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+arXiv:2301.13801v1  [cs.SI]  29 Jan 2023
+Cultural Differences in Friendship Network Behaviors: A
+Snapchat Case Study
+Agrima Seth
+agrima@umich.edu
+School of Information, University of
+Michigan,
+Ann Arbor, Michigan, USA
+Jiyin Cao
+jiyincao@gmail.com
+Stony Brook University
+Stony Brook, New York, USA
+Xiaolin Shi
+Xiaolin@snap.com
+Snap Inc.
+Santa Monica, California, USA
+Ron Dotsch
+rdotsch@snap.com
+Snap Inc.
+Santa Monica, California, USA
+Yozen Liu
+yliu2@snap.com
+Snap Inc.
+Santa Monica, California, USA
+Maarten W. Bos
+maarten@snap.com
+Snap Inc.
+Santa Monica, California, USA
+ABSTRACT
+Culture shapes people’s behavior, both online and offline. Surpris-
+ingly, there is sparse research on how cultural context affects net-
+work formation and content consumption on social media. We an-
+alyzed the friendship networks and dyadic relations between con-
+tent producers and consumers across 73 countries through a cul-
+tural lens in a closed-network setting. Closed networks allow for
+intimate bonds and self-expression, providing a natural setting to
+study cultural differences in behavior. We studied three theoreti-
+cal frameworks of culture - individualism, relational mobility, and
+tightness. We found that friendship networks formed across dif-
+ferent cultures differ in egocentricity, meaning the connectedness
+between a user’s friends. Individualism, mobility, and looseness
+also significantly negatively impact how tie strength affects con-
+tent consumption. Our findings show how culture affects social
+media behavior, and we outline how researchers can incorporate
+this in their work. Our work has implications for content recom-
+mendations and can improve content engagement.
+CCS CONCEPTS
+• Human-centered computing → Social networks; Social me-
+dia; Social network analysis.
+KEYWORDS
+Social media platforms, Cross-cultural analysis, Social ties, User
+Behavior Modeling, relationship modeling, tie strength
+ACM Reference Format:
+Agrima Seth, Jiyin Cao, Xiaolin Shi, Ron Dotsch, Yozen Liu, and Maarten
+W. Bos. 2023. Cultural Differences in Friendship Network Behaviors: A
+Snapchat Case Study. In Proceedings of the 2023 CHI Conference on Human
+Factors in Computing Systems (CHI ’23), April 23–28, 2023, Hamburg, Ger-
+many. ACM, New York, NY, USA, 14 pages. https://doi.org/10.1145/3544548.3581074
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full cita-
+tion on the first page. Copyrights for components of this work owned by others than
+the author(s) must be honored. Abstracting with credit is permitted. To copy other-
+wise, or republish, to post on servers or to redistribute to lists, requires prior specific
+permission and/or a fee. Request permissions from permissions@acm.org.
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+© 2023 Copyright held by the owner/author(s). Publication rights licensed to ACM.
+ACM ISBN 978-1-4503-9421-5/23/04...$15.00
+https://doi.org/10.1145/3544548.3581074
+1
+INTRODUCTION
+In the past two decades, social media platforms have transformed
+how individuals build and maintain their relationships. These plat-
+forms are increasingly becoming the preferred method for initiat-
+ing intimate relationships [53], seeking advice [32], and commu-
+nity building [33]. With social media platforms becoming an inte-
+gral part of social life for many of us (there are 4.26 billion social
+media users as of 2021 [42]), understanding the drivers of user be-
+haviors is imperative.
+Directly engaging with others (e.g., sending messages) and con-
+suming their content (e.g., viewing, replying, and reacting to Sto-
+ries and posts) are often studied to understand behavioral patterns
+on social media platforms. User behavior on online social media
+platforms can be said to be broadly driven by a complex combina-
+tion of (a) user identity (personality, demographics), (b) the norms
+(descriptive and prescriptive) that the users in a network collec-
+tively subscribe to, (c) the relationship between users (friends, ac-
+quaintances, strangers), (d) usage intent; for example, professional
+(LinkedIn) vs. curated self-presentation (Instagram), and (e) plat-
+form affordances. While any particular platform usually provides
+the same affordances to all users on that platform, users bring their
+different backgrounds, experiences, expectations, beliefs, and val-
+ues to the platform. As a result, different behaviors on the same
+platform are culturally influenced [2, 7, 11, 36].
+Most studies on social media user behavior are based on data
+that is west-centric [39, 54], and thus, their results have an implied
+context of western cultural norms. These findings fail to account
+for the heterogeneity in user behavior that arises from different
+cultural contexts [4, 32]. Hence, to further understand how cul-
+tural values affect behavior on these platforms, our work focuses
+on how users from different cultural backgrounds interact differ-
+ently on a platform. Specifically, we use theoretical frameworks of
+cultural values to study the differences in the formation of friend-
+ship networks and the moderation of differential behavior of con-
+tent consumption within these friendship networks. This paper
+uses three theoretical frameworks of cultural values: Hofstede’s
+concept of Individualism [20], Thomson and colleagues’ concept
+of Relational Mobility [46], and Gelfand and colleagues’ concept
+of Tightness [15]. The data we used for our analyses is from the
+
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+Seth, et al.
+camera and messaging platform Snapchat. Snapchat is used in al-
+most 150 countries and has 347 million daily active users world-
+wide [41]. Snapchat is a closed network, meaning that a lot of the
+content shared by individuals on Snapchat is only available to a
+limited set of trusted users. Past work on eliciting the motivations
+for Snapchat usage has shown that Snapchat is used to commu-
+nicate with close relationships and is viewed as a platform with
+a relatively lower emphasis on self-presentation and impression
+management compared to platforms like Instagram [3, 6, 35, 50].
+Because closed networks have less formal pressures and allow for
+intimate bonds and self-expression, they provide us with a cleaner
+setting to study differences in human behavior.
+Specifically, we focus on 1) how culture influences network cre-
+ation and 2) how culture influences content consumption behav-
+iors embedded in the network. In particular, for the second ques-
+tion, we are interested in how culture moderates the effect of tie
+strength (i.e., the closeness between individuals) on content con-
+sumption. Past work has shown that tie strength strongly predicts
+a variety of user behaviors on platform, including what informa-
+tion will be exchanged [31, 51, 54], the likelihood to change one’s
+actions [8], the attention given to content [52], and the preferred
+behavior to signal engagement [5]. We will explore how tie strength
+moderates tie strength’s effect on content consumption. To study
+content consumption behavior, we use the metric of dwell time,
+i.e., the time a user spends consuming content that another user
+creates.
+In sum, we ask the following research questions:
+(1) How do friendship networks differ in countries with differ-
+ent cultural values?
+(2) How do cultural values change the effect of tie strength on
+dwell time?
+To answer our first research question, we studied the network
+properties of friendship networks across 73 countries, which have
+been surveyed by either Hofstede [20], Thomson et al. [46], or
+Gelfand et al. [15], and have different cultural values that lie on a
+continuum of the three cultural values of individualism, mobility,
+and tightness. We analyzed how friendship network size and ego-
+centricity — the extent to which a person’s friends are connected
+with each other — vary across cultures in the closed network set-
+ting of Snapchat. We find that users from more individualistic, mo-
+bile, and loose cultures have a more extensive friendship network
+and are less egocentric. Next, we analyzed within these networks
+how tie-strength between users impacts engagement with content
+(dwell time) and the role of cultural values as a moderator. We
+found that individualism, mobility, and looseness negatively mod-
+erate the effect of tie strength on content consumption.
+Where previous work on culture and social media platforms has
+primarily been limited to a small sample size [2, 39], this paper
+contributes by studying cultural differences in user behavior on a
+large scale, analyzing hundreds of thousands of users across many
+countries. Further, where other quantitative works are usually lim-
+ited to open or broadcast networks, this study explores relatively
+under-studied closed-network settings [23].
+From an HCI and design perspective, our work can advance
+our understanding of behavior patterns across cultures. We dis-
+cuss the implications of understanding users’ engagement with
+content to design better experiences for the user. When applied to
+platform design, our work would help user-retention of platforms
+without compromising the user experience, in turn creating better
+outcomes for both users and platforms. Our work furthers the re-
+search that helps answer the question: What does it mean to under-
+stand and support users from diverse cultures on online platforms?
+[14]. Most of the designs and practices of online platforms have a
+‘one-size-fits-all’ approach and do not actively account for different
+user preferences across geographies. Our results provide evidence
+of differential behavioral patterns in online friendship networks
+across cultures and suggest how algorithm design can be cultur-
+ally inclusive.
+1.1
+Privacy and Ethics
+The data for this study was taken from Snapchat, and the study was
+conducted within Snapchat in accordance with Snapchat’s policies
+and procedures with respect to Snapchat data. This analysis only
+uses the metadata of the user behavior. It does not analyze the ac-
+tual content of the communication between the users.
+2
+RELATED WORK
+2.1
+Ties and user behavior
+Interpersonal relationships make social media platforms social. Like
+in offline social networks, an individual’s online network consists
+of individuals, with each of whom one shares a different type of re-
+lationship. Each dyadic relationship is different based on the close-
+ness and the purpose they serve to the individual. Social network
+analysis literature uses the term tie strength to differentiate be-
+tween relations of different closeness. This term was coined by
+Granovetter[17], who analyzed the role of different ties in differ-
+ent situations. The two types of ties characterized were strong and
+weak. The four dimensions determining a tie’s strength were: the
+amount of time spent on a tie, the intimacy, the intensity, and re-
+ciprocal services [17]. Although researchers have used different
+operationalizations to conceptualize tie strength depending on the
+purpose of the study, many works on social media platforms op-
+erationalize tie strength as proportional to the total number of ex-
+changes in the dyad. This operationalization of tie strength has
+been used to study various phenomena, like promoting mental
+well-being [25], increased diffusion of information, and access to
+novel information [17, 49]. While these works analyze the role of
+tie strength in reaping social benefits, studies have also focused on
+justifying Granovetter’s hypothesis that the two ties elicit differ-
+ent interaction patterns, for which they analyze how information
+from different ties is received [19, 23, 24, 52]. These studies find
+evidence that individuals spend more time on the content received
+from stronger ties.
+2.2
+Cultural values
+One primary aspect of culture is that it’s the normative value sys-
+tem that dictates acceptable practices and helps differentiate one
+group from another. Culture is both a result of the accepted past
+actions and the determinant of acceptable future actions. One of
+the ways to reason about attitudes and actions is to understand
+the culture people are in. Prior studies have shown that an indi-
+vidual’s behavior in the online space is influenced by their culture
+
+Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+in the same way as offline behaviors. With cultural values shaping
+actions, we must first understand how culture can be measured
+and then how culture affects behavior. While prior work usually
+focuses on groups and their specialized culture, we introduce liter-
+ature from cultural psychology in our work. Culture is often opera-
+tionalized through dimensions where a dimension is defined as “an
+aspect of a culture that can be measured relative to other cultures.”
+In this paper, we bring in concepts from three dominant cultural
+psychology theories, namely, individualism-collectivism [20], re-
+lational mobility[46], and tightness-looseness[15], to explore how
+culture impacts network creation and content consumption behav-
+iors within a network. Below, we briefly introduce each of the cul-
+tural dimensions.
+2.2.1
+Hofstede’s Individualism-collectivism. Hofstede [20] analyzed
+data from over 50 countries and identified six critical dimensions of
+national culture. Individualism-collectivism is one dimension that
+has drawn the most research attention. Typically, individualism
+leads to loose ties among the individuals of a society. Individual-
+ists focus on "I" as opposed to "we." Because groups are less im-
+portant to them, individualists also tend to show no difference in
+their behaviors and attitudes toward ingroups versus outgroups. In
+contrast, collectivism leads to a collective identity, and the welfare
+of an individual is implicitly assumed to be linked to the interests
+of the larger group. Hence, collectivists focus on "we." Because of
+their particular focus on "we," collectivists are known to have differ-
+ent norms and behaviors towards ingroups versus outgroups and
+place greater emphasis on harmony.
+Because of the "I" nature, individualists need to constantly reach
+out to build networks and also tend to see relationships as fluid.
+In contrast, collectivists see relationships as given, and thus, they
+are less active in building networks. As a result, we predict that
+individualism will be positively correlatedwith friendship network
+size.
+Individualists are less likely to treat other people based on re-
+lationship strength and group membership, whereas collectivists
+tend to have a strong tendency to favor ingroup members and peo-
+ple they are close to. This should also be manifested in how tie
+strength drives content engagement behavior in different cultures.
+As a result, we predict that individualism will negatively moderate
+the positive effect of tie strength on content engagement, such that
+the effect of tie strength on content engagement will be weaker for
+individualists than for collectivists.
+Hence, we hypothesize that:
+H1a: The friendship network for individualisticcultures is larger than
+the friendship network for collectivistic cultures
+H1b: Individualism negatively moderates the effect of tie strength on
+content engagement.
+2.2.2
+Relational Mobility. Thomson et al. [46] conducted a survey
+across 39 countries using a set of 12 questions to construct their
+dimension of culture. Relational mobility indicates the degree of
+freedom and opportunities the members of a culture have to form
+and terminate relationships. The two opposing poles on this in-
+dex are high and low relational mobility. For example, relational
+mobility is high in North America and low in Japan. Because re-
+lationships in high-mobility cultures are less stable and easier to
+change than those in low-mobility cultures, they are more fragile.
+It also requires more effort to maintain committed relationships.
+Prior work has shown that cultures with higher relational mobil-
+ity tend to share more about themselves (self-disclosure), are more
+active in giving support, and tend to have more trust in the mem-
+bers of the society [46, 55]. Because cultures high in mobility have
+more opportunities to form relationships, it allows individuals to
+have a larger network. In a similar vein, because in high mobil-
+ity cultures, individuals see relationships as more fragile and fluid,
+they are less likely to adjust their interpersonal behaviors based on
+tie strength. As such, we predict that relational mobility will neg-
+atively moderate the effect of tie strength on content engagement,
+such that the effect of tie strength on content engagement will be
+weaker in high-mobility cultures than in low-mobility cultures.
+Hence, we hypothesize that:
+H2a: The friendship network for high mobility cultures is larger than
+the friendship network of low mobility cultures
+H2b: Relational mobilitynegatively moderates the effect of tie strength
+on content engagement.
+2.2.3
+Tightness. Gelfand et al. [15] conducted a survey across 33
+countries using 12 behaviors across 15 situations to construct their
+dimension. Tightness-looseness is about the extent to which a so-
+ciety tolerates norm-deviant behaviors. The two opposing poles
+on this index are tight and loose. For example, Looseness is high
+in North America and low in Japan. Tight cultures have stronger
+norms and are less tolerant of behavior that deviates from the norm.
+In contrast, loose cultures have relatively weaker norms and are
+more tolerant of behavior that deviates from the norm. As such,
+we predict that tightness should be negatively correlated with net-
+work size because a tight culture makes it hard for people to bring
+new members to a social network. Cultural tightness is often con-
+sidered a selection criterion to test whether a new member can fit
+in. In contrast, the level of scrutiny will be much lower in a loose
+culture, making it easier for an individual to expand their network.
