (Xp = Yq) ≃ (X = Y )(¯p � q).
+□
+For the remainder of this section, let A be a central type. We begin by showing that the type of
+A-bands is a delooping of A.
+Proposition 4.4. We have that Ω BAut1(A) ≃ A.
+Proof. We have Ω BAut1(A) ≃ (A = A)(refl) ≃ (A ≃ A)(id) ≃ A, where the first equivalence uses
+Lemma 4.3 and the last equivalence is by centrality.
+□
+
+14
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+Corollary 4.5. The unique H-space structure on A is deloopable.
+□
+Note that this gives an independent proof that it is associative (cf. Proposition 2.33).
+Theorem 4.6. The type A has a unique delooping.
+Proof. We must show that the type ΣB:U* (ΩB ≃∗ A) is contractible. We will use BAut1(A), with the
+equivalence ψ from Proposition 4.4, as the center of contraction. Let B : U* be a pointed type with a
+pointed equivalence φ : ΩB ≃∗ A. Given x : B, consider pt =B x. Since A is connected, B is simply
+connected. Therefore, to give a banding on pt =B x, it suffices to do so when x is pt, in which case
+we use φ. So we have defined a map f : B → BAut1(A), and it is easy to see that it is pointed.
+We claim that the following triangle commutes:
+ΩB
+Ω BAut1(A)
+A .
+φ
+∼
+Ωf
+ψ
+∼
+Let q : pt =B pt. Then (Ωf)(q) is the path associated to the equivalence
+A ≃ (pt =B pt) ≃ (pt =B pt) ≃ A.
+The first equivalence is φ−1 and the last is φ, as these give the pointedness of f. The middle equivalence
+is the map sending p to p�q. The map ψ comes from the evaluation fibration, so to compute ψ((Ωf)(q))
+we compute what happens to the base point of A. It gets sent to refl, then q, and then φ(q). This
+shows that the triangle commutes.
+It follows that Ωf is an equivalence. Since B and BAut1(A) are connected, f is an equivalence as
+well. So f and the commutativity of the triangle provide a path from (B, φ) to (BAut1(A), ψ) in the
+type of deloopings.
+□
+We conclude this section by showing how to deloop maps A →∗ A.
+Definition 4.7. Given f : A →∗ A, define Bf : BAut1(A) →∗ BAut1(A) by
+Bf(Xp) :≡ (X → A)(f ∗◦˜p−1),
+where f ∗ :≡ f ◦id∗, and we have used that f ∗ ◦ ˜p−1 is well-defined as an element of the set-truncation.
+To give a banding of (X → A)(f ∗◦˜p−1) we may induct on p and use Proposition 3.12.
+The same
+argument shows that Bf is a pointed map.
+Note that f(a∗) = f(a)∗ for any a : A, since f is an H-space map by Proposition 2.31, so there’s
+no choice involved in this definition.
+Let g : BAut1(A) →∗ BAut1(A). Given a loop q : pt = pt, the loop (Ωg)(q) is the composite
+pt = g(pt) = g(pt) = pt,
+which uses pointedness of g and apg(q). We will identify (pt = pt) with A and then write
+Ω′g : A ≃∗ (pt = pt)
+Ωg
+−−→∗ (pt = pt) ≃∗ A.
+Proposition 4.8. We have that Ω′Bf = f for any f : A →∗ A.
+Proof. The following diagram describes how Bf acts on a loop p : pt =BAut1(A) pt:
+Arefl
+(A → A)(f ∗)
+A
+Arefl
+(A → A)(f ∗)
+A
+p
+g�→g◦˜p−1
+∼
+∼
+
+CENTRAL H-SPACES AND BANDED TYPES
+15
+Since ˜p is in the component of the identity, we have ˜p(a) = x · a for all a : A, where x = ˜p(pt). So
+˜p−1(a) = x\a. Then the composite A ≃ A on the right is seen to be
+a �→ evf ∗
+��
+a · f ∗(−)
+�
+◦ ˜p−1
+�
+= evf ∗
+�
+a · f ∗�
+x\(−)
+��
+= a · f(x∗∗) = a · f(x).
+The domain Arefl = Arefl is identified with A by sending a path p to ˜p(pt), which in this case is the x
+above. The codomain (A ≃ A)(id) is identified with A using evid, which sends the displayed function
+to pt · f(x), which equals f(x). So we have that ΩBf = f. By Lemma 2.6, they are equal as pointed
+maps.
+□
+Proposition 4.9. We have that BΩ′g = g for any g : BAut1(A) →∗ BAut1(A).
+Proof. Given an A-band Xp, we need to show that g(Xp) = (X → A)((Ω′g)∗◦˜p−1). First we construct
+a map of the underlying types from left to right. For y : g(Xp), define the map
+Gy : X
+∼
+−→ (pt = Xp) ≃ (Xp = pt)
+apg
+−−→ (g(Xp) = g(pt)) ≃ (pt = pt) → A,
+where the second map is path inversion, and the fourth map uses the trivialization of g(Xp) associated
+to y and pointedness of g. The identification pt = g(pt) corresponds to a unique point y0 : g(pt).
+To check that Gy lies in the right component, we may induct on p and assume y ≡ y0 since g(pt) is
+connected. We then get the map
+Gy0 : A
+id∗
+−−→ A ≃ (pt = pt)
+Ωg
+−−→ (pt = pt) → A,
+since path inversion on (pt = pt) corresponds to inversion on A, and y0 corresponds to the pointing
+of g. This map is precisely the definition of (Ω′g)∗, so G lands in the desired component.
+To check that G defines an equivalence of bands we may again induct on p. Write �y0 : pt ≃ g(pt)
+for the equivalence associated to the point y0 : g(pt), which is a lift of the (equivalence associated to
+the) banding of g(pt). It then suffices to check that the diagram
+g(pt)
+(A → A)((Ω′g)∗)
+pt
+�
+y0
+−1
+G
+ev(Ω′g)∗
+commutes. Let y : g(pt), which we identify with a trivialization y′ : pt = g(pt). Chasing through the
+definition of G and using that apg(refl) = refl, we see that
+Gy(pt) = ev(y′ � y0) = �y0
+−1(y′(pt)) ≡ �y0
+−1(y),
+where ev : (pt = pt) → A is the last map in the definition of Gy, which transports pt along a path.
+Thus we see that the triangle above commutes, whence G is an equivalence of bands, as required.
+□
+Theorem 4.10. We have inverse equivalences
+Ω′ : (BAut1(A) →∗ BAut1(A)) ≃ (A →∗ A) : B.
+In particular, the type BAut1(A) →∗ BAut1(A) is a set.
+Proof. Combine Propositions 4.8 and 4.9.
+□
+4.2. Tensoring bands. In this section, we will construct an H-space structure on BAut1(A), where
+we continue to assume that A is a central type. This H-space structure is interesting in its own right,
+and also implies that BAut1(A) is itself central. It that follows that A is an infinite loop space.
+This elementary lemma will come up frequently.
+Lemma 4.11. Let P : BAut1(A) → U be a set-valued type family. Then �
+Xp P(Xp) is equivalent to
+P(pt).
+
+16
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+Proof. Since each P(Xp) is a set, �
+Xp P(Xp) is equivalent to �
+X:U
+�
+p:A=X P(X|p|0). By path in-
+duction, this is equivalent to P(A|refl|0), i.e., P(pt).
+□
+A consequence of the following result is that any pointed A-band is trivial.
+Proposition 4.12. Let Xp be an A-band. Then there is an equivalence (pt =BAut1(A) Xp) → X.
+Proof. By Lemma 4.3, there is an equivalence (pt =BAut1(A) Xp) ≃ (A ≃ X)(˜p). We will show that
+evp : (A ≃ X)(˜p) → X is an equivalence. By Lemma 4.11, it’s enough to prove this when Xp ≡ pt,
+and this holds because A is central.
+□
+We now show that path types between A-bands are themselves banded. This underlies the main
+results of this section.
+Proposition 4.13. Let Xp and Yq be A-bands. The type Xp =BAut1(A) Yq is banded by A.
+Proof. We need to construct a band ∥A = (Xp = Yq)∥0. Since the goal is a set, we may induct on p
+and q, thus reducing the goal to ∥A = (pt =BAut1(A) pt)∥0. Using that (pt =BAut1(A) pt) ≃ (A ≃ A)(id)
+and that A is central, we may give the set truncation of the inverse of the evaluation fibration at
+idA.
+□
+The following is an immediate corollary of Proposition 4.12.
+Corollary 4.14. For any A-band Xp, the A-band (Xp = Xp) is trivial.
+□
+We next define a tensor product of banded types, using the notion of duals of bands.
+Write
+refl∗ : A = A for the self-identification of A associated to the inversion map id∗ (Definition 3.14) via
+univalence.
+Definition 4.15. Let Xp be an A-band. The band p∗ :≡ |refl∗| � p is the dual of p, and the A-band
+X∗
+p :≡ Xp∗ is the dual of Xp.
+Since id∗ is an involution, it follows that taking duals defines an involution on BAut1(A), meaning
+that X∗∗
+p
+= Xp.
+Lemma 4.16. We have pt = pt∗ in BAut1(A).
+Proof. The underlying type of pt∗ is A, which has a base point, so this follows from Proposition 4.12.
+□
+We now show how to tensor types banded by A.
+Definition 4.17. For Xp, Yq : BAut1(A), define Xp ⊗ Yq :≡ (X∗
+p = Yq), with the banding from
+Proposition 4.13.
+It follows from Lemma 4.3 that the type Xp ⊗Yq is equivalent to (X = Y )(p∗ � q). Since taking duals
+is an involution, we also have equivalences Xp ⊗ Yq ≡ (X∗
+p = Yq) ≃ (Xp = Y ∗
+q ) ≃ (X = Y )(p � q∗).
+Moreover, from Corollary 4.14, we see that X∗
+p ⊗ Xp = pt.
+Tensoring defines a binary operation on BAut1(A), and we now show that this operation is sym-
+metric.
+Proposition 4.18. For any Xp, Yq : BAut1(A), there is a path σ(Xp,Yq) : Xp ⊗ Yq =BAut1(A) Yq ⊗ Xp
+such that σpt,pt = 1.
+Proof. By univalence and the characterization of paths between bands, we begin by giving an equiv-
+alence between the underlying types. The equivalence will be path-inversion, as a map
+(X = Y )(p � q∗) −→ (Y = X)(q � p∗).
+To see that this is valid it suffices to show that the inversion of p � q∗ is q � p∗. We have:
+p � q∗ ≡ p � refl∗ �q = refl∗ �q � p = q � refl∗ � p = q � refl∗ �p ≡ q � p∗,
+
+CENTRAL H-SPACES AND BANDED TYPES
+17
+where we have used associativity of path composition, and that refl∗ = refl∗ by Proposition 3.15.
+To prove the transport condition, we may path induct on both p and q which then yields the
+following triangle:
+(A = A)(refl∗)
+(A = A)(refl∗)
+A .
+evrefl∗
+p�→p
+evrefl∗
+Here we are writing evrefl∗ for the composite (A = A)(refl∗) ≃ (A ≃ A)(id∗)
+evid∗
+−−−→ A. The horizontal
+map is given by path-inversion, which is homotopic to the identity by Proposition 3.15, hence the
+triangle commutes.
+Paths between paths between banded types correspond to homotopies between the underlying
+equivalences. Thus σpt,pt = 1 since path-inversion on (A = A)(refl∗) is homotopic to the identity.
+□
+We now use Lemma 2.4 to make BAut1(A) into an H-space.
+Theorem 4.19. The binary operation ⊗ makes BAut1(A) into an abelian H-space.
+Proof. We start by showing the left unit law. Since pt∗ = pt, we instead consider the goal (pt = Xp) =
+Xp. An equivalence between the underlying types is given by Proposition 4.12, which after inducting
+on p clearly respects the bands. Using Proposition 4.18 and Lemma 2.4, we obtain the desired H-space
+structure.
+□
+Corollary 4.20. For a central type A, the type BAut1(A) is again central. Therefore, A is an infinite
+loop space, in a unique way. Moreover, every pointed map A →∗ A is infinitely deloopable, in a unique
+way.
+Proof. That BAut1(A) is central follows from condition (2) of Proposition 3.6, using Theorems 4.10
+and 4.19 as inputs.
+That A is a infinite loop space then follows from Proposition 4.4: writing
+BAut0
+1(A) :≡ A and BAutn+1
+1
+(A) :≡ BAut1(BAutn
+1(A)), we see that BAutn
+1(A) is an n-fold delooping
+of A. The uniqueness follows from Theorem 4.6. That every pointed self-map is infinitely deloopable
+in a unique way follows by iterating Theorem 4.10.
+□
+Note that BAut1(A) is essentially small (Remark 4.2), so these deloopings can be taken to be in
+the same universe as A.
+From Theorem 4.19 we deduce another characterization of central types:
+Proposition 4.21. A pointed, connected type A is central if and only if ΣX:BAut1(A) X is contractible.
+Proof. If A is central, then by the left unit law of Theorem 4.19, we have
+ΣX:BAut1(A) X ≃ ΣX:BAut1(A) (pt∗ =BAut1(A) X) ≃ 1.
+Conversely, if ΣX:BAut1(A) X is contractible, then so is its loop space. But the loop space is equivalent
+to Σf:A→∗A ∥f = id∥, i.e., the fibre of evid over the base point. Thus evid is an equivalence, since A
+is connected.
+□
+4.3. Bands and torsors. Let A be a central type. We define a notion of A-torsor which turns out
+to be equivalent to the notion of A-band from the previous section. Under our centrality assumption,
+it follows that the resulting type of A-torsors is a delooping of A. An equivalent type of A-torsors has
+been independently studied by David W¨arn, who has also shown that it gives a delooping of A under
+the weaker hypothesis that A has a unique H-space structure.
+Definition 4.22. An action of A on a type X is a map α : A × X → X such that α(pt, x) = x for
+all x : X. If X has an A-action, we say that it is an A-torsor if it is merely inhabited and α(−, x) is
+an equivalence for every x : X. The type of A-torsor structures on a type X is
+TA(X) :≡
+�
+α:A×X→X
+(α(pt, −) = idX) × ∥X∥−1 ×
+�
+x:X
+IsEquiv α(−, x),
+
+18
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+and the type of A-torsors is �
+X:U TA(X).
+Since A is connected, an A-action on X is the same as a pointed map A →∗ (X ≃ X)(id). Normally
+one would require at a minimum that this map sends multiplication in A to composition. We explain
+in Remark 4.28 why our definition suffices.
+The condition that α(−, x) is an equivalence for all x is equivalent to requiring that for every
+x0, x1 : X, there exists a unique a : A with α(a, x0) = x1.
+It is also equivalent to saying that
+(α, pr2) : A × X → X × X is an equivalence.
+For any type X, write ev≃ : (A ≃ X) → X for the evaluation fibration which sends an equivalence
+e to e(pt). For a map f, write Sect(f) for the type of (unpointed) sections of f.
+Lemma 4.23. For any X, we have an equivalence
+TA(X) ≃ ∥X∥−1 × Sect(ev≃).
+Proof. This is simply a reshuffling of the data. The map from left to right sends a torsor structure
+with action α : A × X → X to the map X → (A → X) sending x to α(−, x). By assumption, this
+lands in the type of equivalences, and the condition α(pt, −) = idX says that it is a section. We leave
+the remaining details to the reader.
+□
+Lemma 4.24. Let X be an A-torsor. Then X is connected.
+Proof. Since X is merely inhabited and our goal is a proposition, we may assume that we have x0 : X.
+Then we have an equivalence α(−, x0) : A → X. A is connected by Proposition 3.3, so it follows that
+X is.
+□
+Proposition 4.25. Let X be an A-torsor. Then X is banded by A.
+Proof. Associated to the torsor structure on X is a section X → (A ≃ X) of ev≃.
+Since X is
+0-connected, it lands in a component of A ≃ X. By univalence, this determines a banding of X.
+□
+Theorem 4.26. Let X be a type. There is an equivalence TA(X) ≃ ∥A = X∥0. Therefore, there is
+an equivalence between the type of A-torsors and BAut1(A).
+Proof. Proposition 4.25 gives a map f. We check that the fibres are contractible. Let p : ∥A = X∥0 be
+a banding of X. An A-torsor structure t on X with f(t) = p consists of a section s of ev≃ that lands in
+the component (A ≃ X)(˜p), where ˜p denotes the equivalence associated to p. But by Proposition 4.12,
+the evaluation fibration (A ≃ X)(˜p) → X is an equivalence, so it has a unique section.
+□
+Remark 4.27. It follows that TA(X) is a set. One can also show this using Corollary 4.14 and Propo-
+sition 3.6.
+Remark 4.28. Let X be an A-torsor, or equivalently, an A-band. By Corollary 4.14, we have an
+equivalence e : A ≃ (X ≃ X)(id). Since A has a unique H-space structure, this equivalence is an
+equivalence of H-spaces, where the codomain has the H-space structure coming from composition.
+Since A is connected, the A-action on X gives a map α′ : A →∗ (X ≃ X)(id). (In fact, α′ = e, but
+we won’t use this fact.) Using the equivalence e, it follows from Theorem 4.10 that any map with the
+same type as α′ is deloopable in a unique way. That is, it has the structure of a group homomorphism
+in the sense of higher groups (see [BvDR18]). This explains why our naive definition of an A-action
+is correct in this situation.
+5. Examples and non-examples
+We show that the Eilenberg–Mac Lane spaces K(G, n) are central whenever G is abelian and n > 0.
+In addition, we produce examples of products of Eilenberg–Mac Lane spaces which are central and
+examples which are not central. At present, we do not know whether there exist central types which are
+not products of Eilenberg–Mac Lane spaces. Along the way, we use our results to give a self-contained,
+independent construction of Eilenberg–Mac Lane spaces. To this end, we begin by discussing the base
+case K(G, 1).
+
+CENTRAL H-SPACES AND BANDED TYPES
+19
+5.1. The H-space of G-torsors. Given a group G, we construct the type TG of G-torsors and show
+that it is a K(G, 1). Specifically, a pointed type X is a K(G, 1) if it is connected and comes equipped
+with a pointed equivalence ΩX ≃∗ G which sends composition of loops to multiplication in G. (We
+always point ΩX at refl.)
+When G is abelian, we can tensor G-torsors to obtain an H-space structure on TG which is analogous
+to the tensor product of bands of Theorem 4.19. These constructions are all classical and we therefore
+omit some details.
+Definition 5.1. Let G be a group. A G-set is a set X with a group homomorphism α : G → Aut(X).
+If the set X is merely inhabited and the map α(−, x) : G → X is an equivalence for every x : X, then
+(X, α) is a G-torsor. We write TG for the type of G-torsors. Given two G-sets X and Y , we write
+X →G Y for the set of G-equivariant maps from X to Y , defined in the usual way.
+We may write g · x instead of α(g, x) when no confusion can arise. The following is straightforward
+to check:
+Lemma 5.2. Let X and Y be G-torsors. There is a natural equivalence (X =T G Y ) ≃ (X →G Y ).
+In particular, a G-equivariant map between G-torsors is automatically an equivalence.
+□
+Any group G acts on itself by left translation, making G into a G-torsor which constitutes the base
+point pt of both TG and the type of G-sets. Since a G-equivariant map pt →G X is determined by
+where it sends 1 : G, the map (pt →G X) → X that evaluates at 1 is an equivalence. It is clear that
+the type TG is a 1-type, which implies that its loop space is a group.
+Proposition 5.3. We have a group isomorphism ΩTG ≃ G.
+We only sketch a proof since this is a classical result.
+Proof. Since paths between G-torsors correspond to G-equivariant maps, we have equivalences of sets
+(pt =T G pt) ≃ (pt →G pt) ≃ G,
+where the second equivalence is given by evaluation at 1. The first equivalence sends path compo-
+sition to composition of maps, which reverses the order—i.e., it’s an anti-isomorphism. The second
+equivalence evaluates a map at 1 : G. Thus, for φ, ψ : pt →G pt we have
+φ(ψ(1)) = φ(ψ(1) · 1) = ψ(1) · φ(1),
+where · denotes the multiplication in G.
+In other words, evaluation at 1 is an anti-isomorphism,
+meaning the composite (pt =T G pt) ≃ G is an isomorphism of groups.
+□
+The following proposition says that the G-torsors are precisely those G-sets which lie in the com-
+ponent of the base point.
+Proposition 5.4. A G-set (X, α) is a G-torsor if and only if there merely exists a G-equivariant
+equivalence from pt to X.
+Proof. Suppose X is a G-torsor. To produce a mere G-equivariant equivalence pt ≃G X we may
+assume we have some x : X, since X is merely inhabited. Then (−) · x : G → X yields an equivalence
+which is clearly G-equivariant, as required.
+Conversely, assume that there merely exists a G-equivariant equivalence from pt to X. Since being
+a G-torsor is a proposition, we may assume we have an actual G-equivariant equivalence. But then
+we are done since pt is a G-torsor.
+□
+It follows that TG is connected. Thus by Proposition 5.3 we deduce:
+Corollary 5.5. The type TG is a K(G, 1).
+For the remainder of this section, let G be an abelian group.
+Proposition 5.6. For any two G-torsors S and T, the path type S =T G T is again a G-torsor.
+
+20
+BUCHHOLTZ, CHRISTENSEN, FLATEN, AND RIJKE
+Proof. First we make S =T G T into a G-set. This path type is equivalent to the type S →G T. Using
+that G is abelian, it’s easy to check that the map
+(g, φ) �−→
+�
+s �→ g · φ(s)
+�
+: G × (S →G T) −→ (S →G T)
+is well-defined and makes S →G T into a G-set.
+To check that the above yields a G-torsor, we may assume that S ≡ pt ≡ T, by the previous lemma.
+One can check that Proposition 5.3 gives an equivalence of G-sets, where pt →G pt is equipped with
+the G-action just described. Thus pt →G pt is a G-torsor, as required.
+□
+In order to describe the tensor product of G-torsors, we first need to define duals.
+Definition 5.7. Let (X, α) be a G-torsor. The dual X∗ of X is the G-torsor X with action
+α∗(g, x) :≡ α(g−1, x).
+The tensor product of G-torsors is now defined as X ⊗ Y :≡ (X∗ =T G Y ).
+Proposition 5.8. The tensor product of G-torsors makes TG into an H-space.
+Proof. We verify the hypotheses of Lemma 2.4. Thus our first goal is to construct a symmetry
+σX,Y : (X∗ =T G Y ) =T G (Y ∗ =T G X).
+After identifying paths of G-torsors with G-equivariant equivalences, we may consider the map which
+inverts such an equivalence. A short calculation shows that if φ : X∗ →G Y is G-equivariant, then
+φ−1 : Y ∗ →G X is again G-equivariant. We need to check that the map sending φ to φ−1 is itself
+G-equivariant, so let φ : X∗ →G Y and let g : G. Since the inverse of g · (−) is g−1 · (−), we have:
+(g · φ)−1 = φ−1(g−1 · (−)) = g · φ−1(−),
+using that φ−1 : Y ∗ →G X is G-equivariant. Thus inversion is G-equivariant, yielding the required
+symmetry σ.
+Now we argue that σpt,pt = refl, or, equivalently, that maps pt∗ →G pt are their own inverses. Such
+a map is uniquely determined by where it sends 1 : G, so it suffices to show that φ(φ(1)) = 1 for every
+φ : pt∗ →G pt. Fortunately, we have
+φ(φ(1)) = φ(φ(1) · 1) = φ(1)−1 · φ(1) = 1.
+Lastly, it is straightforward to check that the map (pt∗ →G X) → X which evaluates at 1 : G is
+G-equivariant, for any G-torsor X. This yields the left unit law for the tensor product ⊗. As such we
+have fulfilled the hypotheses of Lemma 2.4, giving us the desired H-space structure.
+□
+Using Proposition 3.6, one can check that TG is a central H-space. (See Proposition 5.9.)
+5.2. Eilenberg–Mac Lane spaces. We now use our results to give a new construction of Eilenberg–
+Mac Lane spaces. For an abelian group G, recall that a pointed type X is a K(G, 1) if it is connected
+and there is a pointed equivalence ΩX ≃∗ G which sends composition of paths to multiplication in
+G. For n > 1, a pointed type X is a K(G, n + 1) if it is connected and ΩX is a K(G, n). It follows
+that such an X is an n-connected (n + 1)-type with Ωn+1X ≃∗ G as groups.
+In the previous section we saw that the type TG of G-torsors is a K(G, 1) and is central whenever
+G is abelian. The following proposition may be seen as a higher analog of this fact.
+Proposition 5.9. Let G be an abelian group and let n > 0. If a type A is a K(G, n) and an H-space,
+then A is central and BAut1(A) is a K(G, n + 1) and an H-space.
+The fact that BAut1(A) is a K(G, n + 1) also follows from [Shu], using the fact that BAut1(A) is
+the 1-connected cover of BAut(A).
+Proof. Suppose that A is a K(G, n) and an H-space.
+Then A →∗ ΩA is contractible, since it is
+equivalent to ∥A∥n−1 →∗ ΩA, and ∥A∥n−1 is contractible.
+So Proposition 3.6 implies that A is
+central.
+By Proposition 4.4, Ω BAut1(A) ≃ A, so BAut1(A) is a K(G, n + 1).
+By Theorem 4.19,
+BAut1(A) is also an H-space.
+□
+
+REFERENCES
+21
+We can use the previous proposition to define K(G, n) for all n > 0 by induction. For the base case
+n ≡ 1 we let K(G, 1) :≡ TG, the type of G-torsors from the previous section. When G is abelian, we
+saw that TG is an H-space, which lets us apply the previous proposition. By induction, we obtain a
+K(G, n) for all n. Note that this construction produces a K(G, n) which lives n − 1 universes above
+the given K(G, 1), but that it is essentially small by the join construction [Rij17].
+5.3. Products of Eilenberg–Mac Lane spaces. Here is our first example of a central type that is
+not an Eilenberg–Mac Lane space.
+Example 5.10. Let K = K(Z/2, 1) = RP ∞ and L = K(Z, 2) = CP ∞, and consider A = K × L.
+This is a connected H-space, and
+�
+K × L →∗ Ω(K × L)
+�
+≃
+�
+K →∗ Ω(K × L)
+�
+since K = ∥K × L∥1
+≃
+�
+K →∗ ΩL
+�
+since K is connected
+≃
+�
+Z/2 →Ab Z)
+by [BvDR18, Theorem 5.1]
+≃ 1.
+So it follows from Proposition 3.6(4) that A is central.
+On the other hand, not every product of Eilenberg–Mac Lane spaces is central.
+Example 5.11. Let K = K(Z/2, 1) = RP ∞ and L′ = K(Z/2, 2). A calculation like the above shows
+that K × L′ →∗ Ω(K × L′) is not contractible, so K × L′ is not central.
+As another example, [Cur68, Proposition Ia] shows that K(Z, 1) × K(Z, 2)) (i.e., S1 × CP ∞) has
+infinitely many distinct H-space structures classically. So it is not central, by Proposition 3.3.
+Clearly both of these examples can be generalized to other groups and shifted to higher dimensions.
+By Proposition 3.3, centrality of a type implies that it has a unique H-space structure.
+The
+converse fails, as we now demonstrate. We are grateful to David W¨arn for bringing our attention to
+this example.
+Example 5.12. The type A :≡ K(Z, 2) × K(Z, 3) is not central, by a computation similar to the one
+in the previous example. However, we note that it admits a unique H-space structure. Since A is a
+loop space it admits an H-space structure, and the type of H-space structures is given by A ∧ A →∗ A
+according to Theorem 2.27. Since A is 1-connected, by [CS20, Corollary 2.32] the smash product
+A ∧ A is 3-connected. It follows that A ∧ A ��∗ A is contractible, since A is 3-truncated. In other
+words, the space of H-space structures on A is contractible.
+References
+[AC63]
+M. Arkowitz and C. R. Curjel. “On the number of multiplications of an H–space”. In:
+Topology 2 (1963), pp. 205–207.
+[BR18]
+Ulrik Buchholtz and Egbert Rijke. “The Cayley-Dickson construction in homotopy type
+theory”. In: High. Struct. 2.1 (2018), pp. 30–41. doi: https://doi.org/10.21136/HS.
+2018.02.
+[Bru16]
+Guillaume Brunerie. “On the homotopy groups of spheres in homotopy type theory”.
+PhD thesis. Laboratoire J.A. Dieudonn´e, 2016. arXiv: 1606.05916.
+[Buc19]
+Ulrik Buchholtz. Non-abelian cohomology (Groups, Torsors, Gerbes, Bands & all that).
+Invited talk at the workshop Geometry in Modal Homotopy Type Theory, Carnegie Mellon
+University. 2019. url: https://youtu.be/eB6HwGLASJI.
+[BvDR18]
+U. Buchholtz, F. van Doorn, and E. Rijke. “Higher Groups in Homotopy Type Theory”.
+In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science.
+LICS ’18. Oxford, United Kingdom: ACM, 2018, pp. 205–214. isbn: 978-1-4503-5583-4.
+doi: 10.1145/3209108.3209150.
+
+22
+REFERENCES
+[Cav21]
+Evan Cavallo. Pointed functions into a homogeneous type are equal as soon as they are
+equal as unpointed functions. Agda formalization, part of the cubical library. 2021. url:
+https://agda.github.io/cubical/Cubical.Foundations.Pointed.Homogeneous.
+html#1616.
+[Cop59]
+A. H. Copeland. “Binary operations on sets of mapping classes.” In: Michigan Mathemat-
+ical Journal 6 (1959), pp. 7–23.
+[CS20]
+J. Daniel Christensen and Luis Scoccola. The Hurewicz theorem in homotopy type theory.
+To appear in Algebraic & Geometric Topology. 2020. arXiv: 2007.05833v2.
+[Cur68]
+C. R. Curjel. “On the H-space structures of finite complexes”. In: Comment. Math. Helv.
+43 (1968), pp. 1–17. doi: 10.1007/BF02564376.
+[Han74]
+Vagn Lundsgaard Hansen. “The homotopy problem for the components in the space of
+maps on the n-sphere”. In: Q. J. Math. 25.1 (Jan. 1974), pp. 313–321. eprint: https:
+//academic.oup.com/qjmath/article-pdf/25/1/313/4366416/25-1-313.pdf.
+[Jam55]
+I. M. James. “Reduced product spaces”. In: Ann. of Math. (2) 62 (1955), pp. 170–197.
+doi: 10.2307/2007107.
+[Rij17]
+E. Rijke. The join construction. 2017. arXiv: 1701.07538.
+[Sco20]
+Luis Scoccola. “Nilpotent types and fracture squares in homotopy type theory”. In:
+Mathematical Structures in Computer Science 30.5 (2020), pp. 511–544. doi: 10.1017/
+s0960129520000146.
+[Shu]
+Mike Shulman. Fibrations with fiber an Eilenberg-MacLane space. Blog post at homotopy-
+typetheory.org. url: https://homotopytypetheory.org/2014/06/30/fibrations-
+with-em-fiber/.
+[Uni13]
+Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of Math-
+ematics. Institute for Advanced Study: http://homotopytypetheory.org/book/, 2013.
+[vDoo18]
+Floris van Doorn. “On the Formalization of Higher Inductive Types and Synthetic Ho-
+motopy Theory”. PhD thesis. Carnegie Mellon University, 2018. arXiv: 1808.10690.
+[Whi46]
+George W. Whitehead. “On products in homotopy groups”. In: Annals of Mathematics
+47 (1946), pp. 460–475.
+University of Nottingham, Nottingham, United Kingdom
+Email address: ulrik.buchholtz@nottingham.ac.uk
+University of Western Ontario, London, Ontario, Canada
+Email address: jdc@uwo.ca
+University of Western Ontario, London, Ontario, Canada
+Email address: jtaxers@uwo.ca
+University of Ljubljana, Ljubljana, Slovenia
+Email address: e.m.rijke@gmail.com
+

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+ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 
+WWW.INFOSOC.IIS.RU 
+ 
+_____________________________ 
+© Калужский М.Л., 2022. 
+Производство и хостинг журнала «Информационное общество» осуществляется Институтом развития 
+информационного общества. 
+Данная статья распространяется на условиях международной лицензии Creative Commons «Атрибуция — 
+Некоммерческое использование — На тех же условиях» Всемирная 4.0 (Creative Commons Attribution – NonCommercial - 
+ShareAlike 4.0 International; CC BY-NC-SA 4.0). См. https://creativecommons.org/licenses/by-nc-sa/4.0/legalcode.ru 
+https://doi.org/10.52605/16059921_2022_06_59 
+Цифровая экономика 
+ИНСТИТУЦИОНАЛИЗАЦИЯ ЦИФРОВОЙ ТОРГОВЛИ В РОССИЙСКОЙ 
+ФЕДЕРАЦИИ: ОБРАТНЫЙ ОТСЧЕТ 
+Статья рекомендована к публикации главным редактором Т.В. Ершовой 31.07.2022. 
+Калужский Михаил Леонидович 
+Кандидат философских наук, доцент 
+МОФ «Фонд региональной стратегии развития», исполнительный директор 
+Омский государственный технический университет, каф. «Организация и управление наукоемкими 
+производствами», доцент 
+Омск, Российская Федерация 
+frsr@inbox.ru 
+Аннотация 
+Институционализация цифровой торговли является одним из важнейших направлений формирования 
+информационного общества в Российской Федерации. Исследования отражают наметившееся отставание 
+российского индекса готовности экономики поддерживать онлайн-покупки. Автор анализирует причины 
+отставания в контексте институциональных особенностей развития цифровой торговли. В качестве 
+основного препятствия, снижающего экономическую эффективность и конкурентоспособность цифровой 
+торговли, выделяется недостаточное внимание государства формированию инновационных институций 
+цифрового рынка. 
+Ключевые слова 
+сетевая экономика; цифровая торговля; цифровой рынок; электронная коммерция; институциональная 
+политика; контрактное производство; сетевое предпринимательство; маркетплейсы; логистический 
+провайдинг 
+Введение  
+Цифровизация не просто определяет ключевое направление развития российской экономики, но 
+служит источником институционального роста цифровой торговли. Указом Президента РФ № 203 
+от 09.05.2017 г. определены национальные интересы, затрагивающие сферу цифровой торговли, 
+среди которых следует выделить: 
+1) формирование виртуальных рынков и обеспечение лидерства на них за счет развития 
+российской экосистемы цифровой экономики; 
+2) обеспечение недискриминационного доступа к товарам и услугам российских 
+поставщиков; 
+3) поддержка отраслей, использующих преимущества информационных технологий; 
+4) увеличение экспорта за рубеж несырьевых товаров и услуг;  
+5) создание платежной и логистической инфраструктуры интернет-торговли. 
+Перед Правительством РФ поставлена задача формирования технологической основы 
+цифровой экономики, в том числе через повышение доступности электронных форм коммерческих 
+отношений предприятиям малого и среднего бизнеса [1]. Для ее решения приняты и реализуются 
+«Стратегия развития информационного общества в Российской Федерации на 2017-2030 годы», 
+национальная программа «Цифровая экономика Российской Федерации» и федеральные проекты 
+«Информационная инфраструктура», «Цифровые технологии», «Цифровые услуги и сервисы 
+онлайн» и др.  
+
+KyPHAA
+HOOPMALOHHOE
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+WWW.INFOSOC.IIS.RU 
+60 
+ 
+1 Российская цифровая торговля в мировых рейтингах 
+Рейтинговые оценки цифровой торговли в России свидетельствуют о том, что пока рано говорить 
+о значительных успехах. Согласно аналитическому отчету «Интернет-торговля в России: 2021» 
+компании Data Insight институциональное и инфраструктурное развитие цифровой торговли в 
+Россия пока далеко от совершенства (см. табл. 1) [2, с. 26]. 
+Таблица 1. Рейтинг Российской Федерации в мировой системе цифровой торговли 
+Рейтинг 
+Место 
+Best Countries For Investment In E-commerce And Digital Sector (Ceoworld) 
+– индекс привлекательности страны для инвестирования в электронную 
+коммерцию и цифровой сектор 
+15 
+The Inclusive Internet Index 
+– индекс доступности цифровой инфраструктуры, цен, локального контента, 
+вовлеченности пользователей и культурных факторов 
+25 
+The Ease of Doing Business Index 
+– индекс благоприятности условий предпринимательской деятельности 
+28 
+UNCTAD B2C E-commerce Index Ranking (UNCTAD) 
+–  индекс готовности экономики поддерживать онлайн-покупки 
+41 
+ 
+В целом приведенный рейтинг довольно наглядно отражает сложившееся положение в 
+цифровой торговле. Индекс Ceoworld показывает лучший результат за счет доминирования на 
+рынке крупных торговых сетей, интернет-магазинов и маркетплейсов. Inclusive Internet Index 
+демонстрирует вовлеченность пользователей в среду цифровой торговли. Ease of Doing Business Index 
+отражает отставание предложения отечественных продавцов от покупательского спроса.  
+Хуже всех выглядит индекс UNCTAD (ООН), согласно которому в 2020 г. готовность 
+экономических институтов поддерживать онлайн-покупки в России находилась на 41 месте из 152 
+стран мира [3, с. 14]. Именно этот индекс отражает недостаточную эффективность 
+институциональной политики государства в сфере цифровой торговли. 
+2 Институциональные процессы в цифровой торговле 
+Цифровая торговля являет собой типичный пример технологической инновации, выступающей 
+следствием очередного институционального цикла [4, с. 25]. Институциональный цикл проходит в 
+своем развитии те же этапы, что и любой жизненный цикл в экономике, менеджменте или 
+маркетинге: выход на рынок, рост, зрелость и упадок. На первых этапах институционального цикла 
+доминируют институции (поведенческие шаблоны и традиции), спонтанно возникающие в 
+рыночной среде за пределами влияния государства. На последних этапах государство 
+регламентирует и ставит под свой контроль экономическую активность. Этот процесс и называется 
+институционализацией. 
+Развитие институций всегда опережает развитие институтов, поскольку они возникают 
+вследствие экономической активности рыночных субъектов, пытающихся выжить под гнетом 
+рыночных доминантов и государства. Тогда как институты представляют собой результат 
+реактивной деятельности государства на сокращение налогооблагаемой базы вследствие 
+вытеснения традиционных субъектов рынка его неинституционализированными игроками. 
+Проще говоря, государство озаботилось институционализацией цифровой торговли после того, как 
+покупатели стали отворачиваться от традиционных продавцов, а объем отправлений с AliExpress 
+превысил объем внутренних отправлений через ФГУП «Почта России». 
+Поэтому не следует ожидать синхронного развития институтов и институций цифровой 
+торговли. Это противоречило бы самой природе институционального развития. Речь идет о 
+естественном несовершенстве институциональной политики государства и ошибках при 
+определении ее приоритетов. Внедрение экономических новаций неизбежно связано с высокой 
+вероятностью незапланированного поведения рыночных субъектов. Оно нуждается в мониторинге 
+ситуации и корректировке. 
+
+KyPHAA
+HOOPMALOHHOE
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+WWW.INFOSOC.IIS.RU 
+61 
+ 
+3 Институциональная эволюция цифровой торговли 
+В основе институций рынка цифровой торговли лежат конкурентные преимущества ее субъектов, 
+связанные с экономией транзакционных издержек при совершении сделок [5, с. 8]. Цифровая 
+экономика предоставила им неограниченный доступ к аудитории, автоматизацию продаж и 
+сетевую инфраструктуру логистики. Причем, на различных стадиях институционального цикла 
+цифровой торговли указанные преимущества доминируют в определенной последовательности 
+(см. рис. 1). 
+ 
+Рис. 1. Конкурентные преимущества на разных стадиях институционального цикла цифровой торговли 
+На первой стадии появилась возможность совершать сделки через электронные доски 
+объявлений, в социальных сетях и на интернет-форумах. Начался бум интернет-магазинов, 
+возникли первые интернет-аукционы, сервисы совместных покупок и дропшиппинг. Структурные 
+изменения происходили вне внимания государства, поскольку на потребительском рынке сделки 
+совершались втемную, налоги с них – не выплачивались. Государственная статистика фиксировала 
+лишь кратное увеличение почтовых отправлений. 
+На второй стадии начался взрывной рост электронной коммерции. Увеличение масштабов 
+интернет-продаж 
+привело 
+к 
+появлению 
+на 
+рынке 
+провайдеров 
+логистических 
+услуг, 
+сформировавших 
+альтернативную 
+инфраструктуру 
+цифровой 
+торговли. 
+Сильнее 
+всего 
+пострадали традиционные оптово-розничные посредники, кредитно-финансовые организации, а 
+также обеспечиваемые ими налоговые поступления. Государству пришлось приступить к 
+институциональному регулированию цифровой торговли. 
+Третья стадия знаменуется завершением структурной перестройки экономического 
+ландшафта, доминированием укрупняющихся ключевых игроков рынка и сменой их 
+стратегических приоритетов. Если прежде основными конкурентами субъектов цифровой 
+торговли выступали традиционные оптово-розничные продавцы, то здесь они терпят 
+сокрушительное поражение и вытесняются на задворки рынка. Конкурентная борьба 
+разворачивается 
+за 
+повышение 
+эффективности 
+и 
+оптимизацию 
+бизнес-процессов 
+при 
+возрастающей роли государственного регулирования. 
+Что будет происходить на четвертой, завершающей стадии институционального цикла 
+цифровой торговли пока трудно предсказать, как и сроки ее начала. Сформируются новые 
+институции и их носители, действующие за рамками институционального регулирования 
+государства. Их появление станет очередной попыткой участников рынка вырваться за рамки 
+удушающего влияния действующих доминантов цифрового рынка. Сейчас до этого еще далеко и 
+на повестке дня стоят совсем иные проблемы. 
+4 Смена институционального вектора 
+Российская экономика находится в самом начале третьей стадии институционального цикла 
+цифровой торговли, где ведущим фактором конкурентоспособности становится сравнительная 
+эффективность бизнес-процессов [6, с. 334]. Между участниками рынка обостряется конкуренция, 
+сам рынок структурируется, значение государственного регулирования возрастает [7, с. 8]. Кроме 
+того, доминирующая роль в вопросах ассортимента, ценообразования и сбыта переходит от 
+продавцов к потребителям. Они голосуют рублем, и глобализация предоставляет им для этого все 
+возможности. 
+Институциональная политика государства осложняется новизной стоящих задач. Основная 
+проблема состоит в определении ориентиров и приоритетов институционального строительства. 
+Любая ошибка неизбежно приводит к институциональному тупику, оттоку покупателей и 
+отставанию в развитии цифрового рынка. В результате он переходит под контроль более успешных 
+доступ к 
+аудитории
+Стадия 
+1
+автоматизация 
+продаж
+Стадия 
+2
+сетевая 
+логистика
+Стадия 
+3
+
+KyPHAA
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+WWW.INFOSOC.IIS.RU 
+62 
+ 
+зарубежных конкурентов [8, с. 113-114]. С другой стороны, наличие успешных конкурентов 
+позволяет изучить их опыт и применить его в своей практике. 
+Так, к примеру, на непродовольственном рынке цифровой торговли в России доминируют 
+два противоборствующих течения: розничные сети (Эльдорадо, DNS, Leroy Merlin и пр.) и 
+маркетплейсы (Wildberries, Lamoda, Ozon, Яндекс-Маркет, Сбермаркет и др.). Их противостояние 
+обусловлено разной моделью ведения бизнеса: розничные сети извлекают прибыль из своих 
+продаж, тогда как маркетплейсы получают ее от оказания торговых услуг. Розничные сети 
+стремятся закрыть и защитить каналы сбыта, а маркетплейсы, наоборот, стремятся максимально 
+открыть их. 
+Первоначально пальма первенства была у розничных сетей, довольно успешно 
+лоббировавших свои интересы через АКИТ (Ассоциация компаний интернет-торговли). Их 
+главный 
+интерес 
+состоял 
+в 
+создании 
+институциональных 
+барьеров 
+для 
+неинституционализированной трансграничной торговли. Образно говоря, сделать так, чтобы 
+покупка телефона Xiaomi на Aliexpress обходилась покупателям столь же дорого, как в России.  
+Максимальным 
+успехом 
+лоббирования 
+торговых 
+сетей 
+стало 
+снижение 
+порога 
+беспошлинного ввоза товаров для личного пользования до 200 евро. Институциональный эффект 
+такого 
+решения 
+представляется 
+весьма 
+спорным, 
+поскольку 
+поддержку 
+получили 
+не 
+производители отечественной продукции, а сфера торговли. В убытке оказались частные 
+потребители, чья покупательская способность снизилась. При этом институциональный цикл 
+торговых сетей уже находится в начале четвертой стадии: на рынке цифровой торговли взрывной 
+рост продаж показывают не они, а маркетплейсы [9]. 
+Главный интерес маркетплейсов заключается в росте продаж через привлечение 
+максимального числа потребителей, для которых главным фактором является цена товара. 
+Маркетплейсы не извлекают прибыль от продажи товаров – она формируется в процессе оказания 
+логистических услуг продавцам. В отличие от розничных сетей, низкие цены для них не беда, а 
+источник финансирования и институционального роста.  
+Со сменой рыночного доминанта на глазах меняется и институциональная политика 
+государства. Так, с 28.03.2022 в ЕАЭС порог беспошлинного ввоза товаров физическими лицами 
+(временно) возвращен к 1000 евро. Кроме того, лидирующие позиции членов АКИТ перешли к 
+крупнейшим маркетплейсам (Wildberries, Ozon, Avito, Lamoda, Яндекс-Маркет), что не могло не 
+сказаться на смене приоритетов ее лоббистской деятельности.1 Вектор институционального 
+развития 
+цифровой 
+торговли 
+в 
+России 
+сменился 
+от 
+попыток 
+ограничить 
+свободу 
+неинституционализованных ее участников к формированию ориентированной на них рыночной 
+инфраструктуры.  
+5 Большой разворот 
+Роль маркетплейсов на рынке цифровой торговли трудно переоценить. Первоначально 
+источником их институционального роста был переток покупателей из традиционной торговли с 
+более высоким уровнем трансакционных издержек и розничных цен. Однако к концу 2010-х гг. этот 
+ресурс исчерпал себя. Сегодня на потребительском рынке сопротивление им оказывают розничные 
+сети, с разной степенью успешности осваивающие цифровую торговлю. Наиболее эффективным 
+оружием против них является снижение розничных цен и повышение доступности товаров для 
+покупателей.  
+2021 год ознаменовался тектоническими изменениями ценовой политики ведущих 
+российских маркетплейсов: они кратно снизили комиссию для продавцов. Снижение комиссии 
+маркетплейсов составило у Wildberries до 5-15% (было 38%), у Ozon.ru до 5-8% (было 5-25%), у Яндекс-
+Маркета до 2% (было 3-20%). Правильность принятого решения подтвердилось феноменальным 
+приростом продаж (в 2-3 раза) (см. табл. 2) [10]. 
+ 
+ 
+ 
+1 Стандарты качества / Бизнесу // АКИТ. URL: https://akit.ru/business/standards (дата обращения 12.06.2022). 
+
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+ 
+Таблица 2. Результаты лидирующих маркетплейсов России за 2021 г. 
+Маркетплейс 
+онлайн-продажи 
+заказы 
+средний чек 
+млрд. руб. 
+прирост 
+млн. шт. 
+прирост 
+руб. 
+прирост 
+Wildberries 
+805,7 
++95% 
+771,9 
++153% 
+1040 
+-23% 
+Ozon 
+446,7 
++126% 
+221,2 
++199% 
+2020 
+-24% 
+Яндекс-Маркет 
+132,6 
++180% 
+29,7 
++151% 
+4110 
++12% 
+AliExpress 
+106,1 
++116% 
+48 
++152% 
+2210 
+-14% 
+Lamoda 
+71,2 
++34% 
+14,1 
++15% 
+5050 
++17% 
+ 
+При этом прирост продаж был обратно пропорционален размеру среднего чека: чем 
+меньше сумма покупки, тем больше желающих ее совершить. Из общей картины несколько 
+выбивается Яндекс-Маркет, но только за счет широкого присутствия на нем розничных сетей, 
+наоборот, ориентированных на прирост среднего чека.  
+Особняком на этом фоне стоит маркетплейс Lamoda, демонстративно игнорирующий 
+институциональные тренды цифрового рынка. У него самые высокие тарифы – 35-70% от 
+розничной цены продаваемых товаров. Можно предположить, что его прибыль значительно 
+превосходит прибыль продавцов товара. В этом Lamoda похож на розничные сети. Неудивительно, 
+что прирост его показателей стабильно ниже прироста продаж других лидеров цифрового рынка. 
+Следует особо отметить наличие огромного потенциала продаж, связанного со снижением 
+суммы среднего чека. Все это задает тренд на совершенствование логистических технологий, 
+направленных на уменьшение транзакционных издержек и снижение розничных цен для 
+покупателей. Участники цифровой торговли, действующие в рамках указанного тренда, 
+добиваются наилучших результатов.  
+6 Новые горизонты цифровой торговли 
+Практика показывает, что институции цифровой торговли оказывают решающее влияние на 
+конкурентоспособность ее субъектов [11, с. 60-61]. Отставание E-commerce Index Ranking лишь 
+подтверждает необходимость корректировки институционального регулирования цифровой 
+торговли и смены его приоритетов. Игнорирование рыночных трендов и закономерностей резко 
+снижает эффективность государственной политики и конкурентоспособность российской 
+цифровой экономики в целом. 
+В качестве институциональных ориентиров следует выделить три приоритетных 
+направления развития сетевой экономики и покупательский спрос как движущую силу рыночного 
+механизма. Выделение этих ориентиров связано с наиболее успешными институциями, 
+определяющими вектор институционального развития цифровой торговли.  
+1. Контрактное производство – институция, основанная на изготовлении продукции 
+независимым производителем по техническому заданию заказчика с отгрузкой «под ключ». Такое 
+производство пере��одит из категории работ в категорию услуг. Оно не производит собственную 
+продукцию, 
+оказывая 
+предоплаченные 
+услуги 
+заказчикам, 
+что 
+обеспечивает 
+большую 
+экономическую эффективность.  
+Контрактное производство не нуждается в кредитовании, имеет отрицательную 
+оборачиваемость средств и не несет предпринимательских рисков в торговле. В этой модели 
+инициаторами производства выступают независимые заказчики, отслеживающие конъюнктуру 
+рынка, принимающие на себя предпринимательские риски и финансирующие производство за 
+счет 
+собственных 
+средств. 
+Контрактное 
+производство 
+становится 
+придатком 
+торговли, 
+ориентирующейся на потребительский спрос. 
+Пример: Биржа контрактного производства Московского инновационного кластера.2 
+2. Логистический провайдинг (англ. Third Party Logistics) – институция, основанная на 
+делегировании нестратегических внутрифирменных функций независимым провайдерам 
+ 
+2 Биржа контрактного производства // Московский инновационный кластер. URL: https://i.moscow/contract_exchange (дата 
+обращения: 18.06.2022). 
+
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+64 
+ 
+логистических услуг. Такой провайдинг также переходит из категории работ в категорию услуг. 
+Провайдеры делятся с заказчиками экономией на масштабе оказываемых услуг, за счет своей узкой 
+специализации обеспечивая более высокое качество и эффективность.  
+Практически любая внутрифирменная функция может быть передана независимому 
+провайдеру: бухгалтерский учет, обработка заказов, разработка технической документации, 
+организация продаж, документооборота и т.д. [12, с. 57]. Высший уровень логистического 
+провайдинга (5PL) предполагает делегирование как функции, так и контроля за ее реализацией по 
+принципу «передал и забыл». В идеальной модели заказчик сосредотачивается на стратегическом 
+направлении деятельности, а все сопутствующие функции делегирует внешним провайдерам.  
+Примеры: маркетплейсы, аутсорсинговые и фулфилментовые компании, бухгалтерские 
+сервисы и т.д. 
+3. Сетевое предпринимательство – институция, основанная на использовании преимуществ 
+виртуальной среды, сетевой экономики и цифровой торговли. Они позволяют сократить 
+затратность ведения бизнеса и снижают входной барьер для участников цифрового рынка, 
+сокращая временные затраты на реализацию бизнес-проектов. 
+Идеальная модель сетевого предпринимательства стремится к тому, что называется 
+«виртуальная организация», не имеющая ни офиса, ни постоянного штата сотрудников [13, с. 279-
+281]. Предприниматель здесь выступает в роли организатора и координатора «цепочек создания 
+ценностей», потенциал которых он использует для реализации своего бизнес-проекта. В качестве 
+его сетевых партнеров выступают как контрактные производители, так и провайдеры 
+логистических услуг. 
+Пример: Самодеятельные продавцы маркетплейсов (Ozon, Wildberries и Яндекс-Маркет), 
+продающие контрактные товары под своими брендами. 
+В своей совокупности все институции образуют экосреду цифровой экономики, в которой 
+цифровая торговля инициирует не только процесс товародвижения, но и товарного производства. 
+Покупатель своим спросом инициирует предпринимательскую активность продавца, который на 
+свой страх и риск организует контрактное производство востребованных товаров и привлекает 
+сетевых провайдеров логистических услуг.  
+В корне меняются институциональные роли участников сетевого рынка: 
+Покупатели – получают возможность неограниченного выбора, ставя продавцов в условия 
+совершенной конкуренции. 
+Продавцы – откликаются на запросы покупателей, первичный спрос которых инициирует их 
+вторичную предпринимательскую активность.  
+Сетевые провайдеры – оказывают логистические услуги продавцам (не покупателям!), 
+принимая на себя отдельные функции организации товародвижения. 
+Производители – оказывают услуги контрактного производства продавцам, соревнуюсь 
+между собой в гибкости производства и скорости выполнения заказов. 
+Пока наибольшую эффективность показывает институция, в рамках которой покупатель 
+взаимодействует с маркетплейсом, принимающим на себя все заботы по организации товаропотока 
+(Wildberries, Ozon, AliExpress). Однако уже сегодня многие продавцы продают свои товары 
+одновременно на нескольких маркетплейсах, а нелояльные покупатели, сравнивая цены, покупают 
+там, где дешевле. Свобода потребительского выбора размоет диктат маркетплейсов, как они сегодня 
+размывают диктат розничных сетей.  
+Рано или поздно и маркетплейсы достигнут предела институционального развития и 
+перейдут в категорию «при прочих равных» за счет обострения внутривидовой конкуренции. Если 
+это 
+произойдет, 
+то 
+между 
+продавцами 
+и 
+покупателями 
+сформируется 
+логистическая 
+инфраструктура, в равной мере доступная всем участникам цифрового рынка. Аналогично 
+электричество или компьютеры были когда-то источником рыночной конкурентоспособности, а 
+сегодня воспринимаются как естественная часть рыночного ландшафта. 
+Заключение 
+Приоритетом институциональной политики государства может стать превращение цифровой 
+торговли в один из локомотивов экономического роста. Для этого необходимо сосредоточиться на 
+
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+ 
+снижении транзакционных издержек в сетях товародвижения. В традиционной торговле 
+потребительскими товарами транзакционные издержки (маржа оптово-розничных сетей) 
+составляли 80-100% от конечной цены товара. В маркетплейсах типа Lamoda они и сегодня 
+составляют 30-70% от цены продавца.  
+Вместе с тем, практика институционального развития цифровой торговли задает совсем иной 
+вектор. Более продвинутые м��ркетплейсы (Wildberries, Ozon, Яндекс-Маркет) еще в начале 2021 года 
+инициативно снизили размер своей комиссии до 3-5% и это привело к впечатляющим результатам. 
+Так, например, продажи самозанятых на Wildberries только в первом квартале 2022 года выросли на 
+410% (до 2 млрд руб.), а их численность увеличилась почти пятикратно (до 150 тыс. чел.).3 
+В условиях экономического кризиса и западных санкций снижение транзакционных 
+издержек в цифровой торговле способно компенсировать снижение покупательной способности 
+населения. 
+Важно 
+сохранить 
+доступность 
+товаров 
+массового 
+спроса 
+и 
+поддержать 
+товаропроизводителей. Вытесняя из торговой цепочки посредническое звено за счет ускоренной 
+цифровизации торговли, можно не только способствовать решению социальных задач, но и 
+стимулировать рост предпринимательской активности в производственной сфере. Представляется, 
+что именно эта цель должна стать одним из приоритетов институциональной политики 
+государства в отношении цифровой торговли на ближайшие годы. 
+Литература 
+1. Указ Президента РФ № 203 от 09.05.2017 г. «О Стратегии развития информационного 
+общества в Российской Федерации на 2017-2030 годы» / Документы // Президент России. 
+URL: http://kremlin.ru/acts/bank/41919. 
+2. Интернет-торговля в России 2021: Аналитический отчет. М.: Data Insight, 2022. 156 с.  
+3. The UNCTAD B2C E-commerce Index 2020 / UNCTAD Technical Notes on ICT for 
+Development 2021 // UNCTAD ONU. 2022. № 17. 22 р. 
+4. Блуммарт Т. Четвертая промышленная революция и бизнес: как конкурировать и 
+развиваться в эпоху сингулярности. М.: Альпина Паблишер, 2019. 204 с. 
+5. Данные для лучшей жизни: Обзор доклада о мировом развитии. Washington: 
+Международный банк реконструкции и развития / Всемирный банк, 2021. 39 с. 
+6. Кочетков Е.П. Цифровая трансформация экономики и технологические революции: 
+вызовы для текущей парадигмы менеджмента и антикризисного управления // 
+Стратегические решения и риск-менеджмент. 2019. Т. 10. № 4. С. 330-341. DOI: 
+10.17747/2618-947X-2019-4-330-341. 
+7. Антимонопольное регулирование в цифровую эпоху: как защищать конкуренцию в 
+условиях глобализации и четвертой промышленной революции: монография. М.: ВШЭ, 
+2019. 391 с. 
+8. Борисова В.В., Юань Х., Тан Л. Стратегии развития электронной платформы Aliexpress в 
+России // Вестник Ростовского государственного экономического университета (РИНХ). 
+2020. № 4 (72). С. 110-115.  
+9. Романова Т. Маркетплейсы рвутся вверх: за счет чего выросли обороты крупнейших 
+ретейлеров России / Бизнес // Forbes. [Электронный ресурс]. 7 июня 2022 г. URL: 
+https://www.forbes.ru/biznes/467927-marketplejsy-rvutsa-vverh-za-scet-cego-vyrosli-oboroty-
+krupnejsih-retejlerov-rossii (дата обращения: 15.06.2022). 
+10. Рейтинг ТОП-100 крупнейших российских интернет-магазинов. М.: Data Insight, 2022. URL: 
+https://top100.datainsight.ru (дата обращения: 17.06.2022). 
+11. Слонимская М.А. Сетевые формы организации экономики. Мн.: Беларуская навука, 2018. 
+279 с. 
+12. Tan A., Shukkla S. Digital transformation of the supply chain: a practical guide for. Danvers 
+(USA): World Scientific Publishing, 2021. 152 p. 
+13. Уорнер М., Витцель М. Виртуальные организации. Нов��е формы ведения бизнеса в XXI 
+веке. М.: Добрая книга, 2005. 296 с. 
+ 
+ 
+ 
+3 Продажи самозанятых из России на Wildberries выросли на 410% с января 2022 года. 29.06.2022. / Экономика // ТАСС. URL: 
+https://tass.ru/ekonomika/15065193 (дата обращения 13.07.2022). 
+
+KyPHAA
+HOOPMALOHHOE
+6ECTB0ИНФОРМАЦИОННОЕ ОБЩЕСТВО | 2022 | № 6 
+WWW.INFOSOC.IIS.RU 
+66 
+ 
+INSTITUTIONALIZATION OF DIGITAL TRADE IN THE RUSSIAN 
+FEDERATION: COUNTDOWN 
+Kaluzhsky, Mikhail Leonidovich 
+Candidate of philosophical sciences, associate professor 
+Fund of Regional Development Strategy, executive director 
+Omsk State Technical University, department “Organization and management of science-intensive industries”, 
+associate professor 
+Omsk, Russian Federation 
+frsr@inbox.ru 
+Abstract 
+The institutionalization of digital trade is one of the most important directions in the formation of the information 
+society in the Russian Federation. The studies reflect the emerging lag in the Russian economy readiness index to 
+support online shopping. The author analyzes the reasons for the lag in the context of the institutional features of 
+the development of digital trade. As the main obstacle that reduces the economic efficiency and competitiveness of 
+digital trade, insufficient attention of the state to the formation of innovative institutions of the digital market is 
+highlighted. 
+Keywords 
+network economy; digital trade; digital market; e-commerce; institutional policy; contract manufacturing; network 
+entrepreneurship; marketplaces; logistics providers  
+References 
+1. Ukaz Prezidenta RF № 203 ot 09.05.2017 g. «O Strategii razvitiya informacionnogo obshchestva v 
+Rossijskoj Federacii na 2017-2030 gody» / Dokumenty // Prezident Rossii. URL: 
+http://kremlin.ru/acts/bank/41919. 
+2. Internet-torgovlya v Rossii 2021: Analiticheskij otchet. M.: Data Insight, 2022. 156 s.  
+3. The UNCTAD B2C E-commerce Index 2020 / UNCTAD Technical Notes on ICT for 
+Development 2021 // UNCTAD ONU. 2022. № 17. 22 r. 
+4. Blummart T. Chetvertaya promyshlennaya revolyuciya i biznes: kak konkurirovat' i razvivat'sya 
+v ehpokhu singulyarnosti. M.: Al'pina Pablisher, 2019. 204 s. 
+5. Dannye dlya luchshej zhizni: Obzor doklada o mirovom razvitii. Washington: Mezhdunarodnyj 
+bank rekonstrukcii i razvitiya / Vsemirnyj bank, 2021. 39 s. 
+6. Kochetkov E.P. Cifrovaya transformaciya ehkonomiki i tekhnologicheskie revolyucii: vyzovy dlya 
+tekushchej paradigmy menedzhmenta i antikrizisnogo upravleniya // Strategicheskie resheniya i 
+risk-menedzhment. 2019. T. 10. № 4. S. 330-341. DOI: 10.17747/2618-947X-2019-4-330-341. 
+7. 
+Antimonopol'noe regulirovanie v cifrovuyu ehpokhu: kak zashchishchat' konkurenciyu v usloviyakh 
+globalizacii i chetvertoj promyshlennoj revolyucii: monografiya. M.: VSHEH, 2019. 391 s. 
+8. Borisova V.V., Yuan' KH., Tan L. Strategii razvitiya ehlektronnoj platformy Aliexpress v Rossii 
+// Vestnik Rostovskogo gosudarstvennogo ehkonomicheskogo universiteta (RINKH). 2020. № 4 
+(72). S. 110-115.  
+9. Romanova T. Marketplejsy rvutsya vverkh: za schet chego vyrosli oboroty krupnejshikh 
+retejlerov Rossii / Biznes // Forbes. [Ehlektronnyj resurs]. 7 iyunya 2022 g. URL: 
+https://www.forbes.ru/biznes/467927-marketplejsy-rvutsa-vverh-za-scet-cego-vyrosli-oboroty-
+krupnejsih-retejlerov-rossii (data obrashcheniya: 15.06.2022). 
+10. Rejting TOP-100 krupnejshikh rossijskikh internet-magazinov. M.: Data Insight, 2022. URL: 
+https://top100.datainsight.ru (data obrashcheniya: 17.06.2022). 
+11. Slonimskaya M.A. Setevye formy organizacii ehkonomiki. Mn.: Belaruskaya navuka, 2018. 279 s. 
+12. Tan A., Shukkla S. Digital transformation of the supply chain: a practical guide for. Danvers 
+(USA): World Scientific Publishing, 2021. 152 p. 
+13. Uorner M., Vitcel' M. Virtual'nye organizacii. Novye formy vedeniya biznesa v XXI veke. M.: 
+Dobraya kniga, 2005. 296 s. 
+
+KyPHAA
+HOOPMALOHHOE
+6ECTB0

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@@ -0,0 +1,2919 @@
+arXiv:2301.00801v1  [stat.ML]  2 Jan 2023
+Causal Inference (C-inf) — asymmetric scenario of
+typical phase transitions
+Agostino Capponi ∗ Mihailo Stojnic †
+Department of Industrial Engineering and Operations Research
+Columbia University, New York, NY 10027, USA
+Abstract
+In this paper, we revisit and further explore a mathematically rigorous connection between Causal in-
+ference (C-inf) and the Low-rank recovery (LRR) established in [10]. Leveraging the Random duality
+- Free probability theory (RDT-FPT) connection, we obtain the exact explicit typical C-inf asymmetric
+phase transitions (PT). We uncover a doubling low-rankness phenomenon, which means that exactly two
+times larger low rankness is allowed in asymmetric scenarios compared to the symmetric worst case ones con-
+sidered in [10]. Consequently, the final PT mathematical expressions are as elegant as those obtained in [10],
+and highlight direct relations between the targeted C-inf matrix low rankness and the time of treatment.