+Similarly, we predict that tie strength’s effect on content engage-
+ment will be weaker in loose cultures than in tight cultures. In a
+loose culture, tie strength is less likely to be seen as a criterion
+that individuals rely upon to decide how they approach a person.
+In contrast, in a tight culture, tie strength is a monitoring mech-
+anism that powerfully regulates people. As a result, people draw
+more influence from tie strength, including content engagement
+behavior.
+Hence, we hypothesize that:
+H3a: The friendship networks for tighter cultures are smaller than
+friendship networks of looser cultures
+H3b: Tightness positively moderates the effect of tie strength on con-
+tent engagement.
+Although the three cultural dimensions originated from differ-
+ent theories, they are often conceptually related. Prior work has
+shown that individualism, relational mobility, and looseness are
+often moderately correlated (Thomson et al., 2018, Appendix Ta-
+ble S8, p. 51 [46]). For example, the U.S. is a culture that is in-
+dividualistic, high mobility, and loose at the same time, whereas
+Japan is a culture that is collectivistic, low mobility, and tight. How-
+ever, while Germany ranks higher in individualism and mobility, it
+
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+Seth, et al.
+ranks lower in looseness, whereas Brazil, though less individualis-
+tic, is more mobile and loose. Thus, while the three theories are
+conceptually related and can serve as a robustness check for one
+another, they each touch upon a unique cultural aspect. When re-
+searchers study the effect of one of the cultural values on individu-
+als, they also tend to include the other two as a way of robustness
+check [44, 46]. As a result, although the three dimensions are from
+different theories, we see them as a whole package.
+In sum, culture provides an important context about the shared
+common knowledge to its members on how to behave in a given
+context and how others will interpret their behavior. Comparative
+work on interpersonal relationships across cultures has shown that
+the same relationships elicit different behaviors in different cul-
+tures, implying that the same relationships across cultures are not
+similarly perceived [13, 16, 18, 30, 37, 47]. Our work aims to ana-
+lyze if user behavior on the same online platform provides empiri-
+cal evidence that the impact of tie strength on their behavior varies
+across cultures.
+3
+DATA
+We conduct our study on the Snapchat platform. Snapchat is an
+online messaging platform where content shared between users
+is ephemeral. Like most platforms, Snapchat allows users to ex-
+change content in the form of text, images, and videos. The inter-
+actions between users can be one-to-one, one-to-group, or one-to-
+all friends (a broadcast interaction). Interactions are identified by
+different names and are introduced below:
+• Snaps: A direct or personal interaction of image or video
+content type between users, which may be one-to-one or
+one-to-group. Depending on the receiver’s chosen settings,
+Snaps disappear immediately after viewing or 24 hours later.
+In our analysis, we only consider Snaps that are exchanged
+between dyads (just two users), which are termed ‘direct
+Snaps.’ We do not analyze Snaps sent to groups.
+• Chats: A text message between users. Akin to Snaps, de-
+pending on the receiver’s chosen settings, chats disappear
+immediately after viewing or 24 hours later. In our analy-
+sis, we only consider the chats that are exchanged between
+dyads (just two users), which are termed ’direct chats.’ We
+do not analyze group chats.
+• Stories: A broadcast interaction (with all of one’s friends)
+having an image orvideo as the content type. Users on Snapchat
+(posters) can create Stories for their friends (viewers) to con-
+sume. Stories constitute a pull communication wherein friends
+decide to either engage with a Story in part or whole or ig-
+nore it. Unlike Snaps and chats that disappear after watch-
+ing, Stories are available for 24 hrs after posting and can be
+viewed multiple times.
+We analyze users on Snapchat who share a friend connection.
+Friendships on Snapchat are bidirectional and are unlike the ‘fol-
+low model’ that platforms like Instagram and Twitter allow (i.e.,
+both individuals need to add each other as friends in Snapchat).
+For each of the 73 countries (Refer appendix D), we randomly sam-
+pled 10,000 unique users (egos), their associated Story viewing ac-
+tivity for one month, and their complete one-hop friend network.
+Though users may have friends across geographies, we filtered the
+data only to include those friend pairs where both friends resided
+in the same country. Aggregated over all 73 countries, cross-country
+friendships accounted for 21.8% of the data. The filtering resulted
+in a total dataset of approx 600,000 users per country. Each user
+can view Stories from multiple friends, with each of whom they
+share a different level of closeness. This results in a data set of
+unique dyadic relations between a Story viewer and a Story poster.
+For each dyadic interaction, we calculate aggregated statistics of
+the total time spent by a viewer on each of the poster’s Stories, the
+total number of Stories shared by a poster, and the total number of
+Snaps and chats exchanged between the two in the dyadic commu-
+nication. To avoid noise from users who rarely engage with each
+other, we only keep those dyadic pairs where at least one direct
+chat or Snap has been exchanged by both the Story poster and the
+viewer during the one month we analyzed. To control for effects
+unrelated to the cultural values but caused by the economic devel-
+opment and platform reach in a country, we include each country’s
+GDP [22], which is a measure of a country’s economic standing,
+GINI [45], which is a measure of economic inequality within a na-
+tion, and Snap’s market penetration 1, which measures the user-
+base of Snapchat for a country. Section 4 details the process used
+to answer each research question. The three cross-cultural theo-
+ries that inform our study did not survey all the same countries.
+Thus, while the three theories do not have a perfect overlap with
+each other (Refer appendix D), using all three allows us to cover
+73 unique countries.
+4
+METHOD
+We use the observational data from Section 3 and create statistical
+models to understand the role of culture on users’ network forma-
+tion and content engagement (dwell time). Building on and align-
+ing with prior cross-cultural work, we consider a country a repre-
+sentative unit of one culture [15, 20, 46] and analyze the users at
+the group level of a country.
+4.1
+RQ1: How do friendship networks differ in
+countries with different cultural values?
+We first measured each country’s average friendship network size
+to determine whether people from different cultures have differ-
+ent friendship networks. For this, we calculated the total number
+of friends per user in each country and averaged it over the total
+number of users in the country.
+Next, for each country under study, we reconstruct the ego net-
+work (egonet) for that country’s randomly sampled 10,000 users.
+An ego network consists of the user (the ego), the user’s friends
+(the alters), and the friendship relations between the alters. The
+egonets formed were independent, i.e., the users’ egonets did not
+overlap. We filter out networks that consist of only two nodes
+(users who are only connected to the default Snapbot and do not
+have other friends on the platform) or star graphs (a pattern where
+a user is connected to other users, but none of those other users are
+connected, which is a pattern mainly shown by bots [38, 48]). Since
+all friendships on Snapchat are bidirectional, we convert the graph
+to a simple graph by removing the multiple edges (edges that are
+incident on the same pair of nodes). For each of the egonets, we
+1Internal Snap INC. marketing data
+
+Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+calculate measures of egocentricity - the density, transitivity, and
+the betweenness centrality of the ego using the igraph package in
+R [12].
+Ego betweenness measures the percentage of shortest paths be-
+tween two alters. In a social network setting, it allows us to mea-
+sure the importance of the ego node. The higher the betweenness
+centrality, the more the ego node is the binding factor between its
+friends. Since centrality is sensitive to network size, we normalized
+it by the maximum possible betweenness of the ego node. This ap-
+proach is in line with prior work on measuring betweenness in
+egonets Na et al.,[28].
+Betweenness centrality of node i =
+�
+푖≠푗≠푘
+푔푗푘 (푖)
+푔푗푘
+Where 푔푗푘 is the number of shortest paths that connect node j and
+node k, 푔푗푘 (푖) is the number of these shortest paths that include
+node i.
+Network density is the ratio of the edges in the user’s network
+to the edges of the same user’s hypothetical network where every
+node is connected to every other node. Likewise, transitivity is the
+number of triads relative to the number of possible triads. In our
+setting, density and transitivity measure the tendency of the users
+to cluster or connect. The higher the density and transitivity, the
+more the tendency of the group to cluster.
+Density for an undirected graph =
+�
+푗≠푘 푧푗푘
+푛∗(푛−1)
+2
+Where n is the number of nodes in a network, and 푧푗푘 is equal to
+1 if the alters j and k are connected.
+Transitivity for an undirected graph =
+3 ∗
+number of triangles in the network
+number of connected triples of nodes in the network
+A high density and transitivity are indicative of people connect-
+ing with friends of friends; a low betweenness, on the other hand,
+implies a reduced tendency of nodes to cluster together. Prior work
+by Na et al. [28] on self-reported Facebook networks in East Asia
+and the USA found that users from the USA were more egocen-
+tric than users from East Asia (had higher Ego Betweenness and
+lower Density and Transitivity). We use the same methodology —
+to analyze data across more countries — to explore whether these
+findings generalize across platforms and for data that is not self-
+reported but an individual’s actual network data from a social me-
+dia platform. To maintain consistency with Na et al.,[28], we log-
+transform density and transitivity and then inverse the transforma-
+tion by multiplying minus one; we transform betweenness using
+푙표푔(1 + 푀푎푥(푥) − 푥) and then inverse the transformation by mul-
+tiplying minus one.
+4.2
+RQ2: How do cultural values change the
+effect of tie strength on dwell time?
+Online social media platforms continually aim to remove obsta-
+cles for content creation and consumption; this has allowed for
+a myriad of content to be available for consumption by users on
+all platforms. With the multitude of content available, attention
+from one’s social network has become a valuable and competitive
+resource. Here, we analyze how users allocate their attention to so-
+cial connections with varying degrees of closeness and how this al-
+location is moderated by culture. We study attention in the context
+of Stories posted by friends in one’s network. We examine whether
+tie strength predicts one’s dwell time on a Story and whether cul-
+ture moderates the relationship.
+4.2.1
+Measuring interest. Attention to a poster’s Story is a proxy
+for the interest in the information shared by the user. Attention
+towards a friend who posts Stories (p) is measured by the total
+time they spend on viewing their Story; longer attention (dwell
+time) for a Story indicates a stronger interest towards that friend.
+To measure total time spent on content consumption (TC), we refer
+to the formulation proposed in prior works on measuring content
+dwell time [23].
+푇퐶(푣,푝) =
+�
+푠∈푆푝→푣
+훿(푠)
+where푆푝→푣 denotes the set of Stories postedby p and consumed by
+v, s denotes (without loss of generality) one such Story sample, and
+훿(푠) indicates the time spent by v in viewing the Story. This mea-
+sures the relative difference in the viewer’s interest across different
+posters. However, as pointed out in prior literature, a viewer’s total
+view time on a poster’sStory can be skewed by the frequency of the
+posting activity of the Story creator, i.e., given the equal likelihood
+to consume Stories from different poster’s푇퐶(푣, 푝1) > 푇퐶(푣, 푝2) if
+|푆푝1→푣| > |푆푝2→푣|. Hence, we model dwell time towards a sender
+s as the average time spent by a viewer on the sender’s Stories.
+퐷푇 (푣, 푝) =
+�
+푠∈푆푝→푣 훿(푠)
+|푆푝→푣|
+Dwell time is measured in seconds. While Stories vary in dura-
+tion and can, in turn, influence dwell times, our initial analysis
+of viewing time distribution showed that most viewing activities
+were short and independent of content duration. This finding is in
+line with prior works on dwell time in closed network settings [23]
+- thus, we do not control for this variable.
+4.2.2
+Measuring social tie strength between two users. Tie strength
+between two users is a complex concept, subject to user percep-
+tions and emotions; hence a direct quantitative measure of tie strength
+between users is challenging. However, measuring the activity of
+direct conversations between two users on social media platforms
+has proven to be an effective proxy in estimating tie strength: the
+higher the number of dyadic message exchanges, the closer the
+two users are. Some users send burst messages while others send
+fewer but longer messages; thus, we model tie strength (TS) as the
+total number of direct Snaps and chats exchanged between a pair
+of users.
+푇푆(푣, 푝) = |퐷퐶푝→푣| + |퐷퐶푣→푝| + |퐷푆푝→푣| + |퐷푆푣→푝|
+where 퐷퐶푝→푣 denotes the set of direct chats sent by the Story
+poster to the Story viewer, 퐷퐶푣→푝 denotes the set of direct chats
+sent by the Story viewer to the Story poster, 퐷푆푝→푣 denotes the set
+of direct Snaps sent by the Story poster to the viewer, and 퐷푆푣→푝
+denotes the set of direct Snaps sent by Story viewer to the poster.
+
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+Seth, et al.
+Preliminary analysis of tie strength in each country showed varia-
+tion; hence, we standardize tie strengths within each country and
+use the standardized version for analysis.
+4.2.3
+Measuring culture of each user. We use the results from Hof-
+stede’s Individualism [20], Thomson et al.’s Relational Mobility [46],
+and Gelfand et al.’s Tightness [15] dimensions, discussed in Sec-
+tion 2 as the measure of cultural values (CV) for the country that
+an individual belongs to. These measures have been widely used
+in the literature. Hofstede’s work has attracted over 45,000 cita-
+tions, Thomson et al.’s (more recent) work has already been cited
+178 times, and Gelfand et al.’s work has more than 2000 citations.
+Since each value system is on a different scale (Appendix D)— In-
+dividualism ranges from 6 to 91, Relational Mobility ranges from
+3.886 to 4.607, and Tightness ranges from 1.6 to 12.3 — we indepen-
+dently standardize each value system across countries and use the
+standardized version for analysis.
+4.2.4
+Mixed effects model to analyze dwell time as a function of tie
+strength and cultural values . We used a linear mixed-effects model
+to address the research question of how cultural values moderate
+the impact of tie strength on the time spent consuming content
+(dwell time) in closed network settings. Since the sets of countries
+surveyed by Hofstede [20], Thomson et al. [46], and Gelfand et al.
+[15] do not have perfect overlap, we created three multilevel mod-
+els to understand how cultural values moderate the effect of tie
+strength on Story dwell time. The models included terms for tie
+strength (dyad level), cultural value (country level), and their in-
+teraction as fixed effects, with random intercepts for country and
+viewer, and the number of friends, the GDP, GINI, and Snap’s mar-
+ket penetration (MP) for a country as control variables. We stan-
+dardized each value system across countries and used the standard-
+ized version for analysis. Since we have multiple observations per
+country and a viewer views multiple posters, we include the ran-
+dom effects due to the country and the viewer.
+퐷푇 (푣, 푝) = 푇푆(푣, 푝) 푋 퐶푉 (푣) + |푣푓 | + 퐺퐷푃 + 퐺퐼푁퐼 + 푀푃+
+(1|푐표푢푛푡푟푦) + (1|푉푖푒푤푒푟)
+where |푣푓 | refers to the number of friends a viewer has, 푇푆(푣, 푝) is
+the tie strength between a pair of viewers and a poster, and 퐶푉 (푣)
+is the cultural value of the viewer, which is the same as the cultural
+value of the poster.
+Since each dyad contains the dwell time of multiple Stories, we
+model random effects for the dyad. However, users in a dyad can
+have two roles: sometimes a user is a viewer, and sometimes a
+poster. A user who is a viewer (v) for a poster p can be a poster (푝′)
+for some other node (푣′). This directionality complicates modeling.
+To simplify, we randomly regard one person as the viewer and the
+other as a poster, disregarding the Stories of that dyad where the
+viewer posted and the poster viewed. To ensure that the results 5
+are robust against role assignment, we bootstrapped the analysis;
+on each run, for each dyad, viewer and poster roles were randomly
+assigned before fitting the model. The bootstrapped results are in
+Appendix B.
+5
+RESULTS
+5.1
+RQ1: How do friendship networks differ in
+countries with different cultural values?
+We report zero-order Pearson correlations between cultural values
+and friendship network size in Table 1. We find that countries that
+rank higher in individualism, mobility, and looseness tend to have
+a bigger friendship network than collectivistic, less mobile, and
+tighter countries. This means that people in the higher ranking
+countries are connected to more friends on Snapchat, supporting
+H1a, H2a, and H3a. To check for robustness, we ran the same analy-
+ses with GDP, GINI, and Snapchat’s market penetration as control
+variables. The addition of control variables reduced the sample size
+of countries, but the results corroborate those reported here A.
+Next, the structural analysis of the ego networks of users from
+different cultures (Table 3) shows that the ego centrality of user
+networks on Snapchat varies with cultural values. Akin to Na et
+al.,[28], we find that the individual structural measures, namely
+density, transitivity, and betweenness, are highly correlated (Ta-
+ble 2), and thus we average the standardized values and report the
+results for this averaged index of ego-centrality. The results show
+that mobility and individualism are negatively correlated with ego-
+centricity, and tightness is positively correlated with egocentrality.
+This means that in countries that rank higher on mobility and indi-
+vidualism, people’s friends on Snapchat are more likely to be con-
+nected to each other, and in countries that rank higher on tightness,
+people’s friends on Snapchat are less likely to be connected to each
+other.
+Table 1: Pearson correlation between cultural values and
+friendship network size (∗푝 < 0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 < 0.001)
+Cultural Value
+Correlation
+Number of countries
+Individualism
+0.68**
+65
+Relational Mobility
+0.31*
+37
+Tightness
+-0.37*
+30
+Table 2: Pearson correlation between network structural
+measures for data across different cultural values after con-
+trolling for GDP, GINI, and market penetration (∗푝 < 0.05, ∗∗
+푝 < 0.01, ∗ ∗ ∗푝 < 0.001)
+Cultural
+Value
+Betweenness
+and
+Transitivity
+Betweenness
+and Density
+Density
+and
+Transitivity
+Individualism
+0.74***
+0.504***
+0.92 ***
+Mobility
+0.76 ***
+0.49**
+0.85***
+Tightness
+0.82***
+0.45*
+0.83 ***
+
+Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+Table 3: Pearson correlation between cultural values and
+egocentrality(∗푝 < 0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 < 0.001)
+Cultural Value
+averaged index of ego-centrality
+Individualism
+-0.07 ***
+Relational Mobility
+-0.04***
+Tightness
+0.06***
+5.2
+RQ2: How do cultural values change the
+effect of tie strength on dwell time?
+Given that the friendship network structures are different across
+cultures, using multilevel modeling, we analyzed how cultural val-
+ues moderate the effect of tie strength on the viewer’s dwell time
+(Tables 4, 5, 6). We see that an increase in the strength of ties in-
+creases the dwell time, a result in line with prior works [23, 52].
+Having more friends reduces a viewer’s dwell time on content,
+which is likely because an increase in the number of friends leads
+to more potential Story content to consume. Though the cultural
+values do not have a significant main effect, they significantly mod-
+erate the effect of tie strength on dwell time across all three cultural
+values. We find that tie strength negatively moderates the effect of
+tie strength for more individualistic, mobile, and looser cultures.
+Thus confirming H1b, H2b, and H3b. The bootstrap results from
+100 runs corroborate the findings reported here in Appendix B.
+Our work focuses on understanding (and not predicting) within-
+dyad level dwell time from theories of country-level cultural val-
+ues, which may not fully account for a lot of individual-level vari-
+ation. However, a significant moderation effect allows us to argue
+for a substantiative effect of cultural values on individual-level be-
+havior [27]. Using only the intersection of countries present across
+all three measures of culture, we check for robustness of these re-
+sults (Appendix C), and the results corroborate the results reported
+in Tables 4, 5, 6. Because the effects we found are on the smaller
+side, there is still a lot of unexplained variance, and we can not fully
+account for all individual-level and item (Story) level variation.
+Table 4: Coefficients from Multilevel Modeling for the ef-
+fect of Individualism as a moderator on Dwell Time (∗푝 <
+0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 < 0.001), Sample size: country = 47,
+dyads = 460000, RMSE = 4.9, AIC = 2793115, BIC = 279226, R2
+conditional = 0.04, R2 marginal = 0.01
+Fixed Effects
+Estimate
+Standard Error
+Intercept
+3.741***
+0.078
+Strength of Ties
+0.092***
+0.007
+Individualism
+0.035
+0.074
+Strength of Ties : Individualism
+-0.014***
+0.007
+Control variables
+Number of Friends
+-0.338***
+0.008
+GDP
+-0.036
+0.068
+GINI
+-0.040
+0.060
+Market Penetration
+0.065*
+0.059
+Table 5: Coefficients From Multilevel Modeling for the effect
+of Mobility as a moderator on Dwell Time (∗푝 < 0.05, ∗ ∗ 푝 <
+0.01, ∗∗∗푝 < 0.001) Sample size: country = 26, dyads = 128800,
+RMSE= 3.12, AIC = 1438399, BIC = 1438504, R2 conditional =
+0.27, R2 marginal = 0.01
+Fixed Effects
+Estimate
+Standard Error
+Intercept
+3.835 ***
+0.097
+Strength of Ties
+0.116***
+0.008
+High Mobility
+0.092
+0.071
+Strength of Ties : High Mobility
+-0.012*
+0.006
+Control variables
+Number of Friends
+-0.35***
+0.011
+GDP
+-0.051
+0.108
+GINI
+-0.02
+0.102
+Market Penetration
+0.108
+0.091
+Table 6: Coefficients From Multilevel Modeling for the effect
+of Tightness as a moderator on Dwell Time (∗푝 < 0.05, ∗∗푝 <
+0.01, ∗∗∗푝 < 0.001), Sample size: country = 25, dyads = 100000,
+RMSE=2.19, AIC = 731754.3, BIC = 731850.8, R2 conditional
+= 0.80, R2 marginal = 0.01
+Fixed Effects
+Estimate
+Standard Error
+Intercept
+3.725***
+0.1
+Strength of Ties
+0.129***
+0.010
+Tightness
+-0.060
+-0.082
+Strength of Ties : Tightness
+0.058***
+0.010
+Control variables
+Number of Friends
+-0.283 ***
+0.011
+GDP
+-0.154*
+0.077
+GINI
+-0.171
+0.069
+Market Penetration
+0.179*
+0.077
+6
+DISCUSSION
+Most social media platforms were introduced in the Global North
+before they started gaining a user base in other countries. As a
+result, studies on understanding users on social media platforms
+primarily draw from west-centric populations, which leads to un-
+intended biases. Using data from 10,000 users per country from
+nearly 73 countries, our work studied how individuals across cul-
+tures differ in their behavior on the same platform. We control
+for confounders like the platform’s market penetration, countries’
+GDP, and GINI score, which may have influenced the platform’s
+user base size and composition. Our main findings are:
+Structure of friendship network. The analysis of the egocentrality of
+the friendship networks showed that individualistic, more mobile,
+and looser cultures are negatively correlated with egocentrality.
+This result is unlike the prior survey-based network analysis by Na
+et al. [28], which found that individualism is positively correlated
+with ego centrality. Na et al. [28] recruited individuals through a
+call for survey participants on the Facebook platform, which re-
+sulted in a substantially varied number of respondents from each
+
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+Seth, et al.
+country and thus could be sensitive to selection and conformity
+bias. In our study, we randomly sampled users and analyzed the
+metadata of the user behavior, which provides a relatively cleaner
+signal for a user’s choices. Apart from a more balanced number of
+users from different countries, we also analyzed data from a sub-
+stantially higher number of countries. Apart from data collection
+and sample size differences, another potential source for the dif-
+ferences in findings could arise from who is befriended on these
+platforms.
+Adams and Plaut posited that friendship’s meaning varies sub-
+stantially across cultures [1]. Markus and Kitayama [26] argued
+that familial ties form an important part of a user’s social network
+in collectivist cultures compared to individualistic cultures. With
+the demographics on Snapchat skewing towards a younger popu-
+lation [9, 10] and motivations differing from Facebook [3, 34, 50], it
+is plausible that (a) the ’younger users’ do not ’friend’ familial ties
+due to the difference in how they make sense of ’friendship’ and
+whom they ’friend,’ and (b) the ’elder’ familial members are ab-
+sent from the platform. Since family ties form an important part of
+collectivist cultures, not including them on their Snapchat friend-
+ship network could be the reason for differences in our findings
+when compared to Na et al. [28]. While our results differ from Na
+et al., [28], they agree with the findings from Igarashi et al.[21]
+that user’s from collectivist cultures had more egocentric networks.
+Given that very few studies have explored how culture affects net-
+work structures, future work in this domain will help establish
+a stronger understanding of how culture influences the network
+structures formed on social media platforms.
+Our findings bear important implications for future work that
+aims to study user interaction patterns on a platform. Firstly, stud-
+ies should elicit and validate the network structure formed for their
+population of interest because the network structures vary across
+subpopulations on the same platform and across platforms, and
+relying on metrics from prior work with a mismatched popula-
+tion might lead to incorrect inferences. Next, the differences in
+friendship networks bear importance for context-aware friendship
+recommendation engines, which we discuss under design implica-
+tions.
+Cultural Values and user behavior. Culture is a complex societal-
+level phenomenon that guides individual behavior. Various studies
+have tried to study culture through a system of ’cultural values.’ In
+this project, we chose three dominant theories in cultural psychol-
+ogy, ranging from Hofstede’s dimensions published in 2001 [20] to
+more recent theories on Tightness and Mobility published in 2011
+and 2018 [15, 46], respectively. Consistent with our hypothesis, we
+found that each cultural value (i.e., individualism, looseness, mobil-
+ity) significantly moderates the effect of tie strength on dwell time,
+highlighting the significance of considering culture in understand-
+ing behavior patterns on social media. In addition, we found that
+individualism, looseness, and mobility moderates the relationship
+between tie strength and dwell time in the same direction. Theo-
+retically, it is logical because in societies where people have more
+freedom to make friends and move between different circles (i.e.,
+high relational mobility), a looser norm (i.e., looseness) is likely to
+develop, and a comparatively more self-focused mindset (i.e., indi-
+vidualism) is likely to rise. Indeed, prior work has also predicted
+that these three variables would have a similar impact on individ-
+ual cognition and behavior [46]. Thus, we extend the prior work in
+cultural psychology by adopting a cultural lens in understanding
+user behaviors on social media.
+6.1
+Design Implications
+The diversity of content on platforms has made good recommen-
+dation systems a necessity. While these recommendation systems
+are becoming increasingly personalized, they fail to distinguish the
+varied meanings that different types of social ties have for users
+from different cultures. For example, if we consider the dyadic pair
+of user A, their strongest tie, and user B, their strongest tie, such
+that user A and B belong to different cultures, the influence of the
+respective strongest tie may be different. Our study, through evi-
+dence, argues for treating users and their friendship relations from
+different cultures differently when designing recommendation sys-
+tems. Analyzing users at a cultural level may reduce the complexity
+of recommendation systems and make the recommendation sys-
+tem more culturally sensitive. By doing so, they may be able to
+better rank the content the user is more likely to engage with at
+a reduced cost. For example, our result suggests that when design-
+ing recommendation systems, tie strength should be given greater
+weight for users in less mobile, tighter, and collectivistic countries
+because our results show that tie strength is more strongly corre-
+lated to content dwell time in these countries.
+Friendship recommendation engines that are unaware of ’how’
+and ’why’ network structures differ across cultures run the risk of
+treating friending activities across different cultures as the same,
+resulting in a suboptimal platform experience. For instance, the
+motivations of individuals from tight cultures could differ from
+those from loose cultures, i.e., in contrast to individuals from loose
+cultures, individuals in tight cultures might feel forced to friend
+not only those whom they want to but also those whom they have
+to - say befriending familial ties. A recommendation engine that
+captures behavior from loose cultures might not be able to recom-
+mend users with whom one shares common friends. Similarly, a
+recommendation engine that focuses on tight cultures would ex-
+plore less and over-recommend users with whom one shares com-
+mon friends. Hence, using the behavioral understanding from only
+either of the cultures risks the failure of the algorithms ( and, in
+turn, platform experience) in the other cultures. Thus, while our
+work takes a step in highlighting ’how’ the network structures
+differ, future work that provides insights into ’why’ the network
+structures differ can further enrich the understanding of design-
+ing friendship recommendation algorithms.
+7
+LIMITATIONS
+Our study is subject to a few important limitations. First, our work
+uses data from Snapchat, which encompasses a significant but lim-
+ited amount of people’s online communications. We could only
+use available data for our study, and some of Snapchat’s user data
+is only available for a limited time. Additionally, the actual con-
+tent of Snapchat communications is not available for analysis. The
+Snapchat user group skews young [10], and studies have found
+that younger people have shifted away from traditional values [29,
+43]. Second, recommendation algorithms play an important role
+
+Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+in network formation on the platform. We did not have access to
+the friend recommendation algorithm for this study, and we could,
+therefore, not control for any potential confounding effects. Get-
+ting an insight into the algorithm and its impact on users across
+geographies could further enrich future work. Further, the focus of
+this study was to understand the friendship network and behavior
+on the online social network, which may differ from an individ-
+ual’s offline friendship networks and their interactions on these
+networks. Next, not every country has been equally surveyed in
+prior research on cultural values. There is a non-perfect overlap be-
+tween countries that have been studied for mobility and countries
+that have been studied for tightness. Once data from more coun-
+tries becomes available, our analyses could be extended to include
+those countries. Future work can further build on ours by analyz-
+ing how content type interacts with cultural values and impacts
+dwell time. By using a large random sample of users across coun-
+tries, country-level measures of economic growth, and inequity,
+we tried to limit selection bias and account for variations across
+countries. GDP and GINI measures help us control for country-
+level socioeconomic status. However, it is plausible that a given
+stratum of society is overrepresented on the platform, and country-
+level socioeconomic measures might not fully control for the plat-
+form user’s socioeconomic status. The lack of finer-grained mea-
+sures could be a limitation of the study.