+Our results have strong implications for applications, where C-inf matrices are not necessarily symmetric.
+Index Terms: Causal inference; Random duality theory; Algorithms; Matrix completion; Spar-
+sity.
+1
+Introduction
+Causal inference (C-inf) deals with the design of estimation strategies that allow researchers to draw
+causal conclusions based on data. The overarching goal is to draw a conclusion regarding the effect of a
+causal variable, which is typically referred to as the “treatment” or the “intervention” on some outcome of
+interest. For example, suppose we want to estimate the causal effect of a drug on deadly cancer progression
+(vs no exposure to the drug). Then we want to compare metastasis in the patient’s body one month after
+the drug regime has begun versus metastasis in the absence of exposure to the drug. The main challenge for
+causal inference is that we are not generally able to observe both of these states: at the point in time when
+we are measuring the outcomes, each individual either has had drug exposure or has not.
+The problem of estimating the counterfactual, i.e., what would have been the outcome in the absence
+of a treatement, is central in many disciplines, including economics, health, and social sciences (see, e.g.
+[2,11,13,14,32,33,50]), machine learning and theoretical computer science (see, e.g. [25–28]). Methodological
+developments to estimate causal effects have been based on experimental or observational data. Experimental
+research offers the most plausibly unbiased estimates, but experiments are frequently infeasible because they
+are costly or subject to moral objections.
+Observational data instead are becoming increasing available
+due to technological advancements in the design of sensor and hardware devices. Our focus is on causal
+inference in observational studies, and specifically on the design of efficient algorithmic techniques to estimate
+counterfactuals.
+The C-inf approaches can be broadly classified into three categories: 1) the unconfoundedness (see,
+e.g. [14, 32]); 2) the synthetic control (see, e.g. [1, 2, 11]); and 3) the matrix completion (see, e.g. [3, 4, 15].
+Matrix completion methods build upon the foundation works of [7,9,29]). Perhaps unexpectedly, all three
+methods heavily rely on mathematical, statistical, and ultimately algorithmic concepts with very deep roots
+∗e-mail: ac3827@columbia.edu
+†e-mail: flatoyer@gmail.com
+1
+
+in information theory. Our work is positioned within the third line of work that mathematically resembles
+the matrix completion (MC) problem.
+Along the same lines, our work extends significantly the analysis developed in the companion paper [10].
+Therein, we obtained the exact explicit typical worst case C-inf phase transitions (PT), and further showed
+that these phase transitions are achievable by the symmetric targeted C-inf matrices. In the present paper,
+we consider a generic asymmetric context, to deal with the situation that C-inf matrices are not necessarily
+always symmetric in real applications.
+This allows us improving upon the results from [10] in certain
+scenarios. We build further upon the RDT-FPT synergistic mechanisms considered in [10], and precisely
+characterize the corresponding asymmetric PTs. We also uncover a doubling low-rankness phenomenon,
+which means that exactly two times larger low rankness is allowed in asymmetric scenarios compared to the
+symmetric worst case ones of [10].
+2
+Causal inference mathematical setup
+In this section, we revisit the explicit causal inference (C-inf) ↔ matrix completion (MC) connection,
+established in [10]. Therein, we have discussed the connection between low rank recovery (LRR), matrix
+completion (MC), and the causal inference (C-inf). Mathematically speaking, one has that the MC is a
+special case of the LRR and the C-inf is a special case of the MC itself. Consequently, the mathematical
+models that describe the LRR problems can be used to describe the MC and ultimately the C-inf ones as
+well. Below we present the C-inf mathematical setup developed through such a connection in [10].
+We start with a low rank matrix Xsol ∈ Rn×n with the singular value decomposition (SVD)
+X = UΣV T ,
+(1)
+where
+σ(X) ≜ diag(Σ)
+and
+U T U = In×n
+and
+V T V = In×n,
+(2)
+with In×n being the n × n identity matrix and diag(·) being the operator that creates a column vector of
+the diagonal elements of its matrix argument. We then define ℓ∗
+p(X) to be the so-called ℓp (quasi) norm of
+σ(X) (the vector of the singular values of X), i.e.
+ℓ∗
+p(X) ≜ ℓp(σ(X)), p ∈ R+.
+(3)
+The following limiting ℓp(·) connections are important as well
+ℓ∗
+0(Xsol) ≜ ℓ0(σ(Xsol)) = ∥σ(Xsol)∥0 = lim
+p−→0 ∥σ(Xsol)∥p = lim
+p−→0 ℓp(σ(Xsol)) = lim
+p−→0 ℓ∗
+p(Xsol).
+(4)
+Moreover, we also define the so-called block masking matrix M as (see Figure 1 as well)
+M matrix in block causal inference (C-inf):
+M ≜ M (l) ≜ 1n×11T
+n×1 − I(l)(I(l))T 1n×11T
+n×1I(l)(I(l))T
+and
+I(l) ≜
+� 0l×(n−l)
+I(n−l)×(n−l)
+�
+.
+(5)
+One then has the following two optimization problems that are at the heart of the C-inf ↔ MC connection
+ℓ∗
+0-minimization (C-inf – MMT)
+min
+X
+ℓ∗
+0(X)
+subject to
+Y = M ◦ X.
+(6)
+ℓ∗
+1-minimization (C-inf – MMT)
+min
+X
+ℓ∗
+1(X)
+subject to
+Y = M ◦ X,
+(7)
+2
+
+M =
+1
+Matrix M – block causal inference (C-inf)
+1
+0
+1
+0 and 1 grouped in blocks
+l × l block of all 1s
+l × (n − l) block of all 1s
+(n − l) × (n − l) block of all 0s
+(n − l) × l block of all 1s
+Figure 1: Matrix M ≜ M (l) – block causal inference (C-inf) setup
+where ◦ stands for the component-wise multiplication. Namely, keeping in mind that ℓ∗
+0(X) effectively counts
+the number of the nonzero singular values of X, the optimization problem in (6) is exactly the recovery of
+the C-inf targeted low rank matrix X from the linear observations Y obtained through a masking via M.
+Moreover, the problem in (6) (with a generic M) is a standard matrix completion setup which on the other
+hand is a special case of the LRR problems (expressed in the “masking matrix terminology” (MMT)). On
+the other hand, the optimization problem in (7) is the tightest convex relaxation heuristic typically utilized
+in the matrix completion literature for solving NP-hard problem (6). For more on the origin of these two
+problems and their connection within the LRR and MC context we refer to the introductory LRR/MC
+papers [7,30,36]. More on their importance and different related algorithmic considerations can be found in
+many papers that followed (see, e.g. [8,16–22,31]).
+Here though, we would particularly like to point out reference [4] where the very same C-inf context
+was considered and the very same C-inf ↔ MC connection recognized. Considerations from [4] are in
+fact especially convenient to properly understand in what C-inf contexts the block structure of the matrix
+M might appear. To see that one can connect it to the so-called counterfactuals and the units/treatments
+terminology employed in [4].
+First we note that M can be alternatively defined as
+Mi,j =
+�
+1,
+(i, j)-th element of Xsol is observed
+0,
+otherwise.
+(8)
+It is then rather clear that ones in M allow reading out the corresponding elements of Xsol while zeros block
+(mask) them. Then the context of [4] is roughly as follows. One first assumes that the matrix X contains
+observations about a certain set of, say, n units (e.g. individuals, subpopulations, and geographic regions)
+over a period of say, n, time instances. After that the rows of X are allocated to the units and the columns
+to the time instances and one would like to estimate the effects that a certain treatment may have on the
+treated units. A subset of the units (say those that correspond to the rows i > l) is then at time l exposed
+to an irreversible treatment. Examples of treatments include health therapies, socio-economic policies, and
+taxes. To ensure an appropriate assessment of the resulting treatment effects, in addition to having the
+values of X after the treatment, one would need to have the access to the so-called counterfactuals – the
+values of the treated units – had the treatment not been applied. Relating back to the matrix completion
+terminology, one would basically need to estimate (a presumably low rank) X while not having access to its
+3
+
+portion covered by the block-mask M = M (l). In other words, one would need to solve (6) with M = M (l).
+The above describes the C-inf via counterfactuals and the underlying role of matrix M. Moreover, if
+one views things in the time domain, i.e. if the columns of M represent time axis, then the observations
+in ceratin rows will not be available after a fixed point in time. In the block scenario this point is fixed
+across the affected rows. However, it does not necessarily need to be fixed (for more in this direction we refer
+to [2] (in particular, the California tobacco example), [49] (in particular, the latent factor modeling in the
+context of the simultaneous/staggered treatment adoption), and to [5, 6, 34] (in particular, the health care
+applications) as excellent references for understanding the need of various C-inf scenarios). As this and [10]
+are the introductory papers, where we present the overall methodology, we selected the block causal inference
+scenario as probably the most representative and well-known one. In some of our companion papers we will
+show how the methodology that we are introducing here can be utilized to handle other C-inf scenarios as
+well.
+3
+ℓ∗
+0 − ℓ∗
+1 equivalence
+As mentioned earlier, solving the generic LRR (and consequently the C-inf as its a special case) might be
+difficult due to a highly non-convex objective function in (6). Various heuristics can be employed depend-
+ing on the practical scenarios that one can face. In the mathematically most challenging so-called linear
+regime, the above mentioned ℓ∗
+1-minimization relaxation heuristic (often called nuclear norm minimization)
+is typically viewed as the best known provably polynomial one. We adopt the same view in what follows
+and take it as a current benchmark for the algorithmic handling of the C-inf. As mentioned above, a rather
+remarkable feature of this heuristic is that sometimes it can actually solve the underlying problems exactly.
+When that happens we say that the following ℓ∗
+0 − ℓ∗
+1-equivalence phenomenon occurs.
+ℓ∗
+0 − ℓ∗
+1-equivalence (C-inf): ℓ∗
+0 ⇐⇒ ℓ∗
+1
+Let Xsol be the solution of (6) and let ˆX be a solution of (7) and set
+RMSE ≜ ∥vec( ˆX) − vec(Xsol)∥2.
+If and only if ( ˆX = Xsol and RMSE = 0)
+then
+(ℓ∗
+0 − minimization ⇐⇒ ℓ∗
+1 − minimization).
+(9)
+The above basically means that when the ℓ∗
+0 − ℓ∗
+1-equivalence happens the optimization problems in (6)
+and (7) are equivalent and as such replaceable by each other. We denote such a phenomenon as ℓ∗
+0 ⇐⇒ ℓ∗
+1.
+That would, of course, be an ideal scenario where it would be basically possible to replace the non-convex
+optimization problem with the convex one without losing anything in terms of the accuracy of the obtained
+solutions. Since the mere existence of such a phenomenon is rather remarkable we will in this paper be
+interested in uncovering the underlying intricacies that enable for it ro happen. Moreover, as it will turn
+out that its occurrence is not an anomaly but rather a consequence of a generic property, we will then
+raise the bar accordingly and attempt to provide not only the proof of its existence but also its a complete
+analytical characterization. This will include a full characterization as to how often and in what scenarios it
+might happen. To do so we will combine the Random Duality Theory (RDT) tools from [37–44] and several
+advanced sophisticated probabilistic concepts that we will introduce along the way in the sections that follow
+below.
+We start with some algebraic ℓ∗
+0 − ℓ∗
+1-equivalence preliminaries which are borrowed from the RDT. The
+first one is a generic LRR ℓ∗
+0−ℓ∗
+1-equivalence result (the result is basically an adaptation of the corresponding
+CS equivalence condition from [39–41] (similar adaptation can also be found in [24])).
+Theorem 1. ( [10] ℓ∗
+0 − ℓ∗
+1-equivalence condition (LRR) – general X) Consider a ¯U ∈ Rn×k such that
+¯U T ¯U = Ik×k and a ¯V ∈ Rn×k such that ¯V T ¯V = Ik×k and a rank− k matrix Xsol = X ∈ Rn×n with all of its
+columns belonging to the span of ¯U and all of its rows belonging to the span of ¯V T . Also, let the orthogonal
+spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) be such that U ≜
+� ¯U
+¯U ⊥�
+and V ≜
+� ¯V
+¯V ⊥�
+and
+U T U ≜
+� ¯U
+¯U ⊥�T � ¯U
+¯U ⊥�
+= In×n
+and
+V T V ≜
+� ¯V
+¯V ⊥�T � ¯V
+¯V ⊥�
+= In×n.
+(10)
+4
+
+For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) = Avec(Xsol) ∈ Rm and let ˆX be the
+solution of (7). If
+(∀W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n)
+− tr ( ¯U T W ¯V ) < ℓ∗
+1(( ¯U ⊥)T W ¯V ⊥),
+(11)
+then
+ℓ∗
+0 ⇐⇒ ℓ∗
+1
+and
+RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0,
+(12)
+and the solutions of (6) and (7) coincide. Moreover, if
+(∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n)
+− tr ( ¯U T W ¯V ) ≥ ℓ∗
+1(( ¯U ⊥)T W ¯V ⊥),
+(13)
+then there is an X from the above set of matrices with columns belonging to the span of ¯U and rows belonging
+to the span of ¯V such that the solutions of (6) and (7) are different.
+Proof. The proof is a trivial adaptation of the proof for symmetric matrices given in Appendix A.
+Continuing further in the spirit of the RDT the following corollary is a matrix completion specific variant
+of the above theorem.
+Corollary 1. ( [10] ℓ∗
+0 − ℓ∗
+1-equivalence condition via masking matrix (MC/C-inf) – general X)
+Assume the setup of Theorem 1 with Xsol being the unique solution of (6). Let the masking matrix M ∈ Rn×n
+have m ones and (n2−m) zeros and let A be generated via M, i.e. let A be the matrix obtained after removing
+all the zero rows from diag−1(vec(M))In2×n2. If and only if
+min
+W,W T W=1,M◦W=0n×n
+tr ( ¯U T W ¯V ) + ℓ∗
+1(( ¯U ⊥)T W ¯V ⊥) ≥ 0,
+(14)
+then
+ℓ∗
+0 ⇐⇒ ℓ∗
+1
+and
+RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0,
+(15)
+and the solutions of (6) and (7) coincide.
+Finally, the following spectral oriented corollary was proven in [10] as well.
+Corollary 2. ( [10] ℓ∗
+0−ℓ∗
+1-equivalence condition via mask-modified bases spectra (C-inf) – general
+X) Assume the setup of Theorem 1 with k ≤ l. Let M ≜ M (l) ∈ Rn×n and I(l) ∈ Rn×(n−l) be as defined in
+(5). Set
+ΛV
+≜
+((I(l))T ¯V ⊥)−1(I(l))T ¯V
+ΛU
+≜
+((I(l))T ¯U ⊥)−1(I(l))T ¯U
+Q
+=
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+− I
+Q⊥
+1
+=
+�
+(I(l))T ¯U ⊥( ¯U ⊥)T I(l)�−1
+− I.
+(16)
+C-inf perfectly succeeds: ℓ∗
+0 ⇐⇒ ℓ∗
+1
+and
+RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0
+If and only if
+λmax(ΛT
+V ΛV ΛT
+UΛU) ≤ 1.
+(17)
+Moreover, if
+λmax (Q) λmax
+�
+Q⊥
+1
+�
+≤ 1,
+(18)
+then again ℓ∗
+0 ⇐⇒ ℓ∗
+1 and RMSE = ∥vec( ˆX − vec(Xsol)∥2 = 0 and the C-inf perfectly succeeds as well.
+5
+
+Since we will be working in the mathematically most challenging large n linear regime, we find it useful
+to introduce the following large dimensional scalings
+β ≜ lim
+n→∞
+k
+n
+and
+η ≜ lim
+n→∞
+l
+n
+and
+α ≜ lim
+n→∞
+m
+n2 = lim
+n→∞
+n2 − (n − l)2
+n2
+= 1 − (1 − η)2.
+(19)
+The key highlight result of [10] is the following theorem obtained through an analysis that relied on the
+above corollary and a combination of the Random duality theory (RDT) and Free probability theory (FPT).
+It basically establishes the worst case phase-transition (PT) that ℓ∗
+1, tightest convex relaxation heuristic,
+exhibits when used for solving C-inf in a typical statistical scenario.
+Theorem 2. (ℓ∗
+1 – phase transition – C-inf (typical worst case)) Consider a rank-k matrix Xsol =
+X ∈ Rn×n with the Haar distributed ( not necessarily independent) bases of its orthogonal row and column
+spans ¯U ⊥ ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) (XT
+sol ¯U ⊥ = Xsol ¯V ⊥ = 0n×(n−k)). Let M ≜ M (l) ∈ Rn×n be as
+defined in (5). Assume a large n linear regime with β ≜ limn→∞ k
+n and η ≜ limn→∞ l
+n and let βwc and η
+satisfy the following
+C-inf ℓ∗
+1 worst case phase transition (PT) characterization
+ξ(wc)
+η
+(β) ≜ β − 1
+2 +
+�
+η − η2 = 0.
+(20)
+If and only if β ≤ βwc
+lim
+n→∞ P(ℓ∗
+0 ⇐⇒ ℓ∗
+1) =
+lim
+n→∞ P(RMSE = 0) = 1,
+(21)
+and the solutions of (6) and (7) coincide with overwhelming probability.
+The results obtained based on the above theorem are shown in Figure 2, where one can clearly see that
+the phase transition curve splits the entire (β, η) region into two subregions: 1) the first one (below (or to
+the right of) the curve) where the ℓ∗
+0 − ℓ∗
+1-equivalence phenomenon occurs; and 2) the second one (above (or
+to the left of) the curve) where the ℓ∗
+0 − ℓ∗
+1-equivalence is lacking. This means that one can recover Xsol
+masked by M as in (6) via the ℓ∗
+1 heuristic from (7) with the residual mean square error (RMSE) equal to
+zero. In other words, for the system parameters (β, η) that belong to the subregion below the curve one has
+a perfect recovery with Xsol and ˆX (the respective solutions of (6) and (7)) being equal to each other and
+consequently with RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0. On the other hand, in the subregion above the
+curve, the ℓ∗
+1 heuristic fails and one can even find an Xsol for which RMSE → ∞.
+4
+Analysis of the ℓ∗
+0 −ℓ∗
+1-equivalence – typical asymmetric scenario
+In this section we consider when the conditions given in Corollary 2 are met. As in [10], we will be working
+in a “typical” statistical context. On the other hand, differently from [10], instead of focusing on the worst
+case (symmetric) scenario we here consider a typical asymmetric scenario setup. Practically this means
+two things: 1) as in [10], both ¯V and ¯U will be assumed as Haar distributed; and 2) differently from Theorem
+2 and [10], ¯V and ¯U will now be assumed as independent of each other. In a way one can view the worst case
+scenario from [10] as an extreme where ¯V and ¯U are “not independent at all” (or, in other words, equal to
+each other). Along similar lines, one can then view the scenario that we will consider here as another extreme
+where ¯V and ¯U are “not dependent at all” (or, in other words, completely independent). In situations where
+no particular structure of a low rank nonsymmetric Xsol is favored over any other this one would naturally be
+a most reasonable choice. In other words, it is not only an extreme case, but actually the one that typically
+might most faithfully describe the performance of the underlying C-inf heuristics.
+4.1
+Free probability theory (FPT) – preliminaries
+Below we provide a short preview of the most basic FPT concepts needed for our analysis (we refer to our
+companion paper [10] for a more detail treatment). As is by now well known, the work od Dan Voiculescu
+6
+
+η
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+β
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+(β, η) region of success/failure — C-inf ℓ∗1 PT
+RMSE −→ ∞, ℓ∗1
+fails
+RMSE = 0, ℓ∗1 succeeds
+ℓ∗1’s PT: ξ(wc)
+η
+(β) = β − 1
+2 + �η − η2 = 0
+Figure 2: Causal inference (C-inf) – typical worst case ℓ∗
+1 phase transition
+on group theories (see, e.g. [46–48]) established the foundations of the FPT. As the practical importance
+of FPT became immediately evident a substantial interest for further studying was generated and, in the
+years that followed, quite a few nice results appeared that made the whole theory more approachable and
+ultimately presentable in an easily understandable way. Along the same lines, we follow into the footsteps
+of [10], leave all the abstractions out and focus on the FPT’s key practically applicable components (for
+further details see also, e.g. [12,23,35,45–48]).
+4.1.1
+Basics of FPT – random matrix variables
+We assume large n linear regime and consider two symmetric matrices A = AT ∈ Rn×n and B = BT ∈ Rn×n
+with Haar distributed eigenspaces. We also assume that their individual respective spectral laws are fA(·)
+and fB(·). Three different transforms of these spectral densities will be needed. We start with the first one,
+the so-called Stieltjes (or G) transform
+G(z)
+≜
+�
+If
+f(x)
+z − xdx,
+z ∈ C \ If,
+(22)
+where If is the domain of f(·). The following inverse relation is also well known
+f(x) = lim
+ǫ→0+
+G(x − iǫ) − G(x + iǫ)
+2iπ
+or
+f(x) = − lim
+ǫ→0+
+imag(G(x + iǫ))
+π
+.
+(23)
+For the above to hold it makes things easier to implicitly assume that f(x) is continuous. We will, however,
+utilize it even in discrete (or semi-discrete) scenarios since the obvious asymptotic translation to continuity
+would make it fully rigorous. A bit later though, when we see some concrete examples where things of this
+nature may appear, we will say a few more words and explain more thoroughly what exactly can be discrete
+and how one can deal with such a discreteness. In the meantime we proceed with general principles not
+necessarily worrying about all the underlying technicalities that may appear in scenarios deviating from the
+typically seen ones and potentially requiring additional separate addressing. To that end we continue by
+considering the R(·)- and S(·)-transforms that satisfy the following
+R(G(z)) +
+1
+G(z) = z,
+(24)
+7
+
+and
+S(z) =
+1
+R(zS(z))
+and
+R(z) =
+1
+S(zR(z)).
+(25)
+Let fA(·) and fB(·) be the spectral distributions of A and B and let RA(z)/SA(z) and RB(z)/SB(z) be their
+associated R(·)-/S(·)-transforms. One then has the following
+Key Voiculescu’s FPT concepts [46, 47]:
+C
+=
+A + B
+=⇒
+RC(z)
+=
+RA(z) + RB(z)
+C
+=
+AB
+=⇒
+SC(z)
+=
+SA(z)SB(z).
+(26)
+Now it is relatively easy to see that (22)-(26) are sufficient to determine the spectral distribution of the sum
+or the product of two independent matrices with given spectral densities and the Haar distributed bases of
+eigenspaces. The above is of course a generic principle. It can be applied pretty much always as long as
+one has access to the statistics of the underlying matrices A and B. In the following section we will raise
+the bar a bit higher and show that in the C-inf context one can use all of the above in such a manner that
+eventually all the quantities of interest are explicitly determined.
+4.1.2
+Spectral preliminaries
+We start by recalling on Q from (16) and introducing Q1
+Q1
+≜
+ΛT
+V ΛV
+Q
+≜
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+− I
+Sp(Q1)
+⇐⇒\0
+Sp(Q),
+(27)
+where Sp(·) stands for the spectrum of the matrix argument and ⇐⇒\0 means the equivalence of the parts
+of the spectra outside the zero eigenvalues. It is rather obvious that it will then be sufficient to handle the
+spectrum of
+D
+≜
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l).
+(28)
+Consider Haar distributed ¯U ⊥
+D ∈ Rn×(n−l) with ( ¯U ⊥
+D)T ¯U ⊥
+D = I(n−l)×(n−l) and let
+UD
+=
+� ¯UD
+¯U ⊥
+D
+�
+with
+U T
+DUD = In×n.
+(29)
+Also, we assume that ¯U ⊥
+D (and UD) are independent of ¯V ⊥. After setting
+¯D
+≜
+(I(l))T U T
+D ¯V ⊥( ¯V ⊥)T UDI(l),
+(30)
+we have that the spectra of D and ¯D are statistically identical, i.e.
+Sp(D) ≜ Sp((I(l))T ¯V ⊥( ¯V ⊥)T I(l)) ⇐⇒P Sp((I(l))T U T
+D ¯V ⊥( ¯V ⊥)T UDI(l)) ≜ Sp( ¯D),
+(31)
+where ⇐⇒P stands for the statistical/probabilistic equivalence. Two facts enable the above statistical iden-
+tity: 1) the spectrum of the projector ¯V ⊥( ¯V ⊥)T does not change under pre- and post-unitary multiplications
+on both sides; and 2) the Haar structure of ¯V ⊥ remains preserved. Modulo zero eigenvalues, we then further
+have
+Sp((I(l))T U T
+D ¯V ⊥( ¯V ⊥)T UDI(l)) ⇐⇒P\0 Sp( ¯V ⊥( ¯V ⊥)T UDI(l)(I(l))T U T
+D) ⇐⇒ Sp( ¯V ⊥( ¯V ⊥)T ¯U ⊥
+D( ¯U ⊥
+D)T ), (32)
+where, similarly as above, ⇐⇒P\0 stands for the statistical/probabilistic equivalence in the part of the
+spectrum outside the zero eignevalues (introduced due to the non-square underlying matrices). Clearly, the
+8
+
+key object of our interest below will be
+˜D
+≜
+¯V ⊥( ¯V ⊥)T ¯U ⊥
+D( ¯U ⊥
+D)T ,
+(33)
+where both ¯V ⊥ and ¯U ⊥
+D are Haar distributed and independent of each other. After setting
+V
+≜
+¯V ⊥( ¯V ⊥)T
+U
+≜
+¯U ⊥
+D( ¯U ⊥
+D)T ,
+(34)
+we easily have from (33)
+˜D
+≜
+VU.
+(35)
+The following lemma proven in [10] characterizes the G-transform of ˜D..
+Lemma 1. ( [10]) Let ¯V ⊥ ∈ Rn×(n−k) and ¯U ⊥
+D ∈ Rn×(n−k) be Haar distributed unitary bases of (n − k)-
+dimensional subspaces of Rn. Let V and U be as in (34) and ˜D as in (35), i.e.
+V
+≜
+¯V ⊥( ¯V ⊥)T
+U
+≜
+¯U ⊥
+D( ¯U ⊥
+D)T
+˜D
+≜
+VU.
+(36)
+In the large n linear regime, with β ≜ limn→∞ k
+n, the G-transform of the spectral density of ˜D, f ˜
+D(·), is
+G±
+˜
+D(z) = z − (β + η) ±
+�
+(z − (β + η))2 + 4βη(z − 1)
+2(z2 − z)
+.
+(37)
+4.2
+Asymmetric scenario – FPT analysis of the ℓ∗
+0 − ℓ∗
+1-equivalence
+As in [10], we will again rely on the free probability theory. This time though things will be a bit more
+complicated as we will be determining, so to say, the “joint spectrum” of λT
+V λV λT
+UλU. In other words, based
+on Corollary 2 and (17), we have
+ℓ∗
+0 − ℓ∗
+1 − −equivalence
+⇐⇒
+λmax(λT
+V λV λT
+UλU) ≤ 1,
+(38)
+and consequently determining the upper edge of the “joint spectrum” of λT
+V λV λT
+UλU would be then sufficient
+to establish ℓ∗
+0 − ℓ∗
+1-equivalence. We recall that in [10] we determined only the individual spectra λT
+V λV and
+λT
+UλU (which in the worst case was sufficient to ultimately obtain corresponding C-inf ℓ∗
+1 PT). While the
+calculations and supporting technicalities might on occasion be a bit heavy the overall methodology will be
+fairly similar to what we presented in [10]. In fact, to make things easier to follow we will try to parallel the
+presentation from [10] as much as possible. We start by recalling on Q1 and introducing Q⊥
+1 , and Q1
+Q1
+≜
+λT
+V λV
+Q⊥
+1
+≜
+λT
+UλU
+Q1
+≜
+Q1Q⊥
+1 .
+(39)
+We also recall on the definitions of Q and Q⊥ and introduce Q in the following way
+Q
+≜
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+− I
+Q⊥
+≜
+�
+(I(l))T ¯U ⊥( ¯U ⊥)T I(l)�−1
+− I
+Q
+≜
+QQ⊥.
+(40)
+9
+
+4.2.1
+The spectrum of Q1 ≜ λT
+V λV λT
+UλU – theoretical considerations
+Since (Q1, Q) and (Q⊥
+1 , Q⊥) are statistically identical pairs, we will, for the time being, focus on only one of
+them, say (Q1, Q). To that end, we first recall the statistical relations within the pairs
+Q1
+≜
+ΛT
+V ΛV
+Q
+≜
+�
+(I(l))T ¯V ⊥( ¯V ⊥)T I(l)�−1
+− I = D−1 − I
+Sp(Q1)
+⇐⇒\0
+Sp(Q),
+(41)
+where ⇐⇒\0 stands for the spectral equivalence outside the zeros eigenvalues. We will also find it convenient
+to work with the spectrum of Q. Later on we will make the necessary adjustments so that the results fully
+fit the spectrum of Q1. To start things off we first note
+GQ(z) = GD−1(z + 1).
+(42)
+To see that (42) indeed holds, we first observe that the spectral functions of Q and D−1, fQ(x) and fD−1(x),
+can be connected in the following way
+fQ(x) = fD−1(x + 1).
+(43)
+Then from (22) we have
+GQ(z) =
+� fQ(x)
+z − x dx =
+� fD−1(x + 1)
+z − x
+dx =
+�
+fD−1(x + 1)
+z + 1 − (x + 1)dx =
+�
+fD−1(x)
+z + 1 − xdx = GD−1(z + 1).
+(44)
+From (41) and (42) we also have
+RQ(z) = RD−1(z) − 1.
+(45)
+Namely, (24) first gives
+RQ(GQ(z)) = z −
+1
+GQ(z),
+(46)
+and then
+RQ(z)
+=
+G−1
+Q (z) − 1
+z
+⇐⇒
+z
+=
+GQ
+�
+RQ(z) + 1
+z
+�
+RD−1(z)
+=
+G−1
+D−1(z) − 1
+z
+⇐⇒
+z
+=
+GD−1 �
+RD−1(z) + 1
+z
+�
+.
+(47)
+Combining (42) and (47) we obtain
+GQ
+�
+RQ(z) + 1
+z
+�
+=
+GD−1 �
+RD−1(z) + 1
+z
+�
+⇐⇒
+GQ
+�
+RQ(z) + 1
+z
+�
+=
+GQ
+�
+RD−1(z) + 1
+z − 1
+�
+⇐⇒
+RQ(z) + 1
+z
+=
+RD−1(z) + 1
+z − 1
+⇐⇒
+RQ(z)
+=
+RD−1(z) − 1,
+(48)
+which is exactly (46). From (25) we further have
+SQ(z) =
+1
+RQ(zSQ(z)) =
+1
+RD−1(zSQ(z)) − 1,
+(49)
+and
+RD−1(zSQ(z)) =
+1
+SQ(z) + 1.
+(50)
+10
+
+Relying further on (25) we also have
+RD−1(zSQ(z)) =
+1
+SD−1(zSQ(z)RD−1(zSQ(z)) =
+1
+SD−1
+�
+zSQ(z)
+�
+1
+SQ(z) + 1
+�� =
+1
+SD−1 (z + zSQ(z)).
+(51)
+A combination of (50) and (51) gives a way to connect the S-transforms of D−1 and Q
+1
+SD−1 (z + zSQ(z)) =
+1
+SQ(z) + 1.
+(52)
+From (40) and the key FPT principles (26) we find
+SQ(z) = SQ(z)SQ⊥(z) = (SQ(z))2,
+(53)
+where we used the fact that Q and Q⊥ are statistically identical and as such have the same S-transform.
+One can now rewrite (52) with z → zRQ(z) and utilize (25) to obtain
+1
+SD−1 (zRQ(z)+zRQ(z)SQ(zRQ(z)))
+=
+1
+SQ(zRQ(z)) + 1
+⇐⇒
+1
+SD−1
+�
+zRQ(z)+zRQ(z)√
+SQ(zRQ(z))
+�
+=
+1
+√
+SQ(zRQ(z)) + 1
+⇐⇒
+1
+SD−1
+�
+zRQ(z)+z√
+RQ(z)
+�
+=
+�
+RQ(z) + 1.
+(54)
+Replacing z → GQ(z), (54) can be further rewritten
+1
+SD−1
+�
+zRQ(z)+z√
+RQ(z)
+�
+=
+�
+RQ(z) + 1
+⇐⇒
+1
+SD−1
+�
+GQ(z)RQ(GQ(z))+GQ(z)√
+RQ(GQ(z))
+�
+=
+�
+RQ(GQ(z)) + 1.
+(55)
+From (24) we find
+RQ(GQ(z)) +
+1
+GQ(z)
+=
+z
+⇐⇒
+GQ(z)RQ(GQ(z))
+=
+zGQ(z) − 1.
+(56)
+Combining further (55) and (56) we also have
+1
+SD−1
+�
+GQ(z)RQ(GQ(z))+GQ(z)√
+RQ(GQ(z))
+�
+=
+�
+RQ(GQ(z)) + 1
+⇐⇒
+1
+SD−1
+�
+zGQ(z)−1+√
+GQ(z)√
+zGQ(z)−1
+�
+=
+�
+zGQ(z)−1
+GQ(z)
++ 1.
+(57)
+As in [12] one has for the connection between the S-transforms of the matrix and its inverse
+SD(z) =
+1
+SD−1(−1 − z).
+(58)
+Keeping (58) in mind, one can rewrite (57) in the following way
+1
+SD−1
+�
+GQ(z)RQ(GQ(z))+GQ(z)√
+RQ(GQ(z))
+�
+=
+�
+RQ(GQ(z)) + 1
+⇐⇒
+SD
+�
+−zGQ(z) −
+�
+GQ(z)
+�
+zGQ(z) − 1
+�
+=
+�
+zGQ(z)−1
+GQ(z)
++ 1.
+(59)
+We will make a SD(z) − GD(z) connection below. however, before doing so, we will need to make certain
+adjustments in the GD(z) transform itself.
+i) Adjusting GD(z) for the difference between ˜D and ¯D
+11
+
+We now briefly recall on the connection between ˜D, ¯D, and D. First, from (32) and (19) we have
+˜D
+=
+¯V ⊥( ¯V ⊥)T ¯U ⊥
+D( ¯U ⊥
+D)T
+¯D
+=
+( ¯U ⊥
+D)T ¯V ⊥( ¯V ⊥)T ¯U ⊥
+D.
+(60)
+The spectra of ˜D and ¯D are modulo scalings practically identical. Since ¯D has all the eigenvalues that ˜D
+has with l = ηn zero eigenvalues removed one can connect their G-transforms in the following way.
+G ¯
+D(z)
+=
+1
+1 − η
+�
+G ˜
+D(z) − η
+z
+�
+.
+(61)
+To see that (61) is indeed true we first connect the spectral pdfs of ˜D and ¯D
+f ¯
+D(x)
+=
+1
+1 − η (f ˜
+D(x) − ηδ(x)) .
+(62)
+Then from (22) we have
+G ¯
+D(x) =
+� f ¯
+D(x)
+z − x dx =
+1
+1 − η
+�� f ˜
+D(x)
+z − x dx − η
+�
+δ(x)
+z − xdx
+�
+=
+1
+1 − η
+�
+G ˜
+D(z) − η
+z
+�
+.
+(63)
+Connecting beginning and end in (63) we obtain (61).
+ii) Adjusting GD(z) for the difference between Q1 and Q
+We recall that Q1 has the same eigenvalues as Q minus n − l − k zero eigenvalues (when n − l − k ≤ 0 that
+means that Q1 has all the eigenvalues of Q plus |n−l−k| zero eigenvalues). To account for this difference we
+find it useful to introduce a matrix D1 obtained by removing/adding |n − (l + k)| ones into the spectrum of
+D. As these added ones are inversion invariant they remain in the spectrum after the inversion. This means
+that after the inversion of D1 and subtraction of the identity matrix they become zeros and basically have
+an effect on Q as if |n − (l + k)| zeros were added or removed which is exactly what we need to account for
+the difference between Q1 and Q. To put everything in the right mathematical context, let D1 be a matrix
+with the Haar distributed eigen-space basis and the spectral function defined int he following way
+fD1 =
+1
+1 − η − (1 − (β + η)) (f ˜
+D − ηδ(x) − (1 − (β + η))δ(x − 1)) ,
+(64)
+where we have now taken into the account the above mentioned adjusting between ˜D and ¯D ( ¯D and D have
+identical spectral functions). Utilizing again (22) we similarly to (63) have
+GD1(z) =
+1
+1 − η − (1 − β + η)
+�
+G ˜
+D(z) − η
+z − 1 − (β + η)
+z − 1
+�
+.
+(65)
+Recalling once again on (24) we have
+RD1(GD1(z)) +
+1
+GD1(z)
+=
+z
+⇐⇒
+RD1(z) + 1
+z
+=
+G−1
+D1(z).
+(66)
+After taking z → RD1(z) + 1
+z we can rewrite (65) as
+GD1
+�
+RD1(z) + 1
+z
+�
+= 1
+β
+�
+G ˜
+D
+�
+RD1(z) + 1
+z
+�
+−
+η
+RD1(z) + 1
+z
+−
+1 − (β + η)
+RD1(z) + 1
+z − 1
+�
+,
+(67)
+and after utilizing (66)
+z = 1
+β
+�
+G ˜
+D
+�
+RD1(z) + 1
+z
+�
+−
+η
+RD1(z) + 1
+z
+−
+1 − (β + η)
+RD1(z) + 1
+z − 1
+�
+.
+(68)
+12
+
+After another replacement, z → zSD1(z), (68) becomes
+zSD1(z) = 1
+β
+�
+G ˜
+D
+�
+RD1(zSD1(z)) +
+1
+zSD1(z)
+�
+−
+η
+RD1(zSD1(z)) +
+1
+zSD1 (z)
+−
+1 − (β + η)
+RD1(zSD1(z)) +
+1
+zSD1 (z) − 1
+�
+.
+(69)
+Using (25) we from (69) further find
+zSD1(z) = 1
+β
+�
+G ˜
+D
+�
+1
+SD1(z) +
+1
+zSD1(z)
+�
+−
+η
+1
+SD1 (z) +
+1
+zSD1 (z)
+−
+1 − (β + η)
+1
+SD1(z) +
+1
+zSD1(z) − 1
+�
+,
+(70)
+and
+zSD1(z) = 1
+β
+�
+G ˜
+D
+� z + 1
+zSD1(z)
+�
+−
+η
+z+1
+zSD1 (z)
+− 1 − (β + η)
+z+1
+zSD1(z) − 1
+�
+.
+(71)
+Taking D → D1 in (41) and correspondingly denoting Q → Q1 and Q → Q1, one can repeat all the steps
+between (41) and (59) to arrive at the following
+SD1
+�
+−zGQ1(z) −
+�
+GQ1(z)
+�
+zGQ1(z) − 1
+�
+=
+�
+zGQ1(z) − 1
+GQ1(z)
++ 1.
+(72)
+Setting
+z1(z)
+≜
+−zGQ1(z) −
+�
+GQ1(z)
+�
+zGQ1(z) − 1
+y(z)
+≜
+z1(z) + 1
+z1(z)SD1(z1(z)),
+(73)
+one has from (72)
+SD1(z1(z))
+=
+�
+zGQ1(z) − 1
+GQ1(z)
++ 1.
+(74)
+After taking z → z1 and rewriting (71) one finally obtains
+z1(z) + 1
+y(z)
+= 1
+β
+�
+G ˜
+D(y(z)) −
+η
+y(z) − 1 − (β + η)
+y(z) − 1
+�
+.
+(75)
+We summarize the above results in the following lemma.
+Lemma 2. Let Q1 be as in (39). Then its G-transform, GQ1(z), satisfies
+z1(z) + 1
+y(z)
+= 1
+β
+�
+G ˜
+D(y(z)) −
+η
+y(z) − 1 − (β + η)
+y(z) − 1
+�
+.
+(76)
+with z1(z) and y(z) as in (73), SD1(z1(z)) as in (74), and G ¯
+D(y(z)) as in Lemma 1.
+A combination of Lemma 1 (where G ˜
+D(·) is explicitly given) and (73)-(75) is then sufficient to determine
+GQ1(z) . Utilizing (23) then enables one to fully determine the spectral distribution. This is a generic
+procedure that in principle works. Below we will move things a step further and provide a more detailed
+analysis of the edges of the spectrum as they play a critical role in the ℓ∗
+0 − ℓ∗
+1-equivalence. It will turn
+out that one can provide their a sufficiently explicit characterization so that the explicit closed form for the
+corresponding C-inf phase transitions can again be obtained. Later on we will return to the above described
+procedure for determining the entire spectrum of Q1 and show what type of results such a procedure actually
+produces.
+13
+
+4.2.2
+Explicit characterization of Q1’s spectral edges
+As we have seen earlier, the upper edge of the spectrum of Q (or Q1), λmax(Q) = λmax(Q1) is directly
+related to the success of the ℓ∗
+1-minimization heuristic in causal inference. More precisely, as Corollary 2
+states, one will have the ℓ∗
+0 − ℓ∗
+1-equivalence if and only if λmax(Q) = λmax(Q1) ≤ 1. Clearly, an explicit
+characterization of λmax(Q) = λmax(Q1) will be sufficient to explicitly characterize the ℓ∗
+0 − ℓ∗
+1-equivalence.
+That will then be enough to conclude when ℓ∗
+1 can be used reliable to handle the casual inference.
+To provide an explicit characterization of λmax(Q) = λmax(Q1) ≤ 1 we rely on the results that we
+presented in the previous section. We start by observing that the spectral function of Q1, fQ1(x), can be
+obtained by utilizing (23) and the above discussed GQ1(z) transform. Moreover, at the edge of the spectrum
+GQ1(z) should be real (the edge of the spectrum is actually the breaking point where the GQ1(z) becomes
+complex, i.e. starts having a nonzero imaginary part). That basically means that at the edge of the spectrum
+one should have (76) satisfied for a real GQ1(z). Moreover, since our targeted edge of the spectrum is one
+that means that (76) needs to be satisfied for a real GQ1(1). Rewriting (73)-(75) for z = 1 gives
+z1(1)
+=
+−GQ1(1) −
+�
+GQ1(1)
+�
+GQ1(1) − 1
+y(1)
+≜
+z1(1) + 1
+z1(1)SD1(z1(1)),
+(77)
+and
+SD1(z1(1))
+=
+�
+GQ1(1) − 1
+GQ1(1)
++ 1 = − z1(1)
+GQ1(1),
+(78)
+and
+z1(1) + 1
+y(1)
+= 1
+β
+�
+G ˜
+D(y(1)) −
+η
+y(1) − 1 − (β + η)
+y(1) − 1
+�
+.
+(79)
+From (77) one further finds
+GQ1(1)
+=
+− (z1(1))2
+1 + 2z1(1)
+y(1)
+=
+−(z1(1) + 1)GQ1(1)
+(z1(1))2
+= z1(1) + 1
+1 + 2z1(1).
+(80)
+The second equality then also gives
+z1(1)
+=
+y(1) − 1
+1 − 2y(1),
+(81)
+and
+z1(1) + 1
+=
+y(1)
+2y(1) − 1.
+(82)
+Plugging (82) into (79) one has
+1
+2y(1) − 1 = 1
+β
+�
+G ˜
+D(y(1)) −
+η
+y(1) − 1 − (β + η)
+y(1) − 1
+�
+,
+(83)
+or
+ζ1(y) ≜ −
+1
+2y − 1 + 1
+β
+�
+G ˜
+D(y) − η
+y − 1 − (β + η)
+y − 1
+�
+= 0.
+(84)
+Utilizing G ˜
+D(z) (with the “−” sign as the lower edge in the bulk of the spectrum of ˜D corresponds to the
+14
+
+upper edge in the spectrum of Q) from Lemma 1 we further have
+ζ1(y) = −
+1
+2y − 1 +
+2β − 1
+2β(y − 1) +
+1
+2βy(y − 1)
+�
+−β + η −
+�
+(y − (β + η))2 + 4βη(y − 1)
+�
+= 0,
+(85)
+and
+ζ1(y)
+=
+−2βy(y − 1) + y(2β − 1)(2y − 1) + (2y − 1)
+�
+−β + η −
+�
+(y − (β + η))2 + 4βη(y − 1)
+�
+2βy(y − 1)(2y − 1)
+=
+2(β − 1)y2 + (1 − 2β + 2η)y + β − η − (2y − 1)
+�
+(y − (β + η))2 + 4βη(y − 1)
+2βy(y − 1)(2y − 1)
+.
+(86)
+Setting
+ζ2(y)
+≜
+2(β − 1)y2 + (1 − 2β + 2η)y + β − η − (2y − 1)
+�
+(y − (β + η))2 + 4βη(y − 1)
+ζ(y)
+≜
+(2(β − 1)y2 + (1 − 2β + 2η)y + β − η)2 − ((2y − 1)
+�
+(y − (β + η))2 + 4βη(y − 1))2,
+(87)
+we easily have
+ζ1(y) = 0
+⇐⇒
+ζ2(y) = 0
+⇐⇒
+ζ(y) = 0.
+(88)
+We therefore below focus on ζ(y). After squaring and grouping the terms we have
+ζ(y) = 4β(c3y4 + c2y3 + c1y2 + c0y + c00),
+(89)
+with
+c3
+=
+β − 2
+c2
+=
+5 − 2β − 2η
+c1
+=
+β − 4 + 3η
+c0
+=
+1 − η
+c00
+=
+0.
+(90)
+From (89) we then also have
+ζ(y) = 4βy(c3y3 + c2y2 + c1y + c0).
+(91)
+Since we are interested in an edge or a breaking point of the spectrum ζ(y) should touch zero for certain y
+which means that it should have a stationary point at such y. To find such a stationary point we take the
+derivative
+d
+�
+ζ(y)
+4βy
+�
+dy
+= 3c3y2 + 2c2y + c1 = 0.
+(92)
+Solving over y gives
+y = −c2 +
+�
+c2
+2 − 3c1c3
+3c3
+.
+(93)
+Setting
+r ≜ c2
+2 − 3c1c3 = 1 + β2 + 4η2 − 2β − 2η − βη,
+(94)
+15
+
+we have from (93)
+yopt = −c2 + √r
+3c3
+.
+(95)
+First we set
+ζ3(y) ≜ c3y3 + c2y2 + c1y + c0.
+(96)
+Clearly, from (91) one has
+ζ(y) = 4βyζ3(y).
+(97)
+Then we plug the value for yopt from (95) and after a bit of algebraic transformations obtain
+ζ3(yopt) = −2(√r)3 − c3
+2 + 3rc2 + 27c2
+3c0.
+(98)
+From (90) we first have
+c2
+=
+−2c3 − 1 + 2c0
+c1
+=
+c3 − 3c0 + 1,
+(99)
+and then from (94)
+r = c2
+3 + 1 + 4c2
+0 + c3 − 4c0 + c0c3.
+(100)
+Combining (98)-(100) after a bit of additional algebraic transformations gives
+ζ3(yopt) = −2(√r)3 + 2c3
+3 − 3c3 + 6c3c2
+0 + 3c2
+3 + 3c3c0 + 3c0c2
+3 − 2 − 24c2
+0 + 12c0 + 16c3
+0.
+(101)
+Below we show that
+c2
+3 = −1 − 2c3 + 4c0 − 4c2
+0
+⇐⇒
+ζ3(yopt) = 0.
+(102)
+We first use (102) to systematically linearize ζ3(yopt) in c3 and obtain
+ζ3(yopt) = −2(√r)3 + (−3c3 + 5c3c0 − 2c3c2
+0 − 1 − 8c2
+0 + 5c0 + 4c3
+0).
+(103)
+Transforming further we also have
+ζ3(yopt)
+=
+−2(√r)3 + (−3c3 + 5c3c0 − 2c3c2
+0 − 1 − 8c2
+0 + 5c0 + 4c3
+0)
+=
+−2(√r)3 + (c3(c0 − 1)(3 − 2c0) + (−4c0 + 1 + 4c2
+0)(c0 − 1))
+=
+2(√c3
+√
+c0 − 1)3 + (c3(c0 − 1)(3 − 2c0) + (−4c0 + 1 + 4c2
+0)(c0 − 1))
+=
+�
+2(√c3)3√
+c0 − 1 + (c3(3 − 2c0) + (−4c0 + 1 + 4c2
+0))
+�
+(c0 − 1).
+(104)
+where the third equality follows after noting that with condition (102) in place r in 100) becomes
+c2
+3 = −1 − 2c3 + 4c0 − 4c2
+0
+=⇒
+r = −c3 + c0c3.
+(105)
+We find it useful to rewrite (104) as
+ζ3(yopt)
+=
+ζ(1)
+3 (yopt) + ζ(2)
+3 (yopt),
+(106)
+where
+ζ(1)
+3 (yopt)
+≜
+2(√c3)3√
+c0 − 1
+ζ(2)
+3 (yopt)
+≜
+c3(3 − 2c0) + (−4c0 + 1 + 4c2
+0).
+(107)
+16
+
+We then look at the squared values of these quantities. First we start with ζ(1)
+3 (yopt)
+(ζ(1)
+3 (yopt))2 = 4c3
+3(c0 − 1),
+(108)
+and utilize the condition (102) to systematically linearize in c3. First we remove the cubic c3 term to arrive
+at the following
+(ζ(1)
+3 (yopt))2
+=
+4c3
+3(c0 − 1)
+=
+4c3(−1 − 2c3 + 4c0 − 4c2
+0)(c0 − 1)
+=
+−8c2
+3c0 + 32c3c2
+0 − 16c3c3
+0 − 8 − 12c3 + 32c0 − 32c2
+0 − 20c3c0,
+(109)
+and apply the same procedure again to arrive at a fully linearized form
+(ζ(1)
+3 (yopt))2 = 4(c3((−2c2
+0 + c0 + 1)(2c0 − 3)) − 2(1 − c0)(2c0 − 1)2).
+(110)
+Then we turn to ζ(2)
+3 (yopt)
+(ζ(2)
+3 (yopt))2 = (c3(3 − 2c0) + (−4c0 + 1 + 4c2
+0))2,
+(111)
+and again utilize the condition (102) to linearize in c3. This time the procedure is simpler as there is only a
+quadratic term in c3 and there is no need to apply the procedure from above in two steps. Instead only one
+step suffices and we have
+(ζ(2)
+3 (yopt))2
+=
+(c3(3 − 2c0) + (−4c0 + 1 + 4c2
+0))2
+=
+c2
+3(3 − 2c0)2 + (−4c0 + 1 + 4c2
+0)2 + 2(−4c0 + 1 + 4c2
+0)c3(3 − 2c0)
+=
+(−1 − 2c3 + 4c0 − 4c2
+0)(3 − 2c0)2 + (−4c0 + 1 + 4c2
+0)2 + 2(−4c0 + 1 + 4c2
+0)c3(3 − 2c0)
+=
+−2c3(9 − 12c0 + 4c2
+0 − (−4c0 + 1 + 4c2
+0)(3 − 2c0)) − (−4c0 + 1 + 4c2
+0)(8 − 8c0)
+=
+4(c3((−2c2
+0 + c0 + 1)(2c0 − 3)) − 2(1 − c0)(2c0 − 1)2).
+(112)
+Comparing (110) and (112) we have
+(ζ(1)
+3 (yopt))2
+=
+(ζ(2)
+3 (yopt))2.
+(113)
+Now we will show that one also has (ζ(1)
+3 (yopt))2 = −(ζ(2)
+3 (yopt))2. We again look at the condition in (102)
+and replace the values for c0 and c3 from (91) to obtain
+c2
+3
+=
+−1 − 2c3 + 4c0 − 4c2
+0
+⇐⇒
+(β − 2)2
+=
+−1 − 2(β − 2) + 4(1 − η) − 4(1 − η)2
+⇐⇒
+(β − 2)2 + 2(β − 2) + 1
+=
+4(1 − η) − 4(1 − η)2
+⇐⇒
+(β − 2 + 1)2
+=
+4η(1 − η)
+⇐⇒
+β
+=
+1 − 2
+�
+η(1 − η).
+(114)
+From (107) we then also have
+ζ(1)
+3 (yopt)
+≜
+2(√c3)3√c0 − 1 = 2(
+�
+β − 2)3√−η = 2(2 − β)
+�
+η(2 − β) ≥ 0.
+(115)
+Similarly, we have
+ζ(2)
+3 (yopt)
+≜
+c3(3 − 2c0) + (−4c0 + 1 + 4c2
+0)
+=
+(β − 2)(1 + 2η) + (1 − 2η)2
+=
+(−1 − 2
+�
+η(1 − η))(1 + 2η) + (1 − 2η)2
+=
+−2
+�
+η(1 − η)(1 + 2η) − 6η + 4η2
+17
+
+≤
+−2
+�
+η(1 − η)(1 + 2η) − 6η + 4η
+=
+−2
+�
+η(1 − η)(1 + 2η) − 2η
+≤
+0.
+(116)
+A combination of (106), (107), (113), (115), and (116) finally gives
+(ζ(1)
+3 (yopt))2
+(113)
+=
+(ζ(2)
+3 (yopt))2
+(115),(116)
+⇐⇒
+ζ(1)
+3 (yopt)
+=
+−ζ(2)
+3 (yopt)
+⇐⇒
+ζ(1)
+3 (yopt) + ζ(2)
+3 (yopt)
+=
+0
+(106)
+⇐⇒
+ζ3(yopt)
+=
+0.
+(117)
+Moreover, a combination of (102), (114), and (117) gives
+β = 1 − 2
+�
+η(1 − η)
+⇐⇒
+c2
+3 = −1 − 2c3 + 4c0 − 4c2
+0
+⇐⇒
+ζ3(yopt) = 0.
+(118)
+After combining (97) and (118) one then also has
+β = 1 − 2
+�
+η(1 − η)
+⇐⇒
+c2
+3 = −1 − 2c3 + 4c0 − 4c2
+0
+⇐⇒
+ζ3(yopt) = 0
+⇐⇒
+ζ(yopt) = 0. (119)
+From (88) one then has that for yopt
+ζ1(yopt) = ζ2(yopt) = 0,
+(120)
+which means that y = yopt is indeed a choice for y that ensures that functional equation used to determine
+GQ1(z) is satisfied. Moreover, since the derivative condition is met as well, i.e. since ζ(yopt) = 0, one has
+that not only is yopt a point where ζ(y) crosses zero, it is actually a point where it touches zero. That is
+exactly what is needed to determine an edge of the spectrum. Since we operated using the “−” sign in the
+definition of G ˜
+D(z) that means (based on the considerations from [10]) that we have determined the lower
+edge in the corresponding spectrum of ˜D (or any of ¯D and D) which after the inversion means that we have
+determined the upper edge in the spectrum of Q1 or Q.
+One can even explicitly determine yopt. From (90), (95), (99), and (105) we obtain
+yopt = −c2 + √r
+3c3
+= −(5 − 2β − 2η) +
+�
+η(2 − β)
+3(β − 2)
+.
+(121)
+In Figure 3 we show yopt as a function of η. The whole mechanism of “touching zero” as β decreases is
+shown in Figure for η = 0.9. As can be seen from the figure, for β > 1−2
+�
+η(1 − η) = 0.4 ζ1(y) remains below
+zero one therefore can not be a part of the spectrum. On the other hand, for β ≤ 1−2
+�
+η(1 − η) = 0.4 ζ1(y)
+does intersect zero line which implies that one is now in the spectrum (there is y = y(11) and consequently
+a real GQ1(1) such that ζ1(y) = 0). The borderline or the breaking point happens exactly when the ζ1(y)
+curve touches the zero line. As figure indicates that happens for y = yopt = 0.25, exactly as the theory
+predicts.
+We summarize the above results in the following lemma.
+Lemma 3. Assume the setup of Lemmas 1 and 2 with Q1 as in (39). Then we have for the upper edge of
+the Q1’s spectrum
+β = 1 − 2
+�
+η(1 − η)
+⇐⇒
+λmax(Q1) = 1.
+(122)
+Moreover,
+β ≤ 1 − 2
+�
+η(1 − η)
+⇐⇒
+λmax(Q1) ≤ 1.
+(123)
+Proof. Follows from the above discussion.
+18
+
+η
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+yopt
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+0.45
+0.5
+yopt as a function of η
+yopt =
+√1−η
+√η−√1−η
+η = 0.9
+=⇒
+yopt =
+√1−η
+√η+√1−η = 0.25
+Figure 3: yopt as a function of η
+4.2.3
+The spectrum of Q1 ≜ λT
+V λV λT
+UλU – practical evaluations
+Now that we have fully characterized the upper edge of the Q1’s spectrum we can return to the consideration
+of the entire spectrum. Relying on the above presented machinery we can establish the following lemma.
+Lemma 4. Assume the setup of Lemmas 1 and 2 with Q1 as in (39). Let GQ1(z) be the solution of the
+following system of equations:
+y(z)
+=
+�
+zGQ1(z) − 1
+�
+zGQ1(z) − 1 + z
+�
+GQ1(z)
+G ˜
+D(y(z))
+=
+y(z) − (β + η) ±
+�
+(y(z) − (β + η))2 + 4βη(y(z) − 1)
+2((y(z))2 − y(z))
+1
+β
+�
+G ˜
+D(y(z)) −
+η
+y(z) − 1 − (β + η)
+y(z) − 1
+�
+=
+−(
+�
+zGQ1(z) − 1 +
+�
+GQ1(z))(
+�
+zGQ1(z) − 1 + z
+�
+GQ1(z)).
+(124)
+Then the spectral function of Q1, fQ1(x), is obtained as
+fQ1(x) = − lim
+ǫ→0+
+imag(GQ1(x + iǫ))
+π
+.
+(125)
+Proof. Follows from Lemma 2 through a combination of the results of Lemma 1 (where G ˜
+D(·) is explicitly
+given) and (73)-(75). The following two sequences of identities are then sufficient to prove the lemma
+y(z)
+=
+z1(z) + 1
+z1(z)SD(z1(z))
+=
+((−zGQ1(z) −
+�
+GQ1(z)
+�
+zGQ1(z) − 1) + 1)
+�
+GQ1(z)
+(−zGQ1(z) −
+�
+GQ1(z)
+�
+zGQ1(z) − 1)(
+�
+zGQ1(z) − 1 +
+�
+GQ1(z))
+=
+�
+zGQ1(z) − 1
+�
+zGQ1(z) − 1 + z
+�
+GQ1(z)
+,
+(126)
+19
+
+y
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+ζ1(y)
+-0.25
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+ζ1(y) as a function of y; η = 0.9
+β = 0.41
+β = 0.4
+β = 0.39
+β = 0.4, η = 0.9
+=⇒
+yopt =
+√1−η
+√η+√1−η = 0.25
+Figure 4: ζ1y as a function of y
+and
+z1(z) + 1
+y(z)
+=
+SD(z1(z))
+z1(z)
+=
+(−zGQ1(z) −
+�
+GQ1(z)
+�
+zGQ1(z) − 1)(
+�
+zGQ1(z) − 1 +
+�
+GQ1(z))
+�
+GQ1(z)
+=
+(
+�
+zGQ1(z) − 1 + z
+�
+GQ1(z))(
+�
+zGQ1(z) − 1 +
+�
+GQ1(z)).
+(127)
+In Figure 5 we show the entire spectrum of fQ1(x). We chose β = 0.4 and η = 0.9 and ran the experiments
+with n = 4000. As can be seen from the figure, the obtained numerical results are in a strong agreement
+with what the theory predicts.
+4.2.4
+ℓ∗
+0 − ℓ∗
+1-equivalence via the spectral limit – asymmetric scenario
+From Corollary 2, (17), and (38) one has in the asymmetric scenario
+ℓ∗
+0 − ℓ∗
+1 − equivalence
+⇐⇒
+λmax(λT
+V λV λT
+UλU) ≤ 1
+⇐⇒
+λmax(Q1) ≤ 1.
+(128)
+From (123) and (128) we finally have
+ℓ∗
+0 − ℓ∗
+1 − equivalence
+⇐⇒
+β ≤ 1 − 2
+�
+η − η2.
+(129)
+Analogously to Theorem 2 we can now establish a precise asymmetric scenario location of the phase transition
+in a typical statistical context.
+Theorem 3. (ℓ∗
+1 – phase transition – C-inf (typical asymmetric scenario)) Assume the setup of
+Theorem 2 with rank-k matrix Xsol = X ∈ Rn×n that now has Haar distributed independent bases of its
+orthogonal row and column spans ¯U ��� ∈ Rn×(n−k) and ¯V ⊥ ∈ Rn×(n−k) (XT
+sol ¯U ⊥ = Xsol ¯V ⊥ = 0n×(n−k)).
+Let M ≜ M (l) ∈ Rn×n be as defined in (5). Let βac and η satisfy the following
+20
+
+x
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+fQ1(x)
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2 Spectral distribution fQ1(x); η = .4; β = .9; n = 4000
+simulated
+theory
+fQ1(x)
+Bulk
+Figure 5: fQ1(x) – spectral function of Q1; β = 0.4 and η = 0.9
+C-inf ℓ∗
+1 asymmetric scenario phase transition (PT) characterization
+ξ(ac)
+η
+(β) ≜ β − 1 + 2
+�
+η − η2 = 0.
+(130)
+If and only if β ≤ βac
+lim
+n→∞ P(ℓ∗
+0 ⇐⇒ ℓ∗
+1) =
+lim
+n→∞ P(RMSE = 0) = 1,
+(131)
+and the solutions of (6) and (7) coincide with overwhelming probability.
+Proof. Follows from Lemma 3 and the above discussion.
+The results related to the use of the ℓ∗
+1-minimization heuristic for solving the causal inference problems
+obtained based on the above theorem are shown in Figure 6.
+As in the worst case scenario, the phase
+transition (PT) curve splits the (β, η) region into two separate subregions where the ℓ∗
+0 − ℓ∗
+1-equivalence
+phenomenon either occurs or fails to occur. Basically, below the curve one has a perfect recovery with the
+residual RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0. Contrary to that, above the curve though, there is an Xsol
+for which RMSE → ∞ and ℓ∗
+1 fails.
+The following corollary adapts the above results so that they fit the standard (α, β) representation
+typically used in the compressed sensing (CS), low rank recovery (LRR), and matrix completion (MC)
+literature.
+Corollary 3. (ℓ∗
+1 – phase transition – C-inf (typical asymmetric scenario; standard (α, β) rep-
+resentation)) Assume the setup of Theorem 3. Let m be the total number of ones in matrix M and let
+α ≜ limn→∞ m
+n2 . Let β and αw satisfy the
+C-inf ℓ∗
+1 asymmetric scenario PT (standard (α, β) representation)
+ξ(wc,s)
+β
+(α) ≜ β − 1 + 2
+�√
+1 − α − 1 + α = 0.
+(132)
+21
+
+η
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+β
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+(η, β) region of success/failure — C-inf ℓ∗1 PT; asymmetric scenario
+worst case
+average case
+RMSE −→ ∞, ℓ∗1
+fails
+ℓ∗1’s PT: ξ(ac)
+η
+(β) = β − 1 + 2�η − η2 = 0
+RMSE = 0, ℓ∗1 succeeds
+Doubling low rankness:
+ξ(ac)
+η
+(2β) = 2ξ(wc)
+η
+(β)
+Figure 6: Causal inference (C-inf) – typical asymmetric scenario ℓ∗
+1 phase transition
+If and only if α ≥ αw
+lim
+n→∞ P(ℓ∗
+0 ⇐⇒ ℓ∗
+1) =
+lim
+n→∞ P(RMSE = 0) = 1,
+(133)
+and the solutions of (6) and (7) coincide with overwhelming probability.
+Proof. Follows as a direct consequence of Theorem 3 after noting that m = n2 − (n − l)2 and consequently
+α = 1 − (1 − η)2.
+Figure 7 shows the results obtained based on the above corollary in the standard (α, β) region format.
+As usual in the PT considerations, the entire (α, β) region is split in the part below the curve where
+RMSE = ∥vec( ˆX) − vec(Xsol)∥2 = 0 and the part above the curve where even RMSE → ∞ is achievable.