+Human behavior is complex and subject to factors that have
+individual-level variation. Hence, it is difficult to fully predict hu-
+man behavior in the social sciences. The focus of our work was to
+test the theory of the effect of culture, as measured at the country
+level, on individual behavior. Like prior works, we can not fully
+account for all individual-level and item (Story) level variation. As
+brought out in the Introduction 1, individual behavior is affected
+by a host of other variables, and content engagement is no differ-
+ent. For example, the Story’s content might be an important fac-
+tor; however, we could not study this due to Snap Inc.’s policies
+on not retaining information about the content. Future studies can
+help make the model more complete by operationalizing the type
+of content and other variables that might affect the dwell time on
+content. While the cultural theories used in this study span a large
+geographic region, the identities of the researchers who created
+these measures could be a source of bias for these measures. As
+argued by Shweder [40] (p. 409), these studies can largely benefit
+from a more emic expansion approach, which would help remove
+biases from future empirical studies.
+8
+CONCLUSION
+We examined the friendship network and the dwell time behavior
+of users across 73 cultures on the online platform Snapchat. We
+studied one month’s data from 10K users from each culture. First,
+we found that the friendship networks curated by individuals from
+different cultures vary in size and egocentricity. We found evidence
+that individuals from individualistic, high mobility, and loose cul-
+tures tend to form larger friendship networks. We analyzed how
+cultural values moderate the relation between tie strength and users’
+content engagement behavior. We found that individualism, high
+mobility, and looseness negatively moderate this effect. This pro-
+vides evidence for psychological theories which posit that relation-
+ships are not perceived similarly across different cultures, and thus
+their effect on user behavior is not uniform across cultures. Our
+work could advance the understanding of engagement with con-
+tent on online platforms and how using this insight can improve
+recommendation systems. Incorporating cultural values in the ex-
+perience design can improve the user experience and does better
+justice to the diverse backgrounds of platform users.
+ACKNOWLEDGMENTS
+We thank Dr. Neil Shah (Snap Inc.) for providing valuable advice
+on friendship network creation and analysis.
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+A
+CULTURAL VALUE AND FRIEND
+NETWORK SIZE WITH CONTROL
+VARIABLES
+Table 7: Pearson correlation between cultural values and
+friendship network size with GDP, GINI, and Market Pen-
+etration as control variables (∗푝 < 0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 <
+0.001)
+Cultural Value
+Correlation
+Number of countries
+Individualism
+0.6**
+47
+Relational Mobility
+0.27
+26
+Tightness
+-0.51*
+24
+
+Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+B
+BOOTSTAPPED RESULTS FOR MIXED
+EFFECTS MODEL (ACROSS 100 RUNS)
+Table 8: Bootstrapped Coefficients From Multilevel Model-
+ing for the effect of Individualism as a moderator on Dwell
+Time
+Fixed Effects
+Estimate
+퐶퐼 (95%)
+Intercept
+3.740
+[3.718,3.762]
+Strength of Ties
+0.103
+[0.100,0.106]
+Individualism
+0.033
+[0.028,0.038]
+Strength of Ties : Individualism
+-0.008
+[-0.011,-0.005]
+Control variables
+Number of Friends
+-0.329
+[-0.341,-0.317]
+GDP
+-0.031
+[-0.038,-0.024]
+GINI
+-0.040
+[-0.042,-0.039]
+Market Penetration
+0.56
+[0.038,0.074]
+Table 9: Bootstrapped Coefficients From Multilevel Model-
+ing for the effect of Mobility as a moderator on Dwell Time
+Fixed Effects
+Estimate
+퐶퐼 (95%)
+Intercept
+3.820
+[3.801,3.839]
+Strength of Ties
+0.114
+[0.111,0.117]
+High Mobility
+0.092
+[0.089,0.095]
+Strength of Ties : High Mobility
+-0.010
+[-0.014,-0.007]
+Control variables
+Number of Friends
+-0.347
+[-0.356,-0.338]
+GDP
+-0.058
+[-0.062,0.054]
+GINI
+-0.020
+[-0.020,-0.015]
+Market Penetration
+0.109
+[0.104,0.114]
+Table 10: Bootstrapped Coefficients From Multilevel Model-
+ing for the effect of Tightness as a moderator on Dwell Time
+Fixed Effects
+Estimate
+퐶퐼 (95%)
+Intercept
+3.740
+[3.737,3.743]
+Strength of Ties
+0.116
+[0.111,0.121]
+Tightness
+-0.061
+[-0.061, -0.060]
+Strength of Ties : Tightness
+0.008
+[0.006, 0.012]
+Control variables
+Number of Friends
+-0.291
+[-0.295,-0.285]
+GDP
+-0.156
+[-0.161,-0.150]
+GINI
+-0.17
+[-0.171,-0.162]
+Market Penetration
+0.170
+[0.169,0.170]
+C
+MIXED EFFECTS MODEL FOR THE
+INTERSECTION OF COUNTRIES PRESENT
+ACROSS ALL THREE MEASURES (FOR 1
+RUN)
+Table 11: Coefficients from Multilevel Modeling for the ef-
+fect of Individualism as a moderator on Dwell Time (∗푝 <
+0.05, ∗ ∗ 푝 < 0.01, ∗ ∗ ∗푝 < 0.001), Sample size: country = 18,
+dyads = 82800, RMSE = 2.947, AIC = 45373.1, BIC = 453824.6,
+R2 conditional =0.27, R2 marginal = 0.01
+Fixed Effects
+Estimate
+Standard Error
+Intercept
+3.699***
+0.126
+Strength of Ties
+0.147***
+0.017
+Individualism
+0.085
+0.116
+Strength of Ties : Individualism
+-0.042***
+0.011
+Control variables
+Number of Friends
+-0.322***
+0.018
+GDP
+-0.168*
+0.074
+GINI
+-0.171
+0.063
+Market Penetration
+0.238
+0.128
+Table 12: Coefficients From Multilevel Modeling for the ef-
+fect of Mobility as a moderator on Dwell Time (∗푝 < 0.05, ∗ ∗
+푝 < 0.01, ∗ ∗ ∗푝 < 0.001) Sample size: country = 18, dyads =
+82800 RMSE = 3.07, AIC = 476936.1, BIC = 477029.3, R2 con-
+ditional = 0.09, R2 marginal = 0.01
+Fixed Effects
+Estimate
+Standard Error
+Intercept
+3.969***
+0.123
+Strength of Ties
+0.126***
+0.014
+High Mobility
+0.008
+0.083
+Strength of Ties : High Mobility
+-0.037***
+0.009
+Control variables
+Number of Friends
+-0.30***
+0.017
+GDP
+-0.152
+0.077
+GINI
+-0.141
+0.060
+Market Penetration
+0.099***
+0.010
+
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+Seth, et al.
+Table 13: Coefficients From Multilevel Modeling for the ef-
+fect of Tightness as a moderator on Dwell Time (∗푝 < 0.05, ∗∗
+푝 < 0.01, ∗ ∗ ∗푝 < 0.001), Sample size: country = 18, dyads =
+82800, RMSE = 2.36, AIC = 482736.7, BIC = 482830, R2 condi-
+tional = 0.48, R2 marginal = 0.01
+Fixed Effects
+Estimate
+Standard Error
+Intercept
+3.767***
+0.135
+Strength of Ties
+0.164***
+0.014
+Tightness
+-0.021
+0.084
+Strength of Ties : Tightness
+0.094***
+0.012
+Control variables
+Number of Friends
+-0.383***
+0.024
+GDP
+-0.183
+0.081
+GINI
+-0.180
+0.070
+Market Penetration
+0.191
+0.105
+
+Cultural Differences in Friendship Network Behaviors: A Snapchat Case Study
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+D
+LIST OF COUNTRIES ANALYZED
+Country
+Individualism
+Mobility
+Tightness
+Argentina
+46
+×
+×
+Australia
+90
+4.308
+4.4
+Austria
+55
+×
+6.8
+Belgium
+75
+×
+5.6
+Brazil
+38
+4.419
+3.5
+Canada
+×
+4.404
+×
+Chile
+23
+4.3
+×
+China
+excluded from analysis since Snapchat is banned
+Colombia
+13
+4.483
+×
+Costa Rica
+15
+×
+×
+Czech Republic
+58
+×
+×
+Denmark
+74
+×
+×
+cEcuador
+8
+×
+×
+Egypt
+38
+3.971
+×
+El Salvador
+19
+×
+×
+Estonia
+×
+4.233
+2.6
+Ethiopia
+27
+×
+×
+Finland
+63
+×
+×
+France
+71
+4.451
+6.3
+Germany
+67
+4.194
+7
+Ghana
+20
+×
+×
+Greece
+35
+×
+3.9
+Guatemala
+6
+×
+×
+Hong Kong
+25
+4.043
+6.3
+Hungary
+55
+3.893
+2.9
+Iceland
+×
+×
+6.4
+India
+48
+×
+11
+Indonesia
+14
+×
+×
+Iran
+excluded from analysis since Snapchat is banned
+Iraq
+38
+×
+×
+Ireland
+70
+×
+×
+Israel
+54
+4.336
+3.1
+Italy
+76
+×
+6.8
+Jamaica
+39
+×
+×
+Japan
+46
+3.934
+8.6
+Jordan
+×
+3.96
+×
+Kenya
+27
+×
+×
+Kuwait
+38
+×
+×
+Lebanon
+38
+4.079
+×
+Libya
+38
+4.015
+×
+Malaysia
+26
+3.886
+11.8
+Mauritius
+×
+4.385
+×
+Mexico
+30
+4.607
+7.2
+Morocco
+×
+4.062
+×s
+Netherlands
+80
+4.448
+3.3
+New Zealand
+79
+4.287
+3.9
+Nigeria
+20
+×
+×
+Norway
+69
+×
+9.5
+Pakistan
+14
+×
+12.3
+Panama
+11
+×
+×
+Peru
+16
+×
+×
+Philippines
+32
+4.158
+×
+Poland
+60
+4.415
+6.0
+
+CHI ’23, April 23–28, 2023, Hamburg, Germany
+Seth, et al.
+Portugal
+27
+4.236
+7.8
+Puerto Rico
+×
+4.603
+×
+Saudi Arabia
+38
+×
+×
+Sierra Leone
+20
+×
+×
+Singapore
+20
+4.133
+10.4
+South Africa
+65
+×
+×
+South Korea
+18
+4.089
+10.0
+Spain
+51
+4.415
+5.4
+Sweden
+71
+4.364
+×
+Switzerland
+68
+×
+×
+Taiwan
+17
+4.118
+×
+Tanzania
+27
+×
+×
+Thailand
+20
+×
+×
+Trinidad and Tobago
+×
+4.421
+×
+Tunisia
+×
+3.954
+×
+Turkey
+37
+4.122
+9.2
+Ukraine
+excluded from analysis due to geo-political instability
+United Arab Emirates
+38
+×
+×
+United Kingdom
+89
+4.315
+6.9
+United States
+91
+4.382
+5.1
+Uruguay
+36
+×
+×
+Venezuela
+12
+4.508
+3.7
+Zambia
+27
+×
+×
+Table 14: List of Countries and the cultural values that they were surveyed for; × signifies country not surveyed for that cultural
+value
+

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+arXiv:2301.01479v1  [math.OC]  4 Jan 2023
+Generalizations of R0 and SSM properties; Extended Horizontal Linear
+Complementarity Problem
+Punit Kumar Yadav
+Department of Mathematics
+Malaviya National Instiute of Technology, Jaipur, 302017, India
+E-mail address: punitjrf@gmail.com
+K. Palpandi
+Department of Mathematics
+Malaviya National Instiute of Technology, Jaipur, 302017, India
+E-mail address: kpalpandi.maths@mnit.ac.in
+Abstract
+In this paper, we first introduce R0-W and SSM-W property for the set of matrices which
+is a generalization of R0 and the strictly semimonotone matrix. We then prove some existence
+results for the extended horizontal linear complementarity problem when the involved matrices
+have these properties. With an additional condition on the set of matrices, we prove that the
+SSM-W property is equivalent to the unique solution for the corresponding extended horizontal
+linear complementarity problems. Finally, we give a necessary and sufficient condition for the
+connectedness of the solution set of the extended horizontal linear complementarity problems.
+1
+Introduction
+The standard linear complementarity problem (for short LCP), LCP(C, q), is to find vectors x, y
+such that
+x ∈ Rn, y = Cx + q ∈ Rn and x ∧ y = 0,
+(1)
+where C ∈ Rn×n, q ∈ Rn and ′∧′ is a min map. The LCP has numerous applications in numerous
+domains, such as optimization, economics, and game theory.
+Cottle and Pang’s monograph [1]
+is the primary reference for standard LCP. Various generalisations of the linear complementarity
+problem have been developed and discussed in the literature during the past three decades (see,
+[7, 10, 11, 13, 14, 16]). The extended horizontal linear complementarity problem is one of the most
+important extensions of LCP, which various authors have studied; see [4, 6, 7] and references therein.
+For a given ordered set of matrices C := {C0, C1, ..., Ck} ⊆ Rn×n, vector q ∈ Rn and ordered set of
+positive vectors d := {d1, d2, ..., dk} ⊆ Rn, the extended horizontal linear complementarity problem
+(for short EHLCP), denoted by EHCLP(C, d, q), is to find a vector x0, x1, ..., xk ∈ Rn such that
+C0x0 =q +
+k
+�
+i=1
+Cixi,
+x0 ∧ x1 = 0 and (dj−xj) ∧ xj+1 = 0, 1 ≤ j ≤ k − 1.
+(2)
+If k = 1, then EHLCP becomes the horizontal linear complementarity problem (for short HLCP),
+that is,
+C0x0 − C1x1 = q and x0 ∧ x1 = 0.
+Further, HLCP reduces to the standard LCP by taking C0 = I. Due to its widespread applications in
+numerous domains, the horizontal linear complementarity problem has received substantial research
+attention from many academics; see [13, 14, 16, 18] and reference therein.
+Various writers have presented new classes of matrices for analysing the structure of LCP solution
+sets in recent years; see for example, [1, 2, 4]. The classes of R0, P0, P, and strictly semimonotone
+1
+
+(SSM) matrices play a crucial role in the existence and uniqueness of the solution to LCP. For
+instance, P matrix (if [x ∈ Rn, x ∗ Ax ≤ 0 =⇒ x = 0]) gives a necessary and sufficient condition
+for the uniqueness of the solution for the LCP (see, Theorem 3.3.7 in [1]). To get a similar type of
+existence and uniqueness results for the generalized LCPs, the notion of P matrix was extended for
+the set of matrices as the column W-property by Gowda et al. [4]. They proved that column W-
+property gives the solvability and the uniqueness for the extended horizontal linear complementarity
+problem (EHLCP). Also, they have generalized the concept of the P0-matrix as the column W0-
+property.
+Another class of matrix, the so-called SSM matrix, has importance in LCP theory. This class of
+matrices provides a unique solution to LCP on Rn
++ and also gives the existence of the solution for the
+LCP (see, [1]). For a Z matrix (if all the off-diagonal entries of a matrix are non-positive), P matrix
+is equivalent to the SSM matrix (see, Theorem 3.11.10 in [1]). A natural question arises whether
+the SSM matrix can be generalized for the set of matrices in the view of EHLCP and whether we
+have a similar equivalence relation for the set of Z matrices. In this paper, we would like to answer
+this question.