+We should point out an interesting similarity between what we observed here in the above corollary and
+in Figure 7 on the one side and what is known to hold in generic LRR. Namely, as Corollary 3 states (and as
+is emphasized in Figure 7), for the same value of α one achieves exactly two times larger β in the asymmetric
+case than in the worst case. As the worst case is basically symmetric, one has that the PTs of the symmetric
+and the nonsymmetric scenarios are distinguished by a factor of two. Similar observation was in place when
+it comes to the comparison between the LRR of the symmetric and the general (nonsymmetric) matrices.
+However, one should keep in mind a fundamental difference as well. In LRR the underlying symmetry is a
+priori known and can be utilized in the algorithms design whereas here it is just the choice of the worst case
+problem instance and is not assumed to be known to the algorithm itself. Of course, given the properties of
+the LRR, such a choice is not necessarily very surprising.
+4.3
+Numerical results
+To complement the above theoretical findings and see how successful in characterizing the utilization of the
+ℓ∗
+1-minimization in C-inf problems they indeed are, we conducted a set of numerical experiments and show
+the obtained results in Figure 8. As in [10], we again observe both the PT’s existence and a solid agreement
+between its theoretical prediction and the results obtained through the simulations.
+In the conducted numerical experiments we chose n = 80 and η in the range [0.6, 0.95]. Clearly, such fairly
+small matrix sizes correspond to the settings quite opposite from the ones that we used in the theoretical
+analysis. Still, even though the theory is predicated on the large n assumption, it is not impossible that
+its conclusions remain valid for smaller values of n as well. The results form Figure 8 confirm that this is
+22
+
+α
+0.75
+0.8
+0.85
+0.9
+0.95
+1
+β/α
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+(α, β) region of success/failure — C-inf ℓ∗1 PT; asymmetric scenario
+worst case
+average case
+RMSE = 0, ℓ∗1 succeeds
+RMSE −→ ∞, ℓ∗1
+fails
+ℓ∗1’s PT: ξ(ac,s)
+β
+(α) = β − 1 + 2
+�√1 − α − 1 + α = 0
+ξ(ac,s)
+2β
+(α) = 2ξ(ac,s)
+β
+(α)
+Figure 7: Causal inference (C-inf) – typical asymmetric scenario ℓ∗
+1 phase transition ((α, β) region)
+indeed the case. Moreover, one can then say that the large n regime, needed for the theoretical consideration,
+practically may start ro kick in already for rather small (of order of a few tens!) values of n. This ultimately
+means that the presented results, although theoretical in nature, have in themselves a strong practical
+component as well. Finally, we should also add that for larger values of n an even better agreement between
+the theoretical and the simulated results is to be expected.
+A few additional points regarding the simulations setup might be useful. First, one should emphasize,
+that in order to be in an agreement with the theoretical considerations, we, in all numerical experiments,
+considered the so-called typical behavior. Following further into the footsteps of the theoretical consider-
+ations, the presented simulations results were obtained for the square matrices. As was the case in [10],
+all theoretical considerations can be repeated assuming the non-square scenarios as well. We, however, (as
+in [10]) prioritized the clarity of the presentations over simple generalizations and opted for the square sce-
+narios which are substantially easier to present. Also, all the simulations needed for Figure 8 were done with
+the singular values of the unknown targeted matrices equal to one. While we refer to [10] for a bit more
+complete discussion regarding such a choice, we here briefly mention that choices of this type are known
+to serve as the worst case examples in establishing the reversal ℓ0 − ℓ1-equivalence conditions. As in [10],
+we also ran the simulations where the singular values were randomly chosen with results either identically
+matching or improving on the ones shown in Figure 8.
+5
+Conclusion
+In this paper, we have built on the mathematical Causal inference (C-inf) ↔ low-rank recovery
+(LRR) connection established in [10] to deal with asymmetric PTs phenomena. The results of [10] proved
+that the nuclear norm (ℓ∗
+1) minimization heuristic, when used for solving the low rank recovery C-inf problems,
+exhibits the so-called phase transition (PT) phenomenon. Moreover, in a typical statistical scenario, [10]
+characterized the exact location of the worst case PT. This effectively meant that there are problem in-
+stances where the ℓ∗
+1 predicated behavior might be improved upon. Here we showed that this is indeed true.
+Considering an asymmetric scenario (in contrast with the symmetric worst case one from [10]) we deter-
+mined the underlying exact phase transitions locations. Moreover, we uncovered a doubling low rankness
+phenomenon, which means that, throughout the entire region of allowed system parameters, matrices of
+exactly two times larger rank can be recovered when compared to the worst case scenario from [10]. Such a
+phenomenon also ensures that the simplicity of the worst case PTs from [10] is preserved in the asymmetric
+23
+
+η
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+β
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+(η, β) region of success/failure — ℓ∗1’s PT; simulated/theory
+ℓ∗
+1’s PT – simulated
+ℓ∗
+1’s PT – theory
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Success
+Failure
+Figure 8: C-inf ℓ∗
+1’s asymmetric scenario phase transition (PT)
+scenarios as well. Consequently, one is again able to elegantly pin down the relation between the low rankness
+of the target C-inf matrix and the time when the treatment is applied.
+Throughout the process of creating the theoretical phase transitions characterizations we also established
+several mathematical results that are of independent interest. All of our theoretical findings we supplemented
+with the results obtained from the corresponding numerical experiments. Moreover, in all cases we observed
+a rather overwhelming agreement between what the theory predicts and what the simulations provide.
+To achieve the desired phase transition results we relied on a combination of the ideas from the Random
+duality theory (RDT) and the Free probability theory (FPT). As a result, we obtained a very powerful
+and generic mathematical apparatus that will serve as a theoretical platform in further explorations. As this
+and the companion paper [10] are of an introductory nature we stopped short of showcasing how the created
+theory fairs when utilized for handling more complex problem instances (these among others include the,
+practically very relevant, noisy and approximately low-rank corresponding ones). Our companion papers will
+establish results along these directions relying on the mathematical framework presented here and in [10].
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+A
+Proof of Theorem 1
+As mentioned earlier, the proof of Theorem 1 is conceptually identical to the corresponding proof when
+matrix X is symmetric. A detailed proof for the symmetric matrices is given below. Before being able to
+present the proof we need a couple of technical lemmas.
+Lemma 5. Let C = CT ∈ Rn×n. Also let all eigenvalues of C belong to the interval [−1, 1]. Finally, let the
+first k entries on the main diagonal, Ci,i, 1 ≤ i ≤ k, be larger than or equal to 1. Then the upper k × k left
+block of C, C1:k,1:k, is an identity matrix, i.e.
+C1:k,1:k
+=
+Ik×k.
+(134)
+Proof. Let λmax(C) be the maximum eigenvalue of C. Then
+λmax(C)
+≜
+max
+∥c∥2=1 cT Cc.
+(135)
+Since by assumptions 1 ≤ Ci,i, 1 ≤ i ≤ k and λmax(C) ≤ 1 we also have for any 1 ≤ i ≤ k
+1 ≤ Ci,i ≤ max
+∥c∥2=1 cT Cc ≜ λmax(C) ≤ 1,
+(136)
+which implies C(i, i) = 1, 1 ≤ i ≤ k. The proof that all other elements of C1:k,1:k are equal to zero proceeds
+inductively.
+1) Induction move from l = 1 to l = 2: First we look at the upper block of size 2 × 2, i.e. at C1:2,1:2.
+We then have
+1 ≥ max
+∥c∥2=1 cT Cc ≥
+max
+∥c1:2∥2=1 cT
+1:2C1:2,1:2c1:2
+≥
+max
+∥c1:2∥2=1 (∥c1:2∥2 + 2|c1c2C1,2|)
+≥
+max
+∥c1:2∥2=1 (1 + 2|c1c2C1,2|) ≥ 1,
+(137)
+which implies C1,2 = 0.
+2) Induction move from l to l + 1: Now we look at the upper block of size (l + 1) × (l + 1), i.e. at
+C1:l+1,1:l+1 while assuming that C1:l,1:l = Il×l. We then have
+1
+≥
+max
+∥c∥2=1 cT Cc
+≥
+max
+∥c1:l+1∥2=1 cT
+1:l+1C1:l+1,1:l+1c1:l+1
+≥
+max
+∥c1:l+1∥2=1
+�
+∥c1:l+1∥2 + 2|cT
+1:lC1:l,l+1cl+1|
+�
+≥
+max
+∥c1:l+1∥2=1
+�
+1 + 2|cT
+1:lC1:l,l+1cl+1|
+�
+≥
+1,
+(138)
+which implies C1:l,l+1 = 0l×1 and completes the proof.
+27
+
+Lemma 6. Assume the setup of Lemma 5. Then the upper k × k left block of C, C1:k,1:k, is an identity
+matrix and the upper k × (n − k) right block of C, C1:k,n−k+1:n is a zero matrix, i.e.
+C1:k,1:k
+=
+Ik×k
+C1:k,n−k+1:n
+=
+0k×(n−k).
+(139)
+Proof. The first part follows by Lemma 5. We now focus on the second part. Consider the following partition
+of matrix C
+C
+=
+�
+C1:k,1:k
+C1:k,n−k+1:n
+Cn−k+1:n,1:k
+Cn−k+1:n,n−k+1:n
+�
+=
+�
+Ik×k
+C1:k,n−k+1:n
+Cn−k+1:n,1:k
+Cn−k+1:n,n−k+1:n
+�
+.
+(140)
+Then assuming that the largest nonzero singular value of C1:k,n−k+1:n is equal to b > 0, we have
+1
+≥
+max
+∥c∥2=1 cT Cc
+≥
+max
+∥c1:k∥2=a,cn−k+1:n
+�
+cT
+1:kC1:k,1:kc1:k + 2|cT
+1:kC1:k,n−k+1:ncn−k+1:n| + cT
+n−k+1:nCn−k+1:n,n−k+1:ncn−k+1:n
+�
+≥
+max
+∥c1:k∥2=a,cn−k+1:n
+�
+a2 + 2|cT
+1:kC1:k,n−k+1:ncn−k+1:n| + cT
+n−k+1:nCn−k+1:n,n−k+1:ncn−k+1:n
+�
+≥
+max
+∥c1:k∥2=a,cn−k+1:n
+�
+a2 + 2|cT
+1:kC1:k,n−k+1:ncn−k+1:n| − cT
+n−k+1:ncn−k+1:n
+�
+≥
+max
+a∈[0,1]
+�
+a2 + 2ba
+�
+1 − a2 − (1 − a2)
+�
+=
+max
+a∈[0,1]
+�
+2a2 − 1 + 2ba
+�
+1 − a2
+�
+,
+(141)
+where the fourth inequality follows since the minimum eigenvalue of Cn−k+1:n,n−k+1:n is larger than or equal
+to the minimum eigenvalue of C which is by the lemma’s assumption larger than or equal to -1. Now, we
+further have
+c ≜ 2a
+�
+1 − a2
+and
+2a2 − 1 + 2ba
+�
+1 − a2 =
+�
+1 − c2 + bc,
+(142)
+and
+d(
+√
+1 − c2 + bc)
+dc
+=
+−c
+√
+1 − c2 + b = 0.
+(143)
+From (143) we then easily obtain
+c =
+b
+√
+1 + b2 .
+(144)
+A combination of (141), (142), and (144) gives
+1 ≥ max
+∥c∥2=1 cT Cc ≥ max
+a∈[0,1]
+�
+2a2 − 1 + 2ba
+�
+1 − a2
+�
+=
+√
+1 + b2,
+(145)
+which implies b = 0 and automatically C1:k,n−k+1:n = 0k×1. This completes the proof.
+Now we can consider the above mentioned theorem that adapts the general ℓ1 equivalence condition
+result from [39–41] to the corresponding one for the ℓ1 norm of the singular/eigenvalues (similar adaptation
+can also be found in [24]).
+Theorem 4. (ℓ∗
+0 − ℓ∗
+1-equivalence condition (LRR) – symmetric X) Consider a ¯U ∈ Rn×k such that
+¯U T ¯U = Ik×k and a rank − k a priori known to be symmetric matrix Xsol = X ∈ Rn×n with all of its
+columns belonging to the span of ¯U. For concreteness, and without loss of generality, assume that X has only
+28
+
+positive nonzero eigenvalues. For a given matrix A ∈ Rm×n2 (m ≤ n2) assume that y = Avec(X) ∈ Rm. If
+(∀W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n)
+− tr ( ¯U T W ¯U) < ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥),
+(146)
+then the solutions of (6) and (7) coincide. Moreover, if
+(∃W ∈ Rn×n|Avec(W) = 0m×1, W = W T ̸= 0n×n)
+− tr ( ¯U T W ¯U) ≥ ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥),
+(147)
+then there is an X from the above set of the symmetric matrices with columns belonging to the span of ¯U
+such that the solutions of (6) and (7) are different.
+Proof. The proof follows literally step-by-step the proof of the corresponding theorem in [39–41] and adapts
+it to matrices or their singular/eigenvalues. For experts in the field this adaptation is highly likely to be
+viewed as trivial and certainly doesn’t need to be as detailed as we will make it to be. Nonetheless, to ensure
+a perfect clarity of all arguments we provide a step-by-step instructional derivation. For concreteness and
+without loss of generality we also assume that the eigen-decomposition of X is
+X = UΛU T =
+� ¯U
+¯U ⊥� �
+¯ΛX
+0k×(n−k)
+0(n−k)×k
+¯Λ⊥
+X
+� � ¯U
+¯U ⊥�T .
+(148)
+(i) =⇒ (the if part): Following step-by-step the proof of Theorem 2 in [41], we start by assuming that
+ˆX is the solution of (7). Then we want to show that if (146) holds then ˆX = X. As usual, we instead of that,
+assume opposite, i.e. we assume that (146) holds but ˆX ̸= X. Then since y = Avec( ˆ
+X) and y = Avec(X)
+must hold simultaneously there must exist W such that ˆX = X + W with W ̸= 0, Avec(W) = 0. Moreover,
+since ˆX is the solution of (7) one must also have
+ℓ∗
+1(X + W) = ℓ∗
+1( ˆX)
+≤
+ℓ∗
+1(X)
+⇐⇒
+ℓ∗
+1(
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+≤
+ℓ∗
+1(X)
+=⇒
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥)
+≤
+ℓ∗
+1(X).
+(149)
+The last implication follows after one trivially notes
+ℓ∗
+1(
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+=
+max
+Λ∗=ΛT
+∗ ∈L∗
+tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+≥
+max
+Λ∗=ΛT
+∗ ∈L0∗
+tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+=
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥),
+(150)
+where
+L0
+∗
+≜
+�
+Λ∗ ∈ Rn×n|Λ∗ = ΛT
+∗ , Λ∗ΛT
+∗ ≤ I, Λ∗ =
+�
+Λ∗,1
+0k×(n−k)
+0(n−k)×k
+Λ∗,2
+��
+⊆
+�
+Λ∗ ∈ Rn×n|Λ∗ = ΛT
+∗ , Λ∗ΛT
+∗ ≤ I
+�
+≜ L∗.
+(151)
+The key observation – “Removing the absolute values”:
+Now, the key observation made in [41] comes into play. Namely, one notes that the absolute values can
+be removed in the nonzero part and that the ℓ∗
+1(·) can be “replaced” by tr (·). Such a simple observation
+is the most fundamental reason for all the success of the RDT when used for the exact performance
+characterization of the structured objects’ recovery. From (149) we then have
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥)
+≤
+ℓ∗
+1(X)
+=⇒
+tr ( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (W) ¯U ⊥)
+≤
+ℓ∗
+1(X)
+⇐⇒
+tr ( ¯U T W ¯U) + ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥)
+≤
+0.
+(152)
+29
+
+We have arrived at a contradiction as the last inequality in (152) is exactly the opposite of (146). This
+implies that our initial assumption ˆX ̸= X cannot hold and we therefore must have ˆX = X. This is precisely
+the claim of the first part of the theorem.
+(ii) ⇐= (the only if part): We now assume that (147) holds, i.e.
+(∃W ∈ Rn×n|Avec(W) = 0m×1, W ̸= 0n×n)
+− tr (( ¯U)T W ¯U) ≥ ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥)
+(153)
+and would like to show that for such a W there is a symmetric rank-k matrix X with the columns belonging
+to the span of ¯U such that y = Avec(X), and the following holds
+ℓ∗
+1(X + W) < ℓ∗
+1(X).
+(154)
+Existence of such an X would ensure that it both, satisfies all the constraints in (7) and is not the
+solution of (7). Following the strategy of [39] one can reverse all the above steps from (153) to (149) with
+strict inequalities and arrive at the first inequality in (149) which is exactly (154). There are two implications
+that cause problems in such a reversal process, the one in (153) and the one in (149). If these implications
+were equivalences everything would be fine. We address these two implications separately.
+1) the implication in (152) – particular X to “overwhelm” W: Assume X = ¯UΛx ¯U T with Λx >
+0 being a diagonal matrix with arbitrarily large elements on the main diagonal (here it is sufficient even to
+choose diagonal of Λx so that its smallest element is larger than the maximum eigenvalue of ¯U T W ¯U). Now
+one of course sees the main idea behind the “removing the absolute values” concept from [39,41]. Namely,
+for such an X one has that ℓ∗
+1( ¯U T X + W) ¯U) = tr(ℓ∗
+1( ¯U T X + W) ¯U)) since for symmetric matrices the ℓ∗
+1(·)
+(as the sum of the argument’s absolute eigenvalues) and tr (·) (as the sum of the argument’s eigenvalues) are
+equal. That basically means that when going backwards the second inequality in (152) not only follows from
+the first one but also implies it as well. In other words, for X = ¯UΛx ¯U T (with Λx > 0 and arbitrarily large)
+tr ( ¯U T W ¯U) + ℓ∗
+1(( ¯U ⊥)T W ¯U ⊥)
+≤
+0
+⇐⇒
+tr ( ¯U T (X + W) ¯U ) + ℓ∗
+1(( ¯U ⊥)T (W) ¯U ⊥)
+≤
+ℓ∗
+1(X)
+⇐⇒
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥)
+≤
+ℓ∗
+1(X),
+(155)
+which basically mans that there is an X that can “overwhelm” W (in the span of ¯U) and ensures that the
+“removing the absolute values” is not only a sufficient but also a necessary concept for creating the
+relaxation equivalence condition.
+2) the implication in (149): One would now need to somehow show that the third inequality in (149)
+not only follows from the second one but also implies it as well. This boils down to showing that inequality in
+(150) can be replaced with an equality or, alternatively, that L0 and L are provisionally equivalent. Neither
+of these statements is generically true. However, since we have a set of X at our disposal there might be an
+X for which they actually hold. We continue to assume X = ¯UΛx ¯U T with Λx > 0 being a diagonal matrix
+with arbitrarily large entries on the main diagonal. Then the last equality in (150) gives
+ℓ∗
+1( ¯U T (X + W) ¯U) + ℓ∗
+1(( ¯U ⊥)T (X + W) ¯U ⊥)
+≤
+ℓ∗
+1(X)
+⇐⇒
+maxΛ∗=ΛT
+∗ ∈L0∗ tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+≤
+ℓ∗
+1(X).
+(156)
+Also, one has
+maxΛ∗=ΛT
+∗ ∈L0∗ tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+≤
+ℓ∗
+1(X)
+⇐⇒
+maxΛ∗,i=ΛT
+∗,i,Λ∗,iΛT
+∗,i≤I,i∈{1,2} tr (Λ∗,1 ¯U T X ¯U + Λ∗,2( ¯U ⊥)T W ¯U ⊥)
+≤
+ℓ∗
+1(X)
+⇐⇒
+maxΛ∗,i=ΛT
+∗,i,Λ∗,iΛT
+∗,i≤I,i∈{1,2} tr (Λ∗,1Λx + Λ∗,2( ¯U ⊥)T W ¯U ⊥)
+≤
+tr (Λx).
+(157)
+Now, if at least one of the elements on the main diagonal of Λ∗,1, diag(Λ∗,1), is smaller than 1, then the
+corresponding element on the diagonal of Λx can be made arbitrarily large compared to the other elements
+30
+
+of Λx and one would have
+maxΛ∗,i=ΛT
+∗,i,Λ∗,iΛT
+∗,i≤I,i∈{1,2} tr (Λ∗,1Λx + Λ∗,2( ¯U ⊥)T W ¯U ⊥)
+<
+tr (Λx)
+⇐⇒
+maxΛ∗=ΛT
+∗ ∈L0∗ tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+<
+ℓ∗
+1(X)
+⇐⇒
+maxΛ∗=ΛT
+∗ ∈L∗ tr (Λ∗
+� ¯U
+¯U ⊥�T (X + W)
+� ¯U
+¯U ⊥�
+)
+<
+ℓ∗
+1(X),
+(158)
+where the last equivalence holds since the difference of the terms on the left-hand side in the last two
+inequalities is bounded independently of X. Also, the last inequality in (158) together with the first equality
+in (150) and the first inequality in (149) produces (154). Therefore the only scenario that is left as potentially
+not producing (154) is when all the elements on the main diagonal are larger than or equal to 1. However,
+the two lemmas preceding the theorem show that in such a scenario L0 = L and one consequently has an
+equality instead of the inequality in (150) which then, together with (149), implies (154). This completes
+the proof of the second (“the only if”) part of the theorem and therefore of the entire theorem.
+31
+

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+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE
+MINIATURIZATION FOR NEUROMORPHIC COMPUTING
+A PREPRINT
+A. S. Goossens 1,2,*, M. Ahmadi 1,2, D. Gupta 1,2, I. Bhaduri 1,2, B. J. Kooi 1,2, and T. Banerjee 1,2.*
+1Groningen Cognitive Systems and Materials Center, University of Groningen, Groningen, The Netherlands
+2Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands
+*{a.s.goossens,t.banerjee}@rug.nl
+January, 2023
+ABSTRACT
+The areal footprint of memristors is a key consideration in material-based neuromorophic comput-
+ing and large-scale architecture integration. Electronic transport in the most widely investigated
+memristive devices is mediated by filaments, posing a challenge to their scalability in architecture
+implementation. Here we present a compelling alternative memristive device and demonstrate that
+areal downscaling leads to enhancement in memristive memory window, while maintaining analogue
+behavior, contrary to expectations. Our device designs directly integrated on semiconducting Nb-
+SrTiO3 allows leveraging electric field effects at edges, increasing the dynamic range in smaller
+devices. Our findings are substantiated by studying the microscopic nature of switching using scan-
+ning transmission electron microscopy, in different resistive states, revealing an interfacial layer
+whose physical extent is influenced by applied electric fields. The ability of Nb-SrTiO3 memristors
+to satisfy hardware and software requirements with downscaling, while significantly enhancing
+memristive functionalities, makes them strong contenders for non-von Neumann computing, beyond
+CMOS.
+Keywords Interface memristor, Areal scaling, Beyond CMOS, Neuromorphic computing, Scanning transmission
+electron microscopy (STEM)
+1
+Introduction
+The growing demand for applications such as artificial intelligence and the Internet of Things has given rise to critical
+challenges in the storage and processing of big data using existing computational architectures [1]. The currently
+employed von Neumann architecture, using complementary metal-oxide-semiconductor (CMOS) hardware, suffers
+from limited transmission speed [2, 3, 4] due to a memory throughput bottleneck as well as energy inefficiency and
+limited scalability [4, 5, 6]. Moving away from CMOS technology, towards logic-in-memory chips would alleviate
+some of the above issues but requires us to massively rethink every aspect of computing [7]. The first step towards this
+is identifying novel materials and devices with suitable physical properties. Resistive switching devices, or memristors,
+are one such class of devices where the resistance can be switched between several states. Reported in different ionic
+materials, they are distinguished by the switching mechanism as either occurring through the material bulk between
+two electrodes or interface-type where switching takes place in a localized region underneath the area of the electrodes
+[8]. Their ability to co-locate memory and computation, and exhibit characteristics absent in digital computing makes
+them important for novel computing approaches. Given the robust way in which the human brain is able to process
+large amounts of data with remarkably low power, it is unsurprising that it serves as a source of inspiration to the
+development of computing beyond using CMOS. As the brain utilizes a vast network, downscaling memristive devices
+is a crucial area of research to develop large scale neuromorphic systems.
+arXiv:2301.03352v1  [cs.ET]  9 Jan 2023
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+For this material-driven research, the areal footprint in unconventional computing architectures that seek to integrate
+in-memory computing devices such as memristors is a prime consideration. Considerable research has been devoted to
+this in the realm of non-volatile conventional filamentary devices. The challenges in their implementation in such novel
+architectures, besides the requirement for unfavourable electroforming processes, lie in their switching endurance [9],
+and their efficacy to exhibit discernible analogue resistance states. Memristive devices that exhibit more than two stable
+states also greatly enhance integration density because each device can store multiple data bits in an analogue manner.
+In valence change memristors, where switching originates from filaments, such behavior is observed in large areal
+dimensions but is lost when devices are downscaled and conduction is mediated by a single nanoscale filament causing
+an abrupt transition between the two resistance states [10]. Further, the effects of Joule heating on filaments are
+an important consideration as devices shrink; Joule heating can cause a wide distribution of switching voltages and
+endurance deterioration. These limitations in device stability, endurance and associated enhanced power of operation
+are major roadblocks in the successful implementation of filamentary devices in large scale architectures.
+Memristive devices have the potential to be integrated in large scale architectures, for which they should exhibit large
+memory windows, high endurance and low variability [11]. Herein the areal switching mechanism is a strong contender.
+A model system in which this mechanism is dominant is Schottky contacts on Nb-doped SrTiO3 (Nb:STO), formed at
+the interface with a high work function metal. It is widely accepted that in these material systems it is not the bulk of
+the device, but an area close to the interface that is responsible for the switching, a more detailed discussion on the
+proposed mechanisms is presented in Supporting Information section S3.
+Distinguishing Nb:STO from conventional semiconductors such as Si, widely used in conventional architectures, is its
+dielectric permittivity which is comparatively large (300) and is strongly dependent on electric field. This property
+extends the parameter space for designing functionality: electric fields can be used to tune the barrier height and width
+relevant for memristive device design. We have previously shown that such Schottky contacts form robust memristors,
+exhibiting non-linear transport and continuous conductance modulation [12], and that their behavior can be described
+by a power-law which can be successfully implemented as a learning algorithm [13]. However, for the applicability of
+Nb:STO-based memristors as hardware elements for non-von Neumann computing architecture beyond CMOS, the
+focus should be on establishing their memristive performance with device miniaturization, which has not been shown
+on such semiconducting platforms. In this work, we demonstrate that memristive devices of Co Schottky contacts
+on Nb:STO exhibit an increase in the analogue memristive memory window in devices down to 1 μm, contrary to
+expectations. Ionic defects are at the heart of memristive behavior, hence one of the following two scenarios is expected.
+For a homogeneous areal mechanism, the current density will scale with device area so that the device resistance in
+both the high resistance state (HRS) and the low resistance state (LRS) scales with the electrode size, but the ratio
+between them is area independent. Alternatively, the resistance window can be severely reduced or even vanish with
+downscaling due to insufficient ionic defects. However, we observe an enhancement in the memory window as the
+device area is reduced, with minimal device-to-device variation, an unforeseen finding.
+To understand the microscopic nature of the switching, we conducted scanning transmission electron microscopy
+(STEM) on virgin samples and on samples subjected to either a positive (SET) or negative (RESET) voltage . Using
+integrated differential phase contrast (iDPC) we image oxygen atomic columns next to the heavy metal atomic columns.
+Figure 1: State stability and multilevel memristive operation. a Schematic of the fabricated devices on Nb-doped
+SrTiO3, electrical connections. Black lines are used to represent the varying overall electric fields acting over each area.
+The field strengths at the interface are also indicated by a color gradient, showing the fields are weakest in the central
+area (blue) and strongest around the perimeter (red). b Current read at +0.3 V for device sizes of 100 μm (black), 10 μm
+(blue) and 1 μm (red). c Current read at 0.3 V after switching between a SET voltage of +1 V (black, red and blue) or
++2 V (green, purple and orange) and a RESET voltage of -2 V (black and green), -2.5 V (red and purple) or -3 V (blue
+and orange). Each combination was repeated over 100 cycles.
+2
+
++2 V
++2 V
+2.5
+Nb-doped SrTiO.
+3 VMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure 2: Characterization of memristive devices in the virgin state. Electrical characteristics of virgin devices. The
+compliance current was fixed at 100 mA for all measurements. Results are shown for four devices of each area in a-f.
+Virgin samples show the existence of a layer near the interface with neither the perovskite structure of the substrate nor
+that of the Co electrode. Applying a bias across the interface results in oxygen vacancy movement, which is a key factor
+controlling the resistance states. These new revelations are consolidated with a mathematical model describing the
+kinetics of trapping and de-trapping in dielectric materials and relates experimental results to the effective trapping
+density. Surprisingly, this is found to be larger for smaller junctions, suggesting that an increase in the density of traps
+is responsible for the increased resistance ratio and attributed to inhomogeneous distribution of the electric field due to
+device edges.
+These memristive devices, integrated directly on a semiconducting platform, demonstrate multistate analogue switching
+with remarkably high memory windows with downscaling, as well as high endurance and low device and cycle variation
+down to the smallest devices. Their ability to meet both hardware and software requirements for unconventional
+computing, make Nb:STO memristors strong material contenders for physical computing beyond CMOS.
+2
+Results
+2.1
+Electrical Characterization
+Figure 1a shows a schematic of the device structure used for the electrical measurements. An array of circular Co
+electrodes of varying sizes are fabricated on a semiconducting Nb:STO single crystalline substrate. The bottom of the
+substrate serves as a back contact for the devices. The top electrodes were patterned by a two-step electron lithography
+process using aluminium oxide as an insulation layer to define the contact areas and to prevent electronic cross talk.
+After fabrication, we performed small range voltage sweeps to characterize the virgin states of each device on a chip.
+The results for devices with radial dimension from 100 μm to 800 nm are shown in Fig. 2, where each sweep followed a
+voltage sequence from 0 to +1 V to -1 V and back to 0 V. We show four devices of each area, which are plotted in Fig.
+2a-f.
+The current magnitudes for different devices of the same area show no significant differences down to 1 μm, indicating
+device-to-device variations are minimal. Establishing this is important as this signifies the sole influence of device area
+in determining the resistance ratio and rules out contributions from device-to-device variation. The 800 nm devices
+show a greater degree of variation; this is likely due to small differences in their areas and edges arising from the
+fabrication process and not inherent to the material or due to device fallibility. No significant differences in the current
+3
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure 3: Resistance ratio, cycling endurance and state stability. a-f show 1000 consecutive current-voltage sweeps
+from +2 V to -3 V to +2 V at a rate of 1.52 Vs−1 for devices of 100 μm down to 800 nm. Starting from a SET voltage
+of +2 V, each device is in an LRS, represented by the upper branch reaching the RESET voltage of -3 V and sweeping
+back, the devices are switched to an HRS (represented by the lower branch.
+densities at low bias values are found in the virgin state, confirming that the entire device area contributes to the charge
+transport (Supporting Information Fig. S1). For all the devices, the current gradually increases and exhibits a small
+hysteretic effect from the virgin state, indicating that no forming step is required.
+Figure 3a-f shows 1000 consecutive current-voltage (I-V) sweeps of these devices. Starting from a SET voltage of +2 V,
+each device is in an LRS, represented by the upper branches. After reaching the RESET voltage of -3 V and sweeping
+back, the devices are switched to an HRS (represented by the lower branches). In all device areas both the SET and
+RESET operation remain continuous, indicating the resistive switching retains its analog nature when downscaling. The
+cycling endurance was measured for over 105 switching cycles without device failure, illustrating an endurance of >105.
+The current in the HRSs scales approximately with area at low bias values, while the low resistance current, is less
+closely correlated to the area. As a result, the resistance window increases with decreasing device area in both forward
+and reverse bias. Figure 1b and Supporting Information Fig. S2 show the current and current density at a low read
+voltage of 0.3 V, respectively. Minimal cycle-to-cycle variations at low reading voltages are found with reproducible
+switching between clearly distinguishable states without degradation in device performance. This also establishes
+the low power operation of these devices after downscaling, which is important for memristor operation. As shown
+in Supporting Information Fig. S3, the device-to-device variation remains low down to 1 μm. The variation in the
+resistance ratio in the 800 nm devices is larger (Fig. S4), and will be discussed later.
+The SET and RESET transitions are gradual and highly tunable. To demonstrate this, a 1 μm device was subjected to
+voltage sweeps varying between different positive (SET) and negative (RESET) voltages. Figure 1c shows that a wide
+range of stable states is available at a low read voltage of +0.3 V. The wide dynamic range combined with the large
+number of distinct addressable states ensures device reliability and increased memory storage capabilities. Each state
+maintains a narrow distribution of current values over the 100 cycles shown, reiterating the stability of the switching
+process.
+2.2
+Scanning Transmission Electron Microscopy
+A microscopy study of the Schottky interface was carried out using STEM. Figure 4 shows atomic resolution cross-
+section STEM-integrated Differential Phase Contrast (iDPC) images of the Co/Nb:STO interface for samples in the
+4
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure 4: Visualization of oxygen vacancy migration using STEM. iDPC-STEM images of Co/Nb:STO samples in
+a the virgin (unbiased) state, b the LRS and (c the HRS, highlighting the structure close and far from the interface. The
+perovskite unit cell of STO, showing Sr in green, O in dark red and Ti in light red, viewed along the <110> in d the
+pristine state and (e with oxygen vacancies. The deficiency of O causes Ti atoms to move away from the vacancies as
+shown by the arrows. f shows a schematic representation of how the interfacial layer is affected by biasing.
+unbiased virgin condition (Fig. 4a), the LRS state (Fig. 4b) and the HRS state (Fig. 4c). To image lighter oxygen atoms,
+integrated into a matrix with heavier Sr and Ti atoms, we utilized STEM-integrated Differential Phase Contrast (iDPC)
+instead of the more commonly employed STEM-High-angle annular dark-field (HAADF) imaging technique.[14, 15].
+The STEM images in Fig. 4a show that, apart from a thin interfacial region, the bulk STO consists of a cubic perovskite
+lattice and no defects are observable. All images taken within the bulk did not show any dislocation and possessed
+the expected perovskite structure as shown in Fig. 4d. However, the structure close to the interface deviates from this
+perovskite structure and is deficient in oxygen. The migration of oxygen ions near the interface towards Co causes
+positively charged Ti ions to be displaced so that they no longer sit equidistantly from the Sr ions along <001>. Figure
+4e illustrates how the loss of O ions gives rise to Ti displacements along the <001> direction away from the interface
+as well as along <1-10> (see Supporting Information Fig. S7) and is similar to what was reported in ref. [16] in
+La0.67Sr0.33MnO3/Hf0.5Zr0.5O2. We believe the creation of this thin layer to be related to the formation of a Schottky
+barrier. The analysis for a non-memristive interface with Ti contacts can be found in Supporting Information Fig. S6.
+Figure 4b shows analogous results to Fig. 4a, but now for the sample switched to the LRS, representing the upper
+branch in Fig. 3, after the application of a positive bias voltage of 2 V. Comparing the two figures shows that in the
+LRS state the extent of the interfacial layer has decreased. This suggests that under the influence of a positive voltage,
+the labile bonds between O and interfacial Co atoms are broken and oxygen moves back into the STO substrate. A
+negative bias voltage of -3 V (corresponding to the lower branch in Fig. 3), on the other hand, causes oxygen to move
+from STO to cobalt causing the formation of CoO and more oxygen vacancies in the STO, highlighted by a larger
+region over which Ti ions are displaced (see Fig. 4c). This indicates that the formation of the CoO switches the sample
+to the HRS state. It has been shown [17, 18] that the oxygen vacancy distribution inside the system will determine
+how the oxygen vacancies are affected by the applied voltage. The formation of an oxygen deficient interfacial layer
+confirms that in these samples the oxygen vacancies are concentrated near the interface. In this case, it is expected
+that the application of a positive voltage will cause oxygen vacancies to be repelled from the interface while a negative
+voltage will cause oxygen vacancies to be attracted to the interface, consistent with our findings. After removing the
+5
+
+(a
+middle of STO
+middle of STO
+Virgin state
+Low resistance state
+High resistance state
+(d)
+(f)
+(e)
+<001>
+Virgin state
+LR state
+HR state
+Co
+Co
+Co
+oxygen deficient areal
+oxygen deficient area
+STO
+STO
+STO
+→ <1-10>
+Zone axis: <110>MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+voltage, the interfacial layer did not reform over time, suggesting the presence of an oxygen-migration blocking layer.
+These results are summarized in Fig. 4f.
+Our results directly confirm the existence of a homogeneous oxygen deficient layer at the interface. The homogeneous
+nature of the defect state layer ensures ionic defects are retained with downscaling. We furthermore show that the
+physical extent of the layer is reduced or extended when a positive or negative voltage is applied respectively. Although
+the uniform nature of the ionic contribution to switching is now verified, this does not explain the origin of the
+unexpected enhancement of the resistance window with downscaling. This we discuss next by considering the trapping
+of electronic charges at oxygen vacancy sites.
+2.3
+Model
+In order to understand how the electrical properties of the devices are influenced by these oxygen vacancies, we
+consider the interaction between electrons and defect states. This interaction is most strongly evidenced by the retention
+characteristics, which have a slow decaying component. This behavior is caused by the detrapping of charges. It has
+been shown that this occurs over long timescales and the different states will remain clearly distinguishable for long
+time periods of hours and that the retention time is tunable by the applied stimuli [12]. We utilized short voltage pulses
+to measure the retention characteristics of each device in both an HRS and LRS. This was done by applying alternating
+SET and RESET pulses of +2 V and -3 V respectively, and reading the small-signal current at either +0.3 V or -0.5 V
+after each writing event. The state retention characteristics of the different devices are shown in Fig. 5 for the LRS (red)
+and HRS (black). Over time, the current in both states tends to an intermediate value. For the LRS, the rate of change
+follows a power law that is commonly observed for charge trapping under bias in high-κ dielectrics, referred to as the
+Curie-von Schweidler law.
+This law describes a non-Debye type relaxation in dielectrics. Empirical evidence of this behavior is seen in a wide
+variety of materials, but the precise physical origin remains unclear. Here we consider the effect of injected electrons
+becoming trapped in defects states within the dielectric. The space charge generated by these trapped electrons lowers
+the electric field, in turn reducing the flow of current through the dielectric. In this case the trapping rate can be
+expressed as:
+dn
+dt = n0σ Jvth
+qvd
+e− nh
+V
+(1)
+where n0 is the maximum number of traps available, J/q is the net flux density, vth and vd are the thermal and drift
+velocities respectively, and σ is capture cross-section. Solving this equation yields the following expression for n:
+n = V
+h ln
+� Q
+Q∗ + 1
+�
+(2)
+where Q =
+�
+Jdt is the total injected charge and
+Q∗ =
+V vdq
+n0hvthσ
+(3)
+Expressing the current as J = Jst−α and extending this analysis results in:
+α
+1 − α ln(Js) ≈ mEap + n
+(4)
+where m is a constant.
+We can also directly relate the trapping rate to the current. QT represents the charge that is trapped when charge Q is
+injected into the dielectric. The ratio dQT
+dQ is a function of current. The current can be written as:
+J = Js
+t
+t0
+−1/(α+1)
+(5)
+where α ≥0 and Js depends on the transport mechanism. For conduction following an exponential relation:
+Js ∝ e
+(1−1/(α+1))V
+V0
+(6)
+Here, V0 is a constant. The full derivation is shown in Supporting Information Section S1 and Fig. S8, and is also
+extended to show that it holds for other transport mechanisms.
+Equations 4 and 6 serve as a direct mathematical proof that the exponent α in the power law is related to the effective
+trap density or capacity of the dielectric to trap electrons. This derivation is applicable to a wider range of systems,
+irrespective of the choice of dielectric material. In Table 1 we show the LRS exponents, α for each device. Larger
+values are observed for smaller devices indicating that the trap density is higher in the smallest device compared to the
+larger device.
+6
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure 5: Trapping dynamics and Schottky interface energy landscapes. Retention characteristics of differently
+sized devices read at +0.3 V a-c) and -0.5 V (d-f) after a SET voltage of +2 V (red) or -3 V (black). g shows the energy
+landscape of a Schottky interface in equilibrium when the dielectric constant does not depend on electric field (solid
+line) and when the dielectric constant is field-dependent (dashed line). EF and EC are the Fermi level and conduction
+band respectively. The energy landscapes at the center and edge of a device are compared in h in equilibrium and i in
+reverse bias. Red circles represent oxygen vacancy states and the green arrow indicates electron tunneling.
+area
+���������
+|Exponent|
+Radius (μm)
+Read at +0.3 V
+Read at -0.5 V
+1
+0.85±0.03
+2.17±0.02
+10
+0.47±0.02
+0.987±0.002
+100
+0.041±0.004
+0.626±0.007
+���������
+trapping density
+Table 1: Magnitude of exponents, α, extracted by fitting a power-law to the low resistance states in the graphs in Fig. 5.
+3
+Discussion
+While this model provides a clear correlation between trapping density and device area, it does not give information
+about the traps; we implicitly take all traps to be of the same kind, while in reality, the nature of traps can vary greatly.
+The trapping rate can depend on the spatial location of the traps and new traps can be generated via defect migration.
+For a more precise picture of the mechanism, we need to consider a distribution of traps with respect to their location
+within the dielectric. Evidenced by the STEM study, oxygen vacancies are the most important class of trapping defects
+to consider. They are abundantly present in SrTiO3 due to their low formation (0.51 eV[19]) and migration (0.62
+eV[20]) enthalpies and their locations within the energy landscape are well documented[21].
+7
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+From the discussion above, it is clear that the energy landscape of these Schottky junctions is far more complex than is
+captured by the most commonly used models that are based solely on parameters of the individual materials forming
+the contact [22, 23]. Transport through these junctions is usually described by the thermionic emission equation, which
+includes an ideality factor accounting for the deviating transport from this ideal diode equation. This model furthermore
+does not consider that the interfacial area is not spatially homogeneous and that in devices of finite areas the boundary
+of the device will be relevant. In particular, it is known that near the edges crowding of the field lines leads to an
+enhancement in the field strength which can decrease the barrier width [24, 25]. This is supported by the results of
+the finite element simulations in Supporting Information Fig. S11 and S12, showing a significant enhancement in the
+electric field around the edge and when downscaling. From the simulations it is evident that there is still a clear field
+gradient in the 1 μm devices, indicating that a further increase in ratio with downscaling can be expected, and the areal
+field shows no apparent saturation till around 10 nm (Fig. S13).
+The observed enhancement is especially important in Nb:STO-based memristive devices as the dielectric constant of
+the substrate strongly depends on electric fields [26, 27]. This will further alter the potential landscape of the Schottky
+interface in such memristive devices. In particular, the dielectric permittivity of Nb:STO rapidly decreases in the
+presence of large electric fields which results in a decrease in the effective Schottky barrier width as illustrated in Fig.
+5g. Consequently, a large reduction in the barrier width is expected to occur near the device edges (Fig. 5h). It has also
+been shown that an electric field can modify the defect states and significantly affect trapping parameters[28].
+Given that the charge transport is governed by the potential landscape, this will hugely impact the measured current,
+pictured in Fig. 5i. Tunneling through the barrier will be enhanced near the device edges leading to a larger current near
+the device perimeter. This will be especially important in the LRS where the interface is depleted of trapped charges
+and the Schottky barrier is narrower, leading to more tunneling [29, 12].
+Transport across the interface is comprised of thermionic emission and tunneling. The thermionic current density is
+expected to be independent of area and is the dominant mechanism in the HRS at low bias voltages, giving rise to the
+decreasing current in the HRS around zero with downscaling observed in Fig. 3. At higher voltage values, however,
+tunneling will also contribute to the current; the tunneling current density will increase with decreasing area. In Fig. 1b,
+the current is read at +0.3 V where we expect both thermionic emission and tunneling to contribute to transport, giving
+rise to similar currents measured for the 10 and 1 μm devices in the HRS. The tunneling contribution increases in the
+LRS, especially in smaller devices due to the larger electric fields, resulting in the observed increase in current density
+with reducing area.
+By applying a potential over the Schottky barrier, the Fermi level is shifted such that tunneling electrons sample different
+oxygen vacancy energy levels. As the reverse bias voltage is increased, electrons are gradually exposed to larger ranges
+of states in which they can become trapped. In addition, in reverse bias, the electric field at the interface becomes
+larger leading to a reduction in the dielectric constant and a corresponding decrease of the Schottky barrier widths. This
+decrease in width will be more pronounced in regions closer to the edge due to the local field enhancement. As a result
+of the narrower barrier, electron-electron scattering will be reduced and the trap states will act as the main barrier for
+transport. The stronger edge field may additionally facilitate the migration of oxygen vacancies resulting in a higher
+number of vacancies accumulating around the perimeter. Consequently, the trapping efficiency will be greater near the
+edge than in the center. This is a unique effect enabled by the electric field control of the dielectric permittivity, does
+not occur in conventional semiconductors and is relevant for Nb:STO memristive device design.
+We can express the area and perimeter of a device with radius r as A = πr2 and p = 2πr respectively. The ratio of the
+perimeter to area:
+p
+A = 2πr
+πr2 = 2
+r
+(7)
+indicates that the edge effects become more dominant as the device area is reduced. As a result, current flow at the
+perimeter will constitute a larger percentage to the overall transport behavior in smaller devices. This explains the
+enhanced current densities observed when downscaling after applying large bias voltages as well as the larger effective
+trapping densities for smaller devices. Specifically, this field enhancement around the device edges gives rise to an
+increase in the dynamic range in smaller devices, and explains the unexpected resistance window scaling.
+4
+Conclusions
+As a first demonstration of exploiting edge effect related additional electric fields, our work successfully demonstrates
+the ability to increase the resistance window by device miniaturization of interface memristors from 100 μm down
+to 1 μm, contrary to expectations, with exceptional robustness to device-to-device and cycle variability. Scanning
+transmission electron microscopy images taken in the virgin, high and low resistance states prove the existence of a
+homogeneous interfacial layer, deficient in oxygen, whose physical extent is influenced by applying an electric field.
+8
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+This, however, does not explain the enhancement in the resistance window with device downscaling. A model describing
+the interaction of electrons with oxygen vacancy trap states shows an increase in the effective trapping density with
+downscaling. The advantage of direct integration of devices on a semiconducting platform of Nb-doped SrTiO3 allows
+for the locally enhanced fields to controllably tune the interfacial energy landscape at the interface, leading to a greater
+contribution of edge effects in smaller devices as confirmed by finite element simulations. With rapid advances made in
+the palette of materials and devices available for neuromorphic hardware, the thrust now should be in their efficient
+integration on semiconducting platforms for on-chip applications with substantial reduction in areal footprint. In this,
+our work provides an encouraging direction.
+5
+Experimental Section
+5.1
+Electrical Device Fabrication
+We investigated a series of Co/Nb-doped SrTiO3 devices, where the device area was varied across the series over a
+range spanning five orders of magnitude ranging from 10−12 to 10−8 m2, with radii between 800 nm and 100 μm.
+The devices were fabricated using Nb-doped SrTiO3 (001) substrates with a doping concentration of 0.1 wt% from
+Crystec. SrTiO3 consists of alternating SrO and TiO2 planes along the [001] direction. The as-received substrates
+have a slight miscut from the exact crystallographic direction and as a result, a mixture of both terminations exists at
+the surface. It has been shown that the local properties of Schottky barriers grown on the different terminations may
+differ, hence to minimize the variation of different areas on the substrate a single termination is desired. To ensure that
+the terminating layer is TiO2, a chemical treatment was carried out with buffered hydrofluoric acid (BHF). A further
+annealing treatment at 960 ◦C in an O2 flow of 300 ccmin−1 to facilitate the reorientation of surface atoms to form
+an atomically flat and straight terraced surface. Atomic force microscopy images were taken at different parts of the
+substrate and confirmed the existence of uniform terraces. The substrate was then coated with a negative resist (AZ
+nLOF 2020) and using electron beam lithography circles of different areas were patterned. A thick insulation layer of
+AlOx was deposited using electron beam evaporation and lift-off was carried out to define a set of direct contacts to
+the substrate. By means of a second lithography step with a positive resist (950 K PMMA), square contact pads were
+defined, each covering a hole and part of the surrounding AlOx: the dimensions of these pads were identical for each
+device to minimize spurious effects arising from significantly different contact resistances. Co (20 nm) and a capping
+layer of Au (100 nm) were then deposited using electron beam evaporation in high vacuum (∼10−6 Torr).
+5.2
+Electrical Characterization
+Electrical measurements were conducted using probes connected to two remote-sense and switch units (RSU) of a
+Keysight B1500A Semiconductor Device Parameter Analyzer. During the voltage sweeping measurements, conducted
+using a sweeping measurement unit (SMU), the bottom of the substrate is held at 0 V while a voltage is applied to
+the top electrode. Due to the diodic nature of the devices in conjunction with large degrees of resistive switching,
+the measured currents during a single sweeping measurement span up to 9 orders of magnitude. For this reason, the
+measurements were performed using auto range for the measured current. The effects of this can be observed in the
+endurance cycling measurements which were performed at high sweeping rates in the form of plateaus in the current
+whenever a limit of the SMU range is reached.
+5.3
+Scanning Transmission Electron Microscopy
+The samples discussed in this work use SrTiO3 (001) substrates with an Nb-doping in place of Ti of 0.1 wt% from
+Crystec. The surface was prepared using a chemical treatment with buffered hydrofluoric acid (BHF). Next, the
+substrates were annealed at 960◦C in an O2 flow of 300 ccmin−1. For STEM samples films were deposited by electron
+beam evaporation of 20 nm of Co capped with 20 nm of Au and 20 nm of Pt. From this, three types of STEM lamellae
+were prepared: virgin (unbiased) samples, low resistance state (LRS) samples and high resistance state (HRS) samples.
+Using a probe station, samples are subjected to bias values of +2 V and -3 V to prepare samples in the LRS and HRS
+respectively. STEM lamellae were extracted from samples along the <110> direction using a Helios G4 CX dual beam
+system with a Ga focused ion beam. The lamellae were thinned to make them transparent to electrons using the focused
+ion beam. Imaging was carried out using a Thermo Fisher Scientific Themis Z S/TEM system operating at 300 kV.
+STEM-High-angle annular dark-field (HAADF) images are most widely used, because they are readily interpretable
+with atomic columns being bright spots in a dark surrounding, where the brightness of the spots scale with the average
+atomic number Z (∼Z1.7). This technique is well suited to image heavy elements, but lighter elements, such as oxygen,
+are harder to detect, and cannot be detected properly when integrated into a matrix with much heavier elements (like Sr).
+Therefore, to gain more insight into the important role played here by the oxygen ions, we utilized here STEM-iDPC
+9
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+instead of STEM-HAADF imaging. This technique uses a four-quadrant annular bright field detector and can be used
+to acquire the projected local electrostatic potential of the sample (when thin) and has clear advantages over traditional
+annular bright field (ABF) imaging [14, 15].
+5.4
+Simulations
+Finite element modeling of the electric field profile at the interface was carried out using COMSOL Multiphysics.
+5.5
+Statistical Analysis
+For the |current|-voltage graphs, the absolute value of the measured current is taken; to determine the current density,
+the measured current was divided by the area of the Co contact. The values in Table 1 were derived by iteratively fitting
+the data in Fig. 5a-f using a power-law equation of the form I = I0(t − t0)−α by means of the Levenberg-Marquardt
+algorithm; the reported errors are the standard errors calculated by this method. The fits are shown in supporting Fig.
+S9. The inverse scaling of the exponent and device area was verified for different devices and different reading and SET
+voltages. Plotting and analysis of electrical measurements was done using OriginPro 8.5. Measurements were repeated
+on four devices of each area to check reproducibility and validity of results.
+For STEM images, multiple regions for each one of the three bias conditions were taken to verify the results. The idpc
+images were filtered by applying a high-pass Gaussian filter using Velox.
+Acknowledgements
+A.G. is supported by the CogniGron Center, University of Groningen. Device fabrication was realized using NanoLab
+NL facilities. We acknowledge technical support from J. G. Holstein, H. H. de Vries, A. Joshua, T. Schouten, and H.
+Adema. We thank R. J. E Hueting, P. Nukala, S. de Graaf and T. Kenyon for useful discussions. A.G., D.G, I.B, and
+T.B. benefited from helpful discussions with the members of the Spintronics of Functional Materials group.
+Conflict of Interest
+The authors declare no conflict of interest
+Author Contributions
+A.G. and T.B. conceived the idea and designed the devices. A.G. and D.G. fabricated devices for electrical measurements
+and performed electrical measurements, along with I.B.. D.G. derived the mathematical model discussed in the
+manuscript. M.A. and A.G. fabricated lamalae for STEM and M.A. took STEM images. Finite element simulations
+were done by A.G.. All authors analyzed the data, discussed the results and agreed on their implications. All authors
+contributed to the preparation of the manuscript.
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+11
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Supplementary Data
+Figure S1: Current densities of virgin devices. Results are shown for devices of radial dimensions of 100 μm (black),
+10 μm (blue) and 1 μm (red).
+Figure S2: Cycle-to-cycle variation. The current density read at +0.3 V for device sizes of 100 μm (black), 10 μm
+(blue) and 1 μm (red).
+S1
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure S3: Device-to-device variation. Current-voltage sweeps from +2 V to -3 V to +2 V at a rate of 1.52 Vs−1
+measured between a SET voltage of +2 V and a -3 V RESET voltage. Measurements are shown for different devices to
+demonstrate the low device-to-device variability.
+Figure S4: Measurements of 800 nm devices: device-to-device variation when controlled with a larger voltage range.
+The red graph is the device presented in the main text. These devices show a greater degree of variation, due to small
+differences in their areas and edges arising from the fabrication process. There resistance ratios, however remain high.
+S2
+
+32 μm
+10 μm
+3.2 μm
+1 μm10'3
+10-5
+ICurrentl (A)
+10'
+109
+10
+11
+TT
+-3
+-2
+-1
+0
+1
+2
+Voltage (V)MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure S5: Side wall profile of electrical measurement device: (a) and (b) STEM-HAADF images. The inset in (b)
+marks the interfacial region close to the edge. STEM-energy-dispersive X-ray spectroscopy (STEM-EDX) elemental
+mapping image of (c) Au, (d) Sr, (e) O, (f) Ti, (g) Al and (h) Co.
+S3
+
+(a)
+(b)
+400nm
+200nm
+Au
+Sr
+(c)
+(d)
+0
+Ti
+(e)
+(f)
+(g)
+Al
+(h)
+CoMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure S6: Nb:STO/Ti interface: (a) STEM-EDX elemental map of Sr Ti and O. (b) elemental intensity as a function
+of position along the line scan in (a). STEM-iDPC images of (c) the interface and (d) away from the interface.
+S4
+
+(a)
+Sr
+Ti
+line scan
+2 nm
+(b)
+Net Intensity / Counts
+Intensity / kCounts
+PositionMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure S7: Ti-column displacement: iDPC-STEM image inside Nb:STO substrate close to the interface. Some of the
+Ti ions occupying ideal perovskite positions are marked in yellow while displaced ions are marked in red with arrows
+highlighting the direction of displacement.
+S5
+
+QQ0
+000
+000
+000
+0Q0
+unmMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+S1. Derivation of the relation between trapping density and exponent.
+Figure S8: Schematic of parameters in section S1: Eap and x represent the applied electric field and centroid of
+trapped charge, defined with respect to the interface, respectively. The number density of trapped charges, n, is depicted
+by the black curve as a function of position in the dielectric.
+If we assume that the rate of trapping has no dependence on the location of traps, the electric field, E, can be expressed
+as:
+E = Eap − qnx
+η
+(S1)
+where Eap is the applied electric field, q the electric charge, n is the number density of trapped charges, x is the centroid
+of the trapped charge with respect to the interface and η is the dielectric permittivity. In [S1], charge trapping was
+analyzed on the basis of three mechanisms, namely first-order trapping, first-order trapping with Coulombic interactions,
+and trapping which increases during injection due to the generation of states. The expressions for current they derive
+are qualitatively similar for each mechanism. Hence, for simplicity, we consider the rate of trapping density to be a
+decay in first order with the addition of electron-electron interactions. Coulombic repulsion may inactivate trapping
+sites surrounding a trapped electron. This is included in the rate equation by multiplying a probability factor. If the
+volume of dielectric rendered inactive by a trap is h, then the trapping is reduced by a factor of (1 − h
+V ), where V is the
+volume of the dielectric. For n trapped charges, the factor is (1 − h
+V )n. The trapping rate can be expressed as:
+dn
+dt = (n0 − n)σ Jvth
+qvd
+�
+1 − h
+V
+�n
+(S2)
+where n0 is the maximum number of traps available, J/q is the net flux density, vth and vd are the thermal and drift
+velocities respectively, and σ is capture cross-section. Assuming the total volume of the dielectric to be much larger
+than the volume deactivated by trapping events so that, 1≫ h/V and n0 ≫ n, this expression can be simplified to:
+dn
+dt = n0σ Jvth
+qvd
+e− nh
+V
+(S3)
+S6
+
+E
+apMEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Solving this equation yields the following expression for n:
+n = V
+h ln
+� Q
+Q∗ + 1
+�
+(S4)
+where Q =
+�
+Jdt is the total injected charge and
+Q∗ =
+V vdq
+n0hvthσ
+(S5)
+We express the current in terms of the electric field as:
+ln
+� J
+J0
+�
+= E
+E0
+= 1
+E0
+�
+Eap − V
+h
+qx
+η ln
+� Q
+Q∗ + 1
+��
+(S6)
+The current follows a decaying power law with time, J = Jst−α, and the injected charge as a function of time is given
+by:
+Q(t) =
+�
+Jdt = Jst1−α
+1 − α
+(S7)
+Substituting S7 into S6 when Q/Q∗ ≫ yields and noting β = V qx
+hE0ϵ:
+ln
+� J
+J0
+�
+= 1
+E0
+�
+Eap − β ln
+�
+Jst1−α
+Q∗(1 − α)
+��
+= 1
+E0
+�
+Eap − β ln
+�
+Jst
+Q∗(1 − α)
+�
+− β(1 − α) ln t
+�
+(S8)
+Comparing S8 with J = Jst−α implies
+α = β(1 − α)
+=
+β
+1 + β
+(S9)
+and
+Js ≈ mEap + n − β ln(Js)
+(S10)
+Where m encompasses several material parameters. Writing β in terms of α, and since measured currents are less than
+10−4 A, Js can be neglected in comparison to ln(Js), leading to:
+α
+1 − α ln(Js) ≈ mEap + n
+(S11)
+β is positive, we know from Eq. S9 that α lies between 0 and 1, and is a monotonically increasing function of β.
+Considering that β = V
+h
+qx
+E0 , an increase in either the effective density, V/h or in x gives rise to an increase in α, with
+the former being physically more likely.
+Instead of deriving an explicit expression for the number density of trapped charge, we can also directly relate the
+trapping rate to the current, as was done in for example [S2]. We use QT to denote the charge that is trapped when
+charge Q is injected into the dielectric. The ratio dQT
+dQ is assumed to be a function of current, i.e.
+dQT
+dQ = f
+� J
+J0
+�
+(S12)
+Substituting Eq. S12 into Eq. S1 gives:
+dE
+dt = Jx
+lη
+dQT
+dQ
+(S13)
+where l is the length of the dielectric and J = dQ
+dt . To relate this to the power law, we assume a solution of the form
+dQT
+dQ =
+� J
+J0
+�α
+(S14)
+with α ≥ 0. A general expression for the current assumes the form:
+J(E) = J0(E)eg(E0)
+(S15)
+S7
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+where the specific functions are determined by the relevant conduction mechanisms. Specifically here, using Eq. S14
+we can express the current as
+J = Js
+t
+t0
+−1/(α+1)
+(S16)
+For conduction given by an exponential relation as in Eq. S6:
+Js ∝ e
+(1−1/(α+1)V
+V0
+(S17)
+while for conduction determined by Frenkle-Poole equation, we arrive at:
+Js ∝ V e
+(1−1/(α+1)V
+1
+2
+V0
+(S18)
+and for Fowler-Nordheim conduction we get:
+Js ∝ V 2e
+(1−1/(α+1))V −1
+V0
+(S19)
+Here, V0 is a constant.
+The dominant transport mechanism in a system can be determined by plotting the current versus voltage on a double
+logarithmic scale. Equations S9, S11, S17, S18 and S19 indicate a clear theoretical proof that the exponent in the power
+law is directly proportional to the effective trap density or capacity of the dielectric to trap electrons, independent of
+which transport mechanism is dominant.
+S8
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure S9: Fits of the retention data to extract the exponents α: The model used is |I| = I0(t − t0)−α), where |I|
+and t are the absolute current and time respectively, and I0 and t0 are fitting parameters. The adjusted R2 values of the
+fits are (a) 0.99237, (b) 0.99475, (c) 0.99689, (d) 0.99717, (e) 0.99995, and (f) 0.99996. (g) shows the dependence of
+the exponents on area.
+S2. Modeling the edge effects
+To visualize the field profiles in our devices we used finite element analysis (COMSOL). The modeling geometry is
+shown in Fig S10. In each simulation, the Nb:STO substrate was modelled as a cube with a dielectric constant of 300
+and a thickness of 0.5 mm (along z), corresponding to the thickness used in the experimental study. A circular Co
+electrode of radius 1 μm, 10 μm or 100 μm was placed on the top surface of the substrate (z=0.5 mm). A ground node
+was placed on the bottom of the substrate (z=0), while a voltage was applied to the top Co electrode. For the simulations
+in Fig. S13. the size of the substrate was reduced to improve the resolution of the mesh. This was required to retain the
+circular nature of the electrodes for the 10 nm devices; this was determined not to influence the electric field strength.
+S9
+
+100 μum read at +0.3 V
+10 μum read at +0.3 V
+1 μm read at +0.3 V
+α=0.041±0.004
+α=0.47±0.02
+α=0.85±0.03
+104
+10~5
+105
+I (A)
+[Currentl 
+ICurrentl 
+ICurrentl
+10°
+10
+(a)
+(b)
+107
+(c)
+10°
+10°
+Time (a.u.)
+Time (a.u.)
+Time (a.u.)
+100 μm read at -0.5 V
+10 μm read at -0.5 V
+1 μm read at -0.5 V
+α=0.626±0.007
+α=0.987±0.002
+10~5
+α=2.17±0.02
+10°5
+10°
+10*6
+Currentl (A)
+W10*
+ICurrentl (A)
+10-
+ICurrentl
+10°
+10
+10°
+109
+(d)
+10~8
+(e)
+()
+10°
+10-
+Time (a.u.)
+Time (a.u.)
+Time (a.u.)
+Read at +0.3
+2.0
+Read at -0.5
+1.5
+1.0
+0.5
+0.0
+(g)
+10-12
+10-11
+10-10
+10-9
+108
+Area (m²)MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure S10: Model sample geometry: the Nb:STO substrate is represented by a cube with a thickness of 0.5 mm
+(along z). Circular Co electrode of radii (a) 1 μm, (b) 10 μm and (c) 100 μm is placed on the top surface of the substrate
+(z=0.5 mm). A ground node is placed on the bottom of the substrate (z=0), while a voltage is applied to the top Co
+electrode.
+Figure S11: Electric field at -3 V: along the surface normal (z direction) for (a)+(d) 1 μm, (b)+(e) 10 μm and (c)+(f)
+100 μm devices, The plots on the top row ((a)-(c)) have the same scale bar, with a maximum field value of 3.5 ×106
+Vm−1. For the bottom row, the scale bar has a maximum value of (d) 2 ×107 Vm−1, (e) 2 ×106 Vm−1 and (f) 3 ×105
+Vm−1.
+S10
+
+(a)
+×10*4 m
+(b)
+×10'4 m
+(c)
+×10'4 m
+2
+2
+2
+0
+×104 m
+2
+×10*4 m
+×104 m
+×104 m(a)
+X10°
+(b)
+310°
+(c)
+X10%
+3
+3
+3
+2.5
+2.5
+2.5
+2
+２
+2
+1.5
+1.5
+1.5
+1
+0.5
+0.5
+0.5
+(d)
+X107
+(e)
+(f)
+×106
+X105
+2
+2
+1.8
+1.8
+2.5
+1.6
+1.6
+1.4
+1.4
+1.2
+1.2 
+1
+1
+1.5
+0.8
+0.8 
+0.6
+0.6
+0.4
+0.4 
+0.5
+0.2
+0.2MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+Figure S12: Electric field at +2 V: along the surface normal (z direction) for (a)+(d) 1 μm, (b)+(e) 10 μm and (c)+(f)
+100 μm devices, The plots on the top row ((a)-(c)) have the same scale bar, with a maximum field strength of -3 ×106
+Vm−1. For the bottom row, the scale bar has a maximum value of (d) -1 ×107 Vm−1, (e) -1.5 ×106 Vm−1 and (f) -2
+×105 Vm−1.