+The connectedness of the solution set of LCP has a prominent role in the study of the LCP. We
+say a matrix is connected if the solution set of the corresponding LCP is connected. In [19], Jones
+and Gowda addressed the connectedness of the solution set of the LCP. They proved that the matrix
+is connected whenever the given matrix is a P0 matrix and the solution set has a bounded connected
+component. Also, they have shown that if the solution set of LCP is connected, then there is almost
+one solution of LCP for all q > 0. Due to the specially structured matrices involved in the study of
+the connectedness of the solution to LCP, various authors studied the connectedness of LCP, see for
+example [19, 20, 21]. The main objectives of this paper are to answer the following questions:
+(Q1) In LCP theory, it is a well-known result that the R0 matrix gives boundedness to the LCP
+solution set. The same holds true for HLCP [17]. This motivates the question of whether or
+not the notion of R0 matrix can be generalized to the set of matrices. If so, then can we expect
+the same kind of outcome in the EHLCP?
+(Q2) Given that a strictly semimonotone matrix guarantees the existence of the LCP solution and
+its uniqueness for q ≥ 0, it is natural to wonder whether the concept of SSM matrix can be
+extended to the set of matrices. If so, then whether the same result holds true for EHLCP.
+(Q3) Motivated by the results of Gowda and Jones [19] regarding the connectedness of the solution
+set of LCP, one can ask whether the solution set of EHLCP is connected if the set of matrices
+has the column W0 property and the solution set of the corresponding EHLCP has a bounded
+connected component.
+The paper’s outline is as follows: We present some basic definitions and results in section 2. We
+generalize the concept of R0 matrix and prove the existence result for EHLCP in section 3. In
+section 4, we introduce the SSM-W property, and we then study an existence and uniqueness result
+for the EHLCP when the underlying set of matrices have this property. In the last section, we give
+a necessary and sufficient condition for the connectedness of the solution set of the EHLCP.
+2
+Notations and Preliminaries
+2.1
+Notations
+Throughout this paper, we use the following notations:
+(i) The n dimensional Euclidean space with the usual inner product will be denoted by Rn. The
+set of all non-negative vectors (respectively, positive vectors) in Rn will be denoted by Rn
++
+2
+
+(respectively, Rn
+++ ). We say x ≥ 0 (respectively, > 0) if and only if x ∈ Rn
++ (respectively,
+Rn
+++).
+(ii) The k-ary Cartesian power of Rn will be denoted by Λ(k)
+n
+and the k-ary Cartesian power of
+Rn
+++ will be denoted by Λ(k)
+n,++. The bold zero ’0’ will be used for denoting the zero vector
+(0, 0, ..., 0) ∈ Λ(k)
+n .
+(iii) The set of all n×n real matrices will be denoted by Rn×n. We use the symbol Λ(k)
+n×n to denote
+the k-ary Cartesian product of Rn×n.
+(iv) We use [n] to denote the set {1, 2, ..., n}.
+(v) Let M ∈ Rn×n. We use diag(M) to denote the vector (M11, M22, ..., Mkk) ∈ Rn, where Mii is
+the iith diagonal entry of matrix M and det(M) is used to denote the determinant of matrix
+M.
+(vi) SOL(C, d, q) will be used for denoting the set of all solution to EHLCP(C, d, q).
+We now recall some definitions and results from the LCP theory, which will be used frequently in
+our paper.
+Proposition 2.1 ([8]). Let V = Rn. Then, the following statements are equivalent.
+(i) x ∧ y = 0.
+(ii) x, y ≥ 0 and x ∗ y = 0, where ∗ is the Hadamard product.
+(iii) x, y ≥ 0 and ⟨x, y⟩ = 0.
+Definition 1 ([4]). Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n .
+Then a matrix R ∈ Rn×n is column
+representative of C if
+R.j ∈
+�
+(C0).j, (C1).j, ..., (Ck).j
+�
+, ∀j ∈ [n],
+where R.j is the jth column of matrix R.
+Next, we define the column W-property.
+Definition 2 ([4]). Let C := (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n . Then we say that C has the
+(i) column W-property if the determinants of all the column representative matrices of C are all
+positive or all negative.
+(ii) column W0-property if there exists N := (N0, N1, ..., Nk) ∈ Λ(k+1)
+n×n
+such that C + ǫN := (C0 +
+ǫN0, C1 + ǫN1, ..., Ck + ǫNk) has the column W-property for all ǫ > 0.
+Due to Gowda and Sznajder [4], we have the following result.
+Theorem 2.2 ([4]). For C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n , the following are equivalent:
+(i) C has the column W-property.
+(ii) For arbitrary non-negative diagonal matrices D0, D1, ..., Dk ∈ Rn×n with diag(D0 +D1 +D2 +
+... + Dk) > 0,
+det
+�
+C0D0 + C1D1 + ... + CkDk
+�
+̸= 0.
+(iii) C0 is invertible and (I, C−1
+0 C1, ..., C−1
+0 Ck) has the column W-property.
+3
+
+(iv) For all q ∈ Rn and d ∈ Λ(k−1)
+n,++ , EHLCP(C, d, q) has a unique solution.
+If k = 1 and C−1
+0
+exists, then HLCP(C0, C1, q) is equivalent to LCP(C−1
+0 C1, C−1
+0 (q)). In this
+case, C−1
+0 C1 is a P matrix if and only if for all q ∈ Rn, LCP(C−1
+0 C1, C−1
+0 (q)) has a unique solution
+(see, Theorem 3.3.7 in [1]). Hence we have the following theorem given the previous theorem.
+Theorem 2.3 ([4]). Let (C0, C1) ∈ Λ(2)
+n×n. Then the following are equivalent.
+(i) (C0, C1) has the column W-property.
+(ii) C0 is invertible and C−1
+0 C1 is a P matrix.
+(iii) For all q ∈ Rn, HLCP(C0, C1, q) has a unique solution.
+2.2
+Degree theory
+We now recall the definition and some properties of a degree from [2, 3] for our discussion.
+Let Ω be an open bounded set in Rn. Suppose h : ¯Ω → Rn is a continuous function and a vector
+p /∈ h(∂Ω), where ∂Ω and ¯Ω denote the boundary and closure of Ω, respectively. Then the degree of
+h is defined with respect to p over Ω denoted by deg(h, Ω, p). The equation h(x) = p has a solution
+whenever deg(h, Ω, p) is non-zero. If h(x) = p has only one solution, say y in Rn, then the degree is
+the same overall bounded open sets containing y. This common degree is denoted by deg(h, p).
+2.2.1
+Properties of the degree
+The following properties are used frequently here.
+(D1) deg(I, Ω, ·) = 1, where I is the identity function.
+(D2) Homotopy invariance: Let a homotopy Φ(x, s) : Rn ×[0, 1] → Rn be continuous. If the zero
+set of Φ(x, s), X = {x : Φ(x, s) = 0 for some s ∈ [0, 1]} is bounded, then for any bounded
+open set Ω in Rn containing the zero set X, we have
+deg(Φ(x, 1), Ω, 0) = deg(Φ(x, 0), Ω, 0).
+(D3) Nearness property: Assume deg(h1(x), Ω, p) is defined and h2 : ¯Ω → Rn is a continuous
+function. If supx∈Ω∥h2(x) − h1(x)∥ < dist(p, ∂Ω), then deg(h2(x), Ω, p) is defined and equals
+to deg(h1(x), Ω, p).
+The following result from Facchinei and Pang [2] will be used later.
+Proposition 2.4 ([2]). Let Ω be a non-empty, bounded open subset of Rn and let Φ : ¯Ω → Rn be a
+continuous injective mapping. Then deg(Φ, Ω, p) ̸= 0 for all p ∈ Φ(Ω).
+Note: All the degree theoretic results and concepts are also applicable over any finite dimensional
+Hilbert space (like Rn or Rn × Rn × Rn etc).
+3
+R0-W property
+In this section, we first define the R0-W property for the set of matrices which is a natural generaliza-
+tion of R0 matrix in the LCP theory. We then show that the R0-W property gives the boundedness
+of the solution set of the corresponding EHLCP.
+4
+
+Definition 3. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n . We say that C has the R0-W property if the
+system
+C0x0 =
+k
+�
+i=1
+Cixi and x0 ∧ xj = 0 ∀ j ∈ [k]
+has only zero solution.
+It can be seen easily that the R0-W property coincides with R0 matrix when k = 1 and C0 = I.
+Also it is noted (see, [8]) that if k = 1, then the R0-W property referred as R0 pair. To proceed
+further, we prove the following result.
+Lemma 3.1. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n
+and x = (x0, x1, ..., xk) ∈ SOL(C, d, q). Then x
+satisfies the following system
+C0x0 = q +
+k
+�
+i=1
+Cixi and x0 ∧ xj = 0 ∀ j ∈ [k].
+Proof. As x0 ≥ 0, there exists an index set α ⊆ [n] such that (x0)i =
+�
+> 0
+i ∈ α
+0
+i ∈ [n] \ α . Since
+x0 ∧ x1 = 0, we have (x1)i = 0 for all i ∈ α. From (d1 − x1) ∧ x2 = 0, we get (d1)i(x2)i = 0 ∀i ∈ α.
+This gives that (x2)i = 0 ∀i ∈ α. By substituting (x2)i = 0 ∀i ∈ α in (d2 − x2) ∧ x3 = 0, we obtain
+(x3)i = 0 ∀i ∈ α. Continue the process in the similar way, one can get (x4)i = (x5)i = ... = (xk)i =
+0 ∀i ∈ α. So, x0 ∧ xj = 0 ∀ j ∈ [k]. This completes the proof.
+We now prove the boundedness of the solution set of EHLCP when the involved set of matrices
+has the R0-W property.
+Theorem 3.2. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n . If C has the R0-W property then SOL(C, d, q)
+is bounded for every q ∈ Rn and d ∈ Λ(k−1)
+n,++ .
+Proof. Suppose there exist q ∈ Rn and d = (d1, d2, ..., dk−1) ∈ Λ(k−1)
+n,++ such that SOL(C, d, q) is
+unbounded. Then there exists a sequence x(m) = (x(m)
+0
+, x(m)
+1
+, ..., x(m)
+k
+) in Λ(k+1)
+n
+such that ||x(m)|| →
+∞ as m → ∞ and it satisfies
+C0x(m)
+0
+= q +
+k
+�
+i=1
+Cix(m)
+i
+x(m)
+0
+∧ x(m)
+1
+= 0 and (dj − x(m)
+j
+) ∧ x(m)
+j+1 = 0 ∀j ∈ [k − 1].
+(3)
+From the Lemma 3.1, equation 3 gives that
+C0x(m)
+0
+=q +
+k
+�
+i=1
+Cix(m)
+i
+and x(m)
+0
+∧ x(m)
+j
+=
+0 ∀j ∈ [k].
+(4)
+As
+x(m)
+∥x(m)∥ is a unit vector for all m,
+x(m)
+∥x(m)∥ converges to some vector y = (y0, y1, ..., yk) ∈ Λ(k+1)
+n
+with ||y|| = 1. Now first divide the equation 4 by ∥x(m)∥ and then take the limit m → ∞, we get
+C0y0 =
+k
+�
+i=1
+Ciyi and y0 ∧ yj = 0 ∀j ∈ [k].
+This implies that y must be a zero vector as C has the R0-W property, which contradicts the fact
+that ||y|| = 1. Therefore SOL(C, d, q) is bounded.
+5
+
+3.1
+Degree of EHLCP
+Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n
+and d = (d1, d2, ...., dk−1) ∈ Λ(k−1)
+n,++ .
+We define a function
+F : Λ(k+1)
+n
+→ Λ(k+1)
+n
+as
+F(x) =
+
+
+C0x0 − �k
+i=1 Cixi
+x0 ∧ x1
+(d1 − x1) ∧ x2
+(d2 − x2) ∧ x3
+.
+.
+.
+(dk−1 − xk−1) ∧ xk
+
+
+.
+(5)
+We denote the degree of F with respect to 0 over bounded open set Ω ⊆ Λ(k+1)
+n
+as deg(C, Ω, 0).
+It is noted that if C has the R0-W property, in view of the Lemma 3.1, F(x) = 0 ⇔ x = 0 which
+implies that deg(C, Ω, 0) = deg(C, 0) for any bounded open set Ω contains the origin in Λ(k+1)
+n
+. We
+call this degree as EHLCP-degree of C.
+We now prove an existence result for EHLCP.
+Theorem 3.3. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n . Suppose the following hold:
+(i) C has the R0-W property.
+(ii) deg(C, 0) ̸= 0.
+Then EHLCP(C, d, q) has non-empty compact solution for all q ∈ Rn and d ∈ Λ(k−1)
+n,++ .
+Proof. As the solution set of EHLCP is closed, it is enough to prove that the solution set is non-empty
+and bounded. We first define a homotopy Φ : Λ(k+1)
+n
+× [0, 1] → Λ(k+1)
+n
+as
+Φ(x, s) =
+
+
+C0x0 − �k
+i=1 Cixi − sq
+x0 ∧ x1
+(d1 − x1) ∧ x2
+(d2 − x2) ∧ x3
+.
+.
+.
+(dk−1 − xk−1) ∧ xk
+
+
+.
+Then,
+Φ(x, 0) = F(x) and Φ(x, 1) = F(x) − ˆq, where ˆq = (q, 0, 0, ...0) ∈ Λ(k+1)
+n
+.
+By using the similar argument as in above Theorem 3.2, we can easily show that the zero set of
+homotopy, X = {x : Φ(x, s) = 0 for some s ∈ [0, 1]} is bounded. From the property of degree (D2),
+we get deg(F, Ω, 0) = deg(F − ˆq, Ω, 0) for any open bounded set Ω containing X. As deg(F, Ω, 0) =
+deg(C, 0) ̸= 0, we obtain deg(F − ˆq, Ω, 0) ̸= 0 which implies SOL(C, d, q) is non-empty. As C has
+the R0-W property, by Theorem 3, SOL(C, d, q) is bounded. This completes the proof.
+4
+SSM-W property
+In this section, we first define the SSM-W property for the set of matrices which is a generalization
+of the SSM matrix in the LCP theory, and we then prove that the existence and uniqueness result
+for the EHLCP when the involved set of matrices have the SSM-W property.
+We now recall that an n × n real matrix M is called strictly semimonotone (SSM) matrix if
+[x ∈ Rn
++, x ∗ Mx ≤ 0 ⇒ x = 0]. We generalize this concept to the set of matrices.
+6
+
+Definition 4. We say that C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n
+has the SSM-W property if
+{C0x0 =
+k
+�
+i=1
+Cixi, xi ≥ 0 and x0 ∗ xi ≤ 0 ∀i ∈ [k]} ⇒ x = (x0, x1, .., xk) = 0.
+We prove the following result.