+Figure S13: Electric field at -3 V: along the surface normal (z direction) for (a) 100 nm, (b) 50 nm and (c) 10 nm
+devices. No saturation of the field is observed in (a) and (b) and the field appears to saturate in the 10 nm devices.
+S11
+
+(a)
+(b)
+(c)
+X106
+X106
+0
+0
+-0.5
+-0.5
+-0.5
+-1
+-1
+-1
+-1.5
+-1.5
+-1.5
+-2
+-2
+-2.5
+-2.5
+-2.5
+-3
+-3
+-3
+(d)
+(e)
+(f)
+×107
+X106
+X105
+10
+0
+10
+-0.1
+-0.2
+-0.2
+-0.2
+-0.4
+-0.4
+-0.3
+-0.6
+-0.4
+-0.6
+-0.8
+0.5
+1
+-0.8
+-0.6
+-1.2
+-1
+0.7
+-1.4
+-0.8
+-1.2
+-1.6
+-0.9
+-1.8
+-1.4
+.1
+-2(a)
+(b)
+(c)
+X107
+×107
+X108
+9
+1.3
+8.8
+1.29
+5.5
+8.6
+1.28
+8.4
+1.27
+8.2
+1.26
+5
+8
+1.25
+7.8
+1.24.
+4.5
+7.6
+1.23
+7.4
+1.22
+4
+7.2
+1.21
+3.5
+7
+1.2MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+S3. Literature survey of interfacial switching
+It is often suggested that a layer close to the interface layer is responsible for the switching [29, S4, S5]. Some groups
+have shown that both high and low resistance states show an area-independent current density, eluding to a switching
+mechanism that occurs homogeneously over the entire device area [S6]. Often this is explained in terms of a change
+in the Schottky barrier height and width induced by charge trapping at the interface [S7, 29, S4, S8, 12, S10] and
+movement of oxygen vacancies [S8, S10]. Other explanations are proposed where the barrier profile is unchanged
+and interfacial changes happen at local regions. Explanation of this type includes It has also been proposed that the
+application of a positive bias results in the generation of oxygen vacancies, forming tunnelling paths and giving rise to a
+LRS where tunnelling, rather than thermionic emission dominate charge transport. The application of a negative bias
+results in the accumulated of large amounts of oxygen in the vacancies which prevents tunnelling and gives a HRS
+[S11, S12].
+Rodenbücher et al. used local-conductivity AFM on highly doped Nb:STO to show the presence of nanoscale
+conducting and switchable clusters. Suggesting that in this case switching is a local phenomenon related to the presence
+of conducting clusters with higher Nb content than their surroundings [S13].
+Finally Chen et al. used scanning tunnelling microscopy and spectroscopy to study the resistive switching in Nb-doped
+SrTiO3 without an electrode, demonstrating that oxygen migration is the results in a variation of electronic structure
+during the switching. With a negative voltage, oxygen anions at the interface near the STM tip were oxidised into oxygen
+molecules and left the lattice. Simultaneously, oxygen vacancies diffuse into the sample, which act like donor-like
+levels causing distortions in LDOS near conduction band and enhance the carrier concentration with electron hopping,
+thus increasing the sample’s conducting. With a positive voltage, oxygen anions return into the sample and the influence
+of the donor-like level became weak and the conductivity decreased [S14].
+Despite a large number of contradictory results and explanations, factors of importance that have been identified include
+the semiconductor doping concentration, electrode material and the quality of the interface.
+S12
+
+MEMRISTIVE MEMORY ENHANCEMENT BY DEVICE MINIATURIZATION
+References
+[S1] D. R. Wolters and J. J. van der Schoot. Kinetics of charge trapping in dielectrics. Journal of Applied Physics,
+58(2):831–837, 1985.
+[S2] R. H. Walden. A method for the determination of high-field conduction laws in insulating films in the presence
+of charge trapping. Journal of Journal of Applied Physics, 43(3):1178–1186, 1972.
+[S3] E. Mikheev, B. D. Hoskins, D. B. Strukov, and S. Stemmer.
+Resistive switching and its suppression in
+Pt/Nb:SrTiO3 junctions. Nature Communications, 5(1):1–9, 2014.
+[S4] E. M. Bourim, Y. Kim, and D.-W. Kim. Interface state effects on resistive switching behaviors of Pt/Nb-doped
+SrTiO3 single-crystal Schottky junctions. ECS Journal of Solid State Science and Technology, 3(7):N95, 2014.
+[S5] Z. Fan, H. Fan, L. Yang, P. Li, Z. Lu, G. Tian, Z. Huang, Z. Li, J. Yao, Q. Luo, et al. Resistive switching induced
+by charge trapping/detrapping: a unified mechanism for colossal electroresistance in certain Nb:SrTiO3-based
+heterojunctions. Journal of Materials Chemistry C, 5(29):7317–7327, 2017.
+[S6] H. Sim, H. Choi, D. Lee, M. Chang, D. Choi, Y. Son, E.-H. Lee, W. Kim, Y. Park, I.-K. Yoo, et al. Excellent
+resistance switching characteristics of Pt/SrTiO3 Schottky junction for multi-bit nonvolatile memory application.
+In IEEE International Electron Devices Meeting, 2005. IEDM Technical Digest., pages 758–761. IEEE, 2005.
+[S7] M. C. Ni, S. M. Guo, H. F. Tian, Y. G. Zhao, and J. Q. Li. Resistive switching effect in SrTiO3−δ/ Nb-doped
+SrTiO3 heterojunction. Applied Physics Letters, 91(18):183502, 2007.
+[S8] X.-B. Yin, Z.-H. Tan, and X. Guo. The role of Schottky barrier in the resistive switching of SrTiO3: direct
+experimental evidence. Physical Chemistry Chemical Physics, 17(1):134–137, 2015.
+[S9] A. S. Goossens, A. Das, and T. Banerjee. Electric field driven memristive behavior at the Schottky interface of
+Nb-doped SrTiO3. Journal of Applied Physics, 124(15):152102, 2018.
+[S10] J. Li, G. Yang, Y. Wu, W. Zhang, and C. Jia. Asymmetric resistive switching effect in Au/Nb: SrTiO3 Schottky
+junctions. Physica Status Solidi (a), 215(6):1700912, 2018.
+[S11] T. Fujii, M. Kawasaki, A. Sawa, Y. Kawazoe, H. Akoh, and Y. Tokura. Electrical properties and colossal
+electroresistance of heteroepitaxial SrRuO3/SrTi1−xNbxO3 (0.0002≤x≤0.02) Schottky junctions. Physical
+Review B, 75(16):165101, 2007.
+[S12] D.-J. Seong, D. Lee, M. Pyun, J. Yoon, and H. Hwang. Understanding of the switching mechanism of a
+Pt/Ni-doped SrTiO3 junction via current–voltage and capacitance–voltage measurements. Japanese Journal of
+Applied Physics, 47(12R):8749, 2008.
+[S13] C. Rodenbücher, W. Speier, G. Bihlmayer, U. Breuer, R. Waser, and K. Szot. Cluster-like resistive switching of
+SrTiO3:Nb surface layers. New Journal of Physics, 15(10):103017, 2013.
+[S14] Y. L. Chen, J. Wang, C. M. Xiong, R. F. Dou, J. Y. Yang, and J. C. Nie.
+Scanning tunneling mi-
+croscopy/spectroscopy studies of resistive switching in Nb-doped SrTiO3.
+Journal of Applied Physics,
+112(2):023703, 2012.
+S13
+

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+The Fermi-Dirac staircase occupation of Floquet bands and current rectification inside
+the optical gap of metals: a rigorous perspective
+Oles Matsyshyn,1, 2 Justin C. W. Song,2 Inti Sodemann Villadiego,3, 1, ∗ and Li-kun Shi1, 3, †
+1Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
+2Division of Physics and Applied Physics, School of Physical and Mathematical Sciences,
+Nanyang Technological University, Singapore 637371, Republic of Singapore
+3Institut für Theoretische Physik, Universität Leipzig, Brüderstraße 16, 04103, Leipzig, Germany
+(Dated: January 4, 2023)
+We consider a model of a Bloch band subjected to an oscillating electric field and coupled to a
+featureless fermionic heat bath, which can be solved exactly. We demonstrate rigorously that in the
+limit of vanishing coupling to this bath (so that it acts as an ideal thermodynamic bath) the occu-
+pation of the Floquet band is not a simple Fermi-Dirac distribution function of the Floquet energy,
+but instead it becomes a “staircase” version of this distribution. We show that this distribution
+generically leads to a finite rectified electric current within the optical gap of a metal even in the
+limit of vanishing carrier relaxation rates, providing a rigorous demonstration that such rectification
+is generically possible and clarifying previous statements in the optoelectronics literature. We show
+that this current remains non-zero even up to the leading perturbative second order in the amplitude
+of electric fields, and that it approaches the standard perturbative expression of the Jerk current
+obtained from a simpler Boltzmann description within a relaxation time approximation when the
+frequencies are small compared to the bandwidth.
+INTRODUCTION
+Quantum many body systems that are periodically
+driven in time have garnered attention over the last
+decades as rich platforms to realize novel collective phe-
+nomena and non-equilibrium states beyond those realized
+in equilibrium [1–25]. A phenomenon that can arise in
+such periodically driven systems and which is forbidden
+in equilibrium, is the existence of an average net recti-
+fied particle flow or ratchet effect. In particular for the
+case of electrons in crystals driven by oscillating electric
+fields, these effects have enjoyed recent renewed attention
+due to their interesting interplay with the dispersions and
+Berry phases of band structures and their potential for
+novel opto-electronic devices [26–32]. Despite their nat-
+ural connection, only a handful of studies have studied
+current rectification effects in Bloch bands through the
+lens of Floquet theory [33–35]. This is in part due to the
+difficulty that even for non-interacting systems, there is
+no simple general formula dictating the occupation of
+Floquet bands analogous to the Fermi-Dirac distribution
+that dictates the occupation of bands in equilibrium [36–
+41].
+One of the central goals of our study is to provide
+simple analytical formulae for the occupation of a single
+Floquet band coupled to a “featureless fermionic bath”
+(which is a commonly used model of bath that for exam-
+ple was employed in Refs. [33, 35, 42]). This featureless
+bath is a physical system that has a finite coupling to
+the fermionic system of interest and is characterized by
+a single relaxation scale Γ. The dynamics of the system
+∗ sodemann@itp.uni-leipzig.de
+† shi@itp.uni-leipzig.de
+0.5
+1
+0
+-2
+0
+-1
+1
+2
+3
+4
+(a)
+(b)
+FIG. 1.
+(a) Time independent limΓ→0 pn as a function of
+¯ϵn/ω calculated using Eq. (46) showing a ladder-like occupa-
+tion. Parameters used: β/ω = 50, µ/ω = 1. (b) Schematic of
+the ladder-like occupation for a Bloch band.
+coupled to this bath can be described in an exact man-
+ner thanks to the fact that we will take the combined
+system plus bath as a non-interacting fermionic system.
+As we will see, this featureless fermionic bath behaves
+as an ideal thermodynamic bath in the limit in which
+its coupling to system is vanishingly small (Γ → 0), and
+in particular in this limit it relaxes the system towards
+the equilibrium Fermi-Dirac occupation of the bands in
+arXiv:2301.00811v1  [cond-mat.mes-hall]  2 Jan 2023
+
+2
+the absence of an external periodic drive.
+As we will
+show, however, when the system is periodically driven in
+time, this bath leads to a self-consistent occupation the
+Floquet bands that is sharply different from that of the
+equilibrium Fermi-Dirac distribution, but which we can
+determine analytically with no approximations. This oc-
+cupation is instead a staircase version of the Fermi-Dirac
+distribution with several jumps that occur at copies of
+the chemical potential shifted by all the harmonics of
+the driving frequency [see Fig. 1(a)]. In the limit of an
+ideal bath (Γ → 0), we will show that this distribution
+coincides with the distribution that has been previously
+obtained within the Boltzmann approach to Floquet sys-
+tems (see in particular Eq. (12) of Ref. [38]).
+Another central purpose of our study is to exploit the
+Floquet formalism to further elaborate on our recent find-
+ing [43] that it is indeed possible for time dependent os-
+cillating electric fields with a frequency that lies within
+the optical gap of a metal, to induce a net rectified DC
+electric current. We will see that this is true even when
+the electric field has a single monochromatic frequency
+ω that is much larger than the relaxation rates and this
+current remains finite in the limit when these rates van-
+ish (Γ → 0) (and therefore does not rely on the fre-
+quency difference effect [44] or in the Raman scattering
+effect [45, 46]). We will demonstrate that this is possi-
+ble by choosing a simple model containing a single Bloch
+band with no Berry curvature, which in a simpler relax-
+ation time Boltzmann description would give rise to the
+so-called “Jerk” effect described in Refs. [43, 47].
+Our
+aim is to use this simple model because it will allow us
+to carry out calculations of its response coupled to the
+fermionic bath in a clear and exact analytical manner.
+We are motivated to do this rigorously in order to
+clarify a series of misconceptions that originated from
+the work of Belinicher, Ivchenko and Pikus [48] and that
+have propagated into some of the subsequent literature
+[43, 45, 46, 49, 50]. In Appendix E we comment in more
+detail about some of these previous works and point out
+more specifically some of their imprecisions.
+One of the central messages of our study is that it is in-
+deed possible to have a net rectified current in response
+to a monochromatic oscillating electric field whose fre-
+quency lies within the optical gap of a system, in the limit
+of vanishing carrier relaxation rates. We will demonstrate
+this within a self-consistent picture of the occupation of
+Floquet Bloch band in the steady state of the system.
+More specifically we will show that in the limit of an
+ideal bath (Γ → 0) the average rectified current in the
+non-equilibrium steady state of the system is given by:
+¯j =
+�
+k
+pk∇k¯ϵk,
+(1)
+where ¯ϵk is the Floquet energy of the band, pk is the occu-
+pation of the Floquet band, ¯j is the current averaged over
+one period, and the integral is over the crystal momen-
+tum in the Brillouin zone with the usual normalization
+of 1/(2π)d.
+The crucial difference between the above expression
+and that for the average current in an equilibrium sys-
+tem, is that the occupation function pk is not simply the
+Fermi-Dirac distribution associated with ¯ϵk, but instead
+it is precisely the stair-case occupation function depicted
+in Fig. 1. Crucially, we will show that generically this
+stair-case occupation is a function that depends on all the
+information of the time dependence of the Hamiltonian,
+and cannot be expressed as a function of only ¯ϵk. We
+will then show that as a consequence of this, the rectified
+current is in fact generically non-zero in the optical gap
+of a metal that breaks inversion and time-reversal sym-
+metries. This result remains true even to second order
+in the amplitude of the oscillating electric field, which is
+the leading order at which rectification currents appear,
+and therefore implies a non-vanishing rectification con-
+ductivity within the optical gap of metals, in agreement
+with our previous results [43]. For other previous dis-
+cussions of the possibility of in-gap rectification see also
+Refs. [46, 51–55].
+Our paper is organized as follows.
+In Section I, we
+setup the approach to open quantum systems, obtain ex-
+act occupation functions for diagonal and time-periodic
+Hamiltonians coupled to a featureless fermionic bath. In
+Section II, based on the exact occupation functions, we
+calculate exact linear and rectification conductivities and
+show that there is a net rectified current in response to a
+monochromatic oscillating electric field whose frequency
+lies within the optical gap of a metal, in the limit of van-
+ishing carrier relaxation rates.
+I.
+THE OPEN-SYSTEM SCHRÖDINGER
+EQUATION APPROACH TO OPEN QUANTUM
+SYSTEMS
+In descriptions of quantum open systems it is typically
+natural to view the combined Hilbert space of the “sys-
+tem” and the “bath” as a tensor product of their Hilbert
+spaces in isolation. There are situations, however, where
+it is possible to alternatively cast this separation of sys-
+tem and bath as a direct sum of their individual Hilbert
+spaces. As we will show, such separation into sums of
+Hilbert spaces is extremely powerful and convenient, be-
+cause it allows to integrate out the dynamics of the “bath”
+in an exact manner and to obtain a simple non-Hermitian
+generalization of Schrödinger’s equation for the system
+which captures its coupling to the bath without any ap-
+proximations.
+One example of the class of models which admits such
+direct sum separation into system and bath are those of
+non-interacting particles. To see this let us imagine that
+the system and the bath as a whole can be described by a
+non-interacting model. For concreteness we can imagine
+this to be a tight biding model of particles hopping on
+a lattice. Because the problem is non-interacting, then
+the dynamics can be analyzed by computing the trajecto-
+ries of single individual particles and then adding them
+
+3
+up. However, for a single particle the Hilbert space of
+the “system” and the “bath” can be naturally viewed as
+a direct sum. For example, in the case of a tight-binding
+model, some sites can be viewed as belonging to the sys-
+tem and the remainder sites as belonging to the bath.
+Let us then consider that the Hilbert space of the sys-
+tem and the bath can be decomposed into a direct sum,
+namely their Hamiltonian and states have block form as
+follows:
+H(t) =
+�
+HS(t)
+HSB(t)
+HBS(t)
+HB(t)
+�
+,
+|ψ(t)⟩ =
+�
+|ψS(t)⟩
+|ψB(t)⟩
+�
+,
+(2)
+where HBS(t) = H†
+SB(t).
+From Eq. (2), the coupled
+Schrödinger equations for system and bath states then
+read:
+i∂t |ψS(t)⟩ = HS(t) |ψS(t)⟩ + HSB(t) |ψB(t)⟩ ,
+(3)
+i∂t |ψB(t)⟩ = HBS(t) |ψS(t)⟩ + HB(t) |ψB(t)⟩ ,
+(4)
+where we set ℏ = 1 throughout the paper. By integrating
+Eq. (4) over time and inserting it into Eq. (3) allows
+to formally eliminate the bath state dynamics |ψB(t)⟩
+and to obtain the open-system Schrödinger equation for
+|ψS(t)⟩:
+i∂t |ψS(t)⟩ = HS(t) |ψS(t)⟩ + HSB(t)UB(t, t0) |ψB(t0)⟩
+− iHSB(t)
+� t
+t0
+dt′ UB(t, t′)HBS(t′) |ψS(t′)⟩ ,
+(5)
+where UB(t, t′) is the bath (intrinsic) evolution operator
+satisfying i∂tUB(t, t0) = HB(t)UB(t, t0). This procedure
+is often carried within the Schwinger-Keldysh formalism
+by integrating out part of the action describing the de-
+grees of freedom of the bath (see e.g. Refs. [33, 35, 42]).
+But this is easier and more physically transparent in our
+first quantization notation and the final results would be
+identical.
+A.
+Featureless fermionic bath
+We will now specialize the above equation to a model of
+a “featureless fermionic bath”, which we define as having
+the following characteristics:
+(i) In a featureless fermionic bath every state of the
+system is coupled to a collection of identical sites with
+the same energy spectrum and the same coupling λ [see
+Fig. 2(a)]. If the system states (basis) are denoted by
+|χn⟩ and the bath states (basis) by |ϕn,j⟩, the bath and
+the system-bath coupling are
+HB =
+�
+n,j
+εj |ϕn,j⟩ ⟨ϕn,j| ,
+(6)
+HSB = λ
+�
+n,j
+|χn⟩ ⟨ϕn,j| ,
+(7)
+where εj is the energy for the bath state |ϕn,j⟩. This
+model of the bath is identical to that employed in
+Refs. [35, 43, 56–62].
+(ii) The featureless fermionic bath is prepared in an ini-
+tial condition at t0 with a Fermi-Dirac distribution that
+only has weight on the bath sites, described by
+ρS(t0) = 0,
+ρB(t0) = �
+n,jf0(εj) |ϕn,j⟩ ⟨ϕn,j| ,
+(8)
+f0(εj) =
+1
+exp[β0(εj − µ0)] + 1,
+(9)
+where µ0 is the chemical potential of and β0 = 1/kBT0
+denotes the temperature of the bath, respectively. The
+assumption of the initial density matrix only having
+weight on the bath is useful but it is not strictly nec-
+essary. This is because in the limit in which the bath
+spectrum becomes a dense continuum, the information
+of the initial condition for the component of density ma-
+trix on the system will decay over time and only the in-
+formation of the initial condition for the density matrix
+on the bath will dictate the late time steady state [this
+will become more clear in Eq. (16) which is a subsequent
+version of Eq. (5)]. Notice also that we have equated the
+evolution of the single particle density matrix with that
+of the many-body one-particle density matrix (equal time
+Greens function), which is possible thanks to the fact that
+the system is non-interacting.
+With assumptions (i) and (ii), by evolving the initial
+condition in Eq. (8) under Eqs. (3) and (4), one finds that
+the one-body density matrix projected onto the system
+at time t is given by
+ρS(t) =
+�
+n,j
+f0(εj) |ψ(j)
+n (t)⟩ ⟨ψ(j)
+n (t)| ,
+(10)
+where |ψ(j)
+n (t)⟩ is the component within system Hilbert
+space that evolves out of the initial state |ψB(t0)⟩ =
+|ϕn,j⟩ in the bath at t0. Eq. (10) states that the density
+matrix for the system is the weighted sum of contribu-
+tions from all bath states with their corresponding initial
+occupations.
+Using Eqs. (5), (6), and (7), we obtain the open-system
+Schrödinger equation for |ψ(j)
+n (t)⟩:
+i∂t |ψ(j)
+n (t)⟩ = HS(t) |ψ(j)
+n (t)⟩ + λ exp[−iεj(t − t0)] |χn⟩
+− i
+� ∞
+t0
+dt′ γ(t − t′) |ψ(j)
+n (t′)⟩ .
+(11)
+system
+DoS
+energy
+bath
+(a)
+(b)
+(c)
+FIG. 2. (a) Schematic of the system-bath coupling HSB. (b)
+The bath’s density of states is much wider than that of the
+system, and we simplify it to be flat in the energy range of
+interest. (c) Schematic of the bath acting as a source as well
+as a sink for the system [see Eq. (27)].
+
+4
+Here, λ exp[−iεj(t−t0)] |χn⟩ is a source term for |ψ(j)
+n (t)⟩
+arising from the bath, while the memory function in the
+second line is given by,
+γ(t) = λ2Θ(t)
+�
+j
+exp(−iεjt),
+(12)
+which encodes the memory of decay for |ψ(j)
+n (t)⟩ due to
+the bath. This memory function makes the Schrödinger
+equation for open systems non-local in time, and in gen-
+eral it incorporates decay and renormalizations of the
+system energies due to their coupling to the bath [see
+Fig. 2 for a depiction].
+(iii) To remove the finite memory delay, we impose one
+further property defining the featureless fermionic bath,
+namely that it has an infinitely broad and flat spectrum
+[see Fig. 2(b)], i.e., the bath density of state is constant:
+νB(ωb) = 2π
+�
+j
+δ(ωb − εj) ≡ ν0.
+(13)
+With this simplification, the finite delay or non-local
+memory of the past time t′ in Eq. (11) is lost, the memory
+function becomes:
+γ(t) = λ2Θ(t)
+�
+j
+� +∞
+−∞
+dωb δ(ωb − εj) exp(−iωbt)
+= λ2ν0Θ(t)
+� +∞
+−∞
+dωb
+2π exp(−iωbt) = δ(t) Γ,
+(14)
+where we used Eq. (13) to obtain the second equation
+and defined
+Γ ≡ λ2ν0
+2
+.
+(15)
+With the above simplification of infinitely broad spec-
+trum for the bath, the open-system Schrödinger’s equa-
+tion reduces to:
+i∂t |ψ(j)
+n (t)⟩ =
+�
+HS(t) − iΓ
+�
+|ψ(j)
+n (t)⟩
++ λ exp[−iεj(t − t0)] |χn⟩ .
+(16)
+The above equation is remarkably simple. It is a sim-
+ple non-Hermitian version of the Schrödinger equation in
+which the system Hamiltonian is dressed by a constant
+imaginary part “−iΓ” which captures the decay into the
+bath. Many recent studies of open quantum systems have
+used non-Hermitian Schrödinger equations that only in-
+clude the first line of Eq. (16). However, we see that the
+influence of the bath is not merely to induce decay, but
+it also produces the second term that acts a source and
+makes the equation inhomogeneous. The balance of these
+two terms is what allows the existence of non-trivial late
+time steady states (see Fig. 2 for depiction).
+B.
+Ideal fermionic bath
+To illustrate that our bath leads to the expected equi-
+librium when the system is not driven in time, we first
+consider the the special case in which HS(t) is time in-
+dependent,
+HS(t) → H0 =
+�
+n
+ϵn |χn⟩ ⟨χn| ,
+(17)
+Eq. (16) can be equivalently expressed as
+i∂ts(j)
+n
+= [ϵn − iΓ]s(j)
+n + λ exp[−iεj(t − t0)],
+(18)
+where
+s(j)
+n
+= ⟨χn|ψ(j)
+n (t)⟩ ,
+(19)
+is the amplitude for the system state |χn⟩. Solving the
+above Eq. (18) gives
+s(j)
+n
+= −iλ exp
+�
+− i
+� t
+t0
+dt′ (ϵn − iΓ)
+�
+×
+� t
+t0
+dt′ exp
+�
+i
+� t′
+t0
+dt′′(ϵn − iΓ − εj)
+�
+=
+λ
+ϵn − iΓ − εj
+�
+e−(Γ+iϵn)(t−t0) − e−iεj(t−t0)�
+. (20)
+Then using Eq. (10), we obtain the steady state, diagonal
+density matrix for the system:
+ρS(t → +∞) =
+�
+n
+fΓ(ϵn) |χn⟩ ⟨χn| ,
+(21)
+in which fΓ(ϵn) = limt→+∞
+�
+j f0(εj)|s(j)
+n |2 and reads
+explicitly as
+fΓ(ϵn) =
+�
+j
+f0(εj)
+λ2
+(ϵn − iΓ − εj)(ϵn + iΓ − εj)
+=
+� +∞
+−∞
+dωb f0(ωb)
+λ2 �
+j δ(ωb − εj)
+(ϵn − iΓ − ωb)(ϵn + iΓ − ωb)
+=
+� +∞
+−∞
+dωb
+π f0(ωb)
+Γ
+(ϵn − ωb)2 + Γ2 ,
+(22)
+where we used Eqs. (13) and (15) in obtaining the last
+equation. The above distribution fΓ(ϵn) shows that when
+HS(t) is time independent, the system “thermalizes” by
+approaching a time independent steady state dictated by
+the initial condition of the bath, f0(ωb), while a finite Γ
+accounts for the broadening of the energy levels of the
+system due to its coupling to the bath.
+Importantly, taking the limit in which the coupling to
+the bath vanishes from Eq. (22), we obtain
+lim
+Γ→0 fΓ(ϵn) = f0(ϵn),
+(23)
+i.e., fΓ(ϵn) reduces to the ideal Fermi-Dirac distribution
+in the limit of Γ → 0. We will then call this Γ → 0 limit
+of the “featureless fermionic bath” an “ideal fermionic
+bath”. The fact that the ideal Fermi-Dirac distribution
+
+5
+appears only when the coupling to the bath is vanish-
+ingly weak is consistent with general considerations of
+statistical physics.
+However, Eq. (22) still allow us to obtain analytically
+the modified occupation at finite coupling to the bath,
+which will be used in subsequent manipulations. By in-
+tegrating over ωb in Eq. (22) using Cauchy’s residue the-
+orem, we find that:
+fΓ(ϵ) = 1
+2
+�
+f+(ϵ) + f−(ϵ)
+�
+,
+(24)
+where f+(ϵ) = [f−(ϵ)]∗ and they are given by:
+f±(ϵ) = 1
+2 ± i
+π Ψ(0)
+�1
+2 ± iβ ϵ ∓ iΓ − µ
+2π
+�
+,
+(25)
+with Ψ(0) the 0-th order Polygamma function (or the
+digamma function). f±(ϵ) will also appear repeatedly in
+more general cases.
+C.
+Diagonal and time-periodic Hamiltonians
+1.
+Diagonal system Hamiltonian
+In this work, we will develop the above general for-
+malism to the special case where the system Hamilto-
+nian HS(t) is time dependent but diagonal in the system
+states. Let us then take the following form for the system
+Hamiltonian:
+⟨χn|HS(t)|χm⟩ = δnm[ϵn + Vn(t)] = δnmϵn(t).
+(26)
+With this, Eq. (16) then reduces to
+i∂ts(j)
+n
+= [ϵn(t) − iΓ]s(j)
+n + λ exp[−iεj(t − t0)].
+(27)
+Solving the above Eq. (27) gives
+s(j)
+n (t) = −iλ exp
+�
+− i
+� t
+t0
+dt′ [ϵn(t′) − iΓ]
+�
+×
+� t
+t0
+dt′ exp
+�
+i
+� t′
+t0
+dt′′[ϵn(t′′) − iΓ − εj]
+�
+,
+(28)
+and then with Eq. (10), we obtain the diagonal density
+matrix for the system:
+ρS(t) =
+�
+n
+pn(t) |χn⟩ ⟨χn| ,
+(29)
+pn(t) =
+�
+j
+f0(εj)|s(j)
+n (t)|2.
+(30)
+2.
+Periodic system Hamiltonian
+Now we consider a periodically driven system. Namely,
+we take the diagonal elements of the Hamiltonian to be
+periodic in time:
+ϵn(t + T) = ϵn(t) =
++∞
+�
+l=−∞
+ϵ(l)
+n exp[−ilω(t − t0)],
+(31)
+where T is the period and ω = 2π/T is the frequency,
+and
+ϵ(l)
+n =
+� T
+0
+dt
+T ϵn(t) exp[ilω(t − t0)]
+(32)
+is the l-th Fourier coefficient for ϵn(t). In particular,
+¯ϵn ≡ ϵ(0)
+n
+=
+� T
+0
+dt
+T ϵn(t),
+(33)
+is the time-average of the diagonal element of the Hamil-
+tonian, which as we will show next, coincides with the
+Floquet energy of state n. To see this, notice that the
+wavefunction that would solve the system Schrödinger’s
+equation in the absence of the bath, can be expressed as
+follows:
+exp
+�
+− i
+� t
+t0
+dt′ ϵn(t′)
+�
+= exp
+�
+− i
+� t
+t0
+dt′[ϵn(t′) − ¯ϵn]
+�
+× exp
+�
+− i
+� t
+t0
+¯ϵn
+�
+≡ φn(t) × exp
+�
+− i¯ϵn(t − t0)
+�
+.
+(34)
+The periodicity of the first factor denoted by φn(t) can
+be shown explicitly:
+φn(t + T) = φn(t) × exp
+�
+− i
+� t+T
+t
+dt′[ϵn(t′) − ¯ϵn]
+�
+= φn(t),
+(35)
+where we used Eq. (33) in obtaining the second equation.
+Therefore we see from second factor in the last line of
+Eq. (35), that the time-average of the diagonal element
+of the Hamiltonian is the Floquet energy itself.
+Let us now consider the Fourier expansion of the peri-
+odic part of the Floquet wavefunction:
+φn(t) = exp
+�
+− i
+� t
+t0
+dt′[ϵn(t′) − ¯ϵn]
+�
+=
++∞
+�
+l=−∞
+φ(l)
+n exp[−ilω(t − t0)],
+(36)
+or equivalently,
+φ(l)
+n = 1
+T
+� t0+T
+t0
+dt
+�
+exp[ilω(t − t0)]
+× exp
+�
+− i
+� t
+t0
+dt′[ϵn(t′) − ¯ϵn]
+��
+. (37)
+The above expression makes clear that the amplitude of
+the harmonics of the wavefunction, φ(l)
+n , are functions
+
+6
+of the full time dependence of the instantaneous energy
+ϵn(t), and are independent of the Floquet energy ¯ϵn. This
+property will be crucial later on for purposes of under-
+standing why there is in-gap rectification. In other words,
+Eq. (37) defines φ(l)
+n as a function of all the harmonics of
+the time dependent energy from Eq. (32) as follows:
+φ(l)
+n = φ(l)
+n (ϵ(±1)
+n
+, ϵ(±2)
+n
+, · · · ).
+(38)
+Also from Eq. (36) it can be shown that these amplitudes
+satisfy the following normalization condition:
++∞
+�
+l=−∞
+��φ(l)
+n
+��2 = 1.
+(39)
+With Eqs. (28), (30), (36), and (13), and by taking
+the late-time limit that allows to neglect transient terms
+of the form exp[−Γ(t − t0)] → 0, we obtain the system
+steady state occupation:
+pn(t) =
+� +∞
+−∞
+dωb
+π f0(ωb)
+× Γ
+�����
++∞
+�
+l=−∞
+φ(l)
+n
+exp[−ilω(t − t0)]
+¯ϵn − ωb − lω − iΓ
+�����
+2
+.
+(40)
+Similar to Eq. (22), by integrating over ωb in Eq. (40),
+we find that:
+pn(t) =
++∞
+�
+l,m=−∞
+�
+φ(m)
+n
+�∗φ(l)
+n exp
+�
+i(m − l)ω(t − t0)
+�
+×
+Γ
+2Γ + i(m − l)ω
+�
+f+(¯ϵn − lω) + f−(¯ϵn − mω)
+�
+,
+(41)
+where f±(ϵ) is given in Eq. (25). The Eq. (41) is one of
+the central formulas of our work because it allows to com-
+pute expectation values of any equal-time system observ-
+ables, even at a finite coupling Γ to featureless fermionic
+bath.
+The expression in Eq. (41) captures the steady state
+occupation of the n-th state in the case of featureless
+fermionic bath, and thus it replaces what would be the
+Fermi-Dirac distribution in equilibrium. One important
+feature of this steady state is that it displays “synchro-
+nization”, namely, it is strictly periodic in the drive:
+pn(t + T) = pn(t).
+(42)
+Remarkably, in the limit of an “ideal bath” (Γ → 0) the
+above distribution becomes time independent and it is
+given by:
+lim
+Γ→0 pn =
++∞
+�
+l=−∞
+|φ(l)
+n |2f0(¯ϵn − lω).
+(43)
+Here ¯ϵn is the Floquet energy of n-th state, and φ(l)
+n are
+the Harmonics of the periodic part of the wave-functions
+defined in Eq. (37). The reader is encouraged to contrast
+this occupation function with that in Eq. (23) obtained
+when the Hamiltonian was time independent. Notice also
+that because the occupation function becomes time inde-
+pendent in this limit, there are no time fluctuations of the
+average fermion occupation of each state n.
+Thus the distribution is an infinite sum of sev-
+eral Fermi-Dirac distributions with chemical potentials
+shifted by the various harmonics of the driving frequency
+lω and weighed by amplitudes of the harmonics of the
+Floquet wavefunctions |φ(l)
+n |2. It is therefore clear that
+the occupation of the state is completely different from
+how the state is filled in equilibrium [see Fig. 1(a) for an
+illustration of the non-equilibrium occupation function].
+One recovers an occupation similar to equilibrium when
+one neglects all the higher harmonics of φ(l)
+n
+with l ̸= 0
+and forces by hand the amplitude of the l = 0 term to
+be φ(0)
+n
+→ 1, but this is not justified in general (not even
+perturbatively as we will illustrate in Sec. II B). We note
+that the idea that Floquet states are not filled in the
+same way as equilibrium states has been emphasized in
+several studies, by using a variety of models for the re-
+laxation when the system is coupled to a heat bath [36–
+39, 41, 63]. In fact the expression for the non-equilibrium
+time independent steady states we find in Eq. (43) has
+been reported before, and is in particular the same kind
+of expression shown in Eq. (12) of Ref. [38].
+3.
+Harmonic time dependent driving
+Computing analytically the integral in Eq. (37) that
+relates the harmonics of the Floquet wavefunction to the
+harmonics of the energy is in general involved. There is
+a simple case where these integrals can be computed in
+a simple closed analytical form, which is when the time
+dependent part Vn(t) of the Hamiltonian has a single
+harmonic:
+Vn(t) = Vn cos[ω(t − t0)].
+(44)
+In this case the coefficients φ(l)
+n from Eq. (37) correspond
+to the l-th Bessel function:
+φ(l)
+n = Jl
+�
+Vn/ω
+�
+.
+(45)
+Substitution of Eq. (45) into Eq. (41) leads to the fol-
+lowing non-perturbative expression for the occupation of
+the states in the limit of Γ → 0:
+lim
+Γ→0 pn =
++∞
+�
+l=−∞
+J2
+l
+�
+Vn/ω
+�
+f0(¯ϵn − lω).
+(46)
+We therefore see that the occupation in the case of the
+ideal fermionic bath becomes a sum of several Fermi-
+Dirac distributions boosted by the different harmonics of
+the Floquet quasi-energies ¯ϵn − lω (l ∈ Z). It is interest-
+ing to note that this ladder-like behavior is analogous to
+
+7
+the Tien-Gordon effect that arises in nanostructures that
+are simultaneously subjected to AC and DC drives [64].
+Similarly as in that case, the ladder behaviour becomes
+more pronounced as the driving becomes stronger [see
+Fig. 1(a)].
+II.
+SINGLE BAND MODEL UNDER
+MONOCHROMATIC LIGHT
+In this section we will use the formalism developed in
+the previous ones to determine the self-consistent occupa-
+tion of an electronic band driven by an oscillating electric
+field and demonstrate the existence of in-gap rectifica-
+tion. Because we are primarily interested here in proving
+and clarifying the origin of in-gap rectification, we will
+focus on a simple model of a Bloch band that has vanish-
+ing Berry connections. These bands can display however
+the in-gap Jerk current effect that arises from the energy
+band dispersions [43]. However, other mechanisms driven
+by the Berry phases, such as the non-linear Hall effect,
+can also lead to in-gap rectification as we have recently
+demonstrated [43].
+Let us now consider our system Hamiltonian to be a
+tight-binding model with a single site per unit cell and
+a trivial single Bloch band (with no Berry connections)
+coupled to a uniform monochromatic electric field. The
+time dependent system Hamiltonian is:
+HS(t) =
+�
+k
+ϵk(t) |χk⟩ ⟨χk| ,
+(47)
+ϵk(t) ≡ ϵ(k − A(t)),
+�
+k
+≡
+�
+BZ
+dk
+(2π)d .
+(48)
+The system states are now labelled by the wave vector k
+and ϵ(k) is the unperturbed band dispersion. We assume
+a monochromatic electric field which leads to the periodic
+vector potential using E(t) = −∂tA(t):
+A(t) = − i
+ω Eω exp(−iωt) + c. c.
+(49)
+A.
+Electric current in the steady state
+Since the system Hamiltonian is diagonal in crystal
+momenta k, we can apply the formalism of Sec. I C to
+compute the steady state occupation of each momenta
+k, by replacing the label in previous sections n → k. If
+we denote the occupation of each state by pk(t), then the
+system’s electric current reads as follows:
+j(t) =
+�
+k
+pk(t)∇kϵk(t)
+=
++∞
+�
+s=−∞
+j(s) exp[−isω(t − t0)],
+(50)
+where we set e = ℏ = 1 throughout the paper. By com-
+bining Eqs. (32) and (41), the weight of each oscillating
+mode of the electric current can be written as:
+j(s) =
+�
+k
++∞
+�
+m,l=−∞
+Γ
+2Γ + i(l − s)ω
+�
+φ(m)
+k
+�∗φ(s+m−l)
+k
+×
+�
+f+(¯ϵk − (s + m − l)ω) + f−(¯ϵk − mω)
+�
+∇kϵ(l)
+k . (51)
+Interestingly, as discussed in Sec. I C, in the limit of an
+ideal heat bath Γ → 0, the distribution function pk(t) be-
+comes time independent, and therefore the time averaged
+electric current (also referred to as rectified current), is
+given by:
+¯j =
+� T
+0
+dt
+T j(t) =
+�
+k
+pk∇k¯ϵk,
+(52)
+where ¯ϵk is the Floquet energy of the band and in our
+current simple single-band model, and is given by the
+time averaged band energy (l = 0 component):
+¯ϵk ≡ ϵ(0)
+k
+=
+� T
+0
+dt
+T ϵ(k − A(t)).
+(53)
+Therefore, we see that Eq. (52) has a resemblance to
+how one would compute the current in a time indepen-
+dent equilibrium system, but with the equilibrium Fermi-
+Dirac distribution replaced by occupation function pk,
+and the bare band dispersion replaced by the dressed
+Floquet band energy ¯ϵk.
+At first glance, this point of
+view might suggest that the time averaged rectified cur-
+rent vanishes in the ideal limit of ω ≫ Γ → 0, just in
+the same way it is expected to vanish in a time inde-
+pendent equilibrium system. In fact, several classic and
+more recent works have incorrectly taken this point of
+view that the non-equilibrium steady state occupation pk
+is a Fermi-Dirac distribution of the dressed Floquet band
+energy [45, 46, 48–50] (see Appendix E for detailed com-
+ments on previous works). However, as we have shown
+in Sec. I C, the correct occupation of the states in the
+non-equilibrium steady state is not a simple Fermi-Dirac
+distribution, but it is given by the following expression
+[see Eqs. (38) and (43)]:
+pk(¯ϵk, ϵ(±1)
+k
+, · · · ) ≡
++∞
+�
+l=−∞
+|φ(l)
+k |2f0(¯ϵk − lω).
+(54)
+In the argument of pk in the above expression, we have
+emphasized that pk is not only a function of the Flo-
+quet band energy ¯ϵk, but also of all the higher har-
+monics ϵ(±1)
+k
+, ϵ(±2)
+k
+· · · of the time dependent energy
+ϵk(t) through its dependence on the amplitudes φ(l)
+k [see
+Eqs. (37) and (38)]. Precisely because of this, the recti-
+fication current ¯j can not be expressed as an integral of
+a total derivative over the Brillouin zone and generally
+does not vanish, i.e.,
+¯j ̸=
+�
+k
+∇k ˜P(¯ϵk) = 0,
+(55)
+
+8
+where ˜P(¯ϵk) would be defined through
+∂ ˜P(¯ϵk)
+∂¯ϵk
+≡ ˜pk(¯ϵk),
+(56)
+which would be possible if the occupation depended only
+on the dressed Floquet energy pk → ˜pk(¯ϵk) [but this is
+not the case for Eq. (55)].
+Therefore, we see that in general a non-zero rectified
+current is expected in the non-equilibrium steady state,
+even in the limit of the ω ≫ Γ → 0. As we will show
+in detail in the following section, this finite rectified cur-
+rent remains non-zero within the optical gap of a metal,
+even within the usual second order of perturbation the-
+ory in the amplitude of the electric field for which recti-
+fication currents are typically computed. These findings
+further substantiate our recent work showing the exis-
+tence of in-gap rectification [43] but appear in tension
+with some other statements in the literature [44–46, 48–
+50]. In Appendix E, we comment in more detail on some
+of these other works clarifying some partial agreements
+but also pointing out some of their imprecisions and in-
+correct statements.
+B.
+Perturbative results
+In this subsection we will compute perturbatively the
+electric current in powers of electric field to the currents
+at modes [see Eqs. (50) and (51)]: s = 0 representing
+rectification conductivity, s = 1 representing linear con-
+ductivity. s = 2 representing second harmonic generation
+is discussed in Appendix A. We will show explicitly that
+even to 2nd order in electric fields, the non-equilibrium
+distribution in the steady state for an ideal bath, pk, dif-
+fers clearly from the naive Fermi-Dirac distribution eval-
+uated in the dressed Floquet bands. This will allow us
+to compute analytically the rectification conductivities
+and prove rigorously that they remain finite within the
+optical gap of the metal.
+Although our conclusions and formulae are valid and
+can be used for any single band model (with no Berry
+connections) in arbitrary dimensions, for simplicity we
+will illustrate our results for a simple 1D model with the
+following band dispersion:
+ϵ(kx) = −t1 cos(a0kx) − t2 sin(2a0kx) + ϵ0,
+(57)
+where ϵ0 is a constant that we have added for convenience
+in order to shift the band energy so that it lies within 0
+and ∆ [See Fig. 3(b)], and a0 is the lattice constant.
+Notice that the above band-structure breaks not only
+inversion, which is always needed to have rectification,
+but also time-reversal symmetry, and therefore it has no
+symmetry relating k → −k. As we will see, this is indeed
+crucial in order to obtain a non-zero in-gap rectification
+conductivities for the models without Berry curvature
+that we are considering in this study. More generally, as
+discussed in Ref. [43], in the case of bands with non-trivial
+Berry connections one can alternatively obtain a non-
+zero in-gap rectification, e.g., via the Berry-Dipole effect
+by breaking time reversal symmetry only by having a
+circularly polarized light instead of having a time-reversal
+breaking band-structure.
+1.
+Occupation function to the second order of electric field
+We begin by deriving the explicit perturbative expres-
+sions for ϵ(l)
+k
+and φ(l)
+k
+discussed in the previous sections
+and can be computed from Eqs. (32) and (37) by replac-
+ing n → k. Up to the second order in the electric field,
+it is sufficient to expand the band dispersion up to the
+same second order, namely:
+ϵ(k − A(t)) = ¯ϵk + ϵ(1)
+k e−iω(t−t0) + ϵ(−1)
+k
+eiω(t−t0)
++ ϵ(2)
+k e−2iω(t−t0) + ϵ(−2)
+k
+e2iω(t−t0) + · · · ,
+(58)
+Using Eq. (49), this perturbative expansion leads to the
+following expressions for ϵ(l)
+k :
+¯ϵk ≡ ϵ(0)
+k
+= ϵ(k) + 1
+ω2
+�
+αβ
+∂α∂βϵ(k) Eα
+ωEβ
+−ω + O(|Eω|4),
+ϵ(1)
+k
+= i
+ω
+�
+α
+∂αϵ(k) Eα
+ω + O(|Eω|3),
+ϵ(2)
+k
+= − 1
+2ω2
+�
+αβ
+∂α∂βϵ(k) Eα
+ωEβ
+ω + O(|Eω|4),
+ϵ(−l)
+k
+=
+�
+ϵ(l)
+k
+�∗.
+(59)
+We can use Eq. (37) to perturbatively evaluate φ(l)
+k lead-
+ing to:
+φ(0)
+k
+= 1 − ϵ(1)
+k
+− ϵ(−1)
+k
+ω
++
+�
+ϵ(1)
+k
+�2 +
+�
+ϵ(−1)
+k
+�2 − 4ϵ(1)
+k ϵ(−1)
+k
+− ϵ(2)
+k
++ ϵ(−2)
+k
+2ω2
+,
+φ(1)
+k
+= −ϵ(1)
+k
+ω − ϵ(1)
+k
+�
+ϵ(1)
+k
+− ϵ(−1)
+k
+�
+ω2
+,
+φ(−1)
+k
+= ϵ(−1)
+k
+ω
+− ϵ(−1)
+k
+�
+ϵ(−1)
+k
+− ϵ(1)
+k
+�
+ω2
+,
+φ(2)
+k
+=
+�
+ϵ(1)
+k
+�2 − ϵ(2)
+k
+2ω2
+,
+φ(−2)
+k
+=
+�
+ϵ(−1)
+k
+�2 + ϵ(−2)
+k
+2ω2
+.
+(60)
+The other φ(l)
+k
+with |l| > 2 will scale with higher powers
+of electric fields, and therefore can be neglected to second
+order. The norm squared of those terms above are:
+��φ(0)
+k
+��2 = 1 − 2
+��ϵ(1)
+k
+��2
+ω2
++ O(|Eω|3),
+��φ(1)
+k
+��2 =
+��ϵ(1)
+k
+��2
+ω2
++ O(|Eω|3),
+��φ(2)
+k
+��2 = O(|Eω|4).
+(61)
+
+9
+Therefore the ideal occupation function pk in the limit
+Γ → 0 to second order in electric fields reads as
+pk =
+�
+1 − 2
+��ϵ(1)
+k
+��2
+ω2
+�
+f0(¯ϵk)
++
+��ϵ(1)
+k
+��2
+ω2
+f0(¯ϵk − ω) +
+��ϵ(−1)
+k
+��2
+ω2
+f0(¯ϵk + ω).
+(62)
+The above expansion contains all the correct terms to sec-
+ond order in electric fields, even though it is not strictly
+perturbative, because the Floquet band energy ¯ϵk also in-
+cludes implicitly a correction of order |Eω|2 [see Eq. (59)].
+In other words, if one wants to obtain a strictly pertur-
+bative expansion to order |Eω|2 one simply needs to Tay-
+lor expand the Fermi-Dirac distribution f0(¯ϵk) above as
+well. However we find it convenient to keep the above
+form, with the understanding that we can only trust its
+predictions to order |Eω|2.
+Let us now comment on the significance of Eq. (62).
+We see above that even to second order, the non-
+equilibrium distribution, pk, contains not only the Fermi-
+Dirac distribution evaluated for the Floquet bands,
+f0(¯ϵk), but also several other terms that make it clearly
+deviate from f0(¯ϵk).
+As we will see these additional
+terms, are precisely the ones that lead to a finite in-
+gap rectification in the clean limit Γ → 0.
+In Ap-
+pendix D, we also demonstrate that the above occupa-
+tion function agrees with the one obtained from a sim-
+pler Boltzmann/relaxation-time description in the limit
+ω ≪ ¯ϵk. Notice also that the above occupation differs
+even to up second order |Eω|2 from the naive Fermi-Dirac
+occupation of the Floquet band, f0(¯ϵk), that was pres-
+sumed in Refs. [45, 46, 48–50] (see Appendix Appendix E
+for further comments on previous studies).
+2.
+Linear conductivity
+The linear conductivity is defined from:
+j(1)
+α
+= σαβ
+Γ (ω)Eβ
+ω + O(|Eω|3),
+(63)
+where the sub-index Γ emphasizes a finite coupling of the
+system to the bath. Using Eqs. (51), (37), (32), the exact
+conductivity of our model at finite coupling to the bath
+is found to be:
+σαβ
+Γ (ω) = i
+ω
+�
+k
+fΓ(¯ϵk)∂α∂β¯ϵk
++
+�
+k
+∂α¯ϵk∂β¯ϵk
+ω2
+iΓ
+2Γ − iω L1(¯ϵk, ω),
+(64)
+where ∂γ ≡ ∂/∂kγ, and
+L1(¯ϵk, ω) = f+(¯ϵk) + f−(¯ϵk + ω)
+− f+(¯ϵk − ω) − f−(¯ϵk),
+(65)
+where f± are defined in Eq. (25). Just as for Eq. (62),
+we have kept the dressed Floquet band energy, ¯ϵk, in the
+integrands of Eq. (64), and therefore this is not a strictly
+perturbative expression. But if desired, the strictly per-
+turbative expression can simply be obtained from the
+one above by replacing the dressed Floquet band en-
+ergy dispersion by the bare unperturbed band dispersion:
+¯ϵk → ϵ(k). This also applies to the subsequent formulas
+of this section.
+In the clean limit (ω ̸= 0 and Γ → 0), the above ex-
+pression reduces to the standard Drude form:
+lim
+Γ→0 σαβ
+Γ (ω) = i
+ω
+�
+k
+f0(¯ϵk)∂α∂β¯ϵk.
+(66)
+Therefore, we see that the real part of the linear con-
+ductivity at finite frequency vanishes when Γ → 0. In
+Fig. 3(c) we illustrate this in detail for the simple model
+1D from Eq. (57). The above Drude form follows from
+the fact that to the linear order of the electric field, the
+ideal occupation function pk in the limit Γ → 0 is the
+same with the equilibrium Fermi-Dirac distribution [see
+Eq. (62)].
+In the DC limit ω → 0 the linear conductivity ap-
+proaches a finite Drude-like value (see Appendix A for
+details):
+lim
+ω→0 σαβ
+Γ (ω) = 1
+2
+�
+k
+∂α∂β¯ϵk
+�fΓ(¯ϵk)
+Γ
+− ∂gΓ(¯ϵk)
+∂¯ϵk
+�
+≈ 1
+2
+�
+k
+∂α∂β¯ϵk
+�f0(¯ϵk)
+Γ
++ O(Γ)
+�
+,
+(67)
+in which
+gΓ(ϵ) = 1
+2i
+�
+f+(ϵ) − f−(ϵ)
+�
+(68)
+is the imaginary part of f+(ϵ) defined in Eq. (25). There-
+fore the clean limit of the DC conductivity resembles the
+prediction of the classic Drude theory for τ ≡ 1/(2Γ), and
+has a Drude peak in the DC limit when the chemical po-
+tential of the bath is within the bandwidth of the system
+µ ∈ [0, ∆] [see Fig. 3(c)]. The fact that the conductivity
+is finite when ω → 0 and has the expected Drude behav-
+ior, evidences that our simple bath produces the correct
+behavior for the relaxation of currents.
+In the limit in which the frequency is small compared
+to the bandwidth but much larger than Γ, we obtain
+the usual decay power 1/ω2 associated with the Drude
+behavior [see Fig. 3(e), left-hand side region]:
+lim
+Γ≪ω≪∆ Re
+�
+σαβ
+Γ (ω)
+�
+= −2Γ
+ω2
+�
+k
+(∂α¯ϵk)(∂β¯ϵk)∂f(¯ϵk)
+∂¯ϵk
+.
+(69)
+On the other hand, in the ultra-large frequency regime
+when the frequency greatly exceeds even the bandwidth,
+the real part of the linear conductivity has a different
+scaling from that of Drude theory:
+lim
+ω≫∆ Re
+�
+σαβ
+Γ (ω)
+�
+= Γ
+ω3
+�
+k
+(∂α¯ϵk)(∂β¯ϵk),
+(70)
+decaying as 1/ω3 [see Fig. 3(e), right-hand side region].
+
+10
+0.2
+0.4
+0
+(c)
+0
+1
+2
+(d)
+0
+10
+20
+0.2
+0.4
+0
+(a)
+0
+(b)
+0
+1
+2
+3
+0
+1
+2
+-1
+-2
+(e)
+-10
+0
+(f)
+-5
+0
+5
+0
+-1
+-2
+1
+FIG. 3.
+(a) The 1D tight binding model whose inversion
+and time-reversal symmetries are broken by the next-nearest-
+neighbour hopping ±it2/2, and its (b) dispersion relation with
+0 the band bottom and ∆ the band top. (c) Real part of the
+dimensionless linear conductivity Re σxx
+Γ (ω)/σ(1)
+0
+illustrating
+how it vanishes at finite frequency as Γ → 0 (which defines the
+optical transparency region), and (d) dimensionless rectifica-
+tion conductivity σxxx
+Γ
+(ω, −ω)/σ(2)
+0
+for different Γ illustrating
+the existence of in-gap rectification in the metal, namely that
+it approaches a finite non-zero value in the limit of Γ → 0 at
+finite ω. The characteristic linear and second order conductiv-
+ities in 1D used here are σ(1)
+0
+= a0·e2/ℏ and σ(2)
+0
+= a2
+0τ0·e3/ℏ2
+with τ0 = ℏ/t1. (e) and (f) Log-log plots of Re σxx
+Γ (ω)/σ(1)
+0
+and σxxx
+Γ
+(ω, −ω)/σ(2)
+0
+for different Γ illustrating their power
+dependencies over ω in different frequency ranges. Parame-
+ters used: a0 = 1, t1/t2 = 2, µ = 5t1/7, β0 = 109/t1.
+3.
+Rectification conductivity
+The rectification conductivity is a three-index tensor
+that relates the time averaged current [namely the aver-
+age DC current corresponding to s = 0 in Eq. (50)] to the
+bilinears of electric field amplitudes. Without loss of gen-
+erality, we define it by choosing the following symmetry
+convention for indices of the electric field bilinears:
+j(0)
+γ
+= σγαβ
+Γ
+(ω, −ω)Eα
+ω(Eβ
+ω)∗
++ σγαβ
+Γ
+(−ω, ω)(Eα
+ω)∗Eβ
+ω + O(|Eω|4).
+(71)
+The exact rectification conductivity of our model at finite
+coupling to the bath, Γ, is given by:
+σγαβ
+Γ
+(ω, −ω)
+=
+�
+k
+∂γ¯ϵk∂α¯ϵk∂β¯ϵk
+2ω4
+[fΓ(¯ϵk + ω) + fΓ(¯ϵk − ω) − 2fΓ(¯ϵk)]
++
+Γ
+2Γ − iω
+�
+k
+∂α¯ϵk∂γ∂β¯ϵk
+2ω3
+L1(¯ϵk, ω)
++
+Γ
+2Γ + iω
+�
+k
+∂β¯ϵk∂γ∂α¯ϵk
+2ω3
+L∗
+1(¯ϵk, ω).
+(72)
+The DC limit of the rectification conductivity can be
+shown to be (see Appendix B for details):
+lim
+ω→0 σγαβ
+Γ
+(ω, −ω)
+= 1
+4
+�
+k
+∂α∂β∂γ¯ϵk
+�fΓ(¯ϵk)
+Γ2
+− 1
+Γ
+∂gΓ(¯ϵk)
+∂¯ϵk
+− 1
+3
+∂2fΓ(¯ϵk)
+∂¯ϵ2
+k
+�
+≈ 1
+4
+�
+k
+∂α∂β∂γ¯ϵk
+�f0(¯ϵk)
+Γ2
++ O(Γ0)
+�
+.
+(73)
+The leading term of the above expression in the sec-
+ond line coincides with the Jerk conductivity predicted
+within the relaxation time approximation from a simple
+Boltzmann-relaxation-time formalism [43, 47, 51].
+For
+an illustration see Fig. 3(d). We have also verified that
+the above ω → 0 limit of the rectification conductivity
+is identical to the ω → 0 limit of the second-harmonic
+generation conductivity σγαβ
+Γ
+(ω, ω) (see Appendix C for
+details).
+Let us now focus on the main regime of our interest,
+which is the “clean-limit” in which the relaxation rate
+vanishes (Γ → 0) while the frequency remains finite. The
+exact expression for the rectification conductivity in this
+limit is given by:
+lim
+Γ→0σγαβ
+Γ
+(ω, −ω) =
+1
+2ω4
+�
+k
+(∂γ¯ϵk)(∂α¯ϵk)(∂β¯ϵk)
+×
+�
+f0(¯ϵk + ω) + f0(¯ϵk − ω) − 2f0(¯ϵk)
+�
+.
+(74)
+Notice that the above rectification conductivity would
+vanish under any symmetry that enforces ¯ϵk = ¯ϵ−k, such
+as time reversal or inversion symmetry.
+Therefore, the
+above expression proves one of our central claims, namely
+that the rectification conductivity remains finite at finite
+frequency within the optical transparency region of the
+metal. The “transparency” here refers to the fact that
+the real part of the linear conductivity vanishes in this
+same limit ω ≫ Γ → 0. We illustrate this behavior in
+Fig. 3(d) for our toy 1D model, confirming that the in
+gap rectification is possible. The origin of this finite rec-
+tification conductivity can be traced back to the fact that
+
+11
+(b)
+0
+1
+10
+15
+5
+0
+0
+(a)
+0
+-2
+2
+4
+(c)
+-8
+-4
+0
+0
+1
+-1
+-2
+2
+FIG. 4. (a) Schematic of the original band (denoted by solid
+line l = 0) and the boosted Floquet bands (denoted by dashed
+lines l = ±1).
+Here the chemical potential µ is below the
+original band. The threshold frequency ωt is the minimum
+frequency for boosted Floquet bands to cross the chemical
+potential. (b) and (c) dimensionless rectification conductivity
+σxxx
+Γ
+(ω, −ω)/σ(2)
+0
+and its Log-log plots for different Γ, show-
+ing that rectification conductivity is non-zero when ω > ωt.
+Parameters used are the same with those in Fig. 3.
+to the second order of the electric field, the ideal occupa-
+tion function pk in the limit Γ → 0 is different from the
+equilibrium Fermi-Dirac distribution [see Eq. (62)].
+While the expression of Eq. (74) is the exact clean limit
+of the rectification conductivity in our model, it can be
+shown that this expression reduces to the more famil-
+iar expression for the Jerk current prediction of the sim-
+ple Boltzmann-relaxation-time expression in the limit in
+which the frequency is small compared to the bandwidth,
+namely Γ ≪ ω ≪ ∆, and it is given by:
+lim
+ω→0 lim
+Γ→0 σγαβ
+Γ
+(ω, −ω) = 1
+ω2
+�
+k
+f0(¯ϵk)∂α∂β∂γ¯ϵk,
+(75)
+which coincides with Eq. (21) of Ref. [43] for the
+Jerk mechanism which has a 1/ω2 decaying power [see
+Fig. 3(f), left-hand side region].
+More details of this
+agreement with the simpler Boltzmann approach are dis-
+cussed in Appendix D.
+Interestingly, in the “ultra-high” frequency limit, when
+the frequency is much larger than the bandwidth ω ≫ ∆,
+the clean rectification conductivity transits to a different
+scaling and decays much faster [see Fig. 3(f), right-hand
+side region]:
+lim
+ω≫∆ lim
+Γ→0 σγαβ
+Γ
+(ω, −ω)
+=
+1
+2ω4
+�
+k
+�
+1 − 2f0(¯ϵk)
+�
+(∂α¯ϵk)(∂β¯ϵk)(∂γ¯ϵk).
+(76)
+In contrast to the Boltzmann-relaxation-time result
+where the large frequency regime is controlled by the
+third momentum derivative of the band dispersion, here,
+the large frequency response is controlled by the third
+power of band velocity, which is a different intrinsic prop-
+erty of the band.
+It is interesting to note that the expression in Eq. (76)
+remains finite even when the unperturbed band is either
+fully occupied [f0(¯ϵk) = 1] or fully empty [f0(¯ϵk) = 0],
+namely the system would be nominally an insulator with-
+out a Fermi surface. This behavior is possible because our
+bath does not conserve the total particle number of the
+system, and therefore, there appears a finite occupation
+of the bands when they are driven by the electric field,
+even if the bands were initially empty in the distant past
+before turning on the time dependent drive.
+In other
+words, all our calculations are performed strictly for a
+bath with fixed chemical potential but not fixed density.
+The appearance of a finite occupation of the bands to
+second order of perturbation theory occurs when the fre-
+quency exceeds the threshold so that one of the copies of
+the Floquet bands boosted by ±ω crosses the chemical
+potential, as depicted in Fig. 4.
+III.
+SUMMARY AND DISCUSSION
+We have shown rigorously that the occupation of states
+in a periodically driven fermionic system coupled to a fea-
+tureless fermionic heat bath approaches a time indepen-
+dent occupation function in the limit in which the cou-
+pling to this bath is vanishingly small. This occupation
+function can be computed analytically and differs from
+the naive Fermi-Dirac occupation of the dressed Floquet
+energies. This non-equilibrium steady state occupation
+instead resembles a staircase version of the Fermi-Dirac
+distribution [see Fig. 1(a) for an illustration], and also
+cannot be expressed as a function of the Floquet energy
+alone, but in general contains information on all the har-
+monics encoding the full time dependence of the Hamil-
+tonian.
+We applied these results to the case in which the
+fermionic system has a Hamiltonian corresponding to a
+single Bloch band without Berry connections (e.g. aris-
+ing from a tight-binding model with a single site per unit
+cell) driven by a monochromatic electric field. We showed
+that this staircase Fermi-Dirac distribution leads to a fi-
+nite rectification conductivity within the optical trans-
+parency region of a metal, which at small frequencies
+compared to the bandwidth agrees exactly with the pre-
+diction of the Jerk current effect expected from a simpler
+Boltzmann-relaxation-time description [43, 47]. Because
+
+12
+the oscillating electric field is monochromatic, this rec-
+tification conductivity does not arise because of the fre-
+quency difference effect of Ref. [44] or the Raman-like
+scattering effect of Refs. [45, 46].
+Our results validate our recent findings [43] that in-
+gap rectification within the optical transparency region
+of metals are indeed possible, even in the limit in which
+carrier relaxation rates vanish, and clarify a discussion
+surrounding this matter [44–46, 48–50]. More details of
+the partial agreement with some of these references but
+also the corrections of imprecisions and incorrect state-
+ments in some of them can be found in Appendix E.
+ACKNOWLEDGMENTS
+We would like to thank Yugo Onishi, Naoto Nagaosa,
+Liang Fu, Victor Yakovenko, Sergey Ganichev, Bing-
+hai Yan, and Fernando de Juan for stimulating discus-
+sions and correspondence. We are particularly thankful
+to Mikhail Glazov and Leonid Golub for patiently and
+openly discussing with us their views on Ref. [48] and
+also some of the subtle aspects of the physics of in-gap
+rectification, from which we benefited and gained key in-
+sights for composing Appendix E.
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+14
+Appendix A: Linear conductivity in the DC limit
+In this appendix we show additional details of the linear conductivity in the DC limit discussed in the main text.
+In the DC limit ω → 0 the linear conductivity [see Eq. (64) in the main text] becomes:
+lim
+ω→0 σαβ
+Γ (ω) = 1
+2
+�
+k
+∂α∂β¯ϵk
+�fΓ(¯ϵk)
+Γ
+− ∂gΓ(¯ϵk)
+∂¯ϵk
+�
+= 1
+2
+�
+k
+∂α∂β¯ϵk
+�f0(¯ϵk)
+Γ
++ Γ
+2
+∂3f0(¯ϵk)
+∂¯ϵ3
+k
++ Γ2
+3
+∂3g0(¯ϵk)
+∂¯ϵ3
+k
++ O(Γ3)
+�
+,
+(A-1)
+in which
+gΓ(ϵ) = 1
+2i
+�
+f+(ϵ) − f−(ϵ)
+�
+,
+g0(ϵ) ≡ lim
+Γ→0 gΓ(ϵ),
+(A-2)
+where gΓ(ϵ) is the imaginary part of f+(ϵ) defined in Eq. (25) in the main text, and we used the Cauchy–Riemann
+equations satisfied by fΓ(ϵ) and gΓ(ϵ)
+∂fΓ(ϵ)
+∂Γ
+= ∂gΓ(ϵ)
+∂ϵ
+,
+∂fΓ(ϵ)
+∂ϵ
+= −∂gΓ(ϵ)
+∂Γ
+,
+(A-3)
+and the resulting relation
+fΓ(ϵ) = f0(ϵ) + Γ∂g0(ϵ)
+∂ϵ
++ O(Γ2),
+(A-4)
+to obtaining the second equation of Eq. (A-1).
+Therefore the clean limit of the DC conductivity resembles the
+prediction of the classic Drude theory for τ ≡ 1/(2Γ):
+lim
+Γ→0 lim
+ω→0 σαβ
+Γ (ω) = 1
+2Γ
+�
+k
+f0(¯ϵk)∂α∂β¯ϵk,
+(A-5)
+and linear conductivity has a Drude peak in the DC limit when the chemical potential of the bath is within the
+bandwidth of the system µ ∈ [0, ∆]. The system can still have a finite linear DC conductivity even if the band is
+nominally fully empty or occupied at finite Γ, namely,
+lim
+ω→0 σαβ
+Γ (ω) = 1
+2
+�
+k
+∂α∂β¯ϵk
+�Γ2
+3
+∂3g0(¯ϵk)
+∂¯ϵ3
+k
++ O(Γ3)
+�
+∝ Γ2 + O(Γ3),
+�
+T0 → 0, µ /∈ [0, ∆]
+�
+.
+(A-6)
+This conductance vanishes when Γ → 0.
+Appendix B: Rectification conductivity in the DC limit
+In this appendix we show more details of the rectification conductivity in the DC limit discussed in the main text.
+In the DC limit, the rectification conductivity [see Eq. (72) in the main text] is:
+lim
+ω→0 σγαβ
+Γ
+(ω, −ω) = 1
+4
+�
+k
+∂α∂β∂γ¯ϵk
+�fΓ(¯ϵk)
+Γ2
+− 1
+Γ
+∂gΓ(¯ϵk)
+∂¯ϵk
+− 1
+3
+∂2fΓ(¯ϵk)
+∂¯ϵ2
+k
+�
+= 1
+4
+�
+k
+∂α∂β∂γ¯ϵk
+�f0(¯ϵk)
+Γ2
++ 1
+6
+∂2f0(¯ϵk)
+∂¯ϵ2
+k
++ Γ2
+24
+∂4f0(¯ϵk)
+∂¯ϵ4
+k
++ Γ3
+45
+∂5g0(¯ϵk)
+∂¯ϵ5
+k
++ O(Γ4)
+�
+,
+(B-1)
+where we again used Eq. (A-4) in arriving at the second equation. In the clean limit Γ → 0, this coincides with the
+Jerk conductivity predicted within the relaxation time approximation, but here we also present the sub-leading in Γ
+correction:
+lim
+Γ→0 lim
+ω→0σγαβ
+Γ
+(ω, −ω) = 1
+4
+�
+k
+∂α∂β∂γ¯ϵk
+�f0(¯ϵk)
+Γ2
++ 1
+6
+∂2f0(¯ϵk)
+∂¯ϵ2
+k
+�
+.
+(B-2)
+Therefore, similarly to the linear conductivity, second order rectification conductivity has a Jerk peak at DC limit
+when the chemical potential is within the bandwidth of the system µ ∈ [0, ∆]. When the band is nominally fully
+empty or occupied, for the rectification conductivity we now have
+lim
+ω→0σγαβ
+Γ
+(ω, −ω) = 1
+4
+�
+k
+∂α∂β∂γ¯ϵk
+�Γ3
+45
+∂5g0(¯ϵk)
+∂¯ϵ5
+k
++ O(Γ4)
+�
+∝ Γ3 + O(Γ4),
+�
+T0 → 0, µ /∈ [0, ∆]
+�
+.
+(B-3)
+This finite DC rectification conductivity again vanishes in the clean limit Γ → 0.
+
+15
+Appendix C: Second harmonic generation
+In this appendix we show the second harmonic conductivity mentioned in the main text. The second harmonic
+conductivity is the one that controls the response oscillating at the double frequency of the drive (s = 2), we define
+it as:
+j(2)
+γ
+= σγαβ
+Γ
+(ω, ω)Eα
+ωEβ
+ω + O(|Eω|3),
+(C-1)
+and it is given by the following expression:
+σγαβ
+Γ
+(ω, ω) = − 1
+2ω2
+�
+k
+fΓ∂α∂β∂γ¯ϵk − 1
+ω3
+Γ
+2Γ − iω
+�
+k
+(∂α¯ϵk)(∂β∂γ¯ϵk)L1(¯ϵk, ω)
+−
+1
+2ω4
+Γ
+2Γ − 2iω
+�
+k
+(∂γ¯ϵk)
+�
+(∂α¯ϵk)(∂β¯ϵk)L2(¯ϵk, ω) + ω
+2 ∂α∂β¯ϵkL1(¯ϵk, 2ω)
+�
+,
+(C-2)
+where
+L2(¯ϵk, ω) = f+(¯ϵk − 2ω) − 2f+(¯ϵk − ω) + f+(¯ϵk) + f−(¯ϵk) − 2f−(¯ϵk + ω) + f−(¯ϵk + 2ω).
+(C-3)
+The low frequency limit of second harmonic conductivity coincides with the low frequency limit of rectification
+conductivity from Eq. (73) in the main text:
+lim
+ω→0 σγαβ
+Γ
+(ω, ω) = 1
+4
+�
+k
+∂α∂β∂γ¯ϵk
+�fΓ(¯ϵn)
+Γ2
+− 1
+Γ
+∂gΓ(¯ϵn)
+∂¯ϵn
+− 1
+3
+∂2fΓ(¯ϵn)
+∂¯ϵ2n
+�
+.
+(C-4)
+Interestingly, at large frequencies ω ≫ ∆ the real part of the second harmonic conductivity decays as 1/ω2 in contrast
+to 1/ω4 power decay of the rectification conductivity.
+Appendix D: Relation to the Boltzmann theory
+In this appendix we discuss the relation between our result and that from a simpler Boltzmann/relaxation-time
+approach. We begin by writing a Boltzmann equation for a single band system in the relaxation time approximation:
+∂tf(k, t) + E(t) · ∇kf(k, t) = −[f(k, t) − f0(ϵk)]/τ,
+(D-1)
+where E(t) = Eωe−iωt + c. c. is a monochromatic electric field.
+The above equations are written in a different gauge with respect to the main text: here k is viewed as a gauge
+invariant mechanical crystal momentum, which corresponds to k−A(t) in the main text. In order to obtain expressions
+for occupation functions in the same gauge as in the main text, we convert to a gauge in which we keep track of the
+occupation of canonical crystal momenta, using the following relation:
+p(k, t) ≡ f(k − A(t), t).
+(D-2)
+The occupation function p(k, t) satisfies the following equation
+∂tp(k, t) = ∂tf(k − A(t), t) − ∂tA(t) · ∇kf(k − A(t), t)
+= ∂tf(k − A(t), t) + E(t) · ∇kf(k − A(t), t)
+= −[f(k − A(t), t) − f0(ϵk−A(t))]/τ,
+(D-3)
+where we used Eq. (D-1) in obtaining the last equation. Therefore we see that the distribution function p(k, t) satisfies
+an equation without explicit electric field derivative term:
+∂tp(k, t) = −[p(k, t) − f0(ϵk−A(t))]/τ.
+(D-4)
+Using the fact that the late-time steady state distribution is periodic, we perform Fourier series expansions for both
+p(k, t) and f0(ϵk−A(t)):
+p(k, t) =
++∞
+�
+l=−∞
+p(l)(k) exp(−ilωt),
+p(l)(k) =
+� T
+0
+dt
+T p(k, t) exp(ilωt);
+f0(ϵk−A(t)) =
++∞
+�
+l=−∞
+f (l)
+0 (k) exp(−ilωt),
+f (l)
+0 (k) =
+� T
+0
+dt
+T f0(ϵk−A(t)) exp(ilωt).
+(D-5)
+
+16
+With the above expansions, Eq. (D-4) becomes
+−ilωp(l)(k) = −p(l)(k)/τ + f (l)
+0 (k)/τ,
+(D-6)
+and leads to
+p(l)(k) =
+1
+1 − ilωτ f (l)
+0 (k).
+(D-7)
+The above solution in general requires an explicit calculation of the following mixed harmonics of the distribution:
+f (l)
+0 (k) =
+� T
+0
+dt
+T f0(ϵ(0)
+k
++ ϵ(1)
+k e−iωt + ϵ(−1)
+k
+eiωt + · · · ) exp(ilωt),
+(D-8)
+where
+ϵk−A(t) =
++∞
+�
+l=−∞
+ϵ(l)
+k exp(−ilωt),
+ϵ(l)
+k =
+� T
+0
+dt
+T ϵk−A(t) exp(ilωt).
+(D-9)
+Let us consider however the clean limit τ → +∞.
+Notice that f (l)
+0 (k) is independent of τ, therefore for l ̸= 0
+components we have
+lim
+τ→+∞ p(l̸=0)(k) =
+lim
+τ→+∞
+1
+1 − ilωτ f (l)
+0 (k) = 0.
+(D-10)
+However the l = 0 component, or time averaged component, which is independent of τ and therefore remains finite
+as τ → +∞, is given by:
+p(0)(k) = f (0)
+0 (k) =
+� T
+0
+dt
+T f0(ϵ(0)
+k
++ ϵ(1)
+k e−iωt + ϵ(−1)
+k
+eiωt + · · · ).
+(D-11)
+Therefore, similarly to Eq. (54) obtained from the full formalism with the bath, the distribution from the Boltzmann
+theory becomes time independent in the canonical crystal momentum, but not in the mechanical physical momentum,
+in the analogous ideal limit of τ → +∞. Notice, however, that the above result has to be viewed as a limit of τ → +∞,
+and not as a situation in which there is no relaxation. This is because in the strict absence of relaxation mechanisms
+there is no unique late-time steady state, namely by taking 1/τ = 0 and neglecting altogether the relaxations in the
+right hand side of Eq. (D-4) any time-independent distribution of the canonical momenta would be a solution.
+If we expand up to the second order of electric fields Eq. (D-11) we obtain:
+p(0)(k) =
+� T
+0
+dt
+T
+�
+f0(¯ϵk) +
+�
+ϵ(1)
+k e−iωt + ϵ(−1)
+k
+eiωt + ϵ(2)
+k e−i2ωt + ϵ(−2)
+k
+ei2ωt�
+f ′
+0(¯ϵk)
++ 1
+2
+�
+ϵ(1)
+k e−iωt + ϵ(−1)
+k
+eiωt�2f ′′
+0 (¯ϵk) + O(|Eω|3)
+�
+= f0(¯ϵk) + |ϵ(1)
+k |2f ′′
+0 (¯ϵk) + O(|Eω|3).
+(D-12)
+Interestingly the above distribution function coincides with the asymptotic behavior of the staircase distribution
+function discussed in the main text [see e.g., Eq. (62)] in limit of ∆ ≫ ω ≫ Γ → 0:
+lim
+ω→0 lim
+Γ→0 pk = lim
+ω→0
+��
+1 − 2
+��ϵ(1)
+k
+��2
+ω2
+�
+f0(¯ϵk) +
+��ϵ(1)
+k
+��2
+ω2
+f0(¯ϵk − ω) +
+��ϵ(−1)
+k
+��2
+ω2
+f0(¯ϵk + ω)
+�
+= f0(¯ϵk) + |ϵ(1)
+k |2f ′′
+0 (¯ϵk).
+(D-13)
+Therefore the expectation value of all equal time observables, such as the electric current, coincide with those of the
+more microscopic Floquet-bath theory of the main text, at least to second order in electric fields. In particular one
+obtains the same rectification conductivity in the above limit as that in Eq. (75) of the main text, that we refer to as
+Jerk effect.
+
+17
+Appendix E: Comments and connections to other works in the literature
+There has been a long-standing debate in the literature about the possibility of in-gap rectification which has been
+clouded by previous imprecise and incorrect statements. In this section we will try to clarify some of this. We begin by
+defining precisely what do we mean by in-gap rectification. The optical gap is defined as the region in the frequency
+domain in which the the hermitian symmetric part of the conductivity tensor vanishes in the limit of low temperatures
+and small scattering rates (see Ref. [43] for a review). We then say that a system has in-gap rectification if any of the
+elements of the rectification conductivity tensor that lead to finite DC currents generated by a monochromatic AC
+electric field with a frequency within the optical gap remain non-zero in that same limit. More specifically:
+Definition of “optical gap” :
+lim
+T0→0 lim
+Γ→0
+�
+σαβ(ω) + [σβα(ω)]∗�
+→ 0,
+when ω ∈ optical gap.
+(E-1)
+Definition of “in-gap rectification” :
+lim
+T0→0 lim
+Γ→0 σγαβ(ω, −ω) ̸= 0,
+when ω ∈ optical gap.
+(E-2)
+Therefore our current manuscript and our previous work in Ref. [43], demonstrate rigorously that in-gap rectification
+in the above sense is indeed possible.
+Nevertheless, some confusion in the literature appears to have originated from different interpretations of the work
+of Belinicher, Ivchenko, and Pikus (BIP) in Ref. [48]. That paper contained statements such as “The conclusion
+that a steady-state photocurrent may appear on illumination in the transparency range of a crystal, reached in earlier
+publications, is shown to be in error”. This statement could be read as implying the impossibility of in-gap rectification
+in the sense we defined above. In fact, this reading of the BIP paper appears to have been made in several references
+claiming that in-gap rectification in the above sense is impossible [44, 45, 50]. Even us in our recent work of Ref. [43],
+read the BIP paper as trying to prove that in-gap rectification is impossible in the above sense.
+However, part of the issue with reading the aforementioned BIP paper, is that it left several crucial gaps in its
+discussion and its derivations that can make it hard to know in a precise way what exactly BIP implied at various
+places and the precise framework that BIP used for reaching such conclusions. For example, a crucial point that can
+lead to a different readings of the BIP paper relates to the definition of the term “gn” that appears in the right hand
+side of their Eq. (8) in Ref. [48], which is a central equation from which various conclusions are derived. Unfortunately
+BIP never spelled out an explicit form for this term, but simply wrote that “gn is the generation function, i.e., the
+rate of change of the distribution function due to optical transitions.”. This leaves open to interpretation what exactly
+they had in mind for “optical transitions”. For example, one could read this by interpreting “gn” as associated only
+with inter-band optical transitions, and in this case, one would be lead to read the BIP paper as trying to imply that
+in-gap rectification in the above sense is impossible.
+There is however an alternative way to interpret “gn” and the notion of “optical transitions” in Ref. [48] as a
+more general notion of irreversible “transitions” that can take place even within what would nominally be the optical
+gap defined in the above sense. This more nuanced way of interpreting the BIP paper has indeed been recently
+emphasized by Glazov and Golub in Ref. [46]. For example Golub and Glazov write in Ref. [46] that “... even for
+transparent media, real electronic transitions should occur to enable the photocurrent.” and that “We reiterate that
+in the absence of any real electronic transitions DC current is forbidden. It is obvious from general reasons: If a DC
+current is generated then this current results in a Joule heat in the sample or in the external circuit connected to the
+sample. It is forbidden by the energy conservation law in the absence of real transitions.”. What Golub and Glazov
+are trying to explain there is in line with our recent thermodynamic analysis in Ref. [43], where we emphasized that in
+order to guarantee the positivity of entropy production, specially when the system is connected to an external circuit,
+it is always important to view the scattering rate Γ as possibly arbitrarily small but not strictly zero. This requirement
+means that physically it is important to have always a non-zero absorption within the nominal optical gap of the
+material. In fact Golub, Ivchenko himself and Spivak, have also emphasized a related aspect of this in Ref. [55] where
+they demonstrated that the CPGE effect associated with the Berry dipole term remains finite within the optical in
+the limit of Γ → 0, but also coexists the other contributions that originate from impurity scattering mechanisms that
+scale in the same way with frequency and remain finite inside of the gap in the limit of Γ → 0. One way to state this
+state of affairs, that has been emphasized by Golub and Glazov to us in private communications, is that while the
+real transitions associated with scattering lead to a vanishingly small linear dissipative conductivity in the limit of
+Γ → 0, there are cancellations of the scattering rate that lead to finite rectification conductivity in this limit but the
+“real electronic transitions” are still taking place. These “real electronic transitions” are therefore the more general
+notion of “optical transitions” that can contribute to the term “gn” in the BIP reference. Therefore, within this point
+of view, one can say that the BIP should not be read as implying that in-gap rectification is impossible in the sense
+we defined above. We are in agreement with the physics of this point of view broadly speaking.
+There is however another crucial aspect of the BIP work in Ref. [48] with which we still find ourselves in disagreement
+and that we believe our current paper provides good evidence to be incorrect in general. BIP stated that “... in
+
+18
+the case of continuous illumination the steady-state distribution function is f0(¯ϵk) irrespective of how weak is the
+interaction of electrons with phonons.” In this statement f0 is the “equilibrium distribution function” (the Fermi-
+Dirac occupation function) and ¯ϵk is the Floquet energy of the band. These statement has been echoed in several
+subsequent works [45, 46, 49, 50]. However, our current work demonstrates that in the limit of Γ → 0 the distribution
+function is sharply different from the naive Fermi-Dirac occupation, but becomes instead the non-trivial Fermi-Dirac
+staircase discussed in the main text, even to the leading order |Eω|2 in the driving monochromatic field. Crucially
+the resulting occupation function cannot be expressed as a function of the Floquet energy alone [see Fig. 1, Eq. (54),
+and Eq. (62) of the main text]. Notice that in order to have a unique and well defined steady state at late times, we
+must necessarily view the relaxation rate Γ as being arbitrarily small but not strictly zero. Therefore, the notion of
+the ideal occupation in the steady state has to be necessarily interpreted as the limit of Γ → 0 of the occupation of
+systems with a finite Γ. This is because systems with strictly zero relaxation rate (Γ = 0) do not have a way to erase
+the memory of their initial conditions and therefore their steady state in the presence of the monochromatic light is
+not uniquely defined.
+We have demonstrated rigorously that at least for an ideal fermionic bath the occupation of states in the limit of
+Γ → 0 is not f0(¯ϵk) as Refs. [45, 46, 48–50] presumed. We would like to emphasize that while the fermionic bath
+might appear to be a somewhat artificial approximation to the true mechanisms of relaxation for certain realistic
+physical situations, it behaves as an ideal thermal bath in the limit Γ → 0. In particular, the particle number becomes
+effectively conserved in such limit since the self-consistent occupations at each momentum become a time independent
+function as we have shown. We have in particular demonstrated that in equilibrium this bath leads to the expected
+Fermi-Dirac occupation of the system.
+More generally speaking, in equilibrium one expects a universality of all
+intensive thermodynamic physical properties of the system of interest for a large class of baths regardless of their
+details, which essentially defines the class of “ideal thermodynamic baths”. However, how this universality carries over
+to non-equilibrium settings is still unclear to us. Therefore whether other baths or other relaxation mechanisms such
+as coupling to phonons, impurities or self-thermalization via electron-electron interactions lead to a similar stair-case
+occupation to the one we have found in the limit of vanishing relaxation rates Γ → 0, remains an interesting open
+problem. We note however that none of the aforementioned Refs. [45, 46, 48–50] has provided a rigorous and controlled
+derivation of the self-consistent occupation of Floquet bands based on any microscopically explicit mechanism of
+relaxation, like the one we have provided. Therefore we do not see any rigorous substance to their claim that the
+occupation is f0(¯ϵk) even for other microscopic relaxation mechanisms such as phonons. Moreover, it has become
+abundantly clear in the study of thermalization of Floquet systems in recent years that the self-consistent occupation
+of Floquet bands coupled to baths that are also bosonic differs clearly from the naive Fermi-Dirac distribution of the
+Floquet bands f0(¯ϵk) [36–41].
+

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+Cost-Sensitive Stacking: an Empirical Evaluation
+Natalie Lawrance* ID(�)1, Marie-Anne Guerry ID 1, and George Petrides ID 2
+1Department of Business Technology and Operations, Vrije Universiteit Brussel (VUB), Brussels, Belgium
+2Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus
+Abstract
+Many real-world classification problems are cost-sensitive in nature, such that the misclassification costs vary be-
+tween data instances. Cost-sensitive learning adapts classification algorithms to account for differences in misclassifica-
+tion costs. Stacking is an ensemble method that uses predictions from several classifiers as the training data for another
+classifier, which in turn makes the final classification decision.
+While a large body of empirical work exists where stacking is applied in various domains, very few of these works
+take the misclassification costs into account. In fact, there is no consensus in the literature as to what cost-sensitive
+stacking is. In this paper we perform extensive experiments with the aim of establishing what the appropriate setup
+for a cost-sensitive stacking ensemble is. Our experiments, conducted on twelve datasets from a number of application
+domains, using real, instance-dependent misclassification costs, show that for best performance, both levels of stacking
+require cost-sensitive classification decision.
+Keywords
+Cost-sensitive learning, classification, ensemble learning, stacked generalization, stacking, blending
+1
+Introduction
+Cost-sensitive learning is relevant in many real-world classification problems, where different misclassification errors incur
+different costs. A prominent example is the field of medicine, where misdiagnosing an ill patient for a healthy one (a false
+negative) entails delayed treatment and potentially life-threatening consequences, while an error in the opposite direction (a
+false positive) would incur unnecessary medical examination costs and stress for the patient. Cost-sensitive classifiers can
+account for the differences in costs not only between different classes, but also between data instances, making instance-
+dependent cost-sensitive classification decisions.
+Many cost-sensitive classifiers employ ensemble methods, which combine predictions from several classifiers to obtain
+better generalisation performance. Superiority of ensembles over individual classifiers is very well known and has been
+extensively studied ([8, 37]). Most cost-sensitive classification ensembles are homogeneous in nature, meaning their
+components are instantiated using the same learning algorithm.
+Stacked generalization or stacking [31] is a well known and widely applied heterogeneous ensemble, where the pre-
+dictions of classifiers produced by different learning algorithms (the base-learners) are used as training inputs to another
+learning algorithm (the meta-learner) to produce a meta classifier, which makes the final classification decision. In the
+literature, the base- and meta- levels of stacking are also referred to level-0 and level-1.
+Homogeneous cost-sensitive ensembles such as cost-sensitive boosting and bagging are widely studied and have been
+shown very successful [25]. Examples of cost-sensitive stacking, on the other hand, are scarce and unsystematic, represent-
+ing for the most part applications to single domains, where the classifiers are trained on synthetic, class-dependent costs
+and are evaluated with cost-insensitive performance metrics. For a discussion on the importance of real costs for a proper
+evaluation see the work by [25]. In fact, there is currently no consensus as to how a cost-sensitive stacking ensemble is to
+be composed and at what stage (level-0 or level-1) cost-sensitive decision-making should be used. This can be clearly seen
+in Table 1, which gives an overview of existing cost-sensitive stacking literature. Stacking is typically made cost-sensitive
+simply through the application of a cost-sensitive classifier either at level-0 (CS-CiS), level-1 (CiS-CS) or at both levels of
+*Email: natalie.lawrance@vub.be
+1
+arXiv:2301.01748v1  [cs.LG]  4 Jan 2023
+
+the the ensemble (CS-CS), resulting in a total of three possible stacking setups. To the best of our knowledge, no compari-
+son of all three setups on multiple domains with appropriate evaluation exists in the literature. Previous related work used
+arbitrary artificial costs in model training and evaluated cost-sensitive models using performance metrics that are either
+cost-invariant or that focus on the performance of only the positive class.
+In this work we aim to fill this gap by providing a thorough comparison of various cost-sensitive stacking ensembles on
+multiple domains using real, instance-dependent costs and performance metrics appropriate for cost-sensitive problems.
+1.1
+Our contributions
+• The main contribution of this work is a rigorous empirical comparison of different setups of cost-sensitive stacking
+ensembles over multiple domains. We evaluate using appropriate performance metrics and attempt to establish best
+practice.
+• Secondly, we introduce a novel cost-sensitive classifier combination method, inspired by MEC-voting and stacking,
+which we call MEC-weighted-stacking.
+• Finally, we present a list of publicly available datasets with clearly defined instance-dependent misclassification
+costs. The costs are based either on the literature, or are defined by us based on both the literature and expert
+knowledge of the data providers. We also define instance-dependent costs for a well known ‘credit-g’ dataset from
+the UCI Machine learning repository, for which only class-dependent costs were available to date.
+1.2
+Outline
+The remainder of the paper is structured as follows. Section 2 presents an overview of the relevant literature. MEC-
+weighted stacking is introduced in Section 3. Our hypotheses to be tested, the experimental setup and the datasets used in
+the study are discussed in Section 4. Section 5 details the results of our extensive experiments, while the main outcomes
+and limitations are discussed in Section 6. Section 7 concludes the paper.
+2
+Related work
+While stacking has been widely used in machine learning applications (the interested reader is invited to peruse the survey
+on stacking literature by [27]), few works are dedicated to the study of cost-sensitive stacking.
+We identified in the literature three different cost-sensitive stacking setups: CiS-CS, CS-CiS or CS-CS, where the
+ensemble was made cost-sensitive simply through the application of a cost-sensitive classifier either at level-0, level-1
+or at both levels of the ensemble. In most cases, the method used to make the classification cost-sensitive is the direct
+cost-sensitive decision as introduced by [35], also called DMECC [25].
+One of the first papers to discuss stacking in a cost-sensitive context was [6]. The authors propose cost-insensitive
+level-0 and cost-sensitive level-1 stacking setup (CiS-CS setup), which was compared to a number of different classifier
+combinations schemes on 16 classification problems. The misclassification costs they used were artificially generated by
+randomly and uniformally sampling costs from on the interval [1,10]. Several other studies followed adopting the same
+CiS-CS stacking setup, however none of the studies explicitly reasoned or justified this choice.
+Several more papers demonstrated similar examples of multiple-domain studies of CiS-CS stacking with arbitrary costs
+( [7, 33, 34]). These mainly differ in the type and the number of algorithms that are employed in the ensemble. We note
+that all of them used cost-insensitive metrics for classifier evaluation.
+[19] considers a stacking setup, where level-0 classifiers were cost-sensitive while level-1 was cost-insensitive (CS-
+CiS setup). The misclassification costs were assumed to be equal to the inverse of the class priors. This approach is very
+commonly adopted in the absence of information about real misclassification costs. It is, however, not appropriate, see [25]
+for a discussion. The resulting stacking classifier was compared to known ensemble methods using classification accuracy,
+a metric that by design assumes equal misclassification costs.
+Most examples of stacking use different learning algorithms in level-0, however in his original work Wolpert suggested
+that this must not be the case and the technique can also be applied when a single algorithm is considered. [5] propose a
+cost-sensitive variant of bag-stacking, a method originally proposed by [29], using bagged cost-sensitive decision trees in
+level-0 and using cost-sensitive logistic regression in level-1, thus implicitly proposing a CS-CS stacking setup. To the best
+2
+
+Table 1: Summary of cost-sensitive stacking literature
+Publication
+Stacking
+Level-0
+Level-1
+Real
+Costs
+CS
+setup
+algorithm
+algorithm
+costs
+type
+evaluation
+[6]
+CiS-CS
+DT, KNN, NB
+LR
+c
+✓
+[19]
+CS-CiS
+DT, KNN, NB
+MT
+c
+[5]
+CS-CS
+DT
+LR
+✓
+i
+✓
+[34]
+CiS-CS
+DT, KNN, NB
+LR
+c
+[7]
+CiS-CS
+ExT, GBDT, LDA, LR, RF
+LR
+c
+[33]
+CiS-CS
+DT, KNN, RF, SVM
+DT, KNN, NB, SVM
+c
+[13]
+CiS-CS, CS-CiS, CS-CS
+DT, NB, KNN, SVM
+LR, ExT
+c
+✓
+this paper
+CiS-CS, CS-CiS, CS-CS
+DT, KNN, LR, SVM
+Adab, DT, KNN, LR, RF, SVM
+✓
+i
+✓
+Costs type:
+c: class-dependent, i: instance-dependent
+Algorithms:
+Adab: Adaboost, DT:decision tree, ExT: extremely randomised trees, GBDT: gradient boosted trees, KNN: k-nearest neighbour,
+LDA: linear discriminant, LR: logistic regression, MT: Meta Decision Trees, NB: naive bayes, RF: random forest, SVM: support vector machines
+of our knowledge, this study is the only example where real instance-dependent costs were used in model training. Models
+were evaluated using a cost-sensitive metric called the savings score, proposed in [2].
+The only study to date that considers all three different cost-sensitive stacking setups is one by [13] on the application
+domain of software defect prediction. The misclassification costs were selected based on a literature however the authors
+emphasised that they treated costs as one of the hyperparameters of the classifier, which, we must note, is incorrect, as was
+previously discussed in [15]. The experiments are run on 15 datasets using the same class-dependent cost matrix on all.
+Balanced error-based metrics were used for evaluation together with cost-based evaluation metrics.
+Identifying real misclassification costs is a complex task, which for many applications may prove too difficult to de-
+fine and compute. Most studies resort to artificially generated misclassification costs (see [25] for a discussion on why
+this is inappropriate) and error-based evaluation metrics are typically employed to assess generalisation performance of
+cost-sensitive stacking. Examples of metrics used include the AUC, the arithmetic or geometric mean of class-specific
+accuracies, the F-measure, and the Matthew’s correlation coefficient (MCC). All of these metrics assume equal misclassi-
+fication costs, and the F-measure does not incorporate the performance on the negative class, so using these metrics is not
+compatible with cost-sensitive learning [17].
+One of the challenges of stacking is the choice of the learning algorithms for the ensemble. Earlier studies proposed to
+use linear regression to combine level-0 inputs [30], however Wolpert does not impose any particular restrictions on which
+algorithm to use in level-1, and he believed that his famous ‘No Free Lunch Theorem’ [32] applies to the meta-learner as
+well. For the overview of which learning algorithms were used in cost-sensitive stacking ensembles to date we refer our
+reader to the summary Table 1.
+3
+MEC-weighted stacked generalization
+In the typical supervised classification framework, a learning algorithm A is presented with a set S of data instances
+(xi,yi), each describing some object i. We call xi a feature vector, and yi the class label of that object, drawn from a finite,
+discrete set of classes {1,...,K}. In this paper we will consider the binary classification problem, where yi ∈ {0,1}. The
+learning algorithm A, given S as input, after a process called training, produces a classifier C, whose task is to predict the
+correict class label ˆyC(xj) ∈ {0,1} for a previously unseen feature vector xj.
+Training any number L of learning algorithms on the same set of data instances S , we obtain a set of classifiers
+C = {C1 ...CL}, and for each feature vector xi the corresponding set of predictions
+ˆ
+Y (xi) = {ˆyC1(xi),..., ˆyCL(xi)}. C is
+called an ensemble of classifiers if the predictions from
+ˆ
+Y (xi) are combined, in some way, into a single prediction of the
+class label for the data instance xi.
+Stacking differs from other classifier ensembles in that the predictions from the set
+ˆ
+Y (xi) are combined with the
+original class label yi to form the set Smeta = {(ˆyC1(xi),..., ˆyCL(xi)),yi} of meta level data instances subsequently used in
+another round of algorithm training to produce a new classifier, which is used to obtain the final predictions.
+The novel method we propose in this paper is inspired by the cost-sensitive weights for model votes paradigm described
+in [25], and consequently called MEC-weighted stacking. To each classifier C, we can assign a weight wC based on that
+classifier’s cost-performance on the validation set: wC = f(ε), where ε is the sum of the misclassification costs of all data
+3
+
+Table 2: Characteristics of the datasets used in our experiments
+Application
+Dataset alias
+# instances
+# Attr
+% positives
+Instance-dependent
+domain
+costs source
+1
+Bankruptcy
+bankruptcy (private)
+404999
+221
+3.31
+this publication
+2
+Churn
+churn kgl (Kaggle*)
+7043
+21
+26.54
+[25]
+3
+Churn
+churn AB [3]
+9410
+45
+4.83
+[3]
+4
+Credit risk
+credit kgl (Kaggle*)
+112915
+15
+11.70
+[2]
+5
+Credit risk
+credit de uci [12]
+1000
+20
+30.00
+this publication
+6
+Credit risk
+credit kdd09 [28]
+38938
+39
+19.89
+[2]
+7
+Credit risk
+credit ro vub [24]
+18918
+24
+16.95
+[24]
+8
+Direct marketing
+dm pt uci [12, 22]
+45211
+17
+11.27
+[4]
+9
+Direct marketing
+dm kdd98 [12]
+95412 (train)
+479
+5.08
+[25]
+96367 (test)
+5.06
+10
+Fraud detection
+fraud ulb kgl [21]
+284807
+31
+0.17
+[25]
+11
+Fraud detection
+fraud ieee kgl (Kaggle*)
+590540
+432
+3.50
+[25]
+12
+HR analytics
+absenteeism be (private)
+36853 (train)
+71
+14.50
+[20]
+35884 (test)
+10.76
+* Kaggle: https://www.kaggle.com/
+instances incorrectly classified by C on a validation set and f(ε) is a transformation function, which for example can take
+one of the following forms: f(ε) = ln((1−ε)/ε), f(ε) = 1−ε, f(ε) = exp((1−ε)/ε), and f(ε) = ((1−ε)/ε)2.
+The general stacking procedure is thus modified with the additional step of collecting the MEC-weights for each of
+the predictions from the set ˆ
+Y (xi), yielding the weighted set of predictions ˆ
+YMEC(xi) = {(wC1 ˆyC1(xi),...,wCL ˆyCL(xi)),yi},
+which is used in meta classifier training instead of ˆ
+Y (xi).
+4
+Experimental setup
+4.1
+Data
+In this study we use a collection of 10 publicly available datasets and 2 private datasets, for which misclassification costs
+have either already been defined or will be defined here. This collection of datasets represents a number of application
+domains: credit scoring, customer churn prediction, direct marketing, credit card fraud detection, and HR analytics.
+4.2
+Misclassification costs
+Table 2 presents the references both to the datasets and to relevant publications where the instance-dependent misclassifi-
+cation costs for a given domain were introduced. Most of the datasets are large, the number of instances ranging between
+1000 and almost 600000. The number of input features ranges from 15 to 479. All of these datasets demonstrate a large
+degree of class imbalance, where the percentage of positives reaches at most 30%, and in dataset fraud ulb kgl less than
+1%.
+In this work we propose instance-dependent costs for these two datasets, for which no costs were previously defined.
+The German credit dataset is well known and is referred to as credit de uci in Table 2. Only class-dependent costs
+were available for this data set, where the prediction task is to identify customers that will default on their loan. We define
+instance-dependent costs using the conceptual framework proposed by [2]. For any data instance i, the cost of a false
+negative Ci
+FN is defined as loss given default and constitutes 75% of the credit line, while the cost of a false positive Ci
+FP is
+the loss of the potential profit from rejecting a good customer, plus the sum of the average expected loss and the average
+expected profit estimated on the training sample. We define profits as simply the interests earned on the credit line in the
+current year. The profits are calculated using historic interest rates for the year 2000 in Germany, which we apply randomly
+and uniformly to the whole sample.
+The bankruptcy dataset was provided by the credit risk department of a European utilities-provider, who was interested
+4
+
+in predicting the risk of corporate bankruptcy for new customers. With minor modifications, it readily transfers to the same
+credit risk model described above. Here the credit line is equivalent 90 days of utilities usage by the customer, which, in
+case of default, the provider loses in full, so Ci
+FP equals the credit amount. The profit margins were provided to us and
+are calculated per customer based on the assumption of a 12-month contract. Thus, the Ci
+FP then equals the annual profit
+margins for the potentially good customer plus the expected average loss and expected average profit calculated on the
+given sample.
+4.2.1
+Data preprocessing
+We take care to employ the same preprocessing steps for each of the datasets in the sample, as recommended by the works
+that first published them.
+In addition to that, we apply the following preprocessing steps to all datasets. All numeric variables are rescaled
+using the quantile statistics, which are robust to outliers. Missing values of numeric variables are imputed with sample
+median, and of categorical variables are encoded as a separate category. All categorical variables are transformed using
+weight-of-evidence coding [1].
+4.2.2
+Data partitioning
+The classifier performance estimates are obtained by means of repeated stratified k-fold cross-validation. The 5×2 cross-
+validation suggested by [10] is used to train and evaluate stacking ensembles. This resampling is repeated 5 times using
+different random seeds, and the results are averaged across folds and across iterations. Large datasets with more than
+100000 observations, to keep training times manageable, were split into five disjoint subsets, uniformly at random.
+We note that two datasets in our sample are provided with a separate test set, used to evaluate model performance. In
+this case, for fairness of comparison, we perform the split into folds on each of the training and test datasets using the
+same seed, we then proceed using the training partition of the training set and the test partition of the test set. The training
+partition of the test set remains unused in evaluation. When training and test data sets contain the same observations at
+different time periods (e.g. in bankruptcy prediction) we ensure that training and test datasets are disjoint and do not contain
+overlapping data instances.
+4.3
+Learning algorithms
+The choice of the algorithms for the base- and meta-level of stacking remains one of the challenges of stacked generaliza-
+tion. To the best of our knowledge, no study exists that demonstrates the necessity to use a specific algorithm combination
+in either base- or meta-level of stacking. The main requirement for the base classifiers of any ensemble is that they are
+sufficiently accurate (meaning they predict better than a random guess) and sufficiently diverse (meaning their errors are
+uncorrelated) [11]. In a heterogeneous ensemble, where the decisions of different learning algorithms are combined, the
+number of base-learners need not be large [26]. All algorithms below have previously been described and discussed in
+detail in a number of machine learning textbooks, for example [16], so we refrain from repeating these descriptions here.
+4.3.1
+Base-learners
+The base learners in our experiments are four well known classification algorithms, which are: CART Decision Tree
+(DT), K-Nearest Neighbors (KNN), Support Vector Machines (SVM) and Logistic Regression (LR). Unlike [7] and [33]
+before us, we choose not to use ensembles such as Random Forest or Extremely Randomised Trees in the base level of
+stacking. The reasons for this are two-fold. Firstly, ensembles in general, and stacking in particular are typically built
+on weak base-learners, which these very powerful models, which are themselves ensembles, certainly are not. Secondly
+these methods are based on decision trees and their errors will be correlated with DT. In our choice we also considered
+the recommendations of [8], one of the largest empirical studies known to date comparing algorithm performance on
+121 datasets. Their results on binary problems (55 UCI datasets) demonstrate that Random Forest, SVM, Bagging and
+Decision Trees have the highest probability of obtaining more than 95% of accuracy, while classifiers of the Naive Bayes
+(NB) family are not competitive in comparison. We therefore do not include NB in our experiments, unlike some previous
+studies in cost-sensitive stacking.
+5
+
+4.3.2
+Meta-learners
+The choice of the meta-learner constitutes a challenge as well, as was called ’the black art’ by the original author of stacked
+generalization [31]. To keep the scale of our experiments manageable and to allow for statistical comparison between
+stacking and base classifiers, we use the same four algorithms that were used in the level-0 of stacking. In addition to that
+we also use two homogeneous ensemble methods that, according to [8], perform well on most problems, namely Adaboost
+(Ada) and Random Forest (RF).
+4.3.3
+Cost-sensitive learners
+While many variants of cost-sensitive learning algorithms exist that can incorporate the misclassification costs during
+classifier training [25], in this study we are not interested in comparing cost-sensitive learning algorithms, but in ways of
+combining cost-sensitive and cost-insensitive learners in a single ensemble. For our purposes it is important that the two
+classifiers we compare are different in all but one thing, that is the composition of the ensemble. We therefore choose
+to turn known cost-insensitive classifiers cost sensitive by applying a cost-sensitive threshold adjustment method called
+DMECC [25]. In this method, each data instance is classified according to its individual cost-sensitive decision threshold,
+which is based on the ratio of misclassification costs of that particular data instance. The threshold is calculated as follows:
+T i
+cs =
+Ci
+FP−Ci
+TN
+Ci
+FP−Ci
+TN+Ci
+FN−Ci
+TP , where Ci
+TN and Ci
+TP refer to the costs of correct classification, and Ci
+FN and Ci
+FP refer to the
+misclassification costs of the positive and negative data instances respectively. A given record is classified as positive
+when its estimated probability of being positive exceeds its individual cost-sensitive threshold T i
+cs [35].
+Since some learning algorithms (such as DT or SVM) are known not to produce reliable probability estimates, we
+applied isotonic calibration [36] to all base-learners.
+4.3.4
+Cost-sensitive stacking
+To the best of our knowledge no definition exists of what constitutes cost-sensitive stacking. Based on the insights from
+the literature earlier discussed in Section 2, we see three main possibilities of introducing cost-sensitivity into the ensemble
+structure.
+1. Level-0 classifiers are cost-sensitive, level-1 classifiers are cost-insensitive.
+2. Level-0 classifiers are cost-insensitive, level-1 classifiers are cost-sensitive.
+3. Both level-0 and level-1 classifiers are cost-sensitive.
+We consider 4 functional forms for the MEC-weights as introduced in Section 3, which resulted in a total of 15 stacking
+setups to be compared. The complete list of ensemble compositions is presented in Table 3.
+MEC-weighted stacking renders the level-1 classifier cost sensitive through manipulation of the training data in a
+cost-sensitive way by applying MEC-weights to the training data of the level-1 classifier. We consider it an alternative to
+obtaining an ensemble where both training levels are cost-sensitive, which is the third stacking setup stated above.
+4.3.5
+Software used
+All of our experiments were performed using the Python programming language (version 3.8). Cost-insensitive algorithm
+implementations came from the scikit-learn (version 1.1.1) Python library [23], while the cost-sensitive implementations
+are our own.
+4.4
+Evaluation
+4.4.1
+Evaluation metrics
+Contrary to previous studies in cost-sensitive stacking, we would like to emphasise the importance of using appropriate
+evaluation metrics for cost-sensitive classifiers. Most authors use traditional evaluation metrics such as ROC AUC, Preci-
+sion or F1 metric. ROC AUC is known to be cost-invariant, since it is a measure that aggregates classifier performance over
+all possible class-dependent thresholds, thus implicitly averaging performance over multiple class-dependent costs, which
+6
+
+Table 3: The complete list of stacking setups compared in our study.
+Stacking setup
+Level-0
+Level-1
+Level-1
+alias
+algorithm type
+input weights f(ε)
+algorithm type
+1
+type-1
+CS
+1
+CiS
+2
+type-1 exp
+CS
+exp((1−ε)/ε)
+CiS
+3
+type-1 ln
+CS
+ln((1−ε)/ε)
+CiS
+4
+type-1 sq
+CS
+((1−ε)/ε)2
+CiS
+5
+type-1 acc
+CS
+1−ε
+CiS
+6
+type-2
+CiS
+1
+CS
+7
+type-2 exp
+CiS
+exp((1−ε)/ε)
+CS
+8
+type-2 ln
+CiS
+ln((1−ε)/ε)
+CS
+9
+type-2 sq
+CiS
+((1−ε)/ε)2
+CS
+10
+type-2 acc
+CiS
+1−ε
+CS
+11
+type-3
+CS
+1
+CS
+12
+type-3 exp
+CS
+exp((1−ε)/ε)
+CS
+13
+type-3 ln
+CS
+ln((1−ε)/ε)
+CS
+14
+type-3 sq
+CS
+((1−ε)/ε)2
+CS
+15
+type-3 acc
+CS
+1−ε
+CS
+is not appropriate. Other error-based metrics typically assume equal class-dependent costs, which, again, is not appropri-
+ate, when instance-dependent costs are known at estimation time. Cost-sensitive learning aims to adapt the classification
+decision of a learning algorithm to the differences between misclassification costs assigned to each of the classes. It is
+therefore important that the evaluation metrics used to assess the performance of cost-sensitive classifiers is also adapted
+to account for the difference in misclassification costs. The typical evaluation metric used in cost-sensitive literature is the
+total misclassification cost [14], that simply adds up the errors weighted with their individual misclassification costs, as
+defined on the test set. Another option is to normalise the total misclassification cost over some budget constraint, which
+will depend on the application domain. A more general way to do this is to use the savings score proposed in [2], where
+the total misclassification costs are normalised with the cost of either misclassifying all positives as negatives, or misclas-
+sifying all negatives as positives, which ever is smallest. This gives a metric on the interval between 0 and 1, facilitating
+comparison across different datasets, when necessary.
+Since the majority of comparisons in our study is performed based on average ranks, it requires no commensurability
+of the evaluation metrics, so the models are ranked according to their total misclassification costs, which allows for more
+precise outcome.
+4.4.2
+Multiple classifier comparison
+In order to compare multiple classifiers on multiple datasets, we use the standard approach of the combination of the
+Friedman omnibus test and post-hoc Nemenyi test [18]. The Friedman test is conducted under the null-hypothesis that
+all algorithms in comparison are equivalent in performance. If this null-hypothesis is rejected, the post-hoc test can be
+performed to identify pairs of classifiers whose performance is significantly different, which is measured using the critical
+difference statistic, and can be visualised using the critical differences diagrams [9]. The non-parametric tests, such as the
+Friedman test, are preferred in case where the number of datasets in comparison is less than 30, which is the number of
+datasets necessary to satisfy the normality assumptions of parametric statistical tests, such as ANOVA [9]. The post-hoc
+test is known to be of low power, not rejecting the null even if the null was rejected for the Friedman test. In this case,
+we additionally apply Wilcoxon signed-ranks test, as appropriate, which is used for pairwise comparisons of classifiers on
+multiple datasets. This test ranks differences in performances of a given pair of classifiers, under the null hypothesis that
+the median difference in ranks is zero. It therefore allows establishing whether the observed differences in performance
+between two classifiers are significant. It is considered more powerful than its parametric equivalent, the paired t-test when
+the assumptions of the latter cannot be guaranteed. It is also considered more powerful than the Sign test, which counts
+the number of wins, losses and ties [9].
+7
+
+Ada
+DT
+KNN
+LR
+RF
+SVM
+Level-1 algorithm
+type-1
+type-1_acc
+type-1_exp
+type-1_ln
+type-1_sq
+type-2
+type-2_acc
+type-2_exp
+type-2_ln
+type-2_sq
+type-3
+type-3_acc
+type-3_exp
+type-3_ln
+type-3_sq
+Stacking setup
+75.10
+72.30
+60.70
+74.60
+71.30
+72.10
+72.40
+69.10
+61.20
+74.60
+70.10
+72.70
+71.90
+71.30
+57.50
+70.70
+71.60
+73.50
+71.50
+70.70
+59.30
+70.60
+70.40
+72.30
+70.70
+70.50
+59.20
+69.00
+70.80
+72.30
+42.90
+45.00
+53.30
+44.40
+44.50
+47.60
+43.60
+45.30
+52.70
+47.20
+44.50
+46.10
+45.10
+46.00
+51.90
+42.40
+46.30
+49.50
+46.50
+49.20
+54.00
+46.20
+43.40
+53.70
+45.20
+44.80
+51.90
+45.40
+45.40
+51.30
+8.00
+6.00
+31.00
+7.00
+5.90
+29.20
+13.40
+16.70
+30.90
+9.00
+14.60
+34.30
+16.60
+13.60
+30.40
+10.40
+14.90
+49.40
+14.30
+14.30
+30.30
+9.60
+12.30
+37.50
+12.80
+13.80
+29.90
+8.00
+14.00
+41.60
+20
+40
+60
+80
+Figure 1: Comparing all classifiers by average rank across 10 datasets. Lower numbers correspond to better rank.
+5
+Experimental results
+The purpose of our experiments is twofold. Firstly, we would like to compare the performance of the different cost-
+sensitive stacking setups in order to determine which of them results in the lowest cost-loss and can be recommended to
+practitioners. Secondly, we aim to empirically evaluate MEC-weighted stacking, which is a new cost-sensitive stacking
+method we earlier described in Section 3.
+Despite our best efforts, not all classifiers trained successfully on all 12 datasets. In particular, we were unable to
+collect results for the MEC-weighted stacking where the weights were defined by the logarithmic function on the credit
+scoring problem credit ro vub, and MEC-weighted stacking with exponential weights were missing on the fraud detection
+dataset fraud ulb kgl. The full results for all 15 stacking setups are thus available on 10 datasets, instead of 12. Unweighted
+stacking results, however, are available on all 12 datasets, which we briefly discuss, for completeness.
+5.1
+Finding the best cost-senstive stacking setup
+5.1.1
+Overall comparison
+We begin with an overall comparison, where all classifiers are evaluated and ranked on each of the 10 datasets, and for each
+of them an average rank is computed across all datasets. Figure 1 presents the average ranks for all stacking classifiers,
+where the vertical axis shows the stacking setup and the horizontal axis shows the corresponding level-1 algorithm. The
+comparison consists of a total of 90 classifiers (6 algorithms and 15 stacking setups). For brevity, we adopt the aliases for
+each of the stacking setups earlier presented in Table 3.
+We notice immediately that the ranking demonstrates clusters with stacking ensembles of type-3 ranking the best,
+while type-1 ensembles rank the worst. We note that models built with KNN and SVM algorithms tend to rank lower
+than decision tree based models or logistic regression. However, the general picture of type-3 stacking ranking the best
+and type-1 ranking the worst remains unchanged for KNN and SVM, although the differences in ranks between the three
+groups are smaller than for other algorithms.
+Whether these differences in ranks are statistically significant will be discussed in the following subsection, where we
+demonstrate the outcomes of statistical tests that compare the performance of various stacking classifiers across multiple
+domains.
+5.1.2
+Comparing unweighted stacking setups on 12 datasets
+We begin by testing the null hypothesis that the three unweighted stacking setups show no difference in performance. The
+comparison is performed for each of the six classification algorithms used as level-1 learners. The null hypothesis of the
+8
+
+Table 4: The outcome of the Friedman multiple hypothesis testing.
+Test statistic (χ(k−1))
+Ada
+DT
+KNN
+LR
+RF
+SVM
+Unweighted (k = 3,n = 12)
+6.5
+32.69**
+32.59**
+32.79**
+32.69**
+32.59**
+32.59**
+All (k = 15,n = 10)
+3.94
+132.44**
+132.09**
+132.35**
+132.29**
+132.09**
+132.09**
+** significant at 0.01 level
+k: number of stacking setups in comparison, n: number of datasets in comparison
+(a) Adaboost
+(b) Decision Tree
+(c) K-Nearest Neighbors
+(d) Logistic Regression
+(e) Random Forest
+(f) Support-Vector Machine
+Figure 2: Pairwise comparison of the three unweighted stacking setups on 12 datasets using Nemenyi test at 0.05 significance level.
+Friedman test was rejected for all 6 comparisons, and the test statistics are presented in row 1 of Table 4.
+We proceed with the post-hoc Nemenyi test to evaluate the alternative hypothesis that the performance of three stack-
+ings setups is not equal. Figure 2 presents the results of the post-hoc tests at the 0.05 significance level. We find that type-3
+stacking ranks best and is significantly different from both type-2 and type-1 for all algorithms except SVM, where the
+difference is only significant for the comparison between type-3 and type-1, but no conclusions can be made regarding the
+differences between ensembles of type-3 and type-2. Similarly, no conclusions can be made regarding the differences in
+rank between type-2 and type-1 stacking ensembles.
+Since the outcome of the post-hoc tests are ambiguous in the case of SVM, we also perform the Wilcoxon rank sum
+test under the null hypothesis that the median of the paired differences is zero. For the comparison between type-3 and
+type-2 unweighted stacking the null is rejected at 0.05 level.
+We conclude from these tests that type-3 stacking performs significantly better than the other two stacking setups.
+5.1.3
+Comparing all cost-sensitive stacking setups on 10 datasets
+We proceed to compare all 15 stacking classifiers on 10 datasets. The outcome of the Friedman rank sum test can be found
+in row 2 of Table 4. The null hypothesis of the Friedman test is rejected for every meta-learner at the 1% significance level,
+so we conclude that the performance of all 15 models is not equal and proceed with the post-hoc test. Figure 3 shows the
+outcome of the Nemenyi test at 0.05 significance level.
+These are for the most part consistent with what we observed in Figure 1, where the classifiers tend to cluster by
+stacking setup, type-3 being the leader, type-2 the second-best and type-1 ranking worst. Similar to what we observed
+above with unweighted stacking, we can reject the null that type-3 stacking and its MEC-weighted variants are equal in
+performance to type-1 stacking and variants. This holds for all algorithms except KNN and SVM. For stacking ensembles
+with KNN in level-1 type-3 and type-3 acc classifiers are not significantly different from type-1 exp and type-1 sq, while
+for SVM no significant differences were detected between type-3 exp and type-3 sq and other type-1 ensembles.
+Since Nemenyi post hoc test is not powerful enough to establish whether the differences between the three stacking
+setups are statistically significant, additional testing is required. From the outcomes of the post hoc test we observed that
+type-3 stacking generally tends to rank highest, and is therefore of most interest to us. We therefore perform the Wilcoxon
+rank sum test for all combinations of pairwise comparisons of stacking algorithms of type-3 vs type-1 and of type-3 vs
+type-2 under the null hypothesis that the median of the rank differences between the two groups is equal to zero. The
+complete tables with the obtained test statistics and p-values can be found in the Appendix. We find that the null could be
+confidently rejected for all comparisons between type-3 and type-1 stacking ensembles, we refer the reader to the Table 7
+9
+
+CD
+1
+2
+3
+type-3
+type-1
+type-2CD
+1
+2
+3
+type-3
+type-1
+type-2CD
+H
+1
+2
+3
+type-3
+type-1
+type-2CD
+H
+Y
+1
+2
+3
+type-3
+type-1
+type-2CD
+1
+2
+3
+type-3
+type-1
+type-2CD
+H
+2
+type-3
+type-1
+type-2(a) Adaboost
+(b) Decision Tree
+(c) K-Nearest Neighbors
+(d) Logistic Regression
+(e) Random Forest
+(f) Support-Vector Machine
+Figure 3: Comparing all stacking setups on 10 datasets using Nemenyi post-hoc test at 0.05 significance level.
+in the Appendix for details.
+As for the comparison of stacking type-3 with type-2, the only algorithm where the null could not be rejected was
+SVM. We found that the differences between all type-3 MEC-weighted stacking variants and type-2 stacking ensembles
+were not significant. However, type-3 unweighted stacking was significantly different from all type-2 stacking variants,
+see Table A.2 for details.
+We can therefore recommend type-3 stacking, where both levels of stacking are cost-sensitive, as the winner.
+5.2
+Evaluating MEC-weighted stacking
+Our next research question is whether within the same setup, MEC-weighted stacking offers any improvement over the
+unweighted stacking. To determine whether there is a statistically significant difference in performance between the MEC-
+weighted stacking models and their unweighted counterparts we perform pairwise comparison using Wilcoxon rank sum
+test. The test statistics and corresponding p-values from the 72 comparisons are reported in Table 5. Values that are
+significant at 5% level were highlighted with boldface text, weakly significant values at 10% level were highlighted with
+italics. As previously, the results are reported per learning algorithm used as the level-1 classifier.
+We find almost no significant differences in performance between unweighted and weighted stacking for setups of
+type-1 and type-2 with rare exceptions. Surprisingly, only one comparison of stacking type-1, where the level-1 classi-
+fier is cost-insensitive shows a significant difference in performance. Namely, the stacking setup type-1 sq learned with
+Logistic Regression in level-1. Referring back to the average rankings reported in Figure 1 it happens to be the best
+performing Logistic Regression stacking of type-1, so in this instance MEC-weighted stacking is significantly better than
+its counterpart with equally weighted meta-inputs. It is, however, an exception, and we must conclude that introducing
+cost-sensitivity through MEC-weights into level-1 of stacking has no positive impact on type-1 stacking performance.
+Similar conclusions can be drawn for stacking of type-2, where base classifiers are cost-insensitive but the meta-
+classifier is made cost-sensitive using DMECC. Here we observe only two statistically significant test outcomes, both of
+which have lower average ranks than the unweighted stacking of type-2.
+For the setup of type-3 where both level-0 and level-1 of stacking are made cost-sensitive using DMECC, the null
+could not be rejected for AdaBoost, Logistic Regression and KNN. Most of the MEC-weighted ensembles built with
+Decision Tree, Random Forest and SVM were significantly different from unweighted stacking of type-3. Looking at the
+differences in the average ranks, however we note that unweighted stacking of type-3 ranks noticeably better than any of
+the MEC-weighted models.
+We must therefore conclude that MEC-weighted stacking does not offer a statistically significant improvement over
+conventional stacking.
+10
+
+CD
+12345
+6
+7
+89101112131415
+type-3_sq
+type-1_acc
+type-3_In
+type-1_in
+type-3_exp
+type-i
+type-3_acc
+type-1_exp
+type-3
+type-1_sq
+type-2_exp
+type-2_ln
+type-2_sq
+type-2
+type-2_accCD
+234567
+8
+9101112131415
+type-3
+type-1_acc
+type-3_acc
+type-1
+type-3_sq
+type-1_exp
+type-3_in
+type-i_in
+type-3_exp
+type-1_sq
+type-2_exp
+type-2_sq
+type-2
+type-2_acc
+type-2_InCD
+234567
+8
+9101112131415
+type-3
+type-1_exp
+type-3_In
+type-1
+type-3_acc
+type-1_sq
+type-3_sq
+type-i_in
+type-3_exp
+type-1_acc
+type-2_In
+type-2_exp
+type-2_acc
+type-2_sq
+type-2CD
+2345
+6
+9 101112131415
+type-3
+type-1.
+exp
+type-3_acc
+type-1_acc
+type-3_In
+type-1_In
+type-3_sq
+type-1_sq
+type-2_acc
+type-1
+type-2
+type-3_
+exp
+type-2_exp
+type-2_sq
+type-2_InCD
+1
+23456
+57
+8
+9101112131415
+type-3
+type-1
+type-3_sq
+type-1_acc
+type-3_In
+type-1_ln
+type-3_acc
+type-1_exp
+type-3_exp
+type-1_sq
+type-2
+type-2_in
+type-2_acc
+type-2_sq
+type-2_expCD
+1234567
+9101112131415
+type-3
+type-1
+type-3_exp
+type-1_sq
+type-3_sq
+type-1_ln
+type-3n
+type-1_exp
+type-3_acc
+type-1_acc
+type-2_sq
+type-2_in
+type-2_acc
+type-2
+type-2_expTable 5: Pairwise comparison of unweighted and MEC-weighted stacking using Wilcoxon rank sum test. Statistically significant
+values are marked with boldface (significance 0.05) and italics (significance 0.1).
+Level-1
+Unweighted stacking type-1 vs
+algorithm
+type-1 acc
+type-1 exp
+type-1 ln
+type-1 sq
+Ada
+18.0 (0.3)
+18.0 (0.3)
+18.0 (0.3)
+18.0 (0.3)
+DT
+23.0 (0.63)
+23.0 (0.63)
+23.0 (0.63)
+23.0 (0.63)
+KNN
+27.5 (1.0)
+21.0 (0.56)
+17.0 (0.32)
+26.5 (0.92)
+LR
+27.0 (0.96)
+14.0 (0.15)
+22.0 (0.56)
+10.5 (0.08)
+RF
+23.0 (0.63)
+23.0 (0.63)
+23.0 (0.63)
+23.0 (0.63)
+SVM
+18.0 (0.3)
+27.0 (0.96)
+22.0 (0.56)
+27.0 (0.96)
+Unweighted stacking type-2 vs
+type-2 acc
+type-2 exp
+type-2 ln
+type-2 sq
+Ada
+22.0 (0.56)
+23.0 (0.63)
+23.0 (0.63)
+23.0 (0.63)
+DT
+27.5 (1.0)
+27.5 (1.0)
+26.5 (0.92)
+18.5 (0.35)
+KNN
+25.0 (0.8)
+21.0 (0.5)
+25.0 (0.8)
+21.0 (0.5)
+LR
+10.5 (0.08)
+20.5 (0.47)
+24.5 (0.76)
+16.5 (0.26)
+RF
+27.5 (1.0)
+22.5 (0.61)
+17.5 (0.3)
+26.5 (0.92)
+SVM
+23.5 (0.68)
+19.5 (0.41)
+18.5 (0.36)
+10.5 (0.08)
+Unweighted stacking type-3 vs
+type-3 acc
+type-3 exp
+type-3 ln
+type-3 sq
+Ada
+13.0 (0.16)
+19.0 (0.43)
+19.0 (0.43)
+19.0 (0.43)
+DT
+2.0 (0.01)
+11.0 (0.11)
+0.0 (0.0)
+10.0 (0.08)
+KNN
+24.0 (0.77)
+17.0 (0.32)
+24.0 (0.77)
+16.0 (0.28)
+LR
+13.0 (0.16)
+19.0 (0.43)
+14.0 (0.19)
+24.0 (0.77)
+RF
+4.0 (0.01)
+8.0 (0.05)
+3.0 (0.01)
+9.0 (0.06)
+SVM
+10.0 (0.08)
+0.0 (0.0)
+9.0 (0.06)
+7.0 (0.04)
+5.3
+Comparing cost-sensitive stacking with single cost-sensitive models
+Finally, one might ask whether the effort involved in training the level-1 classifier is worth it. To answer this we compare
+the best stacking classifier with the corresponding single classifier. Having previously determined the best stacking setup,
+where DMECC was applied in both levels, and no MEC-weights are applied, we will omit other classifiers from this
+analysis. We average classifier performance across cross-validation folds using the savings metric (see Section 4.4) for
+commensurability. We also rank the resulting selection of classifiers by savings and average ranks across 12 datasets. The
+results are reported in Table 6, where the winning classifier (per algorithm) is marked with boldface font, the best performer
+per dataset is marked with italics.
+We note that cost-sensitive stacking always achieves positive savings, meaning its total misclassification costs are lower
+than the predetermined budget. Stacking has higher average savings on all algorithms except KNN and Random Forest. In
+terms of average ranks, stacking wins for all level-1 algorithms except KNN, which, we note, is one of the worst ranking
+algorithms in our study.
+6
+Discussion
+Outcome 1: using cost-sensitive models in both levels of stacking is recommended
+The results presented in this paper,
+have demonstrated that there is a statistically significant difference in performance between the three different stacking
+setups considered in our experiments, namely CiS-CS, CS-CiS, and CS-CS. Contrary to the majority of cost-sensitive
+stacking papers that assumed that one level of cost-sensitive decision-making is sufficient, our experiments demonstrate
+that stacking models where the DMECC was applied in both levels of stacking achieved the highest ranking.