+Proposition 4.1. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n . If C has the SSM-W property, then the
+followings hold:
+(i) C−1
+0
+exists and C−1
+0 Ci is a strict semimonotone matrix for all i ∈ [k].
+(ii) (I, C−1
+0 C1, ..., C−1
+0 Ck) has the SSM-W property.
+(iii) (P T C0P, P T C1P, ..., P T CkP) has the SSM-W property for any permutation matrix P of order
+n.
+Proof. (i): Suppose there exists a vector x0 ∈ Rn such that C0x0 = 0. Then we have
+C0x0 = C10 + C20 + ... + Ck0.
+This gives that x0 = 0 as C has the SSM-W property. Thus C0 is invertible.
+Now we prove the second part of (i).
+Without loss of generality, it is enough to prove that
+C−1
+0 C1 is a strictly semimonotone matrix. Suppose there exists a vector y ∈ Rn such that y ≥ 0
+and y ∗ (C−1
+0 C1)y ≤ 0. Let y0 := (C−1
+0 C1)y, y1 := y and yi := 0 for all 2 ≤ i ≤ k. Then we get
+C0y0 = C1y1 + C2y2 + ... + Ciyi + .. + Ckyk,
+yj ≥ 0 and y0 ∗ yj ≤ 0 ∀j ∈ [k].
+Since C has the SSM-W property, yj = 0 ∀j ∈ [k]. Thus C−1
+0 C1 is a strict semimonotone matrix.
+This completes the proof.
+(ii): It follows from the definition of the SSM-W property.
+(iii): Let x = (x0, x1, ..., xk) ∈ Λ(k+1)
+n
+such that
+(P T C0P)x0 =
+k
+�
+i=1
+(P T CiP)xi, xj ≥ 0 and x0 ∗ xj ≤ 0 ∀j ∈ [k].
+As P is a non-negative matrix and PP T = P T P, we can rewrite the above equation as
+C0Px0 =
+k
+�
+i=1
+CiPxi, Pxj ≥ 0 and Px0 ∗ Pxj ≤ 0 ∀j ∈ [k].
+By the SSM-W property of C, Pxj = 0 for all 0 ≤ j ≤ k which implies x = 0. This completes the
+proof.
+In the above Proposition 4.1, it can be seen easily that the converse of the item (ii) and (iii) are
+valid. But the converse of item (i) need not be true. The following example illustrates this.
+Example 4.2. Let C = (C0, C1, C2) ∈ Λ(3)
+2×2, where
+C0 =
+�
+1
+0
+0
+1
+�
+, C1 =
+�
+1
+−2
+0
+1
+�
+, C2 =
+�
+1
+0
+−2
+1
+�
+.
+It is easy to check that C−1
+0 C1 = C1 and C−1
+0 C2 = C2 are P matrix. So, C−1
+0 C1 and C−1
+0 C2 are SSM
+matrix. Let x = (x0, x1, x2) = ((0, 0)T , (1, 1)T , (1, 1)T ) ∈ Λ(3)
+2 . Then we can see that the non-zero x
+satisfies
+C0x0 = C1x1 + C2x2, x1 ≥ 0, x2 ≥ 0 and x0 ∗ x1 = 0 = x0 ∗ x2.
+So C can not have the SSM-W property.
+7
+
+The following result is a generalization of a well-known result in matrix theory that every P
+matrix is a SSM matrix.
+Theorem 4.3. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n . If C has the column W-property, then C has the
+SSM-W property.
+Proof. Suppose there exists a non-zero vector x = (x0, ..., xk) ∈ Λ(k+1)
+n
+such that
+C0x0 =
+k
+�
+i=1
+Cixi, xj ≥ 0, x0 ∗ xj ≤ 0 ∀j ∈ [k].
+Consider a vector y ∈ Rn whose jth component is given by
+yj =
+
+
+
+
+
+
+
+
+
+−1
+if (x0)j > 0
+1
+if (x0)j < 0
+1
+if (x0)j = 0 and (xi)j ̸= 0 for some i ∈ [k]
+0
+if (x0)j = 0 and (xi)j = 0 for all i ∈ [k]
+.
+As x is a non-zero vector, y must be a non-zero vector. Consider the diagonal matrices D0, D1, ..., Dk
+which are defined by
+(D0)jj =
+
+
+
+
+
+
+
+
+
+(x0)j
+if (x0)j > 0
+−(x0)j
+if (x0)j < 0
+0
+if(x0)j = 0 and (xi)j ̸= 0 for some i ∈ [k]
+1
+if (x0)j = 0 and (xi)j = 0 for all i ∈ [k]
+and for all i ∈ [k],
+(Di)jj =
+�
+0
+if (x0)j > 0
+(xi)j
+else
+.
+It is easy to verify that D0, D1, ..., Dk are non-negative diagonal matrices and diag(D0 + D1 + ... +
+Dk) > 0. And also note that
+x0 = −D0y and xi = Diy ∀i ∈ [k].
+(6)
+By substituting the Equation 6 in C0x0 = �k
+i=1 Cixi, we get
+C0(−D0y) =
+k
+�
+i=1
+CiDi(y) ⇒
+�
+C0D0 + C1D1 + ... + CkDk
+�
+y = 0.
+This implies that det(C0D0 + C1D1 + ... + CkDk
+�
+= 0. So, C does not have the column W-property
+from Theorem 2.2. Thus we get a contradiction. Therefore, C has the SSM-W property.
+The following example illustrates that the converse of the above theorem is invalid.
+Example 4.4. Let C = (C0, C1, C2) ∈ Λ(3)
+2×2 such that
+C0 =
+�1
+0
+0
+1
+�
+, C1 =
+�1
+1
+1
+1
+�
+, C2 =
+�1
+1
+1
+1
+�
+.
+Suppose w = (x, y, z) ∈ Λ3
+2 such that
+C0x = C1y + C2z and y, z ≥ 0, x ∗ y ≤ 0, x ∗ z ≤ 0.
+8
+
+From C0x = C1y + C2z, we get
+�x1
+x2
+�
+=
+�y1 + y2 + z1 + z2
+y1 + y2 + z1 + z2
+�
+.
+As x ∗ y ≤ 0, x ∗ z ≤ 0 and from the above equation, we have
+y1(y1 + y2 + z1 + z2) ≤ 0 and y2(y1 + y2 + z1 + z2) ≤ 0,
+z1(y1 + y2 + z1 + z2) ≤ 0 and z2(y1 + y2 + z1 + z2) ≤ 0.
+(7)
+Since y, z ≥ 0, from the equation 7, we get x = y = z = 0. Hence C has the SSM-W property. As
+det(C1) = 0, by the definition of the column W-property, C does not have the column W-property.
+We now give a characterization for SSM-W property.
+Theorem 4.5. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n has the SSM-W property if and only if (C0, C1D1+
+C2D2 + ... + CkDk) ∈ Λ(2)
+n×n has the SSM-W property for any set of non-negative diagonal matrix
+(D1, D2, ..., Dk) ∈ Λ(k)
+n×n with diag(D1 + D2 + ... + Dk) > 0.
+Proof. Necessary part: Let (D1, D2..., Dk) ∈ Λ(k)
+n×n be the set of non-negative diagonal matrix with
+diag(D1 + D2 + ... + Dk) > 0. Suppose there exist vectors x0 ∈ Rn and y ∈ Rn
++ such that
+C0x0 =
+�
+C1D1 + C2D2 + ... + CkDk
+�
+y and x0 ∗ y ≤ 0.
+For each i ∈ [k], we set xi := Diy. As each Di is a non-negative diagonal matrix, from x0 ∗ y ≤ 0,
+we get x0 ∗ xi ≤ 0 ∀i ∈ [k]. Then we have
+C0x0 = C1x1 + C2x2 + ... + Ckxk,
+xi ≥ 0, x0 ∗ xi ≤ 0 ∀i ∈ [k].
+As C has the SSM-W property of C, we must have x0 = x1 = ... = xk = 0. This implies
+x1 + x2 + ... + xk = (D1 + D2 + ... + Dk)y = 0.
+As diag(D1 + D2 + .... + Dk) > 0, we have
+y = 0. This completes the necessary part.
+Sufficiency part: Let x = (x0, x1, ..., xk) ∈ Λ(k+1)
+n
+such that
+C0x0 = C1x1 + C2x2 + ... + Ckxk and xj ≥ 0, x0 ∗ xj ≤ 0 ∀j ∈ [k].
+(8)
+We now consider an n × k matrix X whose jth column as xj for j ∈ [k]. So, X = [x1 x2 ... xk].
+Let S := {i ∈ [k] : ith row sum of X is zero}. From this, we define a vector y ∈ Rn and diagonal
+matrices D1, D2, .., Dk such that
+yi =
+�
+1
+i /∈ S
+0
+i ∈ S
+and (Dj)ii =
+�
+(xj)i
+i /∈ S
+1
+i ∈ S ,
+where (Dj)ii is the diagonal entry of Dj for all j ∈ [k]. It can be seen easily that Djy = xj for all
+j ∈ [k] and each Dj is a non-negative diagonal matrix with diag(D1 + D2 + ...+ Dk) > 0. Therefore,
+from equation 8, we get
+C0x0 =
+�
+C1D1 + C2D2 + ... + CkDk
+�
+y,
+x0 ∗ y ≤ 0.
+From the hypothesis, we get x0 = 0 = y which implies x = 0.
+This completes the sufficiency
+part.
+9
+
+We now give a characterization for the column W-property.
+Theorem 4.6. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n
+has the column-W property if and only if
+(C0, C1D1 + C2D2 + ... + CkDk) ∈ Λ(2)
+n×n has the column-W property for any set of non-negative
+diagonal matrices D1, D2, ..., Dk of order n with diag(D1 + D2 + ... + Dk) > 0.
+Proof. Necessary part: It is obvious.
+Sufficiency part: Let {E0, E1, ..., Ek} be a set of non-negative diagonal matrices of order n such that
+diag(E0 + E1 + ... + Ek) > 0. We claim that det(C0E0 + C1E1 + ... + CkEk) ̸= 0.
+To prove this, we first construct a set of non-negative diagonal matrices D1, D2, ..., Dk and E as
+follows:
+(Dj)ii =
+�
+Ej
+ii
+if �k
+m=1 Em
+ii ̸= 0
+1
+if �k
+m=1 Em
+ii = 0 and Eii =
+�
+1
+if �k
+m=1 Em
+ii ̸= 0
+0
+if �k
+m=1 Em
+ii = 0 ,
+where (Dj)ii is iith diagonal entry of Dj for j ∈ [k] and Eii is iith diagonal entry of matrix E.
+By an easy computation, we have DjE = Ej ∀j ∈ [k] and diag(D1 + D2 + ... + Dk) > 0. From
+diag(E0 + E1 + ... + Ek) > 0, we get diag(E0 + E) > 0. As DjE = Ej ∀j ∈ [k] and (C0, C1D1 +
+C2D2 + ... + CkDk) has column W-property, by Theorem 2.2, we have
+det(C0E0 + C1E1 + ... + CkEk) = det(C0E0 + C1D1E + ... + CkDkE)
+= det(C0E0 + (C1D1 + ... + CkDk)E) ̸= 0.
+Hence C has the column W-property. This completes the proof.
+A well-known result in the standard LCP is that strictly semimonotone matrix and P matrix are
+equivalent in the class of Z matrices (see, Theorem 3.11.10 in [1]). Analogue this result, we prove
+the following theorem.
+Theorem 4.7. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n
+such that C−1
+0 Ci be a Z matrix for all i ∈ [k].
+Then the following statements are equivalent.
+(i) C has the column W-property.
+(ii) C has the SSM-W property.
+Proof. (i) =⇒ (ii): It follows from Theorem 4.3.
+(ii) =⇒ (i): Let {D1, D2, ..., Dk} be the set of non-negative diagonal matrices of order n such
+that diag(D1 + D2 + ... + Dk) > 0. In view of Theorem 4.6, it is enough to prove that (C0, C1D1 +
+C2D2 + ... + CkDk) has the column W-property.
+As C has the SSM-W property, by Theorem 4.5, we have (C0, C1D1 + ... + CkDk) has the
+SSM-W property. So, by Proposition 4.1,
+�
+I, C−1
+0
+�
+C1D1 + ... + CkDk
+��
+has the SSM-W property
+and C−1
+0
+�
+C1D1 + C2D2 + ... + CkDk
+�
+is a strict semimonotone matrix. As C−1
+0 Ci is a Z matrix, we
+get C−1
+0
+�
+C1D1 + C2D2 + ... + CkDk
+�
+is also a Z matrix. Hence C−1
+0
+�
+C1D1 + C2D2 + ... + CkDk
+�
+is a P matrix. So, by Theorem 2.3, (C0, C1D1 + C2D2 + ... + CkDk) has the column W-property.
+Hence we have our claim.
+Corollary 4.8. Let C = (C0, C1, ..., Ck) ∈ Λk+1
+n×n such that C−1
+0 Ci be a Z matrix for all i ∈ [k].
+Then the following statements are equivalent.
+(i) C has the SSM-W property.
+(ii) For all q ∈ Rn and d ∈ Λ(k−1)
+n,++ , EHLCP(C, d, q) has a unique solution.
+Proof. (i)
+=⇒ (ii): It follows from Theorem 4.7 and Theorem 2.2. (ii) =⇒ (i): It follows from
+Theorem 2.2 and Theorem 4.3.
+10
+
+In the standard LCP [3], the strictly semimonotone matrix gives the existence of a solution of
+LCP. We now prove that the same result holds in EHLCP.
+Theorem 4.9. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n
+has the SSM-W property, then SOL(C, d, q) ̸= ∅
+for all q ∈ Rn and d ∈ Λ(k+1)
+n,++ .
+Proof. As C has the SSM-W property, C has the R0-W property. From Theorem 4.1, it is enough
+to prove that deg(C, 0) ̸= 0. To prove this, we consider a homotopy Φ : Λ(k+1)
+n
+× [0, 1] → Λ(k+1)
+n
+as
+Φ(x, t) = t
+
+
+C0x0
+x1
+x2
+x3
+.
+.
+.
+xk
+
+
++ (1 − t)
+
+
+C0x0 − �k
+i=1 Cixi
+x0 ∧ x1
+(d1 − x1) ∧ x2
+(d2 − x2) ∧ x3
+.
+.
+.
+(dk−1 − xk−1) ∧ xk
+
+
+.
+Let F(x) := Φ(x, 0) and G(x) := Φ(x, 1). We first prove that the zero set X = {x : Φ(x, t) =
+0 for some t ∈ [0, 1]} of homotopy Φ contains only zero. We consider the following cases.
+Case 1: Suppose t = 0 or t = 1. If t = 0, then Φ(x, 0) = 0 =⇒ F(x) = 0. As C has the SSM-W
+property, by Lemma 3.1, we have F(x) = 0 ⇒ x = 0. If t = 1, then Φ(x, 1) = 0 =⇒ G(x) = 0.