+While these conclusions hold for this particular post-training method, cost-sensitivity can be introduced either before
+or during training of the learning algorithm. Further experiments are required to investigate how different cost-sensitive
+methods affect our conclusions. Now that we have established how cost-sensitive stacking should be built, future work
+can focus on combining various kinds of cost-sensitive algorithms, including pre-, during- and post-training cost-sensitive
+methods [25]. Another interesting avenue for future research would be investigating homogeneous cost-sensitive stacking,
+an example of which was proposed in [5] using cost-sensitive decision trees as base classifiers and cost-sensitive logistic
+11
+
+Table 6: Comparing single classifiers with type-3 unweighted stacking. Savings score is reported for each classifier, higher is better.
+Best model per dataset is marked with italics. The winning classifier is marked with boldface letters.
+Ada
+DT
+KNN
+LR
+RF
+SVM
+Dataset
+Single
+Stacking
+Single
+Stacking
+Single
+Stacking
+Single
+Stacking
+Single
+Stacking
+Single
+Stacking
+absenteeism be 1
+0.188
+0.225
+0.219
+0.242
+0.199
+0.224
+0.188
+0.23
+0.172
+0.243
+0.188
+0.223
+bankruptcy
+0.03
+0.123
+0.112
+0.123
+0.105
+0.058
+-0.043
+0.126
+0.31
+0.123
+-0.024
+0.024
+churn AB
+0.171
+0.081
+0.052
+0.087
+0.07
+0.019
+0.062
+0.082
+0.1
+0.086
+0.033
+0.04
+churn kgl
+0.311
+0.295
+0.0
+0.297
+0.242
+0.031
+0.302
+0.295
+0.273
+0.296
+0.225
+0.066
+credit de uci
+0.399
+0.391
+0.293
+0.387
+0.337
+0.351
+0.396
+0.388
+0.424
+0.386
+0.405
+0.354
+credit kdd09
+0.318
+0.303
+0.277
+0.302
+0.287
+0.276
+0.312
+0.303
+0.313
+0.302
+0.289
+0.279
+credit kgl
+0.511
+0.411
+0.156
+0.411
+0.408
+0.202
+-0.053
+0.411
+0.499
+0.411
+0.378
+0.175
+credit ro vub
+1.793
+1.796
+1.787
+1.795
+1.786
+1.785
+1.727
+1.793
+1.762
+1.797
+1.773
+1.786
+dm kdd98 train
+0.108
+0.143
+0.033
+0.147
+0.035
+0.061
+0.122
+0.119
+0.059
+0.147
+0.036
+0.044
+dm pt uci
+0.568
+0.558
+0.537
+0.557
+0.551
+0.511
+0.562
+0.558
+0.556
+0.557
+0.551
+0.529
+fraud ieee kgl
+0.109
+0.494
+0.444
+0.505
+0.477
+0.438
+0.372
+0.495
+0.584
+0.506
+0.407
+0.444
+fraud ulb kgl
+-0.062
+0.714
+0.625
+0.701
+0.679
+0.706
+0.75
+0.72
+0.762
+0.728
+-0.124
+0.688
+Avg Savings
+0.37
+0.461
+0.378
+0.463
+0.431
+0.389
+0.391
+0.46
+0.484
+0.465
+0.345
+0.388
+Avg Rank
+5.08
+4.17
+9.42
+4.33
+8.17
+9.17
+6.75
+4.08
+4.58
+3.83
+9.25
+9.08
+regression as level-1 classifier.
+Outcome 2: cost-insensitive classifiers do not perform well when costs are known, even in stacking
+As was pre-
+viously shown in [20] cost-insensitive classifiers, having no way to account for differences in misclassification costs,
+typically perform worse than cost-sensitive models when evaluated using cost-based performance metrics. In our study,
+we observed yet another confirmation to this in the context of heterogeneous ensembles, where base-learners were cost-
+sensitive and meta-learners were cost-insensitive. It is, however, surprising that applying MEC-weights has no positive
+impact on the performance of these cost-insensitive meta-learners. So we conclude that the transfer of cost information
+via cost-sensitive decision-making of the base-classifiers, and via MEC-weights was not sufficient to influence the final
+decision of the meta-learner. And even though application of MEC-weights to the meta-inputs makes the meta-level cost-
+sensitive, the performance of this method is inferior to unweighted stacking models. We can hypothesise that it may be
+different if the meta-learner used misclassification costs internally, but this questions is left for future research.
+Limitations
+Our current work is not without limitations, which we address below. The choice of the algorithms to be
+used in stacking is likely to impact its performance. In order to keep our experiments manageable, we limited ourselves
+to algorithms used previously in the cost-sensitive stacking literature. No parameter tuning was performed to preserve
+the same base-classifier composition across domains. Those familiar with SVM classifiers could remark that not tuning
+this algorithm is a mistake. We are aware of this limitation, which resulted in possibly poor comparative performance of
+SVM-based ensembles, however the purpose of the work was to ensure that the ensembles compared differ in only one
+thing, which is the inclusion of cost-sensitive decision-making into different levels of the ensemble. We are interested
+in relative performance of stacking setups, not in optimal performance of every learning algorithm on every domain. In
+order to perform statistical tests, we had to ensure that the classifiers were the same in every ensemble for every dataset,
+while parameter tuning will have resulted in different parameter settings on different datasets, which would have prevented
+us from performing statistical comparison. In future work we may experiment with homogeneous stacking, where the
+diversity of the ensemble will be created by hyperparameter tuning of the base classifiers.
+7
+Conclusions
+Stacking is a well established state-of-the-art ensemble method, that has been widely applied to many application domains.
+In this work we provide insights into ways to make stacking cost-sensitive. We compare 90 stacking models built with
+15 different compositions of the stacking ensemble using 6 well known classification algorithms. We evaluate on 12 real-
+world cost-sensitive problems with clearly defined, non-synthetic, instance-dependent misclassification costs. In contrast
+12
+
+to the absolute majority of cost-sensitive literature, our experimental results demonstarate that for the best results, not one,
+but two layers of cost-sensitive decision-making are required.
+We also found that applying MEC-weights to the training inputs of the level-1 classifier in stacking did not significantly
+change the performance of stacking models where the level-1 algorithm applied the default decision threshold to classify.
+Moreover, MEC-weighted stacking models where both levels were cost-sensitive performed worse than the unweighted
+stacking of the same type, indicating that two levels of cost-sensitivity is sufficient for good performance.
+Another contribution of our work is the consolidation of all publically available datasets with record-dependent costs
+in one place. In addition to that we derive instance-dependent costs for the well known credit-g dataset from the UCI
+repository.
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+A
+Additional results
+A.1
+Pairwise comparison of stacking setups of type-3 and type-1.
+Table 7: Wilcoxon test statistics (p-values).
+Level-1 algorithm
+type-3
+type-3 acc
+type-3 exp
+type-3 ln
+type-3 sq
+Ada
+type-1
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 acc
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 exp
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 ln
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 sq
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+DT
+type-1
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 acc
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 exp
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 ln
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 sq
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+KNN
+type-1
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 acc
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 exp
+0.0 (0.0)
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+0.0 (0.0)
+type-1 ln
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+0.0 (0.0)
+0.0 (0.0)
+LR
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+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 acc
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+0.0 (0.0)
+0.0 (0.0)
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+0.0 (0.0)
+type-1 exp
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+0.0 (0.0)
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+0.0 (0.0)
+0.0 (0.0)
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+0.0 (0.0)
+RF
+type-1
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+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-1 acc
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+0.0 (0.0)
+0.0 (0.0)
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+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+SVM
+type-1
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+0.0 (0.0)
+8.0 (0.05)
+1.0 (0.0)
+6.0 (0.03)
+type-1 acc
+0.0 (0.0)
+0.0 (0.0)
+8.0 (0.05)
+1.0 (0.0)
+6.0 (0.03)
+type-1 exp
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+0.0 (0.0)
+8.0 (0.05)
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+0.0 (0.0)
+8.0 (0.05)
+1.0 (0.0)
+6.0 (0.03)
+15
+
+A.2
+Pairwise comparison of stacking setups of type-3 and type-2.
+Table 8: Wilcoxon test statistics (p-values).
+Level-1 algorithm
+type-3
+type-3 acc
+type-3 exp
+type-3 ln
+type-3 sq
+Ada
+type-2
+0.0 (0.0)
+0.0 (0.0)
+3.0 (0.01)
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+0.0 (0.0)
+DT
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+3.0 (0.01)
+type-2 acc
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+3.0 (0.01)
+3.0 (0.01)
+3.0 (0.01)
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+type-2 exp
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+2.0 (0.01)
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+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+0.0 (0.0)
+type-2 sq
+0.0 (0.0)
+4.0 (0.01)
+3.0 (0.01)
+3.0 (0.01)
+3.0 (0.01)
+KNN
+type-2
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+8.0 (0.05)
+type-2 acc
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+type-2 exp
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+8.0 (0.05)
+type-2 ln
+5.0 (0.02)
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+type-2 sq
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+7.0 (0.04)
+8.0 (0.05)
+LR
+type-2
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+0.0 (0.0)
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+0.0 (0.0)
+0.0 (0.0)
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+0.0 (0.0)
+0.0 (0.0)
+RF
+type-2
+0.0 (0.0)
+3.0 (0.01)
+4.0 (0.01)
+3.0 (0.01)
+4.0 (0.01)
+type-2 acc
+0.0 (0.0)
+3.0 (0.01)
+3.0 (0.01)
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+3.0 (0.01)
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+4.0 (0.01)
+3.0 (0.01)
+4.0 (0.01)
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+3.0 (0.01)
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+0.0 (0.0)
+3.0 (0.01)
+3.0 (0.01)
+3.0 (0.01)
+4.0 (0.01)
+SVM
+type-2
+6.0 (0.03)
+13.0 (0.16)
+25.0 (0.85)
+13.0 (0.16)
+20.0 (0.49)
+type-2 acc
+6.0 (0.03)
+14.0 (0.19)
+24.0 (0.77)
+14.0 (0.19)
+20.0 (0.49)
+type-2 exp
+6.0 (0.03)
+13.0 (0.16)
+24.0 (0.77)
+13.0 (0.16)
+19.0 (0.43)
+type-2 ln
+6.0 (0.03)
+13.0 (0.16)
+25.0 (0.85)
+13.0 (0.16)
+20.0 (0.49)
+type-2 sq
+6.0 (0.03)
+13.0 (0.16)
+25.0 (0.85)
+13.0 (0.16)
+19.0 (0.43)
+16
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+Gravastar-like black hole solutions in q-theory
+M. Selch, J. Miller, and M.A.Zubkov
+Ariel University, Ariel, 40700, Israel
+(Dated: January 10, 2023)
+We present a stationary spherically symmetric solution of the Einstein equations, with a source
+generated by a scalar field of q-theory.
+In this theory Riemannian gravity, as described by the
+Einstein - Hilbert action, is coupled to a three - form field that describes the dynamical vacuum.
+Formally it behaves like a matter field with its own stress - energy tensor, equivalent to a scalar
+field minimally coupled to gravity. The asymptotically flat solutions obtained to the field equations
+represent black holes. For a sufficiently large horizon radius the energy density is localized within
+a thin spherical shell situated just outside of the horizon, analogous to a gravastar. The resulting
+solutions to the field equations, which admit this class of configurations, satisfy existence conditions
+that stem from the Black Hole no - hair theorem, thanks to the presence of a region in space in
+which the energy density is negative.
+I.
+Introduction
+General Relativity (GR) is based on the axiom that the gravitational field is encoded by the geometry of spacetime:
+in a perfect vacuum spacetime is flat, while massive objects distort the surrounding spacetime from being otherwise
+flat to having a curved geometry. Test particles move along geodesics of the background spacetime in a way such
+that the trajectory depends on the geometry of the spacetime. In flat space the trajectory is a straight line, while in
+curved space it gets accelerated away from straight line motion. This is the equivalence principle: the gravitational
+field is equivalent to acceleration in the background spacetime. In this manner Newton’s ‘action at a distance’ theory
+of gravity is replaced by Einstein’s field theory approach, in which the field is the very geometry of the spacetime.
+The vacuum Einstein field equations contain a cosmological constant [1], which relates the large-scale expansion of
+the universe in cosmological models. In turn the cosmological constant is related to the vacuum energy density [2–6].
+Its value according to astronomical observations lies at a typical energy scale of the order of 10−3 eV [7–9], while its
+range of values as inferred from theoretical models is much larger [10–14].
+Volovik and Klinkhamer have suggested [15, 16] that the smallness of the observed vacuum energy density can
+be explained on the basis of a thermodynamic argument by which the vacuum energy density is exactly canceled in
+equilibrium (perfect quantum vacuum). q-theory appears as a low-energy effective theory [17–20] as opposed to a
+purely fundamental theory [10–14], and as a result contributions to the vacuum energy density become suppressed at
+macroscopic scales and reside in small perturbations of the equilibrium state. In the Klinkhamer Volovik model, the
+effective vacuum energy density that enters the low-energy field equations is described by a vacuum field variable. At
+high energies it may be described by the 3-form field Aβγδ, which is antisymmetric with respect to permutation of
+indices. From this tensor the scalar q-field, the low energy vacuum variable, is composed as q2 = − 1
+24FαβγδF αβγδ,
+where Fαβγδ = ∇[αAβγδ] is the field strength. The equilibrium value of q alters if the vacuum is perturbed towards a
+new equilibrium state. More details can be found in §II.
+Black holes (BHs) are unstable due to several effects. They may evaporate gradually resulting in Hawking radiation
+[21]. Alternatively a BH may undergo a transition to a white hole [22], through quantum mechanical tunneling from
+inside a trapped region to an anti-trapped region. Another possible outcome is the formation of a vacuum star. In this
+model the event horizon takes the form of a boundary between different phases of the quantum vacuum [23]. There
+are a number of similar phenomena between semi-metals and BHs, for example the event horizon emerging on the
+boundary between type I and type II Weyl semimetals. Volovik [24] has discussed an analogous process that occurs
+in Dirac and Weyl semimetals that suggests the viability of the formation of a gravastar or vacuum star after vacuum
+reconstruction, once Hawking radiation has been ended. According to [24] (and unlike conventional gravastars [25])
+the gravastar admits three distinct regions: the vacuum inside the Cauchy horizon with the de Sitter metric, the
+vacuum inside the thin shell between the Cauchy horizon and the event horizon, and the vacuum outside the event
+horizon with the ordinary Schwarzschild metric. Even classically, BHs may be unstable. Regarding it as an ‘excited
+state’, decay due to exponential growth of the metric (plus soliton) perturbations becomes possible [26].
+The geometry of the spacetime in a neighbourhood of a gravastar can be described using the action of a q-field
+coupled to gravity, where the q-field, as mentioned above, is related to the energy density of the quantum vacuum
+[27, 28]. The goal of this work is to show (by numerical means) that (static and spherically symmetric) “scalar-haired”
+BHs exist within q-theory induced by the scalar q-field minimally coupled to gravity. The solutions are discussed
+thoroughly and are interpreted within the context of gravastars. A similar (numerical) calculation by Nucamendi and
+Salgado can be found in refs. [29], in which solutions to the field equations are derived for the case of a scalar field
+arXiv:2301.02914v1  [gr-qc]  7 Jan 2023
+
+2
+coupled minimally as well, in a generic static, spherically symmetric and asymptotically flat spacetime very similar
+but not identical to our considerations within q-theory. There, “scalar-hair” BH solutions were shown to exist.
+As mentioned above, such BH solutions admit non-trivial “hair” associated with the scalar field. A general BH
+“no-hair” conjecture was originally proposed by Ruffini and Wheeler [30] (see also Hawking 1975 [21] for a thorough
+pedagogical overview). A set of conditions arise from no-hair theorems [31] for the existence of a solution to the
+Einstein equations with a scalar field source, one of which is the the no-hair integral, defined in (60), of the solution.
+It is shown in §IV that these criteria are satisfied for the solutions obtained in this work.
+For a given spacetime the presence of an event horizon can be inferred from analyzing ingoing null trajectories,
+as explained in §III. A set of coordinates convenient for both tracking null trajectories and describing the whole
+neighborhood of an event horizon are Painlevé-Gullstrand (PG) coordinates, originally suggested in [32, 33]. Below
+can be found a brief description of how they are derived and the manner in which PG coordinates describe null
+trajectories. More complex spacetime tensors including the stress-energy and Einstein tensors in PG coordinates are
+given in §III A.
+PG coordinates were originally suggested as an alternative to Schwarzschild coordinates for describing radial null
+trajectories in Schwarzschild spacetime. The unique solution of the Einstein equation that is spherically-symmetric,
+stationary, non-spinning with no net charge is the Schwarzschild metric with the form
+ds2 = −fdt2 + 1
+f dr2 + r2dΩ2,
+(1)
+where dΩ2 = dθ2 +sin2 θdϕ2 is the line-element of a unit two-sphere and f = 1−2M/r is the Schwarzschild term with
+M being the mass of the background. The four-velocity U µ = dxµ/dτ ≡ ˙xµ (τ being proper time along the worldline)
+satisfies the normalization condition −1 = gµνU µU ν = −f ˙t2 + f −1 ˙r2 = U 2 �
+−f + f −1(dr/dt)2�
+where U ≡ dt/dτ.
+The quantity ε = −gµνξµU ν = fU is a constant of motion, since ξµ = (∂/∂t)µ is a timelike Killing field. Accordingly,
+the four velocity of a radially outgoing or ingoing spherically-symmetric worldline is
+U µ =
+� ε
+f , −
+�
+ε2 − f, 0, 0
+�
+.
+(2)
+In PG coordinates, the time coordinate, denoted tp to distinguish it from the t in Schwarzschild coordinates, is the
+proper time along the worldline of the geodesic. As such the four velocity now has the more natural form
+U µ
+p =
+� ˙tp, ˙r, ˙θ, ˙φ
+�
+=
+�
+1 , −
+�
+ε2 − f, 0, 0
+�
+≡ (1, −v, 0, 0)
+(3)
+where a ‘dot’ refers to a derivative with respect to tp, and
+v =
+�
+ε2 − f
+(4)
+is the radial component of the velocity on the free-falling trajectory. The PG time coordinate tp is related to the
+Schwarzschild time coordinate as
+dtp = εdts +
+�
+ε2 − f
+f
+dr.
+(5)
+After writing the Schwarzschild metric (1) in PG coordinates the Painlevé-Gullstrand metric is obtained with the
+form
+ds2 = −dtp
+2 + 1
+ε2 (dr + vdtp)2 + r2dΩ2.
+(6)
+Note that as pointed out in [34], the form of the metric in (6) is somewhat analogous to the conserved Newtonian
+energy
+E = 1
+2
+� dr
+dtp
+�2
++ Φ(r)
+(7)
+where Φ(r) = −M/r is a Newtonian type potential and E = (ε2 − 1)/2 is constant. If the particle falls from rest at
+infinity, ε = 1, E = 0, such that (6) reduces to the standard Painlevé Gullstrand metric
+ds2 = −dtp
+2 +
+�
+dr +
+�
+2M
+r dtp
+�2
++ r2dΩ2.
+(8)
+
+3
+Both forms of the metric in (6) and (8) are regular at the horizon r = 2M, ergo the spacetime geometry inside and
+outside the horizon of a black hole can be related without the emergence of any singularities. As explained in [34], the
+Newtonian type energy motivates the following ansatz for the metric in the generalized Painlevé-Gullstrand form:
+ds2 = −dtp
+2 +
+1
+1 + 2E(tp, r)
+�
+dr + v(tp, r)dtp
+�2
++ r2dΩ2,
+(9)
+where
+v(tp, r) =
+�
+2E(tp, r) + 2m(tp, r)
+r
+.
+(10)
+Here E and M are not constant values, rather they are functions of tp and r. The metric in (10) is the one used in
+this paper, but without an explicit dependence on tp, namely only stationary solutions are considered.
+The metric signature is taken to be (−1, 1, 1, 1), and we use natural units, namely c = ℏ = 1 is assumed. In the
+opening section we defer setting the Newtonian constant G = 1 for the purpose of offering clarity in our calculations,
+but later it will be set to unity.
+This paper is structured in the following way. In section II we introduce our model for q-theory, which is effectively
+a scalar field theory with a double-well potential interaction, minimally coupled to Einstein gravity.
+In section
+III the Einstein equations are given for the case of a static spherically symmetric spacetime, in two different sets
+of coordinates: generalized Painlevé-Gullstrand, and those that shall be referred to as generalized Schwarzschild
+coordinates. In section IV the restrictions on solutions to the field equations are explain, which arise due to the no-
+hair theorems that hold for scalar field theories minimally coupled to gravity. Section V contains a detailed discussion
+of one specific static and spherically symmetric q-theory solution to the field equations. Section VI builds on section V,
+containing a local scan of the space of solutions around the solution considered in section V. In section VII instabilities
+of the obtained solutions are addressed, both due to classical perturbations as well as due to Hawking radiation. We
+end our work in VIII with a conclusion of our findings.
+II.
+The model under consideration
+In this paper we consider a gravitating dynamical vacuum of the type introduced in Refs. [15, 16]. One way to
+describe such a system is through a 3-form field Aβγδ, antisymmetric with respect to permutation of indices. From a
+stand point, this system can be considered to be matter described by a field Aβγδ, interacting with the gravitational
+field. The secondary scalar field q, which is the effective degree of freedom at low energies, is composed of a three
+form field Aβγδ as
+q2 = − 1
+24FαβγδF αβγδ,
+Fαβγδ = ∇[αAβγδ],
+(11)
+Fαβγδ = ±q√−gεαβγδ,
+F αβγδ = ±q
+1
+√−g εαβγδ ,
+(12)
+where εαβγδ and εαβγδ are completely antisymmetric, namely ε0123 = 1 and ε0123 = −1. The square brackets denote
+antisymmetrization of indices. From these relations it follows that
+δq
+δgαβ = 1
+2qgαβ .
+(13)
+The action of the model has the form
+S =
+�
+d4x√−g
+�
+R
+16πG − ϵ(q) − 1
+2gαβ∇αq∇βq
+�
+(14)
+where G is Newton’s constant. ϵ is a polynomial function in q that has the form
+ϵ(q) = λ
+4
+�
+q4 − 1
+aGq2
+�
+.
+(15)
+λ and a are real numbers assumed to be O(1)-parameters with λ, a > 0. The scale associated with the potential
+function is due to G and, therefore, it is the Planck scale.
+
+4
+Variation of the action with respect to the metric results in the Einstein equations:
+Rαβ − 1
+2gαβR = −8πG
+�
+gαβ
+�
+ρ(q) + 1
+2∇αq∇αq
+�
+− ∇αq∇βq + 2□q δq
+δgαβ
+�
+(16)
+where
+ρ(q) = ϵ(q) − dϵ
+dq
+�1
+2gµν δq
+δgµν
+�
+= ϵ(q) − dϵ
+dq q
+(17)
+follows directly from (13). The function ρ enters the Einstein equations in the same way as the cosmological constant.
+The shift from ϵ to ρ, as well as the final term on the right hand side of the Einstein equations, follow from the relation
+q = q(gµν). A self-sustained quantum vacuum fulfills
+0 = P = −ρ
+(18)
+in thermodynamic equilibrium, where P refers to pressure and ρ is the energy density. This yields, in our example,
+the equilibrium values qeq = 0, ±
+1
+√
+3aG, which satisfy ρ(qeq) = 0. By further analogy with thermodynamics, a chemical
+potential µ may be defined (up to a constant) as
+µ = dϵ
+dq
+����
+q=qeq
+.
+(19)
+Given that the original potential function is even with respect to q, and assuming that the vacuum has a non-trivial
+equilibrium configuration, it shall be assumed that qeq =
+1
+√
+3aG from now on. The energy density function is then
+given by
+ρ(q) = ϵ(q) − µq = λ
+4
+�
+q4 − 1
+aGq2 +
+2
+(3aG)
+3
+2 q
+�
+.
+(20)
+It has a local minimum at q = qmin = −( 1
+√
+3 + 1)
+1
+√
+4aG, a local maximum at q = qmax = (− 1
+√
+3 + 1)
+1
+√
+4aG and another
+local minimum at the equilibrium value q = qeq =
+1
+√
+3aG with ρ(q = qeq) = 0. Consequently, the equilibrium value is
+also a double root of ρ. The other roots are located at q = 0 and q0 = −
+2
+√
+3aG.
+Vacuum stability requires the vacuum compressibility χvac to be positive :
+χ−1
+vac =
+�
+q2 d2ϵ
+dq2
+� ����
+q=qeq
+=
+λ
+6a2G2 ≥ 0 ,
+(21)
+fulfilled for λ, a > 0.
+The energy density function for λ = 1 and different values of a is plotted in Fig. (1) with the black line marking
+the level of vanishing energy density. The potential has the same basic characteristic of two wells: the well containing
+the equilibrium value of the q-field as a minimum with zero energy density, and the well that is deeper and hence
+allows for negative energy densities. The region of negative energy densities starts at q = q0 and ends at q = 0.
+The action in (14) lacks higher derivative terms of the q-field, which usually appear in the effective field theory
+description without fine tuning.
+These terms are (in the absence of an intermediate high-energy physics scale)
+suppressed by the Planck energy scale. As long as the q-field varies slowly (q′ ≪ 1 in units with G = 1), these
+contributions are negligible. This is the approach adopted in the following part of the discussion.
+Variation of the action with respect to the three - form field Aαβγ yields the generalized Maxwell equation
+∇α(√−g
+�
+−dϵ(q)
+dq
+δq
+δFαβγδ
++ □q
+δq
+δFαβγδ
+)
+�
+= 0
+(22)
+⇔ ϵαβγδ∇α
+�
+−dϵ(q)
+dq
++ □q
+�
+= 0
+(23)
+⇔ dϵ(q)
+dq
+− □q = µ
+(24)
+Here µ is the integration constant. Inserting this into the Einstein equations (16) yields
+Rαβ − 1
+2gαβR = −8πG
+�
+gαβ(ϵ(q) − µq + 1
+2∇αq∇αq) − ∇αq∇βq
+�
+(25)
+
+5
+Figure 1. The energy density of the q-field for λ = 1 and different values of a is depicted. While λ sets the absolute scale, a
+determines the depth and separation width of the potential wells.
+which comprises both the gravitational field and the matter (generalized Maxwell) equations. The fact that the q-field
+is not fundamental, but rather only an effective degree of freedom leads to the effective replacement of ϵ with ρ. In
+equilibrium, the scalar-field value corresponds to a minimum of the energy density ρ, which is located at the value
+zero.
+This shows that the problem reduces to that of solving the (modified) Einstein equations given by (25).
+It is
+equivalent to a scalar field theory minimally coupled to gravity with a scalar field potential ρ.
+III.
+Static and spherically symmetric solutions of the Einstein equations
+The discussion is now about solving the static, spherically symmetric Einstein equations in order to find asymptotically
+flat solutions that describe a BH with a non-trivial q-field behavior. The q-field, as well as all the other functions,
+depend on a single coordinate only, which we choose to be the standard radial coordinate. The q-field is expected
+to relax to the equilibrium value at large values of the radial coordinate, and to deviate from the equilibrium value
+approaching smaller and smaller values for the radial coordinate.
+In the following, two different ansätze are used for the metric in order to solve the Einstein equations:
+Gα
+β = 8πG(Tq)α
+β, Gα
+β = Rα
+β − 1
+2gα
+βR
+(26)
+where the Einstein tensor is Gα
+β and the q-field energy-momentum tensor is (Tq)α
+β. Primes above symbols label
+derivatives with respect to the radial variable, throughout this work.
+A.
+Generalized Painlevé coordinates
+The first ansatz is a generalized Painlevé-Gullstrand metric with the form
+ds2 = −dt2 +
+1
+1 + 2E(r)(dr + v(r)dt)2 + r2dΩ2
+(27)
+and
+v(r) =
+�
+2E(r) + 2Gm(r)
+r
+(28)
+while
+dΩ2 = dθ2 + sin2(θ)dφ2 .
+(29)
+
+energy density p(q)
+energy density pgi,zoomed
+14.0
+a = 0.1
+a = 0.1
+a = 0.2
+7.5
+a = 0.2
+a=0.5
+a=0.5
+50
+a=l
+a=l
+a=2
+5.D 
+a=2
+40
+a= 5
+a= 5
+25
+(b)d
+30
+(b)d
+0.D
+24
+2.5
+5.0
+0
+-7.5
+-10
+-2
+L-
+2
+10.0
+4
+E-
+0
+1
+3
+4
+1.0
+0.5
+0.D
+0.5
+1D
+1'5
+2D
+a
+q6
+As explained in the introductory remarks, the terms E and v in the metric are related to the kinematical quantities
+of a particle in motion in the background. As elucidated in [34], v(r) may be interpreted as the velocity of a freely
+falling test particle as it falls in towards a (spherically symmetric) gravitating object from infinity, while E(r), at least
+asymptotically, can be related to the normalized total energy of a test particle (E(∞) = (e2−1)
+2
+, where e represents
+the total energy per unit rest mass of a test particle at infinity). This motivates labeling v a velocity function, and E
+an energy function.
+For the first metric ansatz, the non-vanishing Einstein tensor components are
+Gt
+t = −2Gm′
+r2
+, Gt
+r =
+−2E′
+r(1 + 2E)
+�
+2E + 2Gm
+r
+,
+(30)
+Gr
+r = −2rE′ + 4E′Gm − 4EGm′ − 2Gm′
+(1 + 2E)r2
+,
+(31)
+Gθ
+θ = Gφ
+φ = (3r2(1 − 2Gm
+r
+)(E′)2 + 3r(1 + 2E)E′Gm′ − (r + m)(1 + 2E)E′
+(32)
+− r2(1 − 2Gm
+r
+)(1 + 2E)E′′ − r(1 + 2E)2Gm′′)/((1 + 2E)2r2)
+The energy momentum tensor (Tq)α
+β for the q-field takes the form
+(Tq)α
+β = −(gα
+β(ρ(q) + 1
+2gαβ∇αq∇βq) − gαγ∇γq∇βq)
+(33)
+with non-vanishing components
+(Tq)t
+t = (Tq)θ
+θ = (Tq)φ
+φ = −(ρ(q) + 1
+2(1 − 2Gm
+r
+)(q′)2),
+(34)
+(Tq)t
+r =
+�
+2E + 2Gm
+r
+(q′)2,
+(35)
+(Tq)r
+r = −(ρ(q) − 1
+2(1 − 2Gm
+r
+)(q′)2).
+(36)
+The Einstein equations can thus be brought into the following form
+Gm′ = 4πGr2(ρ(q) + 1
+2(1 − 2Gm
+r
+)(q′)2))
+(37)
+2E′
+1 + 2E = −8πGr(q′)2
+(38)
+q′′ = (1 − 2Gm
+r
+)−1(−2q′
+r + 2Gmq′
+r2
++ 8πGrρq′ + dρ
+dq ).
+(39)
+The first two equations are obtained from the radial and time components of the Einstein equations. Using these to
+simplify the equation due to the angular components leads to the third equation.
+The second equation can be solved for E. With the definition F = ln(1 + 2E) and E(∞) = lim
+r→∞ E(r) the result is
+F(r) = ln(1 + 2E(∞)) + 8πG
+� ∞
+r
+s(q′(s))2 ds .
+(40)
+This leaves the first and third equations, which are solved numerically.
+B.
+Generalized Schwarzschild coordinates
+An analogous procedure for the second ansatz yields the metric in the form
+ds2 = −f(r)dt2 +
+1
+h(r)dr2 + r2dΩ2,
+(41)
+
+7
+which shall be referred to as generalized Schwarzschild coordinates.
+From this form of the metric the following
+non-vanishing components of the Einstein tensor are derived:
+Gt
+t = rh′ + h − 1
+r2
+, Gr
+r = rhf ′ + hf − f
+r2f
+,
+(42)
+Gθ
+θ = Gφ
+φ = −1
+4(rh(f ′)2 − 2rfhf ′′ − 2fhf ′ − (rff ′ + 2f 2)h′)/(rf 2)
+(43)
+The non-vanishing components of the corresponding energy momentum tensor are
+(Tq)t
+t = (Tq)θ
+θ = (Tq)φ
+φ = −(ρ(q) + 1
+2h(q′)2), (Tq)r
+r = −(ρ(q) − 1
+2h(q′)2).
+(44)
+The Einstein equations can finally be brought into the following form
+1 − h − rh′ = 8πGr2(ρ + 1
+2h(q′)2)
+(45)
+f ′
+f − h′
+h = 8πGr(q′)2
+(46)
+q′′ = 1
+h
+dρ
+dq − h′
+h q′ − 2
+r q′ − 4πGr(q′)3.
+(47)
+The first two equations are obtained from the radial and time components of the Einstein equations, and subsequently
+used to simplify the relation for the angular components. This results in the third equation. The second equation can
+be solved and reads, with k = ln(f) and the assumption k(r = ∞) = 0, as
+k(r) = −
+� ∞
+r
+�h′(s)
+h(s) + 8πGs(q′(s))2)
+�
+ds .
+(48)
+The connection between the two parameterizations is provided by the relation
+h(r) = 1 − 2Gm(r)
+r
+(49)
+according to which the first and third equations of the two ansätze are identical. For convenience and for the purpose
+of being consistent with other literature on this topic, we introduce the notation
+f(r) = h(r)e2δ(r)
+(50)
+and
+˜δ(r1, r2) =
+� r2
+r1
+4πGs q′(s)2 ds .
+(51)
+C.
+Black hole spacetime characteristics
+In §V we will start from the assumptions that there exists an event horizon at a certain radial coordinate value r = rh,
+and that spacetime is asymptotically flat. It then follows that
+δ(r) = −˜δ(rh, ∞) + ˜δ(rh, r),
+F(r) = ln(1 + 2E(∞)) + 2˜δ(r, ∞).
+(52)
+In order to check for the existence of an event horizon, radial null geodesics in generalized Painlevé-Gullstrand
+coordinates or generalized Schwarzschild coordinates need to be discussed. The requirement ds2|θ=θ0,φ=φ0 = 0 leads
+to
+−dt2 + (dr + vdt)2
+1 + 2E
+= 0 ⇔ dr
+dt = ±
+√
+1 + 2E − v
+(53)
+in generalized Painlevé-Gullstrand coordinates and
+−fdt2 + 1
+hdr2 = 0 ⇔ dr
+dt = ±
+�
+fh
+(54)
+
+8
+in generalized Schwarzschild coordinates. The plus sign is associated with outward motion, while the minus sign
+corresponds to inward motion. If dr
+dt < 0 for both signs and every r < r0, then r0 marks an event horizon. More
+precisely, it marks the locus of an apparent horizon. In a static (or more generally in a stationary) spacetime, the
+apparent and the event horizon coincide. Generalized Schwarzschild coordinates are only valid on one side of the
+event horizon, since they become singular at an event horizon (with at least either f = 0 or h = 0). In the vicinity of
+an event horizon, h(r) ≪ 1, and the outward velocity of a massless particle may be approximated by
+(dr
+dt )out =
+�
+1 + 2E(r) − v(r) =
+�
+1 + 2E(r) −
+�
+2E(r) + 2Gm
+r
+(55)
+=
+�
+1 + 2E(r) −
+�
+2E(r) + 1 − h(r) =
+h(r)
+2
+�
+1 + 2E(r)
++ O(h(r)2).
+As a result, r = rh marks an event horizon if h(rh) = 0. Taking a look back at the Einstein equations in the generalized
+Painlevé-Gullstrand ansatz (39), it can be observed that, although the coordinates are regular on the horizon, the
+inclusion of a scalar field in Einstein gravity leads to singular behavior as the horizon is approached. This is manifest
+in the third Einstein equation for both ansätze.
+Of particular interest about solutions in q-theory is the distribution of energy density inside and outside the event
+horizon, as well as the question of whether or not out solutions are singular at the origin.
+To answer these questions it is convenient to define
+−(Tq)t
+t = V (r) + T(r),
+V (r) = ρ(q(r)),
+T(r) = 1
+2h(r)(q′(r))2
+(56)
+such that the energy density into a potential part V and a kinetic part T.
+The latter quantity may be deduced from the Kretschmann invariant
+K(r) = RµνρσRµνρσ
+(57)
+as r → 0, where Rµνρσ is the Riemann curvature tensor. For generalized Schwarzschild coordinates it takes the
+explicit form
+Kq−theory(r) = 1
+r4 [4r4h2(r)(δ′(r))4 + 8r4h2(r)(δ′(r))2δ′′(r) + 4r4h2(r)(δ′′(r))2
++ 8r2h2(r)(δ′(r))2 + r4(h′′(r))2 + (9r4(δ′(r))2 + 4r2)(h′(r))2
++ 4(3r4h(r)(δ′(r))3 + 3r4h(r)δ′(r)δ′′(r) + 2r2h(r)δ′(r))h′(r)
++ 2(2r4h(r)(δ′(r))2 + 2r4h(r)δ′′(r) + 3r4δ′(r)h′(r))h′′(r) + 4(h(r) − 1)2].
+(58)
+For Schwarzschild spacetime with h(r) = f(r) = 1 − 2GM
+r
+and Schwarzschild mass parameter M the Kretschmann
+scalar reads
+KSchwarz(r) = 48G2M 2
+r6
+.
+(59)
+Black hole spacetimes with minimally coupled scalar fields underlie a series of criteria to be fulfilled as dictated by
+the black hole no-hair theorems. These criteria are the topic of the next chapter. From now on we will work in units
+with G = 1.
+IV.
+Black hole no-hair theorems
+In this section the BH no-hair theorems are explained, that restrict the set of allowed solutions of scalar hair black
+holes (SHBH’s) in curved spacetime, which is the same class of solutions considered in this work.
+The BH no-hair theorems, as discussed in [31, 35], assert the following:
+1. In the absence of event horizons there exist no non-trivial, regular scalar soliton solutions, that satisfy the dom-
+inant energy condition but violate the strong energy condition, at every point in asymptotically flat spacetimes.
+2. In the presence of an event horizon in a static, spherically symmetric and asymptotically flat spacetime, no
+non-trivial regular scalar-field solution exists outside the event horizon, if the dominant energy condition holds
+but the strong energy condition is violated.
+
+9
+3. Any SHBH solution must necessarily have V (rh) < 0 where rh denotes the radial location of the event horizon,
+and V is the potential energy density of the scalar field. The vicinity of the event horizon is enveloped by a
+region of negative scalar-field energy density.
+The strong energy condition is defined by the requirement that Tαβkαkβ ≥ 0, where Tαβ is the covariant energy
+momentum tensor and kα is an arbitrary null vector field, and the requirement that −T α
+βpβ must be a future
+pointing causal vector field whenever pα is. The strong energy condition stipulates that for every timelike vector field
+uα, the condition (Tαβ − 1
+2Tµνgµνgαβ)uαuβ ≥ 0 holds.
+If in the first case, spherical symmetry is assumed with a positive scalar field potential, hence the dominant energy
+condition does not need to be imposed in order to infer the absence of non-trivial scalar field solutions. From the
+first two criteria, it is immediate that within the domain of outer communications (the region from the event horizon
+to asymptotically flat infinity), the scalar field energy density must have negative values. The third criterion then
+specifies where this region of negative energy density must be located.
+This leaves as the only possibility for a non-trivial SHBH solution in the considered setup a q-field which asymptot-
+ically relaxes to its equilibrium value but sweeps over field values corresponding to negative potential energy density
+as the horizon is approached, for sure in the horizon proximity.
+Both SHBH’s, and scalar solitons, have to fulfill an integral equation as a necessary condition for existence, as
+derived from a scaling argument [31]. These conditions are written below in our notation convention. A necessary
+condition for the existence of a scalar soliton (scalaron) in curved spacetime, in a non BH geometry is
+� ∞
+0
+4πr2 exp (2δ(r))
+�
+Eflat
+kin (r) + 3V (r)
+�
+dr = 0
+(60)
+It comprises the flat space kinetic energy density Eflat
+kin (r) = 1
+2(q′(r))2, as well as the potential energy density V (r) =
+ρ(q(r)). Analogously, a necessary condition for the existence of a SHBH solution is
+� ∞
+rh
+4πr2 exp(2δ(r))
+�2rh
+r
+�
+1 − m(r)
+r
+�
+− 1
+�
+Eflat
+kin (r) +
+�2rh
+r
+− 3
+�
+Vpot(r))dr = 0.
+(61)
+where rh is the radial coordinate of the event horizon. The fulfillment of this latter condition is taken as a tool to
+fine-tune the shooting parameter q(rh) (introduced and discussed in section V), in finding SHBH solutions. In the
+limit rh → 0 (60) is recovered. For future convenience we introduce the function
+nhf(r) =
+� r
+rh
+s2 exp(2δ(s))
+�2rh
+s (1 − m(s)
+s
+) − 1
+�
+Eflat
+kin (s) +
+�2rh
+s
+− 3
+�
+V (s)) ds
+(62)
+which is then supposed to fulfill lim
+r→∞ nhf(r) = 0.
+V.
+A representative SHBH solution for q-theory
+The existence of SHBH solutions has been known for some time and was first considered numerically in [29] for a scalar
+field minimally coupled to gravity in a double well scalar field interaction potential. Subsequently, these solutions
+have been discussed in the framework of the of isolated horizons [36]. The latter formalism has been treated in detail
+in [26] and references cited therein. The difference between this work and [29] is in the more restrictive potential
+of q-theory. There are only two free parameters, the absolute scale provided by λ and the depth or separation of
+the wells as parameterized by a. A shift in the q-field, accompanied by a corresponding change in the scalar energy
+density function, does not lead to a further quantitative change in the solution as induced by the shift in the q-field
+itself. It can be seen as fixed by the condition ρ(q = 0) = 0. The scalar potential in [29] has three free parameters
+and allows for adjusting the positions of the wells, independently. Consequently, the solutions found and presented
+within that work cannot be used for q-theory, since they lie outside the parameter space spanned by λ and a. In the
+following, we replace the parameter a by the location of the minimum qeq =
+1
+√
+3a of the shallow potential well.
+We use Python for plotting and numerically solving the Einstein equations for the minimally coupled q-field. The
+solver is non-adaptive and makes use of a refined (fourth order) Runge-Kutta method. Refined means that the grid
+size close to the horizon is smaller than further away. This is due to the observation that the equations become
+singular at the horizon. We will nevertheless employ a prescription of how to start “close” to the horizon.
+This comes about since only one of the three boundary conditions of the differential equations (q(r = rbound),
+q′(r = rbound) and m(r = rbound) at radial coordinate boundary position r = rbound) is free for an asymptotically
+flat black hole spacetime regular at the horizon which we are searching for.
+This freedom resides in the radial
+
+10
+Figure 2. A representative SHBH solution for q-theory outside the event horizon is shown for a horizon radius of rh = 1, q-field
+potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6.
+coordinate location of the event horizon for a fixed scalar field potential function, as is the case for Schwarzschild
+spacetime. The dependencies of the boundary values on the horizon radius are known precisely only at the horizon
+except for one parameter, the “shooting” parameter. How the singular behavior at the horizon is circumvented is
+explained further below, together with an account of numerical errors. The results have been confirmed by an adaptive
+routine of the NDsolve-method in Mathematica, which is the most accurate. We will still stick with Python in the
+following discussion, since the difference between the solving routines is negligibly small for tiny grid sizes (of the
+order ∼ 105 − 106), as used in Python.
+Both a discussion of how to numerically avoid the singularity in the third Einstein equation at the horizon, as well
+as a discussion of numerical errors are lacking in [29], but will be provided in the following. We shall start with a
+qualitative discussion of a representative solution in this section before focusing on the space of solutions in the next
+section.
+A.
+Outside the event horizon
+The method of finding solutions for q-theory, makes use of previously discovered solutions, insofar as they are used
+as a starting point in an interpolation of solutions between the respective parameter spaces. For a horizon radius of
+rh = 1, q-field potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6 (to be
+introduced further below) the solution is shown in Fig. (2) for the region outside the event horizon (also termed the
+domain of outer communications). It is in qualitative agreement with that of [29].
+The q-field starts with a negative value such that q0 < q(r = rh) < 0, which is equivalent to V (r) = ρ(q(r)) < 0, as
+is required by the no-hair theorems. More specifically we have qmin < q(r = rh) < 0. The q-field starts in the right
+half of the deep potential well shown in Fig. (1) with the characteristic points of the double well potential named
+as discussed in section I. It increases into the positive energy density domain, passes the local maximum and relaxes
+asymptotically to the equilibrium value qeq.
+The metric function m(r), termed mass function, starts from the horizon with m(rh) = 1
+2rh, initially decreases as
+
+q-field
+potential and kinetic energy densities versus q'
+0.02
+q (r)
+0.1 -
+Vr vs. T(r) vs. q'(r)
+0.D1
+T(r)
+..
+50V(r)
+0.D
+q(r)
+0.00
+0.02
+0.2
+0.03
+0.3
+15
+25
+0.04 
+0.0
+0.5
+1b
+20
+3.D
+0.0
+S0
+15
+2D
+25
+3.D
+logia(r/rh?
+logia(r/rh?
+evolution of the mass function
+metric functions
+3.D
+h(r)
+25
+f(r)
+5 -
+(μ)g
+20
+m(r)
+0
+5-
+LD 
+ OT-
+0.5
+15 +
+0.D 
+0.0
+0.5
+1D
+15
+2D
+25
+3.D
+0.D
+15
+20
+25
+3.D
+logia(r/rh?
+logia(rfrh?
+relabive ermor of the q-field solution
+no-hair function
+2.00
+0
+f.n.d.,5-p.s.m for g
+f.n.d.,5-p.s.m for q
+175
+-2
+Jogia of the relative 
+-4 
+f.n.d.,5-p.s.m for m
+150
+6
+125
+1LDo
+oT-
+0.75
+-12
+0.50 -
+-14
+0.25
+-16
+00
+0.2
+0.4
+0.6
+0.B
+12
+16
+0.DO
+14
+0.D
+0.5
+15
+20
+25
+3.D
+logia(r/rh?
+logia(r/rh?11
+ρ(q) < 0 until reaching a global minimum. It is located at a radial coordinate value shortly advancing that of q = 0,
+since the kinetic energy density T(r) = 1
+2h(r)(q′(r))2 is positive everywhere outside the horizon. The mass function
+increases until finally approaching its asymptotic value, the ADM mass of the spacetime, from below. The potential
+and kinetic energy densities, as introduced in equation (56), are shown together with the derivative of the q-field on
+the upper right. The potential energy density is seen to be of minor size as compared to the kinetic part.
+Further shown are the metric functions in generalized Schwarzschild coordinates in the central plot on the right
+hand side. As is required by asymptotic flatness we have the limits
+lim
+r→∞ f(r) = 1,
+lim
+r→∞ h(r) = 1,
+lim
+r→∞ δ(r) = 0,
+(63)
+while limr→∞ E(r) remains a free parameter for the generalized Painlevé-Gullstrand coordinates. This free parameter
+is mostly irrelevant for the discussion of SHBH solutions and only of interest for test particle motion. Its variation
+will be discussed in Appendix A but is of no crucial importance elsewhere in the discussion.
+The lowermost plot on the left shows the error sizes. We calculate the relative error of the solution by taking the first
+numerical derivative (f.n.d.) of q, q′ and m and comparing it with the solution as given by q′, q′′ and m′ and obtained
+from the non-adaptive, refined (fourth order) Runge-Kutta algorithm. The first numerical derivative is calculated
+using a central 5-point stencil method (5-p.s.m.). On a formal level this method approximates the derivative of a
+five times differentiable function accurately up to O((∆r)4)-corrections where ∆r represents the grid spacing. The
+grid size is 106. The first four peaks mark the radial coordinate values where the grid size changes abruptly and are
+artificial, since the 5-p.s.m. formula we employed is based on equal grid size spacing. The final peaks for the relative
+errors are due to the appearance of the local extrema of q′ as well as m. Apart from these points the relative error
+can be seen to be smaller than 10−7 throughout.
+The lowermost plot on the right shows the function nhf(r) introduced in the last section in equation (62) which
+indeed fulfills limr→∞ nhf(r) = 0. This latter property was taken to fine-tune the shooting parameter q(rh), though,
+which will be discussed shortly.
+It can be seen that the plot of the relative errors do not reach as far out as the other plots. An asymptotic analysis
+of the third Einstein equation with δq(r) = q(r) − qeq yields the linear approximation
+(δq)′′ = −2(δq)′
+r
++ ρ′ = −2(δq)′
+r
++ λ
+2aδq ,
+(64)
+which has the solution
+δq(r) = c1
+1
+r exp
+�
+−
+�
+λ
+2ar
+�
++ c2
+1
+r exp
+�
++
+�
+λ
+2ar
+�
+.
+(65)
+The asymptotic behavior of the relevant functions can be deduced from the first Einstein equation and the condition
+limr→∞ m(r) < ∞. A first order Taylor expansion of the potential energy density and its derivatives around q = qeq is
+well justified for δq ≪
+1
+√a. The approximation of the third Einstein equation is well justified for 2m(r)
+r
+, 4πr2ρ(q(r)) ≪
+1. Taken together we then obtain (64). The asymptotic solution has an exponentially growing and an exponentially
+depleting part.
+In the search of a solution it is found that when the q-field and the mass function approach their asymptotic
+values they leave them again after some critical value. This can not be avoided and is an artifact of the numerical
+approximation. It induces a non-vanishing coefficient c2 and therefore exponential growth due to numerical uncer-
+tainty. Therefore the exponentially depleting part is fitted onto the functions q, q′ and m after they tend to change
+very slowly by approaching their asymptotic values. This fitting of the free parameter c1 takes place at those radial
+coordinate values where the relative error plots end. At this point δq(r)
+qeq , m(∞)−m(r)
+m(∞)
+, nhf(r) ≪ 1 hold.
+We now turn to the near horizon region. We search for solutions regular at the horizon by imposing limr→rh q′′(r) <
+∞. The third Einstein equation as well as the assumptions of the existence of an event horizon and of asymptotic
+flatness then reduce the freedom of the boundary conditions of q, q′ and m to one parameter. This parameter may
+be declared as the horizon radius.
+After fixing the scalar field potential parameters and thereby the theory, the
+space of solutions is one dimensional and parameterized by rh as is Schwarzschild spacetime. Up to one degree of
+freedom, the boundary values are known at the horizon. The event horizon condition h(r) = 0 on the one hand and
+limr→rh q′′(r) < ∞ on the other hand imply
+m(rh) = rh
+2 ,
+q′(rh) = rh
+dρ
+dq
+(q(rh))
+(1 − 8πr2
+hρ(q(rh))) .
+(66)
+
+12
+Figure 3. A representative SHBH solution for q-theory inside the event horizon is shown for a horizon radius of rh = 1, q-field
+potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6.
+The remaining freedom then resides in the value of q(rh). It is adjusted so as to yield ("shoot" towards) an asymptot-
+ically flat solution and therefore termed shooting parameter. By choosing it more and more accurately the approach
+of q, q′ and m to their asymptotic values may be improved. This suggests that there exists exactly one shooting pa-
+rameter which is appropriate for ensuring asymptotic flatness. We can only approach it to within a certain numerical
+accuracy. As soon as the solutions to the first and third Einstein equations are obtained, the horizon values of E(r)
+and δ(r) may be deduced by integration of the second Einstein equation. They are given by
+E(rh) = ((1 + 2E(∞)) exp(
+� ∞
+rh
+r(q′(r))2dr) − 1)/2,
+δ(rh) = −˜δ(rh, ∞).
+(67)
+The singular behaviour of the third Einstein equation at r = rh is avoided by the prescription r → r(1 + iϵ). We call
+ϵ the shift parameter and choose it such that 0 < ϵ ≪ 1. The quantities plotted in Fig. (2) are then understood as
+the real parts of the functions of the (total, complex valued) solution. The accuracy of the numerical calculations due
+to finiteness of grid size as well as shift parameter is analyzed in Appendix B.
+B.
+Inside the event horizon
+In extension of the plots of Fig. 2, the solution for a horizon radius of rh = 1, q-field potential parameters λ = 1
+and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6 is shown in Fig. 3 for the region inside the event
+horizon. In distinction to the corresponding figure for the outside region the lowermost plot on the right hand side
+highlights the energy function. The relative error size remains below 10−5 for all of the functions q′, q′′ and m′. The
+peaks mark again those radial coordinate values where the grid size changes abruptly. Close to the event horizon it
+has been chosen smaller as in the case of the region outside the event horizon. Of interest is the behaviour in the limit
+where the radial coordinate tends to zero. The solution is found to be singular in the q-field in this limit. The mass
+function, in contrary, tends to a constant value of about limr→0 m(r) = 0.503. An expansion of the third Einstein
+
+q-field
+potential and kinetic energy densities versus q
+0.19B
+40
+0.200
+V(r) vs. T(r vs. q'(r)
+log 1o(T(r)
+log 1o(V(r)
+0.202
+因
+0.204
+0.206
+0.20B 
+10
+.210
+5
+4
+-2
+L-
+4
+-2
+0
+logia(r/rh?
+Iogia(rfrh)
+evolution of the mass function
+metric-functions
+0.504
+log 1o(h(r)
+EOS'0
+4
+2
+0.502 
+(sjuu
+0
+0.501
+azis
+0.50
+4
+0.499 
+6
++ 60
+5
+-4
+E-
+-2
+-1
+4
+E-
+-2
+-1
+logia(r/rh?
+logia(rfrn)
+relabive ermor of the q-field solution
+energy function
+0
+fogia of the relative error
+f.n.d.,5-p.s.m. for q
+2
+f.n.d.,5-p.s.m. for q
+f.n.d.,5-p.s.m. for m
+Jogia(E(r) - E(rh))
+6
+9-
+ot-
+-12
+714
+-16
+-3
+4
+E-
+-2
+-1
+-
+logia(r/rh?
+Iogia(rfrh?13
+Figure 4. The Kretschmann invariant for a representative SHBH solution for q-theory inside and outside the event horizon is
+shown for a horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter
+ϵ = 10−6.
+equation for r ≪ 1 yields the approximate equation
+q′′ = −q′
+r
+(68)
+with solution
+q(r) = d1 log10(r) + d2.
+(69)
+It is found that d1 = 0.0010±0.0001 and d2 = −0.2010±0.0001 by a straight line fit. This implies the approximations
+F(r) ≈ ln(1 + 2E(rh)) + 8π
+� rh
+r0
+r(q′(r))2dr + 8πd2
+1(ln(r0) − ln(r))
+(70)
+= C + 8πd2
+1ln(1
+r )
+⇔ E(r) ≈ exp(C)
+2
+r−8πd2
+1 − 1
+2
+δ(r) ≈ δ(rh) −
+� rh
+r0
+4πr(q′(r))2dr − 4πd2
+1(ln(r0) − ln(r))
+(71)
+= D − 4πd2
+1ln(1
+r ) ⇔ exp(2δ(r)) ≈ exp(2D)r8πd2
+1,
+h(r) ≈ 1 − 2m(0)
+r
+(72)
+for r < r0 ≪ 1 with for the further discussion irrelevant constants C and D. Consequently the limits
+lim
+r→0 q(r) = −∞,
+lim
+r→0 q′(r) = ∞
+(73)
+for the q-field and its derivative and
+lim
+r→0 E(r) = ∞,
+lim
+r→0 h(r) = −∞,
+lim
+r→0 δ(r) = −∞
+(74)
+for the metric functions follow. Since d1 ≪
+1
+√
+8π, E(r) and exp(2δ(r)) vary very slowly as compared to h(r). Therefore
+the behavior of the metric as r → 0 will asymptote that of Schwarzschild spacetime. Especially, very close to the
+center the energy density will change sign again and becomes positive. The metric can not be continued to the radial
+coordinate origin. There is a curvature singularity in the limit r → 0 as is well known for Schwarzschild spacetime.
+The Kretschmann invariants for q-theory and Schwarzschild spacetime are shown in Fig. (4). While they differ visibly
+asymptotically far from the horizon where both tend to zero, they show the same
+1
+r6 -divergence as r → 0.
+After discussing one solution in detail we proceed with a local scan of the space of solutions around that just
+presented. Both grid size and shift parameter will no longer be mentioned from now on and chosen to be very large
+in the former case and negligibly small in the latter as has been done within this section.
+
+Kretschmann invariant
+30 
+K5chaz(r)
+25 
+21
+((u)xjDTbo)
+15
+1
+5 -
+ 5-
+-10 
+E-
+-2
+-1
+1
+fogia(rfrh?14
+VI.
+The parameter space of SHBH solutions for q-theory
+The qualitative features of the representative solution presented in the previous section are common to all SHBH
+solutions in q-theory. The space of solutions is parametrized by the horizon radius rh as well as the q-field potential
+parameters λ and qeq (or as well a). In order to understand the differences between solutions we perform a local
+scan around the representative solution in the space of solutions and extract different quantities for each individual
+solution, in part in analogy with [29, 36].
+The ADM mass of a (static and spherically symmetric) solution is given by
+MADM(rh, λ, qeq) = lim
+r→∞ mrh,λ,qeq(r).
+(75)
+We decompose it into two contributions
+MADM(rh, λ, qeq) = M Schwarz
+ADM
+(rh) + Mhair(rh, λ, qeq).
+(76)
+The first contribution is the mass parameter of a Schwarzschild black hole of horizon radius rh, M Schwarz
+ADM
+(rh) = rh
+2 .
+It is independent of the scalar field potential parameters, as it describes a (static and spherically symmetric) black
+hole in vacuum. The dressing by the non-constant q-field outside the event horizon yields the non-vanishing additional
+contribution
+Mhair(rh, λ, qeq) = −
+� ∞
+rh
+4πr2T t
+tdr
+(77)
+=
+� ∞
+rh
+4πr2(ρ(q(r)) + 1
+2h(r)(q′(r))2)dr.
+(78)
+Further quantities of interest are the shooting parameter as well as the radial coordinate location relative to rh and
+absolute depth of the global minimum of the mass function as functions of the parameters of the space of solutions.
+The relative radial coordinate location of the global minimum of the mass function is important insofar as it marks
+the region where both the q-field and the mass function vary significantly.
+A.
+Dependence on the horizon radius
+Fig. 5 illustrates the characteristic quantities introduced above for a horizon radius range 10−3 < rh < 103 and scalar
+field potential parameters coincident with those of the representative solution presented in section V. The lower limit
+has been chosen such that the asymptotic dependence on rh is visible. The upper bound is due to lack of precision of
+fitting the asymptotic part of the q-field and mass function. The asymptotic plateau becomes difficult to identify, since
+both q-field and mass function behave less and less smooth in the fitting region. The asymptotics for large horizon
+radii seem to be deducible from the plots as well, though. We will assume that the plots show the true asymptotics
+in both extreme situations rh ≪ 1 and rh ≫ 1.
+We may then draw the following conclusions:
+1) The shooting parameter qshoot(rh) = q(rh) varies significantly. Its asymptotic values are limrh→0 qshoot(rh) =
+0.913qmin and limrh→∞ qshoot(rh) = 0.500qmin, respectively.
+2) For rh ≪ 1 the ADM mass and scalar hair mass converge towards the value
+limrh→0 log10(MADM(rh)), log10(Mscalar hair(rh)) = 0.817. For rh ≫ 1 both ADM mass and scalar hair mass
+increase linearly with the horizon radius and may be parameterized by
+log10(MADM(rh)) = c1 log10(rh) + c2
+(79)
+log10(Mscalar hair(rh)) = c3 log10(rh) + c4
+(80)
+with c1 = 1.0031 ± 0.0001, c2 = −0.1985 ± 0.0001, c3 = 1.0136 ± 0.0001 and c4 = −0.8747 ± 0.0001 obtained
+by a straight line fit. The q-theory ADM mass is a monotonically increasing function of the horizon radius and
+everywhere larger than the corresponding Schwarzschild spacetime ADM mass (which coincides with the mass
+parameter).
+3) The region outside the event horizon where the q-field and mass function vary significantly approaches the event
+horizon relative to its size for rh ≫ 1. So it seems that in this regime the q-field behaves non-trivially only just
+
+15
+Figure 5. The variation of several characteristic quantities of SHBH solutions for q-theory with respect to the horizon radius
+rh is shown for the q-field potential parameters λ = 1 and qeq = 0.158. These quantities include shooting parameter, ADM
+mass, scalar-hair mass, relative location of the global minimum of the mass function as well as the absolute value of the global
+minimum of the mass function.
+outside the horizon and relaxes to its equilibrium value very quickly in the near horizon regime. This region of
+significant change does not approach the horizon infinitely close but relaxes to the value
+limrh→∞ rmin(rh)/rh = 1 + e−0.916 = 1.400.
+In the opposite limit rh ≪ 1 the region of significant change of both q-field and mass function gets pushed
+further and further away from the horizon region. This implies that for the very limit rh → 0 no scalar soliton
+(scalaron) exists (contrary to the conclusions drawn in [29]). Rather the q-field remains constant outside the
+event horizon with a value of qshoot(0) = limrh→0 qshooot(rh) = 0.913qmin. The corresponding energy density is
+negative resulting in a Schwarzschild-anti de Sitter spacetime with cosmological constant Λ = 8πGρ(qshoot(0)).
+4) The size of the global minimum of the mass function converges to a constant for rh ≪ 1 with value
+limrh→0 log10(−min(m)(rh)) = 1.125.
+For rh ≫ 1 it decreases linearly and faster than the negative ADM
+mass. It may be parametrized by
+log10((−min(m)(rh)) = d1 log10(rh) + d2
+(81)
+with d1 = 3.0268 ± 0.0001 and d2 = −3.5790 ± 0.0002.
+B.
+Dependence on the parameters of the scalar field potential
+We now consider changes in the scalar field potential parameters. The effect of a variation of the scalar field potential
+parameter λ for a horizon radius range
+1
+10 ≤ rh ≤ 10 and fixed scalar field potential parameter qeq = 0.158 for the
+characteristic functions presented previously in Fig. (5) is visualized in Fig. (6). The radial parameters and masses
+have been rescaled in a particular way following [29]. It can be seen that the rescaled quantities seem to depend only
+
+value of the shooting parameter
+ADM-massoftheblackholesolution
+0.9 
+2
+MSchwa2
+0.B
+1
+0
+0.7
+-1
+0.6 
+-2
+-3
+0.5
+-2
+-1
+0
+i
+2
+3
+E-
+-2
+-1
+0
+i
+2
+3
+Iogia(rh)
+fogia(rh)
+relative location of the global minmum of m
+global minimum of m(rh)
+log 1o(rmin/rn)
+log 1o(rmin  rn)/rn)
+Jogia(Fmin/rh) vs. Jogia(Fmin - Fh)/rh)
+5 -
+3
+logia(-min(m(rh)))
+3
+1
+2
+0
+1
+-1 
+0 1
+E-
+-2
+-1
+0
+1
+2
+3
+E-
+-2
+-1
+0
+1
+2
+3
+logia(rh?
+Iogia(rh?16
+Figure 6. The variation of several characteristic quantities of SHBH solutions for q-theory with respect to the scalar field
+potential parameter λ is shown for a horizon radius of rh = 1 and remaining q-field potential parameter qeq = 0.158. These
+quantities include shooting parameter, ADM mass, scalar-hair mass, relative location of the global minimum of the mass
+function as well as absolute value of the global minimum of the mass function.
+on two instead of three parameters. To be more precise about the parameter dependencies define
+�rh =
+√
+λrh,
+�
+rmin =
+√
+λrmin,
+ˆ
+MADM =
+√
+λMADM,
+�
+min(m) =
+√
+λmin(m).
+(82)
+It then follows that
+�
+rmin = 1,
+ˆ
+MADM = ˆ
+MADM( �rh, qeq),
+�
+min(m) =
+�
+min(m)( �rh, qeq).
+(83)
+The effect of a variation of the scalar field potential parameter qeq for a horizon radius range
+1
+10 ≤ rh ≤ 10 and fixed
+scalar field potential parameter λ = 1 for the characteristic functions presented previously is visualized in Fig. (7).
+As qeq increases, the potential wells of the scalar field potential recede from each other and become more pronounced.
+All quantities (with minor exceptions) seem to be monotonically growing (monotonically decreasing in the case of
+the negative valued quantity min(m)) with qeq. The dependence of the different characteristic quantities of SHBH
+solutions on qeq is not so easily deducible. Nevertheless it seems that, with exception of large horizon radius values,
+the dependence on qeq may be factorized from that on �rh.
+The local scan of the parameter space of solutions has revealed that the space of solutions is effectively only two
+dimensional with the dependence on the two parameters factorizing by a monotonically increasing function of qeq to
+good approximation within the represented parameter space area.
+The topic of the following section will be a stability analysis of SHBH solutions due to perturbations of the q-field
+solution.
+VII.
+Stability of the SHBH solutions
+An important question to ask is whether SHBHs are stable. Do perturbation modes of the q-field and the metric
+which grow exponentially in the SHBH spacetime of our q-theory model exist? The question of classical instability
+
+value of the shooting parameter
+ADM-massoftheblackholesolution
+0.95
+14 
+入=4
+ ^=1
+13 
+ ^=1
+0.50
+12 
+0.B5
+(4)o%b
+0.B0
+1f
+●入=4
+^=1
+0.75
+ 入=1
+8:
+ 入=
+0.70 -
+2.0
+1.5
+-1.0
+0.5
+0.0
+0.5
+1'D
+2.0
+1.5
+1.0
+0.5
+0.D
+0.5
+1D
+fogiatVArn)
+logia(VArn)
+relative location of the global minimum of m
+globalminimum ofm.
+●^=4
+-2
+●^=4
+225
++ ^=1
+← ^=1
+入=1
+200
+4 -
+ 入= 4
+175
+→ =2
+6
+150
+125
+1D0
+-10 
+0.75
+0.50
+-12 
+0.25 
+1.00
+0.50
+0.25
+o.bo
+0.25
+0.50
+0.75
+1D0
+2.0
+1.5
+1.0
+d.5
+0.D
+0.5
+1'D
+Iogia(rh?
+ogia( VArh?17
+Figure 7. The variation of several characteristic quantities of SHBH solutions for q-theory with respect to the scalar field
+potential parameter qeq is shown for a horizon radius of rh = 1 and remaining q-field potential parameter λ = 1.
+These
+quantities include shooting parameter, ADM mass, scalar-hair mass, relative location of the global minimum of the mass
+function as well as absolute value of the global minimum of the mass function.
+has been discussed for spherically symmetric, time dependent perturbations of the metric and scalar field (here the
+q-field) in [29]. Following a similar notation to that introduced in [29] (see Eqs. (17)-(19) therein) we define the
+s-wave perturbations by
+˜q(t, r) = q(r) + δq(r, t) ,
+(84)
+˜f(r, t) = f(r)(1 − h1(r, t)) ,
+(85)
+˜h(r, t) = h(r)(1 − h2(r, t))
+(86)
+in generalized Schwarzschild coordinates. The functions q(r), f(r) and h(r) represent the unperturbed solutions for
+q-theory in the generalized Schwarzschild coordinates, while δq(r, t), h1(r, t) and h2(r, t) denote small perturbations
+to the non-perturbed solution. Note that in Eq. (86) h2 is defined as a small correction to h with grr = 1/h in (41),
+whereas in Eq. (19) in [29] it is defined as a small correction to grr directly with a different sign. The two relations
+are equivalent, as is easily checked by substituting h(1 − h2) for h in grr = 1/h and then expanding.
+After linearization of the Einstein equations (45-47) it can be shown that [29]
+∂rh1(r, t) = ���rh2(r, t) − 16πr ( ∂r q(r) ) (∂r δq(r)) ,
+h2(r, t) = 8πr ( ∂r q(r) ) δq(r).
+(87)
+As such the metric perturbations are expressible in terms of the q-field perturbation. The latter may be determined
+by solving a one-dimensional Schrödinger equation of the form [29]
+� 1
+2m(−i∂r∗)2 + V ∗
+eff(r∗)
+�
+ψ(r∗) =
+1
+2m(i∂t)2ψ(r∗),
+dr∗(r)
+dr
+= exp(−δ(r))
+h(r)
+(88)
+where ψ(r∗(r)) ≡ rδq(r) and r∗ is the “tortoise” coordinate. The scalar field mass m is obtained as follows. Insert
+˜q(t, r) into the action functional and expand in δq(t, r) around q(r). Far away from the event horizon when q(r) is
+close to its equilibrium value qeq, the kinetic cross terms as well as the linear potential term in δq are negligible and
+
+value of the shooting parameter
+ADM-massoftheblackholesolution
+22.5
+0.98
+中
+24.0
+O qeg = 0.223
+0.96 
+★ qeg = 0.193
+17.5
+ qeg = 0.173
++ qeg = 0.158
+60
+15.0
+(4.jb
+(usjnavw
+ qeg = 0.129
++qeg=0.112
+12.5
+0.92
+14.0
+qeg = 0.223
+0.90
+qeg = 0.193
+ qeo = 0.173
+7.5
+qeg = 0.158
+0.8 -
+★ qeg =0.129
+5.D
+→ q= 0.112
+1.00
+0.50
+0.25
+0.00
+0.25
+0.50
+0.75
+LDo
+1.00
+0.50
+0.25
+0.0
+0.25
+0.50
+0.75
+LDO
+fogia(rh)
+fogia(rh)
+relative location of the global minimum of m
+globalminimumofm.
+-qeg=0.223
+0+
+25
++ qe = 0.193
+ qeg = 0.173
+→ qea = 0.158
+★ qeg = 0.129
+DOT-
+2D
+← qeg= 0.112
+Jogia(rmin/rh)
+O qeg =0.223
+tusjujuu
+ Qeg = 0.193
+200
+ qeg = 0.173
+15
+ qea = 0.158
+★ qea = 0.129
+← qeg=0.112
+300
+1D
+0.5
+400
+1.00
+0.50
+.25
+0.00
+0.25
+0.50
+0.75
+LDO
+1.00
+0.50
+±.25
+0.DO
+0.25
+0.50
+0.75
+LDO
+fogia(rh)
+Iogia(rh?18
+the action with dynamical field (perturbation) δq has a proper kinetic term with at least quadratic potential terms.
+The quadratic term yields m in the same way as does the real Klein Gordon action with (self-)interactions. The so
+obtained value for m reads
+m =
+�
+d2ρ(q)
+dq2
+������
+q=qeq
+=
+�
+3
+2λq2eq.
+(89)
+It will formally not be needed in the following but has been introduced in order to provide a properly normalized
+Schrödinger problem. The effective potential reads [29] (note that the expression for Veff in this work is defined with
+a factor of
+1
+2m compared to Eq. (22) in [29])
+Veff(r∗(r)) =
+1
+2mh(r) exp(2δ(r))[h(r)
+r
+(δ′(r) + h′(r)
+h
+)
+− 8πrh(r)(q′(r))2(δ′(r) + h′(r)
+h(r) + 1
+r )
++ 16πrq′(r)dρ
+dq (q(r)) + d2ρ
+dq2 (q(r))].
+(90)
+The separation ansatz ψ(r∗(r)) = ξ (r∗(r)) exp
+�
+±i
+√
+2mEt
+�
+leads to the following stationary Schrödinger equation
+for ξ (r∗(r))
+� 1
+2m(−i∂r∗)2 + V ∗
+eff(r∗)
+�
+ξ(r∗) = Eξ(r∗)
+(91)
+with energy eigenvalue E. A sufficient condition for static, spherically symmetric configurations to be unstable [37]
+is that the differential operator on the left-hand side of (91), is negative in the Hilbert space L2(M), where M is the
+spacetime manifold on which the metric is defined. Accordingly, a sufficient condition for unstable solutions is the
+existence of a bound state E < 0 in the Schrödinger problem. In the same manner the existence of bound states
+that correspond to E < 0 in the Schrödinger problem with potential Veff is equivalent to the existence of unstable
+perturbations of q-theory solutions. The exponential factor of the perturbation becomes of order unity after a time
+of order
+τ =
+1
+√−2mEmin
+.
+(92)
+The subscript min on E signifies the lowest bound state energy which is of most importance for giving the scale of
+τ (for a discrete and finite bound state spectrum, as is the case here). This time can be seen as the lifetime of the
+SHBH in the presence of these perturbations.
+In order to determine the value Emin, the lowest eigenvalue of bound state energies of the effective Schrödinger
+equation (91), we solve the eigenvalue problem numerically in the variable r for horizon radii in the range 1 ≤ rh ≤ 1000
+by approximating the equation on a grid of finite size. The differential operator is approximated by a central point
+stencil method accurate to fourth order of the grid spacing. The Runge Kutta solver shares this level of accuracy.
+We employ vanishing boundary conditions for the eigenfunctions in the limits r → 0 as well as r → ∞. To get an
+impression of the shape of the eigenfunction to the eigenvalue Emin, we replace the effective potential, illustrated
+in Fig. (8) for different horizon radii, by an auxiliary potential. This auxiliary potential is a parabola fitted to the
+negative potential well of the effective potential in the region shown in red in the figure (where Veff < 0). Outside
+it is set to zero. This yields a finite depth harmonic oscillator potential. The solutions of the wave equation may be
+determined exactly in this auxiliary potential. Finding the bound state energy eigenvalues is then identical to the
+quantum harmonic oscillator except for them being finite in amount, while the eigenfunctions are compromised due
+to the vanishing of the potential. Nevertheless, as is in concordance with the results in [29], the expectation is that
+the lowest energy bound state eigenfunction is close to a Gaussian in shape which is the exact eigenfunction in this
+case for the quantum harmonic oscillator. That this expectation is also fulfilled in our case is illustrated in Fig. (9).
+We plot the normalized eigenfunctions corresponding to the lowest bound state energy eigenvalues for several horizon
+radii and scalar field potential parameters λ = 1 and qeq = 0.158. A comparison with Fig. (8) shows that the peaks
+of the eigenfunctions are situated almost exactly at the minimum of the effective Schrödinger potential. It becomes
+apparent here and has been observed that for increasing horizon radii the Gaussians loose their shape and tend to
+disappear. In this regime the lowest eigenvalues approach zero very quickly from below. This inspires the conclusion
+
+19
+Figure 8. The effective potential of the scalar perturbation mode ξ(r∗) is shown for different horizon radii and scalar field
+potential parameters λ = 1 and qeq = 0.158. It comprises a negative valued well where the wave function of bound states are
+predominantly located. The well is highlighted in red. A zoom makes this region visible for the horizon radii rh = 100 and
+rh = 1000.
+Figure 9. Normalized eigenfunction solutions to the lowest bound state eigenvalue for the scalar perturbation mode ξ(r∗) are
+shown for different horizon radii and scalar field potential parameters λ = 1 and qeq = 0.158. They are almost perfect Gaussians
+in shape which is the case for the quantum harmonic oscillator.
+that large SHBHs are indeed stable and opposes that found in [29]. In Fig. (10) the lifetime τ = τ(rh) of SHBHs
+as a function of the horizon radius and scalar field potential parameters λ = 1 and qeq = 0.158 is shown. We fit the
+obtained lifetimes τ = τ(rh) corresponding to the solutions for the lowest eigenvalues e = e(rh) to the function
+f(rh) = a · (log10(rh))ntan(c · log10(rh) − b) + d
+(93)
+parameterized by the scale parameters a and c, the horizontal shift parameter b, the vertical shift parameter d and
+the power parameter n with optimal parameters popt and covariance matrix pcov given by
+popt =
+�
+�
+�
+�
+�
+a
+b
+c
+d
+n
+�
+�
+�
+�
+� =
+�
+�
+�
+�
+�
+1.53
+1.14
+1.73
+5.31
+1.23
+�
+�
+�
+�
+� ,
+pcov =
+�
+�
+�
+�
+�
+0.044
+0.082 0.053 0.037 −0.007
+0.081
+0.179 0.115 0.085
+0.014
+0.053
+0.115 0.074 0.055
+0.009
+0.037
+0.085 0.055 0.043
+0.010
+−0.007 0.014 0.009 0.010
+0.028
+�
+�
+�
+�
+� .
+(94)
+The lifetime then becomes infinite at the finite horizon radius r0
+h = π+2b
+2c
+and the SHBHs are therefore stable beyond
+the threshold r0
+h for the chosen scalar field potential parameters.
+
+effective Schrodinger potential for the q-field perturbation
+0.125
+0.002
+(u)"PA
+000'0
+0.05
+0.002
+0.025
+0.105
+0.110
+0.115
+log 1o(r/rn)
+0.00
+rn = 1
+In = 5
+.025
+F = 10
+Tn = 25
+.050
+rn = 100
+rn= 1000
+0.0
+0.5
+1D
+15
+2D
+25
+3.D
+fogia(rtrh)eigenfunctionwith corresponding eigenvalue E-E
+0.6
+ T = 1
+- rh= 5
+0.5
+- rn = 25
+.
+0.4 
+!!
+tts)*
+*3
+0.3
+!!
+:1
+0.2
+0.1 
+0.D .
+0.D
+0.5
+LD
+15
+20
+25
+3.D
+logia(r/rn?20
+Figure 10. The lifetime of SHBHs due to s-wave q-field and metric perturbations represented by the scalar perturbation mode
+ξ(r∗) is shown as a function of the horizon radius with scalar field potential parameters λ = 1 and qeq = 0.158. The lifetime is
+finite for small SHBHs below a certain radius or mass threshold value, whereas SHBHs are stable beyond this threshold. The
+threshold is highlighted in red.
+The stability analysis has revealed that SHBH are unstable due to classical s-wave perturbations of both the q-
+field and the metric below a certain size or equivalently mass threshold and stable beyond (at least for the chosen
+parameters).
+VIII.
+Conclusion
+In the present paper we consider q-theory comprising a scalar field q minimally coupled to gravity.
+The q-field
+describes a dynamical gravitating vacuum. According to the estimates proposed in [24], this theory may contain BH
+solutions that resemble that of a gravastar, i.e. a configuration with energy concentrated inside a thin spherical shell.
+Contrary to the conventional gravastar, the state proposed in [24] contains an event horizon. Using direct numerical
+calculations, we confirm the existence of similar BH configurations, with some reservations, though. Namely, inside
+the event horizon space - time resembles the interior of the Schwarzschild BH, and does not contain the de Sitter - like
+domain. Besides, the mentioned thin shell is located outside the event horizon, not inside. Furthermore, there should
+exist a region in space, where the energy density is negative. This is required to satisfy the “no-hair” theorems. As
+a result, the thin shell situated just outside of the horizon contains both a piece of negative energy and a piece with
+positive energy density. The integration of the energy density inside the shell yields a total positive energy resulting
+in the ADM mass perceived by the distant observer.
+According to Eq. (30) the energy density is proportional to the derivative of the mass function m(r). The latter
+function is represented within Fig. 2. One can see that for the given example solution, the spherical shell of finite
+thickness exists and is situated just outside the horizon. Inside this shell the essential variation of m(r) is localized.
+Close to but outside the event horizon, the energy density is negative, then it passes through zero, and becomes
+positive in the second piece of the shell. The shell ends where m(r) exponentially approaches its asymptotic value,
+the constant that represents the black hole mass seen by the infinitely distant observer.
+The results of section VI demonstrate that the spherical shell approaches the horizon relative to its size when the
+horizon radius is increased. In the limit of very large BHs, virtually the entire energy due to the q-field is localized
+in the thin shell situated outside the horizon and close to it. It does not approach the event horizon ifinitely close,
+but stops, such that the energy density sign transition is positioned around 7
+5rh, where rh denotes the event horizon
+radius.
+A stability analysis with respect to the s-wave metric and q-field perturbations shows that the BH solutions of the
+type considered in the present paper may be classically unstable . However, the corresponding configurations are
+stable for sufficiently large BHs. We therefore claim that stable heavy SHBHs do exist.
+We do not discuss here questions related to the stability of the considered configurations on the quantum level.
+This issue remains outside of the scope of the present paper.
+In conclusion, in this work we confirm the supposition of [24] about the existence of BH solutions in q-theory that
+look similar to gravastars. These states escape the conditions of the no-hair theorem, due to the region in space with
+negative energy density. At spatial infinity these solutions approach the Schwarzschild solution, but differ from it
+
+lifetime of aSHBH duetoperturbations
+120
+fit
+threshold
+data
+140
+8+ 
+40
+24
+0.+
+0.D
+0.5
+1b
+15
+2D
+25
+3.D
+logiatrh?21
+Figure 11. The effect of different choices of E(∞) on the SHBH solution for q-theory outside the event horizon is shown for a
+horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.158, a grid size of 106 and shift parameter ϵ = 10−6.
+The inward and outward massless particle velocities are shown for both Schwarzschild spacetime and q-theory spacetime with
+Schwarzschild mass parameter M = mq−theory(rh).
+essentially close to the horizon. Inside the horizon, the vacuum density is negative and changes sign very close to the
+center. The singularity of curvature at r = 0 is the same as that of the Schwarzschild solution.
+The authors are grateful to G.E.Volovik for the proposition to consider the given problem, and for useful discussions
+during the initial stage of the work.
+Appendix A. Test particle characteristics.
+We discuss here the properties of test particles contained in the free parameter E(∞) = limr→∞ E(r). It is related
+to the total energy per unit rest mass of a test particle e by E(∞) = (e2 − 1)/2 which moves towards the SHBH
+horizon starting at infinity with initial velocity v(∞) =
+�
+2E(∞) where v(∞) = limr→∞ v(r). Different values for
+E(∞) correspond to different initial kinetic energies of a test particle. The choice E(∞) = 0 corresponds to a particle
+at rest, while E(∞) > 0 corresponds to a particle initially moving towards the SHBH. E(∞) < 0 is not possible, since
+then v(r) would become imaginary while approaching asymptotically flat infinity. The motivation for the introduction
+of generalized Painlevé-Gullstrand coordinates as well as their relation to test particle motion are presented in more
+detail in [34].
+The numerical solution presented in Fig. 2 for different initial values of the free parameter E(∞) is shown in Fig. 11.
+The function E(r) is monotonically decreasing with r as is v(r) in q-theory, whereas E(r) is constant for Schwarzschild
+spacetime while v(r) is also decreasing. This is in concordance with the increase of the kinetic energy of a test particle
+as it moves towards the event horizon of the black hole. The inward and outward velocities of a massless particle
+( dr
+dt )in/out are monotonically increasing with r, as the gravitational pull of the black hole decreases by further recession
+from the horizon. This is true for both Schwarzschild and q-theory spacetime with one exception in q-theory. In the
+region of large change of the q-field the velocity of outward moving massless particles has a small dip before increasing
+again. In this region the energy density of the q-field becomes positive. As expected, the outward motion tends to
+
+energy function
+velocity function
+3.5
+6
+E() = 0
+E() = 0
+3.D
+E(∞) = 0.01
+E() = 0.01
+E(∞) = 0.05
+5
+E() = 0.05
+25
+E() = 0.1
+E() = 0.1
++2E(r)
+E() = 0.5
+.
+E() = 0.5
+2D
+E(∞) = 1
+- E(∞) = 1
+foge(1
+15
+LD
+2 -
+0.5 -
+1 -
+00
+0 +
+0.D
+0.5
+15
+2D
+25
+3.0
+0.D
+0.5
+15
+2D
+25
+3.D
+Iogia(rfrh?
+logia(r/rh?
+massless particle outward velocity (Schwarzschild)
+massless particle inward velocity (Schwarzschild)
+LD
+1.0
+0.B 
+1.5
+0.6 
+E(∞) = 0
+2.0
+E(∞) = 0.01
+0.4
+E() = 0.05
+E(∞) = 0.1
+E(o) = 0
+2.5
+E(∞) = 0.5
+0.2
+E() = 0.01
+E() = 1
+E() = 0.05
+E() = 0.1
+3.0
+0.D
+E(∞) = 0.5
+E(∞) = 1
+0.2 
+3.5
+0.D
+0.5
+15
+2D
+25
+3.D
+0.0
+0.5
+1D
+15
+2D
+25
+3.D
+Iogia(rfrh)
+Iogia(rfrh)
+massless particle outward velocity (q-theory)
+massless particle inward velocity (g-theory)
+LD
+E() = 0
+E() = 0
+0.B 
+E(∞) = 0.01
+E() = 0.01
+E() = 0.05
+E(∞) = 0.05
+E(∞) = 0.1
+E(∞) = 0.1
+0.6 
+E() = 0.5
+E(∞) = 0.5
+E(∞) = 1
+E(∞) = 1
+0.4
+0.2 
+日
+0.0
+-10
+0.2 
+0.0
+0.5
+15
+2D
+25
+3.D
+0.0
+0.5
+LD
+15
+2D
+25
+3.D
+Iogia(rfrh)
+logia(rfrh)22
+Figure 12. The effect of different shift parameters on SHBH solutions for q-theory outside the event horizon is shown for a
+horizon radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.158 and a grid size of 106.
+zero as the event horizon is approached. We chose the Schwarzschild mass parameter M = mq−theory(rh). Choosing
+M = mq−theory(∞) instead would have implied that the graphs for ( dr
+dt )out in the Schwarzschild case cross zero
+already outside the event horizon as set by q-theory. Close to the event horizon both inward and outward massless
+particle velocities are smaller for q-theory spacetime as compared to Schwarzschild spacetime. This suggests that
+black hole absorptivity is greater for q-theory spacetime as compared to Schwarzschild spacetime.
+Appendix B. Accuracy estimates.
+The numerical solution presented in Fig. (2) is approximate both because of finite grid size and finite shift parameter.
+The solution with rh = 1 and q-field potential parameters λ = 1 and qeq = 0.158 is analyzed in Fig. (12) and Fig.
+(13) with respect to variations of the shift parameter and grid size, respectively.
+The plots in (12) show that absolute differences of q-fields and mass functions for different neighboring shift parameters
+ϵ are almost exactly coincident with the absolute size of the imaginary part of the q-fields and mass functions for
+the larger shift parameter present in the corresponding absolute difference plots. The maximal value of the absolute
+size of the imaginary parts of the represented q-fields and mass functions shrinks by one order of magnitude for each
+decrease of the shift parameter by one order of magnitude as expected. It is about two order of magnitude smaller
+than the shift parameter for the q-fields and about one order of magnitude larger than that of the shift parameter
+for the mass functions, though. We choose a shift parameter of ϵ = 10−6 in most of our plots, as it is seen to be
+negligibly small to have any effect. The same choice argument will be applied for the grid size to which we now turn.
+The plots in (13) show the absolute differences of q-fields and mass functions for different neighboring grid sizes as well
+as relative error estimates for the functions q′, q′′ and m′ analogous to the lowermost left plot in Fig. (2) for different
+grid sizes. The expectation that the differences between the q-fields and mass functions as well as the error estimates
+decrease with increasing grid size are fulfilled. The differences of the q-fields and mass function decrease by about
+one order of magnitude for a grid size increase of one order of magnitude. The error estimates are comparable for
+the different grid sizes close to the horizon. This indicates that no mayor improvement may be achieved with further
+increase of the grid size. Further away from the horizon the error estimates indeed decrease visibly with increasing
+grid size.
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+(Math. Phys. ), 1917:142–152, 1917.
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+
+difference of q-fields with respect to 
+absolute size of im(qtr) with respect to 
+4
+4
+Jog1alqs (r) - qz (r3)
+6
+fog1a(im(qe(r)l)
+8
+-10
+-10
+-12 
+=10-8+=10-9
+-12
+ =10-8
+E=10-7 +E= 10-8
+ = 10~7
+-14
+E= 10-°+E= 10-7
+-14
+++++.
+ = 106
+-16
+ = 10-5 = 10-6
+-16
+ = 10~5
+-1B 
+= 10-4+= 10-5
+-1B
+= 10-4
+E=10-3 += 10-4
+ = 10~3
+20
+LD
+12
+20
+0.D
+0.2
+0.4
+0.6
+0.B
+14
+16
+0.0
+0.2
+0.4
+0.6
+0.B
+LD
+12
+14
+16
+Iogia(rfrh)
+Iogia(rfrh)
+difference of the mass functions with respect to 2
+absolute size of im(mtr) with respect to z
+2
+-2 
+JogialIme, (r) - me, (rl)
+logia(m(me(r))
+-6
+6
+-8
+-10
+=10-8+=10
+ =10-8
+-10
+= 10-7
+-12
+E = 106 +
+E= 10°
+-12 
+= 10-6
+14
+14
+ = 10-5
+16
+E=10-4+=10
+ = 10-4
+=10-3+=10-4
+~16 
+= 10~3
+0.D
+0.4
+0.6
+0.B
+12
+14
+16
+00
+0.2
+t0
+0.6
+0.B
+1D
+12
+14
+16
+Iogia(rfrh)
+Iogia(rfrh)23
+Figure 13. The effect of different grid sizes on SHBH solutions for q-theory outside the event horizon are shown for a horizon
+radius of rh = 1, q-field potential parameters λ = 1 and qeq = 0.158 and shift parameter ϵ = 10−6.
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+4
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+grid size = 5 · 10° ++ grid size = 2.5 · 105
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+7.5
+ grid size = 5 · 105
+10.0
+grid size = 106
+grid size = 2.5 - 106
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+0.B
+LD
+12
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+ogiatr/rh?
+relative size of the solution error (5-p.s.m.) for q" with respect to grid size
+0.D 
+-2.5
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+Jog1a(/Agtel)
+0'5-
+grid size = 105
+grid size = 2.5 · 105
+-7.5
+grid size = 5 · 105
+10.0
+grid size = 106
+grid size = 2.5 · 100
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+0.2
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+0.B
+LD
+12
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+logia(rfrn)
+difference of the mass functions with respect to grid size
+m(.(rl)
+4
+grid size =5. 10° + grid size = 2.5 - 104
+-6 :
+grid size = 105 +grid size = 5 - 104
+Jogia(Im(ms(r) -
+grid size = 2.5 - 105 +grid size = 105
+.- grid size = 5 - 105 + grid size = 2.5 - 105
+grid size = 10° + grid size = 5 - 105
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+5.0
+grid size = 2.5 · 105
+7.5
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+15.0
+0.D
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+0.6
+0.B
+1D
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+14
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+logia(rfrh?24
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+

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@@ -0,0 +1,2199 @@
+Astronomy & Astrophysics manuscript no. SOAP_GPU
+©ESO 2023
+January 12, 2023
+SOAP-GPU: Efficient Spectral Modelling of Stellar Activity Using
+Graphical Processing Units
+Yinan Zhao1 and Xavier Dumusque1
+Department of Astronomy of the University of Geneva, 51 chemin de Pegasi, 1290 Versoix, Switzerland
+e-mail: yinan.zhao@unige.ch
+January 12, 2023
+ABSTRACT
+Context. Stellar activity mitigation is one of the major challenges for the detection of earth-like exoplanets in radial velocity mea-
+surements. Several promising techniques are now investigating the use of spectral time-series, to differentiate between stellar and
+planetary perturbations. In this context, developing a software that can efficiently explore the parameter space of stellar activity at the
+spectral level is of great importance.
+Aims. The goal of this paper is to present a new version of the Spot Oscillation And Planet (SOAP) 2.0 code that can model stellar
+activity at the spectral level using graphical processing units (GPUs).
+Methods. We take advantage of the computational power of GPUs to optimise the computationally expensive algorithms behind the
+original SOAP 2.0 code. For that purpose, we developed GPU kernels that allow to model stellar activity on any given wavelength
+range. In addition to the treatment of stellar activity at the spectral level, SOAP-GPU also includes the change of spectral line
+bisectors from center to limb, and can take as input PHOENIX spectra to model the quiet photosphere, spots and faculae, which allow
+to simulate stellar activity for a wider space in stellar properties.
+Results. Benchmark calculations show that for the same accuracy, this new code improves the computational speed by a factor of 60
+compared with a modified version of SOAP 2.0 that generates spectra, when modeling stellar activity on the full visible spectral range
+with a resolution of R=115’000. Although the code now includes the variation of spectral line bisector with center-to-limb angle,
+the effect on the derived RVs is small. We also show that it is not possible to fully separate the flux from the convective blueshift
+effect when modeling spots, due to their lower temperature and thus the appearance of molecular absorption in their spectra. Rather
+negligible for the Sun, this degeneracy between the flux and convective blueshift effect become more important when we move to
+cooler stars, however, this issue does not impact the estimation of the total effect (flux plus convection), and therefore users can trust
+this output.
+Conclusions. The publicly available SOAP-GPU code allows to efficiently model stellar activity at the spectral level, which is essential
+to test further stellar activity mitigation techniques working at the level of spectral timeseries not affected by other sources of noise.
+Besides a huge gain in performance, SOAP-GPU also includes more physics and is able to model different stars than the Sun, from
+F to K dwarfs, thanks to the use of the PHOENIX spectral library. We however note that due to the limited understanding of stellar
+convection and activity on other stars than the Sun, the more we go away from the solar case, the more the output of the code should
+be taken with care.
+Key words. Methods: data analysis – Techniques: radial velocities – Techniques: spectroscopic - Stars: activity
+1. Introduction
+The radial velocity (RV) method has been proven to be one of
+the most successful method to detect exoplanets since the dis-
+covery of the first exoplanet orbiting a solar-type star (Mayor
+& Queloz 1995). In order to detect earth-like planets orbiting in
+the habitable zone of its parent star, a precision of a few dozens
+of cms−1 must be reached. Although the state-of-the-art spec-
+trographs such as ESPRESSO, and EXPRESS are not far from
+that precision (50 and 58 cms−1, respectively Pepe et al. 2021;
+Brewer et al. 2020), the main limitation to detect Earth-like plan-
+ets with the RV technique is stellar activity. Two major physical
+processes dominating stellar activity on a time scale of the host
+star’s rotational period are the flux imbalance due to the temper-
+ature difference and therefore contrast between active and quiet
+regions (hereafter flux effect. (e.g. Saar & Donahue 1997; Du-
+musque et al. 2014; Donati et al. 2017)) and the inhibition of
+convective blueshift (hereafter CB effect). The CB effect is due
+to the presence of strong local magnetic fields inside active re-
+gions, which suppress the CB inside those regions and leads to
+positive RV variations (e.g. Cavallini et al. 1985a; Meunier et al.
+2010).
+Many methods have been proposed to mitigate activity-
+induced variations using photometric and spectroscopic time se-
+ries. In the one-dimensional time series space, many parametric
+models based on analytic forms or different Gaussian process
+(GP) frameworks have been developed to model stellar activ-
+ity using photometry or spectroscopic activity indicators (e.g.
+Aigrain et al. 2012; Rajpaul et al. 2015; Aigrain et al. 2016;
+Gilbertson et al. 2020b; Barragán et al. 2022). Jointly modeling
+the data with Keplerians to model planets in addition to a GP to
+model stellar activity may significantly reduce the stellar activity
+but may also lead to overfitting when the GP kernel or priors are
+not wisely set. This is particularly dangerous when the planetary
+properties are not constrained from transit observations.
+Due to inherent problems in modeling stellar activity in one-
+dimensional time series, the community is now shifting toward
+modeling it in a two-dimensional space. Collier Cameron et al.
+Article number, page 1 of 17
+arXiv:2301.04259v1  [astro-ph.SR]  11 Jan 2023
+
+A&A proofs: manuscript no. SOAP_GPU
+(2021) calculated the autocorrelation function (ACF) of cross-
+correlation function (hereafter CCF Baranne et al. 1996), to iso-
+late Doppler shift from shape shift variations and applied prin-
+ciple component analysis (PCA) on the obtained ACFs to model
+shape changes related to stellar activity. A planet signal of am-
+plitude ∼ 40 cm/s can be recovered when the algorithm is ap-
+plied to the HARPS-N solar data (Dumusque et al. 2015; Col-
+lier Cameron et al. 2019; Dumusque et al. 2021). Zhao et al.
+(2022a) projected CCFs time series onto the Fourier basis func-
+tions and modelled line variability using different basis. Results
+on simulated data show a 48% reduction in RV rms. de Beurs
+et al. (2022) trained a convolutional neural network (CNN) on
+both simulated CCFs and HARPS-N solar CCFs and were able
+to significantly reduce stellar activity effects.
+The idea behind building the CCF is to extract with the best
+precision the RV information contained in a spectrum. How-
+ever, key variations at the spectral level related to stellar activ-
+ity may be lost when performing the dimensionality reduction
+imposed by the CCF. Therefore, several methods have been pro-
+posed to disentangle stellar activities at the spectral level. Davis
+et al. (2017) applied PCA to simulated spectral time series and
+demonstrated that eigen-vectors are spectral line dependent. Ra-
+jpaul et al. (2020) used GP to directly derive RV information
+from spectral time series. Jones et al. (2017) also applied mul-
+tivariate GP to model stellar activity on PCA-reduced spectral
+dataset. Cretignier et al. (2022), based on the knowledge that the
+impact of stellar activity is line-depth dependant (Cretignier et al.
+2020a), used PCA to model stellar activity in the flux-flux gra-
+dient space (named the “shell” space) and results on HD10700
+(τ Ceti) and HD12861 (α Cen B) indicates the method can
+successfully remove variations from non-Doppler origin. Last
+by not least, Binnenfeld et al. (2020, 2022) are developing the
+unit-sphere representation periodogram (USuRPER), to seper-
+ate Doppler from other RV variations. This technique is based
+on representing spectra as unit vectors in a multidimensional hy-
+perspace.
+The spectral time series used to evaluate the performance of
+the algorithms developed to mitigate stellar activity at the spec-
+tral level are either obtained from simulated data or real obser-
+vations. The major issue with simulations, is that most of them
+only model the RV activity effect at the CCF level (e.g. Du-
+musque et al. 2014; Herrero et al. 2016) due to computational
+inefficiency. A few other libraries of simulated spectra affected
+by stellar activity exist, but generating them takes hours to run,
+which is not convenient when exploring the parameter space in
+stellar activity and properties (e.g. Gilbertson et al. 2020a; Du-
+musque 2016). Regarding real observations, solar data obtained
+by the HARPS-N solar telescope (Collier Cameron et al. 2019;
+Dumusque et al. 2021), HELIOS on HARPS1 and more recently
+the solar feed of NEID (Lin et al. 2022) are the best we can get, in
+terms of S/N and sampling. However, those spectra corresponds
+for the most part to quiet activity phases of the Sun (end of cy-
+cle 24 end beginning of cycle 25) and can only used to mitigate
+stellar activity for star very similar to the Sun. When moving to
+stellar observations, the recent Extreme precision Spectrograph
+(EXPRES) Stellar Signals Project (ESSP) shared some valuable
+data. However, due to the small number of stars and the rather
+small number of spectra available, it was rather difficult to com-
+pare different activity mitigation techniques together (Zhao et al.
+2022b). As a conclusion of this discussion, it is essential for the
+community to have access to a code that can simulate efficiently
+1 https://www.eso.org/public/announcements/ann18033/
+stellar activity at the spectral level, and for a wide range of stellar
+properties.
+In this paper, we present a new code, Spot Oscillation And
+Planet Graphical Process Unit (SOAP-GPU) based on GPU
+computation that can efficiently model simplified and realistic
+stellar activity at the spectral level. In Sect 2, we revisit the ar-
+chitecture of the SOAP 2.0 code it is based on (Dumusque et al.
+2014) and discuss about its limitations. The algorithms behind
+SOAP-GPU are presented in Sect 3. In Sect 4, we explore the
+physical parameters of stellar activity and simulation of different
+cases are presented. Finally, we draw our conclusion in Sect 5.
+The SOAP-GPU code is publicly available on Github and Zen-
+odo2 along with a brief manual and some examples.
+2. Revisiting SOAP 2.0
+In this section, we revisit the code Spot Oscillation And Planet
+(SOAP 2.0 Dumusque et al. 2014). This code aims at modeling
+both the flux effect and the CB effect of active regions affecting
+RV measurements. Although the public version of the SOAP 2.0
+code can only simulate stellar activity at the level of the CCFs,
+modeling the effect at the spectral level follow the same ideas. In
+this section, we first discuss the basic algorithms behind SOAP
+2.0 and demonstrate the limit of the code, in terms of computa-
+tional efficiency, when we want to model stellar activity at the
+spectral level.
+Fig. 1. The stellar disk is initialized with velocity and intensity fields.
+Left: The intensity in each cell is computed depending on a limb dark-
+ening law. Right: The velocity in each cell is computed considering ro-
+tational period, stellar inclination and radius of the star. As we can see,
+iso-velocity lines are not vertical as we implemented differential rota-
+tion in SOAP-GPU, which was not the case in SOAP 2.0.
+2.1. The structure of SOAP 2.0
+SOAP 2.0 first computes the “quiet” (without any active region)
+emission spectrum of the star. To do so, a 2-dimension stellar
+disk containing N × N cells is initialized (N being the resolu-
+tion of the disk, the same parameter called “grid” in Boisse et al.
+2012). Velocity and intensity of all disk cells are computed based
+on the physical configuration of the star (rotational period, stel-
+lar inclination and radius of the star) and a limb darkening law
+(as shown in Fig. 1). In each cell, the quiet photosphere spectrum
+is injected, weighted by the cell intensity (limb-darkening), and
+Doppler-shifted to the projected velocity of that cell (rotation).
+Linear interpolation is applied at this step to project the Doppler-
+shifted spectrum into the original wavelength grid, to make sure
+that spectra in different cells are on a common wavelength grid.
+We note that the public version of SOAP 2.0 was using the CCF
+2 code available here https://github.com/YinanZhao21/SOAP_
+GPU and https://doi.org/10.5281/zenodo.7499461
+Article number, page 2 of 17
+
+Light ratio field
+Velocity field (km/s)
+300
+1.0
+300
+2.0
+1.5
+250
+0.9
+250
+1.0
+200
+0.8
+200
+0.5
+0.7
+150
+150
+0.0
+0.6
+-0.5
+100
+100
+-1.0
+0.5
+50
+50
+-1.5
+0.4
+0 +
+0 +
+2.0
+0
+50
+100
+150
+200
+250
+300
+0
+50
+100
+150
+200
+250
+300Yinan Zhao et al.: SOAP-GPU
+of the high-resolution Kitt Peak Observatory Fourier Transform
+Spectrograph (FTS) quiet photosphere spectrum (S quiet(λ), Wal-
+lace et al. 1998) as approximation of the quiet Sun to increase
+computational speed. However, injecting the original spectrum is
+possible, with the only difference that the dimension of the input
+is ∼500000, compared to 400 for the CCF, and that we will need
+to apply a Doppler-shift each time we want to change the veloc-
+ity of this spectrum, while a simple translation was sufficient in
+the case of the CCF. After injecting the quiet photosphere spec-
+trum in each cells, the integrated quiet solar spectrum is obtained
+by summing the content of all the cells together. All those pro-
+cesses are summarized in the pseudo code below (Algorithm 1).
+Algorithm 1 Quiet spectrum integration
+1: for Xlocation = 1, 2, . . . N do
+2:
+for Ylocation = 1, 2, . . . , N do
+3:
+Shift S quiet(λ) with velocity velX,Y. S quiet(λ)
+→
+S quiet(λ
+′).
+4:
+Do linear interpolation to project the spectrum back
+to the original wavelength grid. S quiet(λ
+′) → S
+′
+quiet(λ)
+5:
+Weight S
+′
+quiet(λ) by limb-darkening intensity IX,Y and
+integrate spectra along disk surface. Squiet+ = IX,Y ×S
+′
+quiet(λ)
+6:
+end for
+7: end for
+The next step consists in initializing the active regions us-
+ing the following parameters: the number of active regions, their
+size, their corresponding latitudes and longitudes, their types (ei-
+ther spot or faculae) and the resolution of the active region con-
+tour. An active region spectrum is also needed at this step to
+model the CB effect. The original SOAP 2.0 code uses the CCF
+of the observed spot spectrum in the visible obtained from the
+Kitt Peak Observatory FTS (S active(λ) with λ the same as for
+S quiet(λ) Wallace et al. 2005). The spectrum used for faculae re-
+gions is the same, with the difference that the contrast of such
+a region follow what is observed in the Sun (e.g. Fig. 3 in Me-
+unier et al. 2010), thus brighter than the quiet sun and with a
+center-to-limb brightening. Other groups use synthetic spectra at
+different temperature to model the quiet photospehere, spots and
+faculae, and include the effect of CB using results from magneto-
+hydrodynamical simulations (e.g. the STARSIM 2 code3 Herrero
+et al. 2016). We note that injecting observed or synthetic spectra
+have their advantages and drawbacks. Using observed spectra
+allows to better model the inhibition of convection inside ac-
+tive regions, but we note that if we use the observed spectra of
+a spot to model a facula (because an observed spectrum of a
+facula across the entire visible spectral range does not seem to
+exist), molecular features will be present in the facula spectrum
+despite the temperature being higher. Using synthetic spectra on
+the contrary allows to better model the temperature, and there-
+fore spectral features, of the injected spectra. The choice of the
+spectra will be addressed in the later sections.
+In order to simulate a spectral time series, we need to calcu-
+late the disk location of active regions at each timestamp. As
+shown in Equation 1 and Equation 2 of (Boisse et al. 2012),
+active regions are first put in the center of the disk and their
+initial configuration is obtained using a rotation matrix. Next,
+to get the position of those active regions as a function of
+time, another rotation matrix is used. At each timestamp, the
+code evaluates which active regions are visible, and which ones
+3 code available here https://github.com/rosich/starsim-2
+are hidden behind the star. This is performed by the function
+Localize(lat, long, i, ph), where lat and long are the latitude and
+longitude of the active region center, i is the inclination angle of
+the stellar disk and ph is the rotational phase. The output of this
+function is a binary; one if visible, zero otherwise. If a region is
+visible, the code proceed to estimate the difference between the
+quiet solar spectrum and active spectrum at the location of the
+active regions.
+The difference for the flux effect, the CB effect and the com-
+bination of the two (total) in each cell can be calculated using
+the following equations:
+∆S
+′
+flux(X, Y) = S
+′
+quiet(X, Y) − Iratio × S
+′
+quiet(X, Y),
+(1)
+∆S
+′
+bconv(X, Y) = S
+′
+quiet(X, Y) − S
+′
+active(X, Y),
+(2)
+∆S
+′
+tot(X, Y) = S
+′
+quiet(X, Y) − Iratio × S
+′
+active(X, Y),
+(3)
+where Iratio is the contrast of the spot or faculae region. The code
+then integrates over all the cells covered by active regions to get
+final difference between the quiet spectrum and active spectrum.
+The final spectrum of each effect at each timestamp can be cal-
+culated by:
+Sintegrated, final = Sintegrated,quiet − ∆Sintegrated,quiet−active.
+(4)
+Once a spectrum for each timestamp is obtained, the code then
+lowers the resolution of the integrated spectrum to match the
+resolution provided in the configuration file. The pseudo code
+that describes how active regions are included, and how the final
+integrated spectra is obtained is summarized below.
+2.2. The limitation of SOAP 2.0
+The structure of SOAP 2.0 provides an efficient way to estimate
+stellar activities on spectroscopic measurement by simulating
+CCFs at different timestamps. The major drawback when chang-
+ing the input from CCFs to spectra is the dimension of the data.
+The dimension of the input CCFs in SOAP 2.0 was 400 in veloc-
+ity space while the input high-resolution spectra we want to use
+have a dimension of ∼ 500000 in the wavelength domain. From
+Algorithm 1 and Algorithm 2, we clearly see that the linear in-
+terpolation is repeatedly called when injecting the spectrum in
+each cell, which is computationally expensive for an array with
+dimension of ∼ 500000. For example, SOAP 2.0 takes ∼ 800
+seconds to calculate an integrated quiet sun spectrum using a
+300×300 disk-grid. Another issue is how the code handles multi-
+ple active regions. Each active region is modeled independently,
+without information from other regions. This algorithm cannot
+handle the case in which some active regions overlap with each
+other. From real observations, we know that some active regions
+have complicated configurations. For example, most of active re-
+gions are a combination of a large faculae presenting a small spot
+in its center. In this context, a more computationally efficient and
+generalized algorithm is needed.
+3. Description of SOAP-GPU
+In the previous section we’ve demonstrated the limitation of
+SOAP 2.0 when modeling stellar activity at the spectral level.
+Here, we present a new version of SOAP, based on Graphical
+Processing Unit (GPU) computing, that is much more efficient
+in term of computational speed, but also that adds some physical
+complexity.
+Article number, page 3 of 17
+
+A&A proofs: manuscript no. SOAP_GPU
+Algorithm 2 Active region updates
+1: for nregion = 1, 2, . . . N do
+2:
+for ttimestep = 1, 2, . . . , T do
+3:
+Localize(lat, long, i, ph).
+4:
+if Localize = 1 then
+5:
+Shift S quiet(λ) with velocity velX,Y. S quiet(λ) →
+S quiet(λ
+′).
+6:
+Do linear interpolation to project the spectrum
+back to the original wavelength space. S quiet(λ
+′) → S
+′
+quiet(λ)
+7:
+Weight S
+′
+quiet(λ) by limb-darkening intensity IX,Y
+8:
+Shift S active(λ) with velocity velX,Y. S active(λ) →
+S active(λ
+′).
+9:
+Do linear interpolation to project the spec-
+trum back to the original wavelength space. S active(λ
+′) →
+S
+′
+active(λ)
+10:
+Weight S
+′
+active(λ) by limb-darkening intensity
+IX,Y
+11:
+Use Equations 1 to 3 to calculate the difference
+of each effect
+12:
+end if
+13:
+Compute summation of ∆S
+′(X, Y) for each effect.
+14:
+end for
+15: end for
+16: for ttimestep = 1, 2, . . . , T do
+17:
+for nregion = 1, 2, . . . N do
+18:
+Use Equation 4 to update final spectrum at tT.
+19:
+Lower the resolution of final spectrum at tT to match
+the HARPS-N observation.
+20:
+end for
+21: end for
+3.1. The basic concept of GPU computing
+The popularity of artificial intelligence has in recent year sig-
+nificantly increased due to the programmability of graphic hard-
+wares. GPU computing uses graphical card as a co-processor for
+parallel computing. Compared with CPU, GPU solves problems
+by breaking them into separate tasks and processing them simul-
+taneously. The basic computational unit that can independently
+perform simple calculation in a graphic card is called a thread.
+A group of threads that communicate and share memory with
+each other is called a block.
+The new version of SOAP presented here, SOAP-GPU,
+is written using the Compute Unified Device Architecture
+(CUDA). CUDA is a compiler and toolkit for programming
+NVIDIA GPUs, and is an extension of the C/C++ programming
+language. CUDA invokes kernel functions by using the syntax of
+<<< Nblocks, Nthreads >>>. This syntax allows the user to define
+the thread hierarchy before launching in parallel the same pro-
+gram function called kernel to many threads. In order to launch
+the computation at the level of the GPU, a host function defined
+in CPU controls the data transfer between CPU and GPU and
+can execute the kernel function inside the GPU.
+Threads in the same block can be accessed as 1D, 2D or
+3D structures. In order to perform the thread level calculation,
+the index of individual thread and block need to be accessed.
+The index of each thread in the same block can be expressed
+as threadIdx. If the block is launched as the 1D structure, each
+thread in the same block can be accessed as threadIdx.x. The
+number of the treads used in each 1D block can be obtained
+as blockDim.x. Grid is a group of blocks. It can be either 1D,
+2D or 3D. For the 1D grid, the index of each block in the
+grid can be expressed as blockIdx.x. Since the input spectra
+of quiet sun and active region are both 1D, we used the con-
+figuration of 1D grid with 1D block and the global index is
+index = blockIdx.x ∗ blockDim.x + threadIdx.x.
+3.2. Fast linear interpolation with GPU
+As mentioned in previous sections, the major limitation in SOAP
+2.0 is the way it handles linear interpolation in each disk cell. A
+GPU provides thousands of cores which can be implement for
+linear interpolation for large data array. Considering that both
+quiet sun and spot spectra are evenly sampled in the wavelength
+domain, then the input wavelength can be described as:
+λn = λ0 + nk,
+(5)
+where n is the pixel number and k is the step size. When a
+Doppler shift is applied, the wavelength array is modified as fol-
+low:
+λ
+′
+n = λn + λn f(β),
+(6)
+where f(β) = −
+�
+1 −
+�
+(1+β)
+(1−β)
+�
+and β = v/c. The variable v is the
+velocity for each cell and c is the speed of light. Since we need
+to project the shifted spectrum back to the original wavelength
+space S (λ
+′) → S
+′(λ), we have to find the index m which satisfies
+λ
+′
+n < λm < λ
+′
+n+1. For the left side, we have:
+λ
+′
+n < λm,
+λ
+′
+n = λn + λn f(β) = λ0 + nk + λ0 f(β) + nk f(β) < λ0 + mk,
+so we have:
+n(1 + f(β)) + f(β)λ0
+k
+< m.
+(7)
+For the right side, we have:
+m < (n + 1)(1 + f(β)) + f(β)λ0
+k
+.
+(8)
+Once the integer m is known, we can estimate the flux for λm
+using the spectrum derivative:
+S
+′
+m =
+∆S n
+(λ
+′
+n+1 − λ
+′
+n) × (λm − λ
+′
+n) + S n,
+(9)
+where S n = S (λn) and ∆S n = S n+1 − S n.
+Equations 7 to 9 can be parallelised using GPU. We launch
+1D grid of 1D blocks with <<< Nblocks, Nthreads >>> to per-
+form the linear interpolation mentioned above and the number of
+blocks and threads satisfies Diminput_spectrum = Nblocks × Nthreads.
+The pseudo code for this part is summarised in Algorithm 3 and
+the quiet sun spectra integration can be rewritten as Algorithm
+4.
+Algorithm 3 Fast interpolation with GPU
+1: index = blockIdx.x ∗ blockDim.x + threadIdx.x
+2: indextarget = ceil(index ∗ (1 + f(β)) + f(β) ∗ λ0/k)
+3: S
+′
+indextarget =
+∆S index
+(λ′
+index+1−λ′
+index) × (λindextarget − λ
+′
+index) + S index.
+Article number, page 4 of 17
+
+Yinan Zhao et al.: SOAP-GPU
+Algorithm 4 Quiet spectrum integration with GPU
+1: for Xlocation = 1, 2, . . . N do
+2:
+for Ylocation = 1, 2, . . . , N do
+3:
+Apply Doppler shift with velocity velX,Y and derive
+S
+′
+quiet(λ) using GPU fast interpolation
+4:
+Weight S
+′
+quiet(λ) by limb-darkening intensity IX,Y and
+integrate spectra along disk surface. Squiet+ = IX,Y ×S
+′
+quiet(λ)
+5:
+end for
+6: end for
+3.3. Active region updates
+As addressed in the previous section, one of the disadvantage of
+SOAP 2.0 is that each active region is modeled independently,
+which makes the code unable to handle complicated active re-
+gion configurations: some active regions may overlap with each
+other; spots may be surrounded by facualae regions. Here we
+propose a revised algorithm to update active regions: an empty
+disk map called In foMap is allocated in the GPU first. At each
+timestamp, A list of active regions with their properties is up-
+loaded and the code calculates the location of active regions pro-
+jected on the disk map. If some regions are visible, we update
+the corresponding pixels with their active region types in the
+information map. For example, if a faculae region is visible at
+(xn, yn), In foMap(xn, yn) = 1.0. If there are multiple regions
+with the same type overlapping with each other, the overlapping
+region in the information map will remain the same. This will
+avoid the over-calculation for the overlapping region issue in the
+SOAP 2.0 since each acitve region is calculated independently.
+This algorithm can also simulate complicated active region con-
+figurations. For example, a spot surrounded by a large faculae
+can be simulated by updating the information map with a fac-
+ulae first. If the spot region is embedded inside the faculae, the
+overlapping region in the information map will be updated with
+the type of the spot. The pseudo code of this part is summarised
+in Algorithm 5.
+Algorithm 5 Active region updates with GPU
+1: for ttimestep = 1, 2, . . . , T do
+2:
+for nregion = 1, 2, . . . N do
+3:
+Localize(lat, long, i, ph).
+4:
+if Localize = 1 then
+5:
+Updating InfoMap(xn, yn) = the type of active
+region.
+6:
+end if
+7:
+end for
+8:
+Inject velocity velX,Y with GPU fast interpolation for the
+active regions in the information map. and derive S
+′
+active(λ).
+9:
+Weight S
+′
+active(λ) by limb-darkening intensity IX,Y
+10:
+Use Equations 1 to 3 to calculate the difference of each
+effect
+11:
+Compute summation of ∆S
+′(X, Y) for each effect.
+12:
+Use Equation 4 to derive the final spectrum at tT.
+13:
+Lower the resolution of final spectrum at tT to match the
+HARPS-N observation.
+14: end for
+3.4. Differential rotation
+In the original SOAP 2.0 code, there is no differential rotation
+implemented. In order to better model stellar activity, differential
+rotation is included when the stellar disk is initialized according
+to the equation ω = ω0 + ω1 sin2(θ), where ω0 = 14.371◦/day
+and ω1 = −2.587◦/day for the Sun (Borgniet et al. 2015). To
+generalise this for other stars, the user can select in the configu-
+ration of SOAP-GPU a rotation period and a differential rotation
+rate. ω0 is then equal to 360/PROT and ω1 to DIFF_ROT*PROT
+(PROT=25.05 and DIFF_ROT=-0.18 for the solar case to repro-
+duce the above equation).
+4. Results
+Fig. 2. The computation speed comparison between SOAP 2.0 and
+SOAP GPU: The integrated quiet sun spectrum is calculated with dif-
+ferent disk resolutions. When disk resolution is below 10, SOAP 2.0
+is faster than SOAP-GPU since the communication between CPU and
+GPU in SOAP-GPU is time-consuming. For resolutions above 10,
+SOAP-GPU is significantly faster than SOAP 2.0. With a typical res-
+olution value of 300, the quiet disk spectrum integration in SOAP-GPU
+is 100 times faster than in SOAP 2.0
+4.1. Performance and precision comparison with SOAP 2.0
+We examined the performance of SOAP-GPU in two aspects:
+computational speed and accuracy. SOAP-GPU code is executed
+on a Nvidia RTX-3090 card while we run the modified SOAP 2.0
+that generates spectra in a MacBook Pro with 2.6 GHz 6-Core
+Intel Core i7. We analysed the speed performance of SOAP-
+GPU by calculating the time it takes to obtain an integrated quiet
+sun spectrum. The input quiet sun spectrum has a dimension of
+547840, thus the kernel function fast interpolation is launched
+with <<< Nblocks, Nthreads >>>=<<< 1070, 512 >>>. We note
+that Nthreads is fixed to 512 and Nblocks is an adaptive number
+based on the dimension of the input. SOAP 2.0 is executed with
+the same simulation configuration on a single CPU. We com-
+puted the integrated quiet sun spectrum with different disk res-
+olution and their computational time is show in Figure 2. When
+the disk resolution is very low, smaller than 10, SOAP 2.0 is
+faster than SOAP-GPU. This is not surprising since the data
+transfer between GPU and CPU in SOAP-GPU is the dominating
+factor. When the disk resolution increases, SOAP-GPU is signif-
+icantly faster than SOAP 2.0. When the resolution is above 100,
+the quiet sun spectrum integration of SOAP-GPU is 100 times
+faster than SOAP 2.0 and both computational curves linearly in-
+crease in log-log space.
+Article number, page 5 of 17
+
+103
+SOAP 2.0
+SOAP GPU
+time (seconds)
+102
+101
+Comuputation
+100
+10-1
+100
+101
+102
+Disk resolution (N)A&A proofs: manuscript no. SOAP_GPU
+Boisse et al. (2012) found no significant change in their re-
+sults with resolution beyond 300, therefore, we used this disk
+resolution for the following of the paper. For the typical disk
+resolution of 300, a spot at disc center with an area of 1% of the
+entire disk will be contained in a grid of 34×34 cells. Due to the
+small size of the grid for such a configuration, the fast interpo-
+lation algorithm (see Algorithm 3) is only able to gain a factor
+of ∼10 in computation time. If the spot size increases to 9% of
+the entire disk, the simulation can then gain almost the full speed
+boost from fast interpolation (100 time faster). Fast interpolation
+at the level of the active region modelisation makes therefore sig-
+nificant improvements in computational speed when considering
+high-resolution simulations or simulations with large active re-
+gions.
+Fig. 3. Comparison of the RVs derived from the simulated spectra mod-
+eled by SOAP2.0 and SOAP-GPU. A single equatorial spot with 1%
+area of the entire disk surface is simulated. It took 1749.3 seconds to
+simulate those spectra with SOAP 2.0 while only 27.9 seconds with
+SOAP-GPU on a Nvidia RTX-3090 card. The computation speed is im-
+proved by a factor of 63.
+We also examined the accuracy of SOAP-GPU. We simu-
+lated a single equatorial spot with a 1% area of the entire disk
+surface using a disk resolution of 300. It took 1749.3 seconds to
+simulate those spectra with SOAP 2.0 for 100 timestamps while
+only 27.9 seconds using SOAP-GPU, which corresponds to a
+gain of a factor 63. The modeled RVs relative to the flux effect,
+the CB effect and the total effect are derived from the simulated
+spectra by cross-correlating them with the same mask originally
+used in SOAP 2.0, and measuring the RV as the mean of a Gaus-
+sian profile fitted to the obtained CCFs. Figure 3 illustrates that
+the simulated spectra from SOAP-GPU provides the same RVs
+as the spectra from SOAP 2.0.
+4.2. Exploration of active region properties
+The dynamics of active regions plays an important role for un-
+derstanding the stellar activity-induced RVs. Most of previous
+study aimed at investigating these effects with real observations.
+For example, Meunier et al. (2010) derived the stellar activity
+induced RVs by using Michelson Doppler Imager/Solar and He-
+liospheric Observatory (MDI/SOHO) magnetograms images. At
+the simulation level, Gilbertson et al. (2020a) investigated the ef-
+fect of spot evolution on the long-term and at the spectral level,
+using a modified version of SOAP 2.0. However, they only con-
+sidered spots, and only their decaying phase. In order to illustrate
+the effects of active region dynamics, we discuss in this section
+the photometric and RV variations observed when an active re-
+gion changes in size, when different number of active regions are
+present and when the active region configuration changes.
+4.2.1. The size evolution of active region
+To explore different active region evolution scenarios, we de-
+veloped and included an evolution module in SOAP-GPU. This
+module can model evolution in three different ways: i) a linear
+growing phase, ii) a linear decaying phase or iii) a growing and
+decaying phase modeled by an an asymmetric Gaussian func-
+tion (Muraközy et al. 2014). Other user-defined functions can be
+added to this module if desired. We show in Fig. 4 the impact
+of active region evolution on the light-curve and on the differ-
+ent RVs derived (flux, CB and total effects). For the asymmet-
+ric Gaussian evolution phase, the maximum size is set to 10000
+millionths of solar hemisphere (MSH) equivalent to 1% of the
+visible hemisphere, the FWHM to 10 days and an asymmetry
+factor of 0.09. For the growing only, or decaying only evolution
+phases, the initial size is set to 10000 MSH and the growth or
+decay rate is set to 400 MSH/day. We found that both flux and
+CB effects are sensitive to the evolution of active regions.
+4.2.2. Complex active regions
+SOAP-GPU also allows users to simulate complex active region
+configurations. From the observational point of view, facluae and
+spots are not independent from each other. The facula distribu-
+tion is based on the spot distribution. This leads to a complex
+configuration in which spots may overlap faculae (Borgniet et al.
+2015; Chapman et al. 2001). In order to model such a configu-
+ration, the SOAP-GPU config file allows users to define the dis-
+tribution of active regions, as a sequence of spots and faculae
+with given properties (size, initial longitude, initial latitude). For
+example, in order to simulate a spot surrounded by a facula, the
+user can define the location and the size of the large facula first
+and then define a smaller spot at the same location. The region of
+overlap will be replaced by the spot as mentioned in Algorithm
+5. A simulation of this case is illustrated in Figure 5, with a 1%
+spot surrounded by a 9% facula. Since the spot has a higher con-
+trast than the faculae, the light curve and RVs of the flux effect
+is dominated by the spot while the RVs of the CB effect is domi-
+nated by facula. Overall, the CB RV effect induced by the facula
+dominates all the other contributions, and thus the total RVs is
+affected mainly by the facula, as it was already demonstrated in
+several studies (e.g. Meunier et al. 2010; Dumusque et al. 2014;
+Milbourne et al. 2019).
+4.3. Exploration of spectral properties
+In this section, we explore the input spectra properties and
+demonstrate how the derived RV behaves depending on the
+wavelength domain.
+4.3.1. Chromatic effects of different wavelength coverage
+To explore the effect induced by different wavelength coverage,
+we injected into SOAP-GPU only the red or only the blue part of
+the quiet sun and spot spectra (see Fig. 6). The red and blue parts
+have the same dimension of 204800, which is different from the
+full spectra. As the fast interpolation kernel function depends
+on the dimensions of the input spectra, the code automatically
+configures the kernel with the option <<< 400, 512 >>>.
+Article number, page 6 of 17
+
+CPU Tot
+CPU FIux
+CPU Bconv
+4
+GPU Tot
+GPU FIuX
+GPUBconv
+2
+[s/u] 
+RV
+0
+-2 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+PhaseYinan Zhao et al.: SOAP-GPU
+Fig. 4. SOAP-GPU simulation of different active region evolution curves: A single spot with a latitude of 30◦and longitude of 180◦is simulated.
+Three spot size evolution types are demonstrated: i) a fast growth and slow decay evolution is shown in the first column, ii) a linear decay evolution
+curve in the second column and iii) a linear growth curve in the third column. The evolution curves and the simulated light curves are shown in the
+first two rows. The RVs of the total effect, the flux effect and the CB effect are present in the rest of the rows. The simulation of an non-evolving
+spot (red dashed line ) is also shown in each figure for comparison.
+The measured RVs are shown in Figure 7. The RVs of the CB
+effect are different between the blue and red parts. One notable
+thing is that the RVs of the CB effect simulated from the red
+inputs goes below zero, while we expect the CB effect to only
+be positive, as it corresponds to an inhibition of CB. To confirm
+that nothing was wrong at the level of the code, we injected for
+the spotted region the same spectrum as the quiet Sun, but we
+red-shifted it by 300 m/s to model at first order the inhibition of
+CB. In this idealist case, the CB effect does not provide negative
+values. After further investigation, those negative values comes
+from the fact that the spot temperature is lower than the quiet
+photosphere. Thus, spectral lines will change in depth, which
+will induce a flux effect even when not considering the contrast
+of the active regions when estimating the CB effect (see Eq. 2).
+In the case of the Sun, this flux effect seen in the CB derived RVs
+is mainly coming from the red part of the spectrum due to molec-
+ular absorption that can be seen in the spot spectrum, but not in
+the quiet photosphere spectrum. As we will see in Sect. 4.3.3,
+we do not obtain negative values when injecting PHOENIX solar
+equivalent spectra for the quiet and active Sun instead of the Kitt
+Peak solar quiet and active atlases, however we still see a slight
+asymmetry in the derived CB RV effect, pointing toward a small
+flux effect contribution. This is likely because PHOENIX spectra
+are not able to model all the absorptions coming from molecular
+bands, and thus the flux effect seen in the CB estimation only is
+stronger for the real solar spectra than for the synthetic spectra.
+This issue prevent us of fully separating the flux from the CB
+effect, however, we note that the total effect (flux + CB) should
+be modeled properly. We note that this feature was not visible in
+SOAP 2.0, as after computing the CCF for the quiet and actives
+regions, we were renormalising them.
+We note that in SOAP 2.0, we used a fixed contrast to model
+the flux effect of active regions. This contrast was derived by
+comparing the Planck function of the quiet Sun effective tem-
+perature and of the spot or facula temperature4, at the average
+wavelength of the input spectra 5293 Å. Now that we use spectra
+4 the config file in SOAP 2.0 allows to give an effective temperature for
+the quiet photosphere, 5778 K for the Sun, and a temperature difference
+with respect to this former value for the spot spectrum (663 K as the
+default value in SOAP 2.0). For a facula, the temperature is dependent
+on the center-to-limb angle and was following what is observed on the
+Sun (e.g. Fig. 3 in Meunier et al. 2010).
+Article number, page 7 of 17
+
+20000
+Gauss
+Decay
+Growth
+Non
+Non
+Non
+area
+15000
+Active region 
+10000
+5000
+0
+1.000
+Light
+0.996
+0.994
+Non
+Non
+Non
+Gauss
+Decay
+Growth
+7.5
+Non
+Non
+Non
+5.0
+Gauss
+Decay
+Growth
+RV (m/s)
+2.5
+0.0
+Tot
+2.5
+-5.0
+Non
+Non
+Non
+4
+Gauss
+Decay
+Growth
+RV (m/s)
+2
+0
+Flux
+-2 
+-4
+-6
+4
+Non
+Non
+Non
+Gauss
+Decay
+Growth
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Phase
+Phase
+PhaseA&A proofs: manuscript no. SOAP_GPU
+Fig. 5. Three different active region configurations are simulated. A sin-
+gle spot located at the latitude of 30◦and the longitude of 180◦with a
+fixed size of 1% of the entire solar disk is present in black. A single fac-
+ula with the same coordinates and a fixed size of 9% of the entire solar
+disk is shown in blue line. The 1% spot region surrounded with 9% fa-
+clua region is labeled in red dashed line. The top panel demonstrates the
+light curves of different configurations and the rest of the panels shows
+the RVs of the total effect, the flux effect and the CB effect.