+Again by C has the SSM-W property, C−1
+0
+exists, which implies that G is a one-one map. So,
+G(x) = 0 ⇒ x = 0.
+Case 2: Suppose t ∈ (0, 1). Then Φ(x, t) = 0 which gives that
+
+
+C0x0 − �k
+i=1 Cixi
+x0 ∧ x1
+(d1 − x1) ∧ x2
+(d2 − x2) ∧ x3
+.
+.
+.
+(dk−1 − xk−1) ∧ xk
+
+
+= −α
+
+
+C0x0
+x1
+x2
+x3
+.
+.
+.
+xk
+
+
+,
+where α =
+t
+1 − t > 0.
+(9)
+From the second row of above equation, we have
+x0 ∧ x1 = −αx1 =⇒ min{x0 + αx1, (1 + α)x1} = 0.
+By Proposition 2.1, we get x1 ≥ 0 and (x0 + αx1) ∗ (1 + α)x1 = 0 which implies that x0 ∗ x1 ≤ 0.
+Set ∆ := {i ∈ [n] : (x1)i > 0}. So, we have
+(x0)i =
+�
+≤ 0
+if i ∈ ∆
+≥ 0
+if i /∈ ∆
+and (x1)i =
+�
+> 0 if i ∈ ∆
+= 0 if i /∈ ∆
+.
+(10)
+From third row of the equation 9, we have (d1 − x1) ∧ x2 = −αx2 which is equivalent
+min{d1 − x1 + αx2, (1 + α)x2} = 0.
+This gives that x2 ≥ 0 and (d1 − x1 + αx2) ∗ (1 + α)x2 = 0. As d1 > 0 and from the last term in
+equation 10, we have
+(x2)i =
+�
+≥ 0 if i ∈ ∆
+= 0 if i /∈ ∆
+.
+11
+
+This leads that x0 ∗ x2 ≤ 0. By continuing the similar argument for the remaining rows, we get
+xj ≥ 0 and x0 ∗ xj ≤ 0 ∀j ∈ [k].
+From the first row of the equation 9, the vectors x = (x0, x1, ..., xk) satisfies
+C0(1 + α)x0 =
+k
+�
+i=1
+Cixi and xj ≥ 0, x0 ∗ xj ≤ 0, j ∈ [k].
+So, x = 0 as C has the SSM-W property.
+From both cases, we get X contains only zero. By the homotopy invariance property of degree
+(D2), we have deg(Φ(x, 0), Ω, 0) = deg
+�
+Φ(x, 1), Ω, 0
+�
+for any bounded open set containing 0. As G
+is a continuous one-one function, by Proposition 2.4, we have
+deg
+�
+C, 0
+�
+= deg
+�
+Φ(x, 0), Ω, 0
+�
+= deg
+�
+F, Ω, 0
+�
+= deg
+�
+G, Ω, 0
+�
+̸= 0.
+This completes the proof.
+We now recall that a matrix A ∈ Rn×n is said to be a M matrix if it is Z matrix and A−1(Rn
++) ⊆
+Rn
++. We prove a uniqueness result for EHLCP when q ≥ 0 and d ∈ Λ(k−1)
+n,++ .
+Theorem 4.10. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n
+has the SSM-W property. If C0 is a M matrix.
+then for every q ∈ Rn
++ and for every d ∈ Λ(k−1)
+n,++ , EHLCP(C, d, q) has a unique solution.
+Proof. Let q ∈ Rn
++ and d = (d1, d2, ..., dk−1) ∈ Λ(k−1)
+n,++ . We first show (C−1
+0 q, 0, ..., 0) ∈ SOL(C, d, q).
+As C0 is a M matrix and q ∈ Rn
++, we have C−1
+0 q ≥ 0. If we set y = (y0, y1, ..., yk) := (C−1
+0 q, 0, ..., 0) ∈
+Λ(k+1)
+n
+, then we can see easily that (y0, y1, ..., yk) satisfies that
+C0y0 = q +
+k
+�
+i=1
+Ciyi, y0 ∧ y1 = 0 and (dj − yj) ∧ yj+1 = 0 ∀j ∈ [k − 1].
+Hence (C−1
+0 q, 0, ..., 0) ∈ SOL(C, d, q).
+Suppose x = (x0, x1, ..., xk) ∈ Λ(k+1)
+n
+is an another solution to EHLCP(C, q, d). Then,
+C0x0 = q +
+k
+�
+i=1
+Cixi, x0 ∧ x1 = 0, (dj − xj) ∧ xj+1 = 0 ∀j ∈ [k − 1].
+(11)
+From the Lemma 3.1, we have
+C0x0 = q +
+k
+�
+i=1
+Cixi and x0 ∧ xj = 0 ∀ j ∈ [k].
+(12)
+We let z := x − y, then z = (x0 − C−1
+0 q, x1, x2, .., xk). By an easy computation, from Equation 12,
+we get
+C0(x0 − C−1
+0 q) =
+k
+�
+i=1
+Cixi
+and
+xj ≥ 0,
+(x0 − C−1
+0 q) ∗ xj = x0 ∗ xj − C−1
+0 q ∗ xj = −C−1
+0 q ∗ xj ≤ 0 ∀j ∈ [k].
+Since C has the SSM-W property, z = 0 which implies that (x0, x1, ..., xk) = (C−1
+0 q, 0, ..., 0). This
+completes the proof.
+12
+
+5
+Connected solution set and Column W0 property
+In this section, we give a necessary and sufficient condition for the connected solution set of the
+EHLCP.
+Definition 5. Let C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n . We say that C is connected if SOL(C, d, q) is
+connected for all q ∈ Rn and for all d ∈ Λ(k−1)
+n,++ .
+We now recall some definitions and results to proceed further.
+Definition 6. [22] A subset of Rn is said to be a semi-algebraic set it can be represented as,
+S =
+s�
+u=1
+ru
+�
+v=1
+{x ∈ Rn; fu,v(x) ∗uv 0},
+where for all u ∈ [s] and for all v ∈ [ru], ∗uv ∈ { >, =} and fu,v is in the space of all real polynomials.
+Theorem 5.1 ([22]). Let S be a semi-algebraic set. Then S is connected iff S is path-connected.
+Lemma 5.2. The SOL(C, d, q) is a semi-algebraic set.
+Proof. It is clear from the definition of SOL(C, d, q).
+The following result gives a necessary condition for a connected solution whenever C0 is a M
+matrix.
+Theorem 5.3. Let C0 ∈ Rn×n be a M matrix. If C = (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n
+is connected, then
+SOL(C, d, q) = {(C−1
+0 q, 0, ..., 0)} for all q ∈ Rn
+++ and for all d ∈ Λ(k−1)
+n,++ .
+Proof. Let q ∈ Rn
+++ and d = (d1, d2, ..., dk−1) ∈ Λ(k−1)
+n,++ . It can be seen from the proof of The-
+orem 4.10 that x = (C−1
+0 q, 0, ..., 0) ∈ SOL(C, d, q). We now show that x is the only solution to
+EHLCP(C, d, q).
+Assume contrary. Suppose y is another solution to EHLCP(C, d, q). As SOL(C, d, q) is con-
+nected, by Lemma 5.2 and Theorem 5.1, it is path-connected. So, there exists a path γ = (γ0, γ1, ..., γk) :
+[0, 1] → SOL(C, d, q) such that
+γ(0) = x, γ(1) = y and γ(t) ̸= x ∀t > 0.
+Let {tm} ⊆ (0, 1) be a sequence such that tm → 0 as m → ∞. Then, by the continuity of γ,
+γ(tm) → γ(0) = x as m → ∞. Since
+�
+γ0(tm), γ1(tm), ...γk(tm)
+�
+∈ SOL(C, d, q),
+C0γ0(tm) = q +
+k
+�
+i=1
+Ciγi(tm),
+γ0(tm) ∧ γ1(tm) = 0 and
+�
+dj − γj(tm)
+�
+∧ γ(j+1)(tm) = 0 ∀j ∈ [k − 1].
+Now we claim that there exists a subsequence {tml} of {tm} such that
+�
+γj(tml)
+�
+i ̸= 0, for some j ∈ [k] and for some i ∈ [n].
+Suppose the claim is not true. This means that for given any subsequence {tml} of {tm}, there exists
+m0 ∈ N such that for all ml ≥ m0, we have
+�
+γj(tml)
+�
+i = 0 ∀i ∈ [n] ∀j ∈ [k].
+13
+
+So, γj(tm) is an eventually zero sequence for all j ∈ [k]. This implies that there exists a natural
+number m0 such that
+γ1(tm) = γ2(tm) = ... = γk(tm) = 0 ∀m ≥ m0.
+As
+�
+γ0(tm), γ1(tm), ...γk(tm)
+�
+∈ SOL(C, d, q), we get γ0(tm) = C−1
+0 (q)
+∀m ≥ m0. This gives us
+that γ(tm) = x for all m ≥ m0 which contradicts the fact that γ(tm) ̸= x for all m. Therefore, our
+claim is true. No loss of generality, we assume a sequence {tm} itself satisfies the condition
+�
+γj(tm)
+�
+i ̸= 0, for some j ∈ [k] and for some i ∈ [n].
+We now consider the following cases for possibilities of j.
+Case 1 : If j = 1, then (γ0(tm))i(γ1(tm))i = 0 which leads to (γ0(tm))i = 0. This implies that
+0 = lim
+m→∞ γ0(tm)i = (C−1
+0 q)i.
+But (C−1
+0 q) > 0 as C0 is a M matrix. This is not possible. So, j ̸= 1.
+Case 2 : If 2 ≤ j ≤ k, then we have (dj−1 − γj−1(tm))i(γj(tm))i = 0 which gives that (dj−1 −
+γj−1(tm))i = 0. By taking limit m → ∞,
+0 = lim
+m→∞(dj−1 − γj−1(tm))i = (dj−1)i − (γj−1(0))i = (dj−1)i > 0.
+This is not possible.
+From both cases, there is no such a j exists. This contradicts the fact. Hence x = (C−1
+0 q, 0, ..., 0)
+is the only solution to EHLCP(C, d, q).
+The following result gives a sufficient condition for a connected solution to EHLCP.
+Theorem 5.4. Let C := (C0, C1, ..., Ck) ∈ Λ(k+1)
+n×n
+has the column W0-property. If SOL(C, d, q) has
+a bounded connected component, then SOL(C, d, q) is connected.
+Proof. If SOL(C, d, q) = ∅, then we have nothing to prove. Let SOL(C, d, q) ̸= ∅ and A be a con-
+nected component of SOL(C, d, q). If SOL(C, d, q) = A, then we are done. Suppose SOL(C, d, q) ̸=
+A. Then there exists y = (y0, y1, .., yk) ∈ SOL(C, d, q)\ A. As A is a bounded connected component
+of SOL(C, d, q), we can find an open bounded set Ω ⊆ Λ(k+1)
+n
+which contains A and it does not
+intersect with other component of SOL(C, d, q). Therefore y /∈ �� and ∂(Ω) ∩ SOL(C, d, q) = ∅.
+Since C has the column W0-property, there exists N := (N0, N1, ..., Nk) ∈ Λ(k+1)
+n×n
+such that
+C + ǫN := (C0 + ǫN0, C1 + ǫN1, ..., Ck + ǫNk) has the column W-property for every ǫ > 0.
+Let z = (z0, z1, ..., zk) ∈ A and ǫ > 0, we define functions H1, H2 and H3 as follows:
+H1(x) =
+
+
+C0x0 − �k
+i=1 Cixi − q
+x0 ∧ x1
+(d1 − x1) ∧ x2
+.
+.
+.
+(dk−1 − xk−1) ∧ xk
+
+
+,
+H2(x) =
+
+
+(C0 + ǫN0)x0 − �k
+i=1(Ci + ǫNi)xi + (�k
+i=1 ǫNiyi − ǫN0y0 − q)
+x0 ∧ x1
+(d1 − x1) ∧ x2
+.
+.
+.
+(dk−1 − xk−1) ∧ xk
+
+
+,
+14
+
+H3(x) =
+
+
+(C0 + ǫN0)x0 − �k
+i=1(Ci + ǫNi)xi + (�k
+i=1 ǫNizi − ǫN0z0 − q)
+x0 ∧ x1
+(d1 − x1) ∧ x2
+.
+.
+.
+(dk−1 − xk−1) ∧ xk
+
+
+.
+By putting x = y in H2(x), and x = z in H1(x) and H3(x), we get
+H1(z) = H2(y) = H3(z) = 0.
+For ǫ is near to zero, deg(H1, Ω, 0)= deg(H2, Ω, 0)= deg(H3, Ω, 0) due to the nearness property
+of degree (D3). As z ∈ Ω is a solution to H3(x) = 0 and C + ǫN has the column W-property,
+we get deg(H3, Ω, 0) ̸= 0 by Theorem 4.3 and 4.9. Since deg(H2, Ω, 0)= deg(H3, Ω, 0), we have
+deg(H2, Ω, 0) ̸= 0. This implies that if we set q2 := q+ǫN0y0−�k
+i=1 ǫNiyi, then EHLCP(C + ǫN, d, q2)
+must have a solution in Ω. As C + ǫN has the column W-property, by Theorem 2.2, EHLCP(C + ǫN, d, q2)
+has a unique solution which must be equal to y. So, y ∈ Ω. It gives us a contradiction. Hence
+SOL(C, d, q) = A. Thus SOL(C, d, q) is connected.
+6
+Conclusion
+In this paper, we introduced the R0-W property and SSM-W properties and then studied the
+existence and uniqueness result for EHLCP when the underlying set of matrices has these properties.
+Last, we gave a necessary and sufficient condition for the connectedness of the solution set of the
+EHLCP.
+Declaration of Competing Interest
+The authors have no competing interests.
+Acknowledgements
+The first author is a CSIR-SRF fellow, and he wants to thank the Council of Scientific & Industrial
+Research(CSIR) for the financial support.
+References
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+

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+2019 22nd International Conference on Computer and Information Technology (ICCIT), 18-20 December 2019 
+978-1-7281-5842-6/19/$31.00 ©2019 IEEE 
+ 
+Prognosis and Treatment Prediction of Type-2 
+Diabetes Using Deep Neural Network and Machine 
+Learning Classifiers  
+Md. Kowsher 
+Dept. of  Applied Mathematics 
+Noakhali Science and Technology 
+University, Noakhali-3814,Bangladesh. 
+ga.kowsher@gmail.com 
+ 
+Mahbuba Yesmin Turaba 
+Dept. of Information and 
+Communication Technology 
+Comilla University 
+Comilla, Bangladesh 
+mahbuba.yesmin11@gmail.com 
+ 
+M M Mahabubur Rahman 
+Dept. of CSTE 
+Noakhali Science and Technology 
+University Noakhali-3814, Bangladesh 
+toufikrahman098@gmail.com 
+ Tanvir Sajed 
+Dept. of Computing Science 
+University of Alberta 
+Edmonton, Canada 
+tsajed@ualbarta.ca 
+ 
+ 
+    Abstract—Type 2 Diabetes is a fast-growing, chronic 
+metabolic disorder due to imbalanced insulin activity. As lots 
+of people are suffering from it, access to proper treatment is 
+necessary to control the problem. Most patients are unaware of 
+health complexity, symptoms and risk factors before diabetes. 