+Fig. 6. Injecting different size of spectra in SOAP-GPU input. Three
+different sets of quiet sun and spot spectra, with different wavelength
+ranges are used as input to SOAP-GPU: The entire spectra with length
+547840 is labeled in black. The blue and red parts of the spectra with
+length 204800 are over plotted in blue and red, respectively.
+as input, and not CCFs, we implemented a contrast that is wave-
+length dependant to model the chromatic effects of stellar activ-
+ity. To do so, we introduced a new GPU kernel function called
+SOAPcontrast <<< Nblocks, Nthreads >>>. This new kernel al-
+lows to perform the wavelength dependent contrast calculation:
+each wavelength pixel is first accessed by the global index of the
+kernel. Next on each thread, it derives the contrast by calculating
+the ratio of two Planck functions at two different effective tem-
+peratures. The absolute value of the contrast in the blue part of
+the spectrum is higher than in the red part. This implies that the
+Fig. 7. The chromatic effect of RVs. SOAP-GPU is initialized with three
+different spectra: entire wavelength coverage, and only red and blue
+spectral parts (see Sect. 6). A single spot with a size of 1%, a latitude of
+30◦and a longitude of 180◦is modeled by the code. The measured RVs
+with different input spectra are labeled with black, red and blue, respec-
+tively. Top: The measured RVs of the total effect. Middle: The measured
+RVs of the flux effect. An offset of 1 ms−1 is added in the red and blue
+RVs. Bottom: The measured RVs of the CB effect.
+flux effect for the blue part is stronger than in the red part, which
+can be seen in the middle panel of Fig. 7.
+4.3.2. Convection as a function of center-to-limb angle
+Solar spectral line profiles become asymmetric due to convective
+motions varying with physical depth inside the solar photosphere
+(e.g. Dravins et al. 1981; Gray 2009). This effect also leads to a
+change in shape of the bisector of spectral lines from disk-center
+to the limb, as photons are coming from different physical depths
+(e.g. Cavallini et al. 1985b). In order to better model the effect of
+convection in SOAP-GPU, we derived this effect from very-high
+spatial and spectral resolution observations of the Sun (Löhner-
+Böttcher et al. 2018; Stief et al. 2019; Löhner-Böttcher et al.
+2019).
+We note that the varying shape of spectral line with center-
+to-limb µ angle is also modelled in the STARSIM 2 code (Her-
+rero et al. 2016) by fitting a fourth-order polynomial function
+on magneto-hydrodynamic CIFIST 3D models (Ludwig et al.
+2009). However, this fifth-order polynomial is only valid for line
+depth as strong as ∼ 0.5 (see Fig. 6 in Herrero et al. 2016)5. This
+was enough to model the shape change of the CCFs in STAR-
+SIM 2, which does not go deeper. However, when working at the
+spectral line level, this polynomial will give completely wrong
+estimate for the core of deep lines, due to extrapolation. We
+therefore used the quiet sun observations at different µ angles
+provided in Löhner-Böttcher et al. (2019). We first measured the
+bisectors of all the available iron deep lines at different µ angle in
+5 The parameters mentioned here are derived from the published code
+( https://github.com/rosich/starsim-2)
+Article number, page 8 of 17
+
+Light curve
+1.002
+1.000
+0.998
+Spot only
+Faculae only
+0.996
+Combined
+Spot only
+Tot RV (m/s)
+20
+Faculae only
+Combined
+10
+0
+Flux RV (m/s)
+4
+Spot only
+Faculae only
+2
+Combined
+-2
+Bconv RV (m/s)
+Spot only
+20
+Faculae only
+Combined
+10
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Phase1.0
+Iflux
+0.8
+Normalized
+0.6
+0.4
+0.2
+Quiet
+Quiet_blue
+1:8:
+Quiet_red
+I flux
+0.8
+Normalized
+0.6
+0.4
+0.2
+Spot
+Spot_blue
+0.0
+Spot_red
+4000
+4500
+5000
+5500
+6000
+6500
+Wavelength (A)5.0
+Full
+Blue
+Tot RV (m/s)
+2.5
+Red
+0.0
+-2.5
+-5.0
+4
+Full
+Flux RV (m/s)
+Blue
+2
+Red
+0
+2
+-4 
+4 
+Full
+Bconv RV (m/s)
+Blue
+2
+Red
+0
+-2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+PhaseYinan Zhao et al.: SOAP-GPU
+Löhner-Böttcher et al. (2019)6 , and fitted them using polynomial
+functions. For µ angle smaller then 0.5, we used a straight line
+to fit the bisectors of the selected deep lines, to prevent strong
+divergence when extrapolating the fit towards very large depths.
+For µ angle larger or equal to 0.5, 3rd order polynomial func-
+tions are used to capture the curvature of the bisectors around
+disk center. The final bisectors, shown in Fig. 8 are obtained by
+interpolating and extrapolating those bisectors from depth 0 to
+1.
+To obtain the dependency of the spectral line bisector as a
+function of µ in an active region, we use the observations pre-
+sented in Cavallini et al. (1985a). We parameterised the bisec-
+tors of the FeI at 6301.5008Å, for the disk center (µ = 1) and
+different center-to-limb angles (µ = 0.82, 0.66 and 0.44). The
+bottom part of the bisectors, below 0.5, is fitted using a straight
+line, the upper part for which we have data, using a 5th-order
+polynomial. Rather than extrapolating the fitted polynomial to-
+wards very shallow depths, which can give unrealistic redshifted
+values, we decided to use the more redshifted data value of the
+top bisector for extrapolation. We show in Fig. 8 the obtained
+active bisectors from depth 0.0 to 1.0.
+Once we have our model for line bisectors at different µ an-
+gles, we can use the Python module Convec.model to apply those
+bisectors to the original spectra, and thus obtain different spec-
+tra for different µ angles. Each cell in the stellar disk takes the
+bisector that has the closest µ angle. However, the code first has
+to remove the original bisector from the spectral lines of the in-
+put quiet and active Kitt Peak solar spectra. To do so, we select
+the same lines as in Löhner-Böttcher et al. (2019) in the input
+spectra and measure their individual bisectors. To model the av-
+erage bisector of the lines selected in the quiet spectrum, we use
+a second-order polynomial. For the active spectrum, due to the
+lower effective temperature, the wings of certain lines fitted are
+blended, which strongly impact the bisector measurement. We
+therefore rejected bisector points that are significantly off. Then,
+we model the average active bisector of the lines by fitting the
+regions below and above a depth of 0.5 with two different linear
+models. Fitting a higher-order polynomial for those active bi-
+sectors was giving unrealistic values when extrapolating to very
+small or very large depths. The measured individual bisectors
+with our models are shown in Fig. 9. To finally obtain quiet and
+active spectra with proper bisector shape as a function of µ an-
+gles, we remove the original bisectors of the quiet and active
+Kitt peak solar spectra, and then add the bisectors measured for
+different µ angles. This is done by shifting each point in those
+spectra depending on their normalised depth.
+To inject the proper bisectors for different µ angles in our
+original spectra, we first remove the original bisector, which
+changes any CB difference between the quiet and active solar
+spectra. We therefore need to impose a shift between the bisec-
+tors of quiet and active regions in order to properly model the in-
+hibition of CB inside active regions. We here make two assump-
+tions: i) the CB is fully inhibited at µ=0.2 in the quiet Sun, and
+ii) it is also fully inhibited for magnetic regions, and this at all µ
+angles. Using the first assumption, we measure for the quiet Sun
+the maximum shift between the bisector at µ=0.85, which is the
+bisector that is the most blueshifted, and the bisector at µ=0.2.
+This maximum happens at a depth of 0.58 and equals to 375 m/s.
+This value is extremely similar to the 340 m/s CB value derived
+from a fit to the data of Liebing et al. (2021) (see Sect. 4.3.3).
+6 We use the following lines: FeI 5250.2084Å, FeI 5250.6453Å, FeI
+5434.5232Å, FeI 5432.9470Å, FeI 5576.0881Å, FeII 6149.2460Å, FeI
+6173.3344Å, FeI 6301.5008Å and FeI 6302.4932Å
+To match the CB relation derived in Sect. 4.3.3, we rescale the
+maximum difference between the µ = 0.85 and µ = 0.2 to be
+340 m/s. We note that this rescaling is negligible in the case of
+the Sun, however, it will be really needed in Sect. 4.3.3 when
+using PHOENIX spectra as input. Using the second assumption,
+we impose that at the same depth of 0.58, the difference in veloc-
+ity between the quiet bisector at µ=0.85 and all active bisectors
+is also 340 m/s. We show the proper shift between the quiet and
+active bisectors in Fig. 8.
+We show the RV impact of considering the µ angle depen-
+dency on the observed solar spectra in Fig. 10. As we can see,
+the impact is not significant when looking at the shape of the
+signal as a function of phase. This come from the fact that due
+to limb-darkening and the projection of active regions on the
+limb, most of the signal comes from larger µ angles (close to
+disc center). The only significant difference is for the amplitude
+of the CB effect. This is because we forced the CB difference be-
+tween the quiet and active sun to be 340 m/s, while the CB differ-
+ence between the quiet and active Kitt peak solar spectra is less
+than 300 m/s when measuring the average difference between the
+quiet and active CCF bissectors (see Fig. 2 in Dumusque et al.
+2014). We note that the complexity of modifying the bisectors
+depending on the center-to-limb angle is not strongly justified
+when using real solar spectra as input due to the small difference
+observed in the estimated RVs, however, this step is critical when
+working with synthetic spectra that does not include the proper
+bisectors, as described in Setc. 4.3.3.
+We are conscious that depending on the magnetic field of an
+active region, the inhibition of the CB will be different and there-
+fore the bisectors more or less redshifted compared to the quiet
+Sun, as seen in Fig. 1 in Cavallini et al. (1985a). Also, faculae
+tend to have weaker magnetic fields than spots and in our case,
+we model those two active regions with the same bisectors and
+the same CB inhibition. It is therefore likely that the CB effect
+for faculae is slightly overestimated, and this will translate in
+larger RV amplitudes when modeling the CB effect for faculae.
+In summary, in this subsection we present a framework to
+model the Sun but also other stars (see also next subsection).
+Different bisectors at different µ are derived from the quiet pho-
+totsphere (Löhner-Böttcher et al. 2019) and facuale (Cavallini
+et al. 1985b) and are injected into the input spectra for which we
+have removed any variation in line bisector from a vertical line.
+4.3.3. Simulation based on the PHOENIX spectral database
+The implementation of convective motions described in the pre-
+ceding section allow us to use synthetic spectra as input, since
+the effect of convection can be injected using the Convec.model
+module. In order to study stellar activity affecting the data used
+in RV, a high resolution spectral library is needed. For SOAP-
+GPU, we decided to make it easy for the user to use as input
+PHOENIX high-resolution spectra (Husser et al. 2013). We note
+however that SOAP-GPU can accept other spectral libraries, but
+it might be a little more difficult for the users to properly setup
+the inputs since the parameters to remove bisectors of input spec-
+tra are only optimized for the PHOENIX spectra and the so-
+lar atlas from the Kitt Peak Observatory FTS (Wallace et al.
+2005). The PHOENIX library propose a collection of spectra
+with the wavelength coverage from 500Å to 5.5µm with reso-
+lutions of 500,000 in the optical. The library covers stellar ef-
+fective temperature from 2300K to 12000K. Since the spectra
+in the PHOENIX library are not normalized, which is critical to
+perform the injection of CB described in the preceding section,
+Article number, page 9 of 17
+
+A&A proofs: manuscript no. SOAP_GPU
+Fig. 8. Average bisectors of quiet and active solar regions from the disk center (µ = 1.0) to the limb (µ = 0.2). Continuous lines: Fifth-order poly-
+nomial fit to the quiet sun bisectors of the FeI 5250.2084Å, FeI 5250.6453Å, FeI 5434.5232Å, FeI 5432.9470Å, FeI 5576.0881Å, FeI 6149.2460Å,
+FeI 6173.3344Å, FeI 6301.5008Å and FeI 6302.4932Å lines as measured by the Laser Absolute Reference Spectrograph (LARS) at the German
+Vacuum Tower Telescope (Löhner-Böttcher et al. 2019). Dashed lines: Fit of the bisectors of the FeI 6301.5008Å spectral line inside a faculae
+region, as measured by the Fabry-Perot interferometer at the Donati Solar Tower (Cavallini et al. 1985a). Below a depth of 0.5, a linear fit is per-
+formed, while a fifht-order polynomial is used to model the top part of the bisector. To prevent unrealistic value when interpolating the polynomial
+above a normalised flux of 0.9 where no measurement exists, we selected the most redshifted part of the top bisector, explaining the vertical values
+for very shallow depths. The two vertical lines are shifted by 340 m/s which corresponds to the solar convective blueshift value derived from a fit
+to the data of Liebing et al. (2021) (see Sect. 4.3.3). The active bisectors at different µ angles are all shifted by those 340 m/s at a depth of 0.58 as
+we make the hypothesis that convection is fully suppressed in magnetic regions (see Sect 4.3.2 for more information).
+Fig. 9. Bisectors of the FeI lines used in Löhner-Böttcher et al. (2019) and fitted model to account for the CB. Left: Bisectors from the quiet Kitt
+Peak solar spectrum. Right: Bisectors from the active Kitt Peak solar spectrum. We rejected the bottom part of the 6301.5008Å bisector because it
+was significantly off by 2500 m/s due to strong contamination by other weak lines.
+we used the open source package “Rassine” (Cretignier et al.
+(2020b)) to perform spectral normalization first. The continuum
+or spectral energy distribution (SED) of input spectra derived
+from “Rassine” are denoted as SEDquiet and SEDactive respec-
+tively.
+The inhibition of CB when working with PHOENIX spectra
+can be rewritten as:
+∆S
+′
+bconv(X, Y) = S
+′
+quiet,n(X, Y) − S
+′
+active,n(X, Y),
+(10)
+where S
+′
+quiet,n and S
+′
+active,n are the normalized quiet and active
+spectra. Since the contrast between the quiet and active region
+Article number, page 10 of 17
+
+1.0
+μ = 0.20
+μ=0.30
+0.8
+μ=0.40
+μ=0.50
+μ=0.60
+Normalized flux
+0.6
+μ= 0.70
+μ=0.80
+μ=0.85
+μ=0.90
+0.4
+μ= 0.95
+μ = 1.00
+Active region μ =0.44
+Active region μ =0.66
+0.2
+Active region μ =0.82
+Active region μ =l.0
+0.0
+-200
+0
+200
+400
+Doppler velocity (m/s)1.0
+1.0
+0.8
+0.8
+Fit
+Binned data
+= 0.6
+Fel5250.2084A
+0.6
+Normalized 
+Normalized 
+Fel 5250.6453A
+Fel5434.5232A
+Fel 5432.9470 A
+Fit
+0.4
+0.4
+Binned data
+Fel5576.0881A
+Fel 5250.2084A
+Fel 6301.5008A
+Fel5250.6453A
+0.2
+0.2
+Fel 5434.5232A
+Fel5432.9470A
+Fel5576.0881A
+0.0
+0.0
+Fel 6301.5008 A
+-200
+-100
+0
+100
+200
+300
+-300-250-200-150-100
+-50
+0
+50
+Doppler Velocity (m/s)
+Doppler Velocity (m/s)Yinan Zhao et al.: SOAP-GPU
+Fig. 10. Comparison of the RVs derived using two different configu-
+rations for the input spectra. A single equatorial spot with 1% area of
+the entire disk surface is simulated in both cases. Red line: RVs derived
+from simulated spectra with observed quiet sun and spot spectra without
+µ dependent bisector injection. Dotted line: RVs derived from simulated
+spectra with µ dependent bisector injection. The input spectra with µ-
+angle dependency are generated with the Python module Convec.model.
+The RVs of the flux, CB and total effect don’t change significantly when
+the µ-dependent CB is introduced.
+is naturally included in the continuum of input PHOENIX spec-
+tra, the flux effect can be rewritten as:
+∆S
+′
+flux(X, Y) = S
+′
+quiet,n(X, Y) × SEDquiet
+− LB(X, Y) × S
+′
+quiet,n(X, Y) × SEDactive,
+(11)
+where LB is the function of limb brightening. For simulation of
+spot regions, LB = 1.0 along the disk. For simulation of faculae
+regions, SOAP 2.0 was using ∆T f = 250.9 − 407.7µ + 190.9µ2
+to model the limb brightening in temperature domain (Meu-
+nier et al. 2010). Since this equation is only valid for the Sun,
+we use the empirical equation derived from 3D MHD simula-
+tions to model other spectral types. 3D MHD simulations mod-
+elling faculae on the Sun can reproduce extremely well the limb-
+brightening observed for faculae (Norris et al. 2017). Using sim-
+ilar simulations, Johnson et al. (2021a) model what would be
+the limb-brightening on other stars (see Figure 3 and Table 1 in
+Johnson et al. (2021a)). Using the parametrisation in Johnson
+et al. (2021a), we derived the limb brightening curves of faculae
+for a G2, K0 and M0 dwarfs. We then linearly interpolated be-
+tween the G2 and K0 and K0 and M0 simulations to obtain the
+limb-brightening dependence for a G8 and G9 dwarf, and a K2
+dwarf, respectively. To obtain the dependence for a F9 dwarf, we
+linearly extrapolated from the G2 and K0 models. The derived
+limb brightening curves from F9 to K2 are shown in Figure 11.
+Finally, the total effect from flux and convection can be de-
+rived using the following equation:
+∆S
+′
+tot(X, Y) = S
+′
+quiet,n(X, Y) × SEDquiet
+− LB(X, Y) × S
+′
+active,n(X, Y) × SEDactive.
+(12)
+Convection velocity changes with spectral type, and de-
+creases towards cooler stars than the Sun. This is a well known
+Fig. 11. Faculae intensity contrast as a function of µ for different spec-
+tral types. The limb brightening curves for the G2, K0 and M0 dwarfs
+are derived from the parametrisation of MHD simulations for 500G fac-
+ulae, shown in Table 1 in Johnson et al. (2021a). The limb brightening
+curves for other spectral types are linearly interpolated by using the two
+closet limb brightening curves. For spectral types hotter than G2, we ob-
+tain limb brightening by performing linear extrapolation using the G2
+and K0 curves. The limb brightening derived from Meunier et al. (2010)
+is labeled with orange.
+effect, that comes out from observations (e.g. Meunier et al.
+2017; Liebing et al. 2021) and magneto hydrodynamic mod-
+els of stellar phostophere (e.g. Allende Prieto et al. 2013). As
+in Liebing et al. (2021), we show in the top panel of Fig. 12
+that the CB is a cubic function of effective temperature in the
+range 4800 to 6300 K. The parametrisation that we obtain is
+CBvel = 95.2388 × ((Teff − 4400)/1000)3 + 91.2791. Once we
+have this relation to measure the velocity of CB as a function of
+effective temperature, we simply have to rescale the difference
+between the quiet Sun bisectors for µ = 0.85 and µ = 0.2 at
+depth 0.58 to be equal to the value given by our relation, and
+we also impose that the active bisectors are shifted by the same
+value, at the same depth (see Sect. 4.3.2 for justification). This
+allows us to model properly the change of convective velocity as
+a function of stellar effective temperature. We note that as in the
+case of the Sun, before injecting the proper bisector for the quiet
+and active regions, we first have to remove the bisector present
+in the PHOENIX spectra that we use. This is a necessary step as
+the PHOENIX spectral library is obtained from 1D atmospheric
+models and cannot properly reproduce line bisector shape. We
+show in the appendix, like for Fig. 9 in the case of the Sun, how
+the bisector of the original PHOENIX spectra for the quiet stellar
+region, a faculae and a spot, are fitted before being removed.
+In Fig. 13, we show the results of a few simulations consid-
+ering µ dependent input spectra and a single 1% equatorial spot
+(left panel) or a single 1% facula (right panel). We highlight the
+RV outputs of SOAP-GPU when using PHOENIX spectra, with
+the quiet Sun temperature set to Teff = 5778K, the spot temper-
+ature set to Teff = 5115K, 5015K and 5215K and the facula tem-
+perature set to Teff = 5928K, 6028K and 6128K. We also show
+the result when inputting the Kitt Peak solar quiet and spot or
+facula spectra (Teff = 5778K and 5115K or 6028K at disk cen-
+ter, respectively) as modified in Sect. 4.3.2, and including (us-
+ing Eq. 11, 10 and 12) the SED from corresponding PHOENIX
+spectra. As we can see, the amplitude of the RV flux effect in-
+creases with an increase in temperature difference between the
+quiet photosphere and the spot or facula. The CB effect does not
+change in amplitude with temperature difference and thus the
+larger amplitude observed for larger difference in temperature
+Article number, page 11 of 17
+
+6
+With μ-conv
+Tot RV (m/s)
+Without μ-conv
+2
+-2
+-4
+4
+Withμ-conv
+Flux RV (m/s)
+2
+Without μ-conv
+0
+-2
+4
+Withμ-conv
+Bconv RV (m/s)
+3
+Without μ-conv
+2
+1
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Phase0.25
+Spectral type: F9
+Spectraltype: G8
+Spectral type: G9
+0.20
+Spectral type: K2
+G2 Meunier et al. 2010
+Intensity Contrast
+G2 500G
+0.15
+K0 500G
+M0500G
+0.10
+0.05
+0.00
+0.2
+0.4
+0.6
+0.8
+1.0
+μA&A proofs: manuscript no. SOAP_GPU
+Fig. 12. CB velocity as a function of spectral type. Top: Data from
+Liebing et al. (2021) and cubic fit to them, giving as relation CBvel =
+95.2388 × ((Teff − 4400)/1000)3 + 91.2791. We show with coloured
+stars the CB velocity value for different spectral types that we model in
+the paper. Bottom: The bisector of the quiet photosphere measured from
+Löhner-Böttcher et al. (2019) at µ = 0.85 (thin lines) and the bisector of
+an active region measured from Cavallini et al. (1985a) at µ = 1.0 (thick
+lines) for different spectral types. We impose that the maximum differ-
+ence between the quiet bisector at µ = 0.85 and µ = 0.2 (not shown
+here), happening at a depth of 0.58 is equal to the CB velocity given by
+the relation found in the top panel. We also force the difference between
+the quiet bisector at µ = 0.85 and the active bisectors, independent of
+their µ angle, to be equal to the same value at a depth of 0.58.
+between quiet and active regions is solely driven by the change
+in contrast of the active regions with temperature. While for the
+flux effect, the simulation using PHOENIX spectra or observed
+solar spectra as input gives the same results, which is not surpris-
+ing as we use the same SED, this is not the case for the CB ef-
+fect. The amplitude derived show a small discrepancy of ∼20%,
+and the maximum has a slight phase shift. This likely comes
+from a different flux effect contribution seen in the derived CB
+RV, due to molecular absorption not perfectly modeled in the
+PHOENIX spectra compared to solar real observations (see dis-
+cussion in Sect. 4.3.1). We note that this asymmetry was already
+something seen in the original SOAP 2.0 paper (see Fig. 6 in Du-
+musque et al. 2014). Something also interesting to note, that we
+see when using as input both the PHOENIX and solar spectra is
+the bump in the CB RV effect when the active region crosses the
+center of the disk (phase=0.5). This is induced by the fact that
+CB is maximum at µ = 0.85 and not at disk center, as was shown
+in Löhner-Böttcher et al. (2019).
+In Fig.14 we show the result of the estimated CB RV effect
+for an equatorial 1% spot or facula for stars of different tempera-
+ture (i.e. spectral type): 6050 K (F9), 5778 K (G2), 5480 K (G8),
+5380 K (G9) and 5100 K (K2). For the spot, we see a positive
+only effect for the F9 and G2 simulations, which is expected, but
+for later spectral type, we start to see the emergence of a flux
+effect. This effect, as already discussed in Sect.4.3.1 comes from
+the absorption of molecules, that change significantly over a few
+hundreds of Kelvin for the quiet photosphere at 5480 K and a
+spot at 4817 K for the G8 simulation for example. Also, we see
+that the more we go towards cooler stars, the more the CB ef-
+fect show a flux contribution, and this comes from the fact that
+molecular absorption is not linear with effective temperature. Al-
+though molecular absorption prevent us of clearly separating the
+flux effect of spots from the CB effect like in the case of the
+F9 or G2 star, the total RV effect including both contributions is
+still properly estimated. Therefore, users should be careful when
+interpreting the estimated RV induced by the inhibition of con-
+vection for spots on stars with spectral type later than the Sun,
+however, they can trust the total RV effect estimated. Regarding
+the facula, we observe a positive only effect for all spectral type,
+and thus the CB effect is properly modeled for this type of active
+regions.
+4.3.4. Limitation of SOAP-GPU when moving away from
+solar twins
+On a careful note, we want to warn the user that a lot of the
+physics included in SOAP-GPU is based on solar observations,
+like for example the variation of the bisector of the quiet photo-
+sphere with respect to the center-to-limb angle (Löhner-Böttcher
+et al. 2019), or the bisector of an active region extracted from
+solar observations (Cavallini et al. 1985a). Therefore, although
+we try to correct for some effects, like the variation of CB veloc-
+ity as a function of spectral type, the more we go away from the
+solar case, the more we should be careful about interpreting the
+results coming out of SOAP-GPU.
+We note that when modifying the bisector shape of solar or
+PHEONIX spectra for the disk center, to account for different
+convection velocity across spectral type, we model the effect of
+the “third signature” of granulation (Gray 2009). The inherent
+shape of the bisector (known as the “second signature” of granu-
+lation) and how it varies with µ (Löhner-Böttcher et al. 2019) is
+still the one observed for the Sun and it is well known in the liter-
+ature that the bisector shape of disc-integrated observations, and
+therefore by analogy at disk center, varies significantly among
+luminosity class and spectral type (see Figure 17.15 in Gray
+2008; Ba¸stürk et al. 2011). In those papers, we see that inside the
+small range from early G’s dwarfs to early K’s dwarfs, the bisec-
+tor shapes does not change drastically and therefore, although
+the integrated spectra for those stars won’t be realist in terms of
+line bisector, the way SOAP-GPU models stellar activity should
+still be quite realistic as in the end, what counts is the differential
+between the activity and quiet phases. As we can see in the pre-
+ceding subsection, besides SOAP-GPU not being able anymore
+to separate the inhibition of the CB effect from the flux effect
+for early K’s, the tests we performed show that the code seems
+to behaves quite well in estimating the total RV effect induced
+by spots and faculae. However, outside of this small range from
+G to K dwarfs, bisectors are completely different, and we warn
+the users that the present version of SOAP-GPU might give very
+unrealistic stellar integrated spectra and estimation of stellar ac-
+tivity. There is perhaps only one exception for dwarfs cooler than
+early K’s, for which the velocity of convection decreases to level
+that are very difficult to measure. For those stars, stellar activity
+is dominated by the flux effect from spot and faculae. Therefore,
+for such stars, users should ignore the output from the CB effect
+and only consider the flux effect.
+To better model stellar activity for stars different than the
+Sun, we would require disk-resolved spectra for other stars.
+Those could come from 3D MHD simulations (e.g. Johnson et al.
+2021b; Dravins et al. 2021), or thanks to spatially resolved spec-
+troscopic observation across stellar surfaces thanks to transiting
+Article number, page 12 of 17
+
+700
+Datafrom Liebing etal.2021
+Cubic fit
+600
+Spectral type: F9
+Spectraltype:G2 (Sun)
+Spectraltype:G8
+500
+Spectral type: G9
+RV (m/s)
+Spectraltype:K2
+400
+300
+CB
+200
+100
+4800
+5000
+5200
+5400
+5600
+5800
+6000
+6200
+Teff (K)
+1.0
+Spectral type: F9
+Spectral type: G2 (Sun)
+Spectraltype:G8
+Spectral type: G9
+0.8
+Spectral type:K2
+I flux
+Normalized
+0.6
+0.4
+0.2
+0.0
+-400
+-200
+0
+200
+400
+Doppler velocity (m/s)Yinan Zhao et al.: SOAP-GPU
+Fig. 13. Simulation of an equatorial 1% spot (left) and 1% facula (right) with different temperatures using PHOENIX spectral library. The quiet
+Sun spectrum is extracted from PHOENIX spectral library with log(g) of 4.5 and Teff = 5778K. The effective temperature of the spot spectra
+are 5015K, 5115K and 5215K, and for the facula spectra 5928K, 6028K and 6128K. We also show the results of using the observed Kitt Peak
+solar spectra including the PHOENIX SEDs (using Eq. 11, 10 and 12). The µ dependent CB is included in the simulations. Left: For the spot we
+see that when the difference in temperature between the spot and the quiet photosphere increases, the RV flux effect becomes larger. We note a
+rather large discrepancy in the CB effect between the observed solar and PHOENIX input spectra. This is likely due to the fact that the observed
+solar spectra are not well normalized (see Sect 4.3.3). Right: For the facula, we note the same problem of negative values for the CB effect, which
+expected as the same badly normalised solar active spectrum is used. We also see that considering the corresponding PHOENIX SED when using
+the observed solar spectra significantly change the flux contribution (see Sect 4.3.3). Considering the PHOENIX SED gives results much closer to
+the PHOENIX simulations.
+Fig. 14. Simulation of the CB RV effect of an equatorial 1% spot (top)
+and 1% facula (bottom) for different temperature of the quiet photo-
+sphere (i.e. different spectral type) using the PHOENIX spectral library.
+The temperature difference between the quiet photosphere ∆Teff for spot
+and facula is fixed to 663K and 250 K, respectively. Due to the strong
+absorption effect of molecules in spots on stars cooler than the Sun, we
+see the appearance of a flux effect in the derivation for the CB effect
+only (see Sect 4.3.3).
+planets (e.g. Dravins et al. 2017). Both approach are challeng-
+ing, however could led to a much more realistic modelisation
+of stellar activity from other stars than our Sun. We note that
+it is rather easy to input different spectra in SOAP-GPU than
+the ones provided (the Sun and PHEONIX spectra), and there-
+fore if users have access to disk-resolved spectra with more re-
+alistic line-bisectors, SOAP-GPU could still be used to obtain
+efficiently disk-integrated spectra and model the corresponding
+stellar activity effect. We note that the CB injection part is an
+individual module in SOAP-GPU that users can easily modify.
+5. Conclusions
+In this paper, we present a GPU-based improvement to SOAP
+2.0, named SOAP-GPU, that allows to efficiently model stellar
+activity at the spectral level. With the implementation of GPU in-
+terpolation and summation, benchmark calculations demonstrate
+that SOAP-GPU improves the computational speed by a factor of
+60 when modeling stellar activity on a full visible spectral range
+at R=115’000 of resolution, while having the same accuracy.
+Beside the huge gain in speed, SOAP-GPU also provides
+a more complex modelisation of stellar activity compared to
+SOAP 2.0. Complex active region scenarios, with regions over-
+lapping is now handled by the code. This is mainly useful when
+modeling active phases of a star like the Sun, with hundreds of
+active regions. The contrast of the active regions is now wave-
+length dependant, and therefore change for each wavelength
+of the modeled spectra. The dependence of line bisector with
+center-to-limb angle µ, following the work of Löhner-Böttcher
+et al. (2019) and Cavallini et al. (1985a), is also now accounted
+for for the Kitt Peak observed quiet and active atlases, but also
+for PHOENIX spectra. Although the induced RV effect is rather
+negligible when using the observed solar spectra as input, includ-
+ing the framework to change line bisector is crucial to properly
+model convection when injecting PHOENIX spectra as input.
+The use of PHOENIX spectra allows us now to model a wide
+variety of stars with different stellar and active region properties,
+and allows us as well to better model faculae, as the correspond-
+ing spectrum now has the proper effective temperature (SOAP
+2.0 was using the Kitt Peak sunspot atlas to model faculae).
+Article number, page 13 of 17
+
+6
+SolarspectrawithPhoenixSED
+Phoenix spectra,spot Teff=5015K
+4
+Phoenix spectra,spot Teff=5115K
+: RV (m/s)
+Phoenixspectra,spotTeff=5215K
+2 
+Tot
+0
+-2 
+4
+Flux RV (m/s)
+2
+0
+-2
+4 
+RV (m/s)
+3
+2
+Bconv F
+1
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Phase6
+Solar spectra with Phoenix SED
+Phoenixspectra,faculaeTeff=5928K
+4
+Phoenix spectra,faculae Teff=6028K
+Tot RV (m/s)
+Phoenix spectra,faculaeTeff=6128K
+2
+0
+-2 
+4
+Flux RV (m/s)
+2
+0
+-2
+-4
+4
+RV (m/s)
+3
+2
+Bconv
+1
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+PhaseF9Teff=6050K
+G2 Teff=5779K
+Spot RV (m/s)
+G8Teff=5480K
+G9Teff=5380K
+2
+K2Teff=5100K
+0
+4
+Faculae RV (m/s)
+3
+2
+1
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+PhaseA&A proofs: manuscript no. SOAP_GPU
+When modeling the inhibition of CB effect using as input
+the solar Kitt Peak quiet and active spectral atlases, we noticed
+that the derived RVs go negative, which is not expected. This
+comes from molecular absorption that can be seen in the spot
+spectrum due to lower temperature compared to the quiet Sun.
+Even though we do not include the contrast of the active region
+when modeling the RV CB effect only (see Eqs. 2 and 10), the
+difference in flux at the level of molecular absorption bands will
+show up as a flux effect in the estimated CB RVs. Positive val-
+ues will be added to the CB RV effect before the spot crosses
+the stellar center, and negative values after, therefore creating an
+asymmetry.
+When modeling other stars than the Sun using PHOENIX
+spectral library, users should be aware that a lot of physics in-
+cluded in SOAP-GPU are based on solar observations, and al-
+though the code tries to correct for known effects like the vari-
+ation of CB velocity as a function of effective temperature (i.e
+spectral type, see Sect. 4.3.3), the more we go away from the
+Sun, the more the results should be interpreted with caution (see
+discussion in Sect. 4.3.4). The modelisation of stellar activity
+for other stars than the Sun is currently limited by the knowl-
+edge we have about disk-resolved bisectors for such stars. Such
+information is very challenging to obtain, however, 3D MHD
+simulations (e.g. Johnson et al. 2021b; Dravins et al. 2021) and
+resolved spectroscopic observation of other stars due to plane-
+tary transits (e.g. Dravins et al. 2017) could significantly help.
+Also, when modeling stars of later spectral type than the Sun,
+we are not able anymore to separate clearly the inhibition of
+CB effect from the flux effect due to the strong absorption of
+molecules. However, the output for the total RV effect (flux plus
+inhibition of CB effects) should be modeled properly. SOAP-
+GPU have been tested up to a K2 star (Teff = 5100 K) with spots
+663 K cooler and give satisfactory results. Modeling later spec-
+tral type is challenging mainly due to continuum normalisation
+of the PHOENIX spectra and injection of spectral line bisector
+due to line blending, and users should be very careful about the
+interpretation of the results for such stars with the present code.
+There are still some improvements that could be made to bet-
+ter model the physics at play. Although the spot and facula spec-
+trum used when considering PHOENIX spectra as input are of
+different temperature, and therefore in spectral content, we still
+associate to those regions the same active bisector as measured
+for the Sun on a facula (Cavallini et al. 1985a). Spots are induced
+by stronger magnetic fields than facula, and thus it is likely that
+the bisector of spectral lines will be slightly different. Although
+it is possible to know what is the bisector of a few spectral lines
+inside a spot at disk center, to our knowledge, no measurement
+of spot line bisectors for different µ angles are published. Cav-
+allini et al. (1985a) also show in their Fig. 1 that depending on
+the facula observed, the bisector shape changes due likely to dif-
+ferent magnetic field strength and therefore different level of CB
+inhibition. As can be seen in Fig. 10, the effect of inducing µ
+dependant spectral line shape in the quiet and active regions is
+rather small, and although with more solar data about spots and
+faculae we could better model the physics at play, results in terms
+of RV derivation would be rather similar. This likely comes from
+the fact due to limb-darkening, most of the weight is put on the
+disk center, where spectral lines does not change significantly in
+shape.
+With the performance of SOAP-GPU, it is now possible to
+model activity at the spectral level for complex stellar surfaces
+with many active regions and for a long period of time. A so-
+lar activity simulator, either based on statistical properties of so-
+lar active regions (similar to Borgniet et al. 2015) or on the ob-
+served distribution of those (similar to e.g. Meunier et al. 2010)
+will be published in a forthcoming paper. We encourage any per-
+son working on techniques to separate the activity effect from
+planetary signals at the spectral level, to test their framework on
+SOAP-GPU simulations, where photon-noise, instrumental and
+telluric systematics are not perturbing the spectral timeseries.
+Acknowledgements. We thank the anonymous referee for the insightful and con-
+structive comments on this paper. We thank Michael Crerignier for his help in
+normalizing PHOENIX spectra with RASSINE. We also thank Xiang Gao for
+the constructive comments on GPU computing. This project has received fund-
+ing from the European Research Council (ERC) under the European Union’s
+Horizon 2020 research and innovation programme (grant agreement SCORE No
+851555). This work has been carried out within the framework of the National
+Centre of Competence in Research PlanetS supported by the Swiss National Sci-
+ence Foundation. The authors acknowledge the financial support of the SNSF.
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+
+A&A proofs: manuscript no. SOAP_GPU
+Appendix A: Line bisectors of PHOENIX spectra
+As discussed in Sects. 4.3.2 and 4.3.3, before injecting the µ
+dependant bisector for solar or PHOENIX spectra to properly
+model CB and its inhibition close to the limb and in active re-
+gions, we need to remove any bisector shape already present
+in the input spectra. As PHOENIX spectral library is gener-
+ated from 1D spectral synthesis, the line bisectors cannot include
+properly the CB effect and therefore should be close to straight.
+In Fig. A.1, we show for each simulation of different spectral
+types the bisector of a few iron lines that are used in Löhner-
+Böttcher et al. (2019). For each spectral type simulated, we show
+the bisectors for the quiet photosphere, but also for simulated
+spot and faculae, 663 K cooler or 250 K hotter, respectively. As
+expected, most of the bisector are close to straight lines. We how-
+ever fitted the average bisector with a second order polynomial to
+remove the small curvatures observed before injecting the proper
+bisectors at different µ angles (see Sect. 4.3.2). It is not clear if
+those curvatures are real effect in the spectral synthesis, or sim-
+ply due to blends. The correction performed is small compared
+to the bisectors that we inject afterward, therefore if only due to
+blends, this process does not significantly change the outputs.
+Article number, page 16 of 17
+
+Yinan Zhao et al.: SOAP-GPU
+Fig. A.1. Bisector of PHOENIX spectra. For each input seed spectrum using PHOENIX spectral library, we use five strong iron lines: FeI
+5250.2084Å (green), FeI 5250.6453Å (cyan), FeI 5434.5232Å (purple), FeI 6173.3344Å (orange) and FeI 6301.5008Å (yellow) to measure the
+average bisector of the input spectra. Bisector outliers outside a window of 0.1Å around each line center are rejected to avoid those points, certainly
+affected by line blending, to bias our measurement of line bisector. Each line correspond to a different spectral type, and from left to right, we can
+see the bisector of the spectrum used for the quiet photosphere, a spot region (663 K cooler) and a facula region (250 K hotter). We average those
+line bisectors at certain depth (as shown by the red dots) and fit the obtained data with a second order polynomial. The fitted bisector is used to
+remove the bisector of input seed spectrum.
+Article number, page 17 of 17
+
+F9 Teff = 6050 K
+F9 Teff = 5387K
+F9 Teff = 6300 K
+1.0 -
+1.0 
+1.0
+0.8
+0.8
+8'0
+0.6 -
+0.6 -
+0.6
+0.4 -
+0.4 
+0.4
+0.2 
+0.2 -
+0.2
+0.0 -
+00
+00
+-200
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+-150
+-100
+-50
+50
+100
+150
+200
+G2 Teff = 5778 K
+G2 Teff = 5115 K
+G2 Teff = 6028 K
+1.0
+1.0 
+1.0
+0.8
+0.8
+8'0
+0.6 -
+0.6 
+0.6
+0.4 -
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+0.2 
+0.2 -
+0.2 
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+-200
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+-50
+0
+50
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+G8 Teff = 5480 K
+G8 Teff = 4817 K
+G8 Teff = 5730 K
+1.0 
+1.0 
+1.0 
+0.8
+80
+0.8
+0.6
+0.6 
+0.6
+0.4 -
+0.4 
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+0.0
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+0
+50
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+150
+200
+200
+-150
+-100
+-50
+50
+100
+150
+200
+G9 Teff = 5380 K
+G9 Teff = 4717 K
+G9 Teff = 5630 K
+1.0 -
+1.0 
+1.0 
+0.8
+0.8
+0.8
+0.6
+0.6 -
+0.6
+0.4
+0.4 
+0.4
+0.2 -
+0.2 -
+0.2
+0.0 -
+0'0
+0.0
+200
+-150
+-100
+-50
+50
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+200
+-200-150-100
+-50
+0
+50
+100
+150
+200
+-200
+-150-100-50
+50
+100
+150
+200
+K2 Teff = 5100 K
+K2 Teff = 4437 K
+K2 Teff = 5350 K
+1.0 
+1.0 
+1.0 
+0.8
+80
+0.8
+0.6 -
+0.6 
+0.6
+0.4 -
+0.4 
+0.4 
+0.2 -
+0.2 
+0.2
+0.0 -
+0'0
+0'0
+-200
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+100
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+0
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+-200
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+200
+150
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+-50
+0
+50
+100
+150
+200

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+arXiv:2301.05133v1  [cs.AI]  12 Jan 2023
+Is AI Art Another Industrial Revolution in the Making?
+Alexis Newton, Kaustubh Dhole
+Department of Computer Science
+Emory University
+{annewto, kdhole}@emory.edu
+Abstract
+A major shift from skilled to unskilled workers was one of
+the many changes caused by the Industrial Revolution, when
+the switch to machines contributed to decline in the social
+and economic status of artisans, whose skills were dismem-
+bered into discrete actions by factory-line workers. We con-
+sider what may be an analogous computing technology: the
+recent introduction of AI-generated art software. AI art gener-
+ators such as Dall-E and Midjourney can create fully rendered
+images based solely on a user’s prompt, just at the click of a
+button. Some artists fear if the cheaper price and conveyor-
+belt speed that comes with AI-produced images is seen as
+an improvement to the current system, it may permanently
+change the way society values/views art and artists. In this ar-
+ticle, we consider the implications that AI art generation in-
+troduces through a post-industrial revolution historical lens.
+We then reflect on the analogous issues that appear to arise
+as a result of the AI art revolution, and we conclude that the
+problems raised mirror those of industrialization, giving a vi-
+tal glimpse into what may lie ahead.
+Introduction
+The industrial revolution caused a major shift from skilled to
+unskilled labor when machines contributed to a massive lay-
+off of artisans in favor of factory-line workers. William Pelz
+describes how prior to the industrial revolution people had
+been working in much the same way for thousands of years,
+producing goods through human labor with some assistance
+via animals or water power. However, he says, “All of this
+changed with the rise of the machine: tools would no longer
+serve people, but rather people would serve machines” (Pelz
+2016). Pelz argues that due to this massive change in goods
+production, the people became an “appendage” to the ma-
+chine, as humans were suddenly at the mercy of machines
+using them to make goods. As factory life took hold, com-
+pensation was no longer based on paying for skill, but rather
+on paying for time. This helped to usher in a different kind of
+day-to-day experience for most commoners–that of a time-
+work discipline. Maxine Berg (2014) notes the change to a
+mechanized factory sector from pre-industrial handicrafts.
+Many traditional workers had to transition to this type of
+work, or risk being left behind by a quickly transforming
+industry, leading to the devaluation and destruction of busi-
+nesses that could not compete with the cheaper and faster
+production that industrialization had to offer.
+Creative AI Across Modalities 2023 (creativeai-ws.github.io),
+Thirty-Seventh
+AAAI
+Conference
+on
+Artificial
+Intelligence
+(www.aaai.org), February 7-23, Washington, D.C., USA
+With the recent introduction of AI-based art models
+(Ho, Jain, and Abbeel 2020; Ramesh et al. 2022; Ruiz et al.
+2022), we argue that a shift largely reminiscent of the post-
+industrial revolution is unfolding. As AI-based models be-
+come more and more common, issues artisans experienced
+in the mid 1800s are reemerging, which similarly question
+the very existence of artists today. As we are on the precipice
+of this revolution, it is imperative for all stakeholders, viz,
+policymakers, ethicists, computer scientists and artists to un-
+derstand what such a shift would entail in order to manage
+the consequences of that shift.
+In this article, our aim is to view recent developments in
+the art industry due to the introduction of artificial intelli-
+gence models through a post-industrial lens. In the follow-
+ing sections, we first discuss how technologies influenced
+views on independent artisans in the aftermath of the indus-
+trial revolution. Next, we discuss the positive and negative
+implications of the post-industrial view on AI generated art.
+Finally, we consider current issues raised by these models,
+and conclude by reflecting on the analogous issues seen in
+the industrial revolution.
+A Historical Shift
+From Individual to Factory Worker
+The switch from an individualist working environment to a
+factory-centered one would permanently change the way so-
+ciety viewed independent artisans, bringing in a new age of
+commercialization that was predicated on the swift produc-
+tion of machine-made products over man-made ones. This
+difference on workforce style in is how the Industrial Rev-
+olution led to a change in “people’s relationship to crafts-
+manship, time, community and their own role in society as a
+whole”.
+Such a change in working style was caused that it is hard
+to imagine a time where machines were not at the fore-
+front of civilization. As the industry transformed resemble
+today’s working world, small-scale artisans were pushed out
+of people’s minds in favor of production that was faster and
+cheaper. The individual craftsman all but disappeared from
+the forefront of the business world. By the 1850s, most in-
+dependent shoemakers had been replaced by shoe factories,
+independent weavers had gone out of business, and women
+with hand looms were quickly outstripped by the factories
+and machines bringing more people cheaper goods of higher
+average quality (Smail 1992).
+However, a plethora of small-scale and skill-intensive sec-
+tors, like those in the metal trades and textile industries,
+
+managed to develop alongside the rise of factories (Berg
+2014). Parts of the world also still value individually-made
+fine arts objects, especially cultures in the east like China
+and India. Berg points to the idea of luxury fashion in France
+or small-scale building restoration which is popular in Eu-
+rope (Berg 2014). Though factorization still prevails, it is
+also important to note that the since the early 2000s demand
+for “niche” artisans has actually shown an upward trend.
+The Alliance for Artisan Enterprise (2012) reports that the
+global market for artisan-made products has increased by
+more than 8% per year since 2002, and is worth more than
+$32 billion. One of the reasons for such a trend has been an
+increased willingness to pay a premium for distinctive vis-
+`a-vis mass-produced, goods.
+Machines Have Politics
+Winner (1996) provides an illuminating example of trans-
+forming industry in “Do Artifacts Have Politics?” when
+he looks at the industrial mechanization of Chicago in the
+1880s where the switch to factory production hurt skilled
+workers. When pneumatic molding machines were imple-
+mented by Cyrus McCormick’s reaper manufacturing plants,
+it drove individuals out of business (Winner 1996). In many
+other industries, this happened because the factory process
+was cheaper, but that was not the case here. In this instance,
+McCormick and the National Union of Iron Molders were
+at war, so even though the iron molding machines were not
+cheaper, they were used to push the previous workers away
+from unionization and out of business. Therefore, while the
+addition of industrial manufacturing hurt many skilled la-
+borers naturally through cheaper replacements, it also hurt
+them unnaturally through the furthering of political agendas
+(Winner 1996).
+Unarguably, Winner concludes that the molding machine
+thus has politics in that its technical arrangements have be-
+come a form of order. Instead of subscribing to the use of
+pneumatic molding machines to speed up or cheapen pro-
+duction, these machines expressed human motives in their
+use towards achieving authority over others. It is undeniable
+that even if artifacts and machines do not have politics, they
+do indeed have power.
+In his famous essay “The Work of Art in the Age of
+Mechanical Reproduction,” Benjamin (1935) discusses the
+change in perception of art in the age of mass production.
+Before mechanical reproduction, art was unique and valued
+for its “aura” – which was derived from its authenticity and
+its physical and cultural context. However, with the ability
+to mechanically reproduce works of art, the traditional bases
+of cultural authority and hierarchy were challenged. As art
+moved to being based on politics instead of tied to rituals,
+it began to serve as a tool for political activism and resis-
+tance, ultimately bringing about social and political change
+(Benjamin 1935).
+Historically, it seems fair to say that the introduction of
+new technologies in the industrial revolution had many im-
+pacts on the individual craftsmen. Some of these impacts
+might be considered a natural course of action, but it is
+imperative as we move forward to acknowledge that other
+forms of use can be exacted through the introduction of new
+technology–use beyond that of merely what a machine phys-
+ically produces.
+AI Art Generation
+We focus on the recent introduction of AI-generated art soft-
+ware to the current art world. AI-based art generators such as
+Dall-E (Ramesh et al. 2021, 2022) and Midjourney are rel-
+atively new pieces of technology that can now create fully
+rendered images based solely on a user’s prompt, often pro-
+ducing impressive and intricate results. If the industrial rev-
+olution changed the way society viewed artisans and crafts-
+men, how might the AI art revolution do the same? We now
+examine AI image generators as a computing technology
+that has the potential to cause this analogous shift in the art
+industry.
+The Perception of AI Art
+The most general fear associated with AI-generated art is
+that it could drastically reduce the amount of jobs available
+to working commercial artists today in areas such as illustra-
+tion, animation, and graphic design. However, some others
+are concerned with the idea that the more “traditional” no-
+tion of art may also be modified by AI-generated art.
+In “The Culture Industry: Enlightenment as Mass De-
+ception,” Adorno and Horkheimer (1944) introduce the term
+“culture industry,” and compare technological advancement
+of mass media and creation to factory production of goods.
+They refer to the “assembly-line character of the culture in-
+dustry, the synthetic, planned method of turning out its prod-
+ucts” as creating a passive society that is being manipulated
+into being satisfied by the products of capitalism, rather than
+by way of true psychological needs such as freedom, creativ-
+ity and happiness (Adorno and Horkheimer 1944).
+In 2018, the art-collective Obvious, produced an art piece
+“Edmond de Belamy,” via a generative adversarial network
+(GAN) software package. The artwork was printed onto a
+canvas and sold at auction for $432, 500, over 43 times
+its pre-auction estimated value (Cohen 2018). Adorno and
+Horkheimer might argue that artwork created by an AI
+is created to be an ideal of art, and that AI artwork is
+merely feeding into mass-media culture industry that threat-
+ens “high arts.”
+Oppositionally, the idea that AI-created art could be
+worthwhile in itself as an art piece, as seen in its high price
+valuation, points heavily towards James Moor’s prediction
+that technology is shifting the questions we ask from “How
+well does a computer do such and such an activity?” to
+“What is the nature and value of such and such an activity?”
+(Moor 1985). In our case, the question shifts from “How
+well can computers make art?” to “What is art?”.
+What is Art?
+The current literature on human attitudes towards AI-
+generated art presents some evidence as to what people
+might feel of this shift in viewpoint. In two recent studies
+(Hong and Curran 2019; Mikalonyt 2022) on attitudes to-
+wards artwork produced by humans versus by artificial intel-
+ligence, researchers found that while most test subjects felt
+
+that AI-generated art could be considered “art,” they were
+much less inclined to consider the art to be produced by an
+“artist”. This is a significant distinction because it implies
+that while artwork contains artistic value just by existing,
+artwork is not made just by putting pen to paper.
+Thus we might push the shift Moor describes even further,
+from the question of “What is art?” to “What is an artist?”.
+“When judging whether an object falls under the category
+of ‘artwork,’ the intent of the creator is seen as more impor-
+tant than even the appearance of the object in question” -
+Mikalonyt (2022) seem to think, this is the reason that their
+participants were not unwilling to consider art created by a
+“robot” to be “art,” but were significantly more at odds with
+calling a “robot” an “artist.”
+In shifting the question from the object to the character-
+istic identity, we come to a yet unsolved question about AI
+art: If artwork generated by AI can be called “art”, but the
+model is arguably not an “artist,” then who is the author of
+such a piece? Mikalonyte and Kneer posit that this question
+has not yet been solved, suggesting that the lack of answer is
+reflected in the fact that autonomously generated AI artwork
+has yet to be copyrighted, with proposals to give copyright
+to the human designers of the artificial intelligence, as well
+as to redefine “authorship” so as to include robots in the def-
+inition (Mikalonyt 2022).
+Human Art from Ends to Means
+Besides raising questions in moral philosophy, such an attri-
+bution to the functional aspects of art vis-`a-vis the aesthetics
+could mean a lot of hope for traditional artists, especially
+those who were dependent on the techniques rather than the
+aesthetics of the final product. Artists who would only be
+differentiated by techniques might resort to promoting un-
+usual techniques which are beyond the current scope of AI
+art models, or at least AI art models at present (e.g painting
+on paper towels or woodblocks, using the back of the paint-
+brush, using fingernails to paint). It wouldn’t be surprising
+if artists would resort to differentiating factors relying on
+“means” rather than “ends”. Artists would especially want
+their work to be intrinsically different - earlier that could be
+achieved via both unique propositions of outcomes and of
+methodology, but now it would largely be the latter. It won’t
+be a surprise to witness more conservative forms being re-
+inforced (Browne 2022). Analogous possible trends of in-
+creased interest in niche artworks as against mass-produced
+ones would actually further promote such resorting to tradi-
+tional artistry and differentiating means.
+Fast-Paced Computational Creativity
+Pelz’s viewpoint that people became appendages to ma-
+chines during the industrial revolution clearly has art
+analogues today. Over the years, a large section of the
+art industry has already resorted to computational meth-
+ods (Li, Hashim, and Jacobs 2021; Feldman 2017) after wit-
+nessing the benefits of generative art. In generative art, new
+concepts, forms, shapes, colors, or patterns are created algo-
+rithmically. Artists or programmers first establish some cri-
+teria, post-which a computer creates new art forms adhering
+to those criteria. Such generative art is considered more aes-
+thetically pleasing than functional (Ball 2019). Hence wher-
+ever the functional aspects of art would be irrelevant, like
+in branding and advertising or the larger design sector, this
+transition towards computational art can accelerate people’s
+dependence on art designs which can be quickly iterated.
+Reduced Dependency on Traditional Artists
+Suddenly, a piece of art that may have taken days to produce
+by a professional can be done with a handful of suggested
+words by almost anyone. It must be noted that if the emer-
+gence of this technology follows a similar path to that of
+mechanization following the industrial revolution, it could
+devalue commercial artists significantly and create massive
+job loss in an industry that was previously known to require
+a deeply human touch. Everything from storyboarding, con-
+cept art, and movie creation to advertising work and social
+media would be significantly different, causing a massive
+problem for the individual artists who may have trained for
+years to perfect their skill sets.
+Benefits to Business
+Art has previously been far from a process one could au-
+tomate, but these AI art generators might be the cause of a
+grand shift from skilled to unskilled labor in the free-lance
+art world. In the past few months, several online publications
+have tested using tools like Dall-E or MidJourney to provide
+art to accompany their written content (Warzel 2022).
+Access to AI art generation tools could be seen as an im-
+provement to the current system for many business owners
+where there is a strong demand for commercial art. Being
+able to generate content for websites, branding, marketing
+and sales in a virtually cost-free manner could help small-
+business owners to reach bigger audiences. Online publica-
+tions have largely stepped away from free-lance artists any-
+ways, with many publications hosting content that was cre-
+ated elsewhere (i.e. embedded tweets or stock photography).
+Stock photography businesses sell royalty-free pictures
+for personal or commercial use, usually paying 15 − 40%
+to the creator of the photo for each license (Vincent 2022).
+Recently, AI have been breaking into this market, with the
+distinct art styles of Dall-E or Midjourney popping up on
+stock photo websites (Edwards 2022). Shutterstock, one of
+the largest stock photo retailers, announced that along-with
+OpenAI, they would be banning artwork from other AI gen-
+erators from being uploaded to their site, and it would also
+create a “Contributor Fund” to help pay the artists whose
+work was used to train the AI software (Vincent 2022).
+Art Democratization
+Platforms like YouTube, Instagram and TikTok, which have
+become hosts of content creation, have been able to at-
+tract millions of content creators who make money us-
+ing their services by providing them with fame and mon-
+etary rewards. This has largely happened with the reduced
+technical and social barriers that these arguably demo-
+cratic platforms have provided. AI art models could also
+tread the same path. Democratization of art would mean
+
+almost anyone can produce artistic creations, including a
+person without limbs or someone with a neurological dis-
+order that affects their ability to draw or paint. AI-art
+models might be exclusively seen as potential attackers
+on the most talented segments of the artistic society, but
+they will doubtlessly open up a level playing field for
+those who considered art out of reach. Besides, such de-
+mocratization would also be reflected in crowd-sourced
+efforts (Bigham, Kulkarni, and Lasecki 2017; Kittur et al.
+2013, 2019; Dhole et al. 2021; Srivastava et al. 2022) which
+would seek contributions to aid in developing large creative
+models fairly.
+Increased Amount of Plagiarism
+The Industrial Revolution was replete with examples of in-
+dustrial espionage (Harris 1985; Christopher Klein 2019)
+where large businesses often stole the work and ideas of
+people who had historically performed it. Governments reg-
+ularly encouraged individuals to steal ideas, especially from
+abroad, since it hardly required applicants to be inventors,
+especially if the invention was abroad. The current AI art
+models already have been exposed to the artworks of many
+artists without giving them proper attribution or even seek-
+ing their permission. Millions of generations of artwork have
+already been utilizing these styles. Besides artists’ work
+being plagiarized, it is unclear how credit would get di-
+vided amongst the artists, model trainers and users writing
+prompts. While there have attempts to legally delineate the
+complete generation pipeline (Fjeld and Kortz 2017; Kim
+2020), the black box nature would make it hard for fair credit
+attribution.
+Increased Carbon Emissions
+The dramatic increase in coal and gas usage, which sky-
+rocketed pollution levels across major cities and indus-
+trial zones, was an unfavorable side consequence of the
+Industrial Revolution. Today, with large groups of peo-
+ple expected to move towards careers of computational
+art, it would be inevitable that training these AI-art mod-
+els would also be performed more frequently, via re-
+searchers as well as artists raising concerns of carbon
+emissions. Strubell, Ganesh, and McCallum (2020)’s lifecy-
+cle assessment of training popular large language models re-
+vealed that a typical training process took nearly five times
+the lifetime carbon emissions of the average American car.
+Besides, the largeness of these models also necessitates GPU
+usage during inference time.
+Furthering Political Agendas
+Winner (1996) has been helpful for giving convincing argu-
+ments that artifacts and machines in general can have biases.
+The 9ft clearance levels of Long Island bridges were a de-
+sign decision made by urban planner Robert Moses in order
+to restrict buses filled with low-income people and racial mi-
+norities from accessing parkways. As a result of the biases
+implicit in these designs, people were given limited access
+to parts of their own city (Winner 1996).
+Benjamin (1935) had argued that as art’s authenticity di-
+minishes due to the ease of mechanization, it begins to be
+based on politics rather than on rituals. Just like the Long
+Island bridges, political agendas intertwined with design
+have had crucial consequences. Therefore, it wouldn’t be a
+surprise if biases mirrored in the AI art generation of to-
+day (Bansal et al. 2022) were exploited to further political
+agendas. Hassine and Neeman (2019) revealed that AI gen-
+erated art skews mostly white, both in depiction and in rep-
+resentation. Unless age, sex or race is specified, prompts to
+the system have built in biases towards young white men
+(Srinivasan and Uchino 2020). This unconscious bias could
+have a manifestation in the real world, just as Robert Moses’
+designs manifested for racial minorities in New York City.
+Questions of Concern
+Finally, we consider the issues to pubic welfare and society
+that AI-art generation introduces through its effects on the
+artists of today. We also consider the scope of involvement
+that computer scientists and AI have had in creating or con-
+tributing to these issues.
+Is the threatened change in the status of artists
+characterized by the primary and essential
+involvement of AI models?
+Industrial “sweatshops” mass-producing art for commercial
+consumption have been a constant long before the computer
+became an element in the equation. From the comic strips of
+the 1880s - 1960s to the comic books of the 1930s - present
+day, to the mass produced landscape art created for furniture
+stores in Asian factories since the mid-1950s, art has been
+commoditized long before the computer (Hersch 2021). In
+these assembly lines, one person would sketch the outlines
+of image, another would pencil in details of the people, an-
+other would ink those images, yet another would draw in the
+backgrounds, and a final hand would color the image. Pro-
+duced by an assembly line usually called a “studio,” the art
+would be signed either by an arbitrarily selected worker or
+even by a completely fictitious artist (Hersch 2021). More to
+the point — this was art created by “factory workers” who
+acted the same role in production as today’s graphics pro-
+grams do (Campbell 2022). Therefore, one could also argue
+that AI models might not be essential to the problem. How-
+ever, what distinctively stands out with the usage of AI art
+models, is the rapid pace of artwork creation and prolifera-
+tion, unlike what was witnessed before the arrival of com-
+puters or the internet.
+Does the threatened change in the status of artists
+occur because of exploiting some unique property
+of AI models?
+The primary difference created by technology is mass ac-
+cess. To staff a “studio” with bit-work artists requires a sub-
+stantial investment in infrastructure, equipment, and labor
+costs. Such programs as are available today are much more
+economical and widely available to individuals — from hob-
+byists to serious artists to mercenary corporations — than
+at any time in our history. So it is legitimate to argue that
+AI models have uniquely driven the scale and creativity of
+
+the problem far more broadly than earlier technology could
+have.
+Could this issue have even arisen without the
+involvement of AI models?
+The answer depends heavily on the question of whether AI-
+art generation programs are considered as tools or entities.
+Motion-picture technology created entirely new art forms.
+One could create art with motion across space and time in
+a way that entirely changed how our culture thought about
+visual art. But the cameras themselves have never — for all
+their technological sophistication — been more than tools.
+Cameras do not set out with purpose to make movies, and
+AI art generation programs do not set out independently to
+create images. Both must be employed by users.
+Conclusion
+Perhaps what AI-art generation software is doing is forc-
+ing our society to confront a much larger issue about artists.
+Instead of threatening the status of artists, advances in com-
+puter technology require us to confront the idea that the view
+of artists has already changed, and has been changing.
+While on the surface AI-generated art seems unique,
+many of the issues that it is raising concerning society’s
+views of art and artists are merely more complex callbacks
+to the mechanization of artisan’s projects in the industrial
+revolution. Specifically, the art generated by AI-generation
+algorithms creates problems that mirror those of industrial-
+ization. In the same way that the artisan was pushed out, so
+too is the artist today. In the same way intellectual property
+was compromised during the 1800s, legal loopholes may en-
+sure many artists are not duly credited without proper AI
+regulation.
+Our study of the industrial revolution analogues serves as
+a warning to look backwards at the past treatment of creators
+and consumers of industry in the wake of newly introduced
+technologies. We pose that this may be an important step to
+take before experiencing the consequences of the technical
+revolutions that unfold before us today.
+But these analogies should not be taken as a discourage-
+ment against developing large models or to undermine the
+efforts of the field of AI in general. We should actively
+strive to improve technical parameters of these models, by
+accounting for the possibility of potential damage early on,
+as these models have and already display tremendous poten-
+tial for business as well as for democratization.
+Limitations
+How AI art generation tools will affect artists is an extremely
+subjective and multifaceted subject, and forecasting it pre-
+cisely will not be easy. We have provided comparisons based
+on events that occurred post the industrial revolution. How-
+ever, we think that empirical evidence would be helpful in
+better understanding many of the issues raised. Our objec-
+tive was to present as thorough and comprehensive an anal-
+ysis as possible by considering the technical, political and
+industrial implications of art. Our section on “What is Art?”
+is quite limited due to the rich history concerning this ques-
+tion from a philosophical point of view, but we still feel it
+was important to include. Nonetheless, we believe that our
+work will serve as a crucial gateway for both the engineering
+and humanities disciplines to facilitate dialogue and advance
+debate about the impact of AI art tools.
+Acknowledgments
+We thank Dr. Kristin Williams and Dr. Steve Newton for
+their crucial thoughts and feedback in numerous drafts. We
+also thank the anonymous reviewers and meta reviewer for
+their invaluable suggestions.
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