+The motion of this research is a comparative study of seven 
+machine learning classifiers and an artificial neural network 
+method to prognosticate the detection and treatment of 
+diabetes with a high accuracy, in order to identify and treat 
+diabetes patients at an early age. Our training and test dataset 
+is an accumulation of 9483 diabetes patients’ information. The 
+training dataset is large enough to negate overfitting and 
+provide for highly accurate test performance. We use 
+performance measures such as accuracy and precision to find 
+out the best algorithm deep ANN which outperforms with 
+95.14% accuracy among all other tested machine learning 
+classifiers. We hope our high performing model can be used by 
+hospitals to predict diabetes and drive research into more 
+accurate prediction models. 
+Keywords—Artificial Neural Network, Type 2 diabetes, Support 
+Vector Machine, Decision Tree, Naive Bayes, LDA, Random 
+forest classifier 
+I. 
+INTRODUCTION 
+Diabetes Mellitus (DM) is a very common metabolic 
+disorder that affects millions of people worldwide. It occurs 
+when the concentration of blood glucose reaches excessive 
+level due to lack of production of insulin by the pancreas 
+organ (Type 1 Diabetes) or due to insulin resistance (Type 2 
+Diabetes) [1]. It has been published that 422 million people 
+are suffering from diabetes approximately in 2014 and it is 
+expected to rise to 438 million in 2030[2, 3]. Among them, 
+90% of cases are Type 2 diabetes (T2DM) [4]. It may arise 
+at an early childhood because of the failure of cells to 
+respond to insulin appropriately [5]. So, patients have to 
+face excessive tiredness, visual disorders, excessive thirst, 
+skin infection recurrence, delayed wound healing and 
+frequent discharge of urine [6]. It has been pointed out by 
+Diabetes Research Center that 80 percent of cases of 
+diabetes can be prevented or delayed if it is detected early 
+[7]. Also, by controlling blood sugar, it is possible to lessen 
+the T2DM effect. A healthy diet, physical exercise, 
+sufficient nutrition for pregnant women, proper medication, 
+weight at a necessary level are crucial to maintaining a safer 
+sugar level. 
+When the diabetes is diagnosed with medical tests, it 
+shows significantly dangerous symptoms but these methods 
+do not perform well because of clinical complexity, time-
+consuming process and very high expense. However, using 
+automated machine learning algorithms, a researcher can 
+predict a disease like diabetes with reduced cost and time. In 
+the field of Artificial Intelligence, classification is 
+considered a supervised technique that analyses patient data 
+and classifies whether or not the patient is suffering from a 
+disease. Researchers have created different AI and machine 
+learning techniques to automate prognosis of various 
+diseases. Machine learning techniques studies algorithm and 
+statistical model that has the capability for accurate 
+prediction by using implicit programming. In medical 
+science, they take the concept of the human brain as it 
+contains millions of neurons to complete tasks of the human 
+body. It is called nonlinear modelling and they are 
+interconnected like brain cells although the neuron creation 
+is done by program [8]. 
+In this paper, first we have discussed various procedures 
+and existing works about the prognosis of T2D , though we 
+emphasized various classification algorithms known as 
+Logistic Regression, KNN, Decision Tree, Naive Bayes, 
+SVM, Linear Discriminant Analysis and Random forest 
+classifier and Artificial Neural Network (ANN) for T2DM 
+prediction. Our selected model is an Artificial Neural 
+Network is found to be superior among all of them. 
+Feedforward neural network contains the signal in one 
+Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.  Restrictions apply. 
+
+direction from the input to the output. It is used in different 
+medical diagnostic applications such as nephritis disease, 
+heart disease, myeloid leukemia etc. [ref]. 
+We have taken a medical dataset from Noakhali Medical 
+College, Bangladesh, consisting of 9483 samples and 14 
+symptoms per sample. The 80% data and 20% data are 
+chosen to be training dataset and testing dataset 
+respectively. Machine learning classification algorithms are 
+applied to dataset and some elements may be missed. Then, 
+the mean and median method is applied in order to detect it. 
+The contributions of this paper are summarized as 
+ 
+● 
+We have proposed a prediction model for T2D 
+using Artificial Neural Network machine learning 
+classifier 
+● 
+We have exerted seven classifier techniques and 
+ANN on T2D data and provided comparison of 
+accuracy among them. 
+● 
+The improvement systems of the model, as well as 
+accuracy, are mentioned in this work. 
+ 
+The remaining of the discussion is organized as follows: 
+Section-II explains related work of various classification 
+techniques for prediction of diabetes, Section-III describes 
+the methodology and materials used, Section-IV discusses 
+evaluated Results and Section-V delineates the conclusion 
+of the research work. 
+ 
+II. 
+RELATED WORK 
+ 
+In recent years, several studies have been published using 
+multiple machine learning classifiers, ANN techniques and 
+various feature extraction methods. These have a drastic 
+change in potential research and some works are discussed 
+related to T2DM. Ebenezer et al. used the backpropagation 
+feature of ANN in order to diagnose diabetes. It finds out 
+the error by juxtaposing input and output number. Here, the 
+preceding round error is greater than the present error each 
+time by means of changing weight to minimize gradient of 
+errors using a technique known as gradient descent [9]. 
+Nongyao et al. delineated risk prediction by using various 
+machine learning classification algorithms such as Decision 
+Tree, Neural Network, Random Forest algorithms, Naïve 
+Bayes, Logistic Regression. All of them followed Bagging 
+and Boosting approaches to improve robustness except RFA 
+[10].  Deepti et al. proposed a model to identify diabetes at a 
+premature age by applying Decision Tree, SVM and Naïve 
+Bayes on Pima Indians Diabetes Database (PIDD) datasets. 
+They chose sufficient measures for accuracy including 
+precision, ROC, F measure, Recall but Naïve Bayes beat 
+them by acquiring the highest accuracy [11]. Su et al. 
+applied decision tree, logistic regression, neural network,and 
+rough sets to assess accuracy through various features like 
+age, right thigh circumference, left thigh circumference, 
+trunk volume and illustrates thigh circumference as a better 
+feature than BMI in anthropometrical data [12]. Al-Rubeaan 
+et al. has presented T2DM based on diabetic nephropathy 
+(DP), then defined high impact risk factors; age and diabetes 
+duration for microalbuminuria, macroalbuminuria and end-
+stage renal disease(ESRD) classifications[13]. Vijayan V. 
+examines various types of preprocessing techniques which 
+includes PCA and discretization. It increases the accuracy of 
+Naïve Bayes classifier and Decision Tree algorithm but 
+reduces SVM accuracy [14]. 
+Micheal et al. proposed Multi-Layer Feed Forward 
+Neural Networks (MLFNN) in order to diagnose diabetes by 
+considering activation units, learning techniques on Pima 
+Indian Diabetes (PID) data set and achieved 82.5% 
+accuracy. It performs better than Naïve Bayes, Logistic 
+Regression (LR) and Random Forest (RF) classifier [15].  
+Sadri et al. chose data mining algorithms like Naive Bayes, 
+RBF Network, and J48 to diagnose T2DM for Pima Indians 
+Diabetes Dataset that has 768 samples. Each sample has 
+nine features as the total number of Pregnancy, Plasma 
+Glucose Concentration, Diastolic Blood Pressure and 2-
+Hour Serum Insulin. Among them, the Naive Bayes 
+algorithm is unbeatable and has 76.95% accuracy [16]. 
+Pradhan et al. devised a classifier for diabetes detection 
+using Genetic programming (GP) at low cost. Simplified 
+function pool consists of arithmetic operations that are used 
+in lower validation [17]. Yang Guo et al. applied Naïve 
+Bayes classifier by using WEKA tool in order to predict 
+Type2 diabetes and obtained remarkable accuracy [18]. 
+ 
+Unlike these works, we have introduced diabetes‟s 
+medication detection system using machining learning and 
+deep ANN that will act like a doctor to choose the right 
+medication of a patient suffering from diabetes. 
+ 
+III.MATERIALS AND METHODS 
+ 
+In order to categorize diabetes therapy and drugs system for 
+patients, the whole workflow is separated into four parts 
+such as data collection, data preprocessing, training data via 
+the proposed algorithms, and predictions.  We have exerted 
+seven machine learning classifiers and deep neural networks 
+into the pre-processed data set. 
+ 
+ 
+Fig.1. System Diagram of T2D analysis 
+Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.  Restrictions apply. 
+
+Data
+Processing Data
+Training Model
+T2Dpatients
+-ML Classifiers:Logistic
+80% Training
+1.Missing Value Check
+Regression, KNN
+2.Handling Categorical
+Decision Tree, Naive
+Bayes, SVM, Linear
+20%Testingset
+3.Value
+4.Features selection
+Discriminant Analysis
+5.Feature Scaling
+and Random Forest
+Classifier
+6.Dimension Reduction
+·DeepANN
+D=Diet & Lifestyle
+I=Insulin
+10 Fold Cross
+M=Bigreanides
+Validation
+S=Secretagogues
+PredictionThe source of our data came from Noakhali Medical 
+College, Bangladesh and the data set is separated into two 
+parts such as training and test set. The training data are 
+manipulated to the diagnostic system and 13 factors have 
+been taken to determine therapy in order to apply machine 
+learning and multilayer ANN. The dataset is tested from the 
+trained machine learning classifiers and artificial neural 
+network. 
+A. Dataset 
+As discussed before our data set contains information about 
+9483 diabetes patients and formatted in comma-separated-
+file (CSV). The dimension of the data set is 9483*14. It 
+preserves 14 different kind of information of a  diabetes 
+patient such as „Name of patient‟,  “Fasting”, “2 h after 
+Pressure” “BMI”, “Duration”, “Age”, “Sex”, “Blood 
+pressure”, “High Cholesterols”, “Heart Diseases”, “Kidney 
+Diseases”, and , “Medications”. The first 13 columns are 
+considered independent variables and the last one is the 
+dependent variable.  It contains kinds of basic medicine 
+name of diabetes such as Diet and Lifestyle Modification, 
+Secretagogues, Biguanides, and Insulin. 
+When datasets consist of enough variables, it increases the 
+accuracy of prediction. Here, “Fasting” measures blood test 
+just before taking food, “2 h after glucose load” provides a 
+blood test after two hours of eating. “BMI” refers to the 
+weight and height of patients in kg/m2. “Medication” 
+indicates proper drugs and therapy. People who recovered 
+T2DM at early stage follow some features: age group 30-75 
+years, diabetes of diagnosis duration is more than half years, 
+glucose level at fasting plasma is higher than 125 mg/dl, 
+creation of plasma indicates equal or greater than 1.7 mg/dl, 
+plasma glucose after two hours is 11.  
+When a patient suffers from kidney problems, it may be a 
+symptom of T2DM as higher sugar level may damage 
+nephron. Even bleary eyesight is considered as a side effect 
+for patients as eye‟s retina and the macula is affected.  Bad 
+cholesterol may lead to Diabetic dyslipidemia which can 
+increase heart diseases and atherosclerosis. Here, we suggest 
+treatment for kidney and v---ision problems. In order to 
+categorize diabetes therapy and drugs system for patients, 
+we applied seven machine learning classifiers and eight 
+deep neural into a data system of Noakhali Medical College, 
+Bangladesh. Training data are manipulated to diagnostic 
+system and twelve factors have been taken to determine 
+therapy in order to apply machine learning and multilayer 
+ANN. For the training and testing of the systems, we 
+divided the data set into 80% training and 20% test set. The 
+training dataset is used to find out the appropriate model and 
+best hyper-parameters and testing data set contains unseen 
+data to predict the performance. 
+ 
+B. Data preprocessing 
+Data preprocessing involves raw data converting into a 
+recognizable format from various sources. The well-
+preprocessed data aids for the best training of algorithms. 
+Multi pre-processing training is held in our presented 
+systems.  
+1.  Missing Value Check 
+Usually, 
+missing 
+values 
+may 
+occur 
+due 
+to 
+data 
+incompleteness, missing field, programming error, manual 
+data transfer from a database and so on. We may ignore 
+missing values but it causes problems in parameter 
+calculation and data accuracy for features such as age, 
+wages and fare. We need to inspect whether a dataset has 
+any missing value or not. There are many ways to handle 
+missing values such as delete rows, missing values 
+prediction, mean, median, mode and so on. But the most 
+prominent policy for missing value replacement is the mean 
+method and also it is used to exchange the approximate 
+results in the dataset [19].  Mean is written in this way in 
+mathematics, 
+                                  (1) 
+ 
+Where, denotes the mean and provides the average number 
+of n. 
+ 
+2. Handling Categorical value 
+Categorical encoding identifies data type and transfers 
+categorical features into numerical numbers as the majority 
+of machine learning algorithms could not cope up with label 
+data directly.  Then numerical values are fed into the 
+specific model. In our data set, there are five categorical 
+variable names as „Name of patients‟, “Heart Diseases”, 
+“Kidney Diseases”, “Sex”, and “Medications”. There are 
+two popular ways of transforming categorical data into 
+numerical data such as Integer encoding and one-hot 
+encoding.  In the label encoder, categorical features are an 
+integer value and contain a natural order relationship, but 
+the multiclass relationship will provide different values for 
+various classes. One hot encoding maps categorical value 
+into binary vectors. Firstly, it is obvious to assign binary 
+value to an integer value of female and male is 0 and 1. 
+Then converting it to a 2 size of 2 possible integers in a 
+binary vector. Here, a female is encoded as 0 and 
+represented as [1, 0] in which index 0 has value 1 and vice 
+versa. It chooses this value as a feature to influence model 
+training [20]. 
+ 
+3. Features Selection  
+Feature selection incorporates the identification and 
+reduction of unnecessary features that have no impact on the 
+objective function and high impact features are kept. Our 
+dataset contains 14 types of elements and we have checked 
+p-value which is a statistical process for finding out the 
+probability for the null hypothesis. The features are taken 
+out whose p-value indicates less than 0.05.  
+Moreover, multicollinearity refers to determine the high 
+correlation which exists between two or more independent 
+features and features that are influential to each other. It is 
+called redundancy when two features are highly correlated. 
+As we have to handle redundancy, it is essential to choose 
+some methods such as χ2Test and Correlation Coefficient. 
+Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.  Restrictions apply. 
+
+x
+KThe Correlation Coefficient can be calculated by numerical 
+data. Assume that A and B are two features and it can be 
+defined as, 
+ 
+          
+  
+ 
+(2)                                                                                              
+ 
+After performing both p-value and multicollinearity test, we 
+could come forward with seven features among thirteen 
+independent features. Those are “Fasting”, “2 Hours after 
+Glucose Load”  “Duration”, “BMI”, “High Cholesterols”, 
+“Heart Diseases”, and “Kidney Diseases”. 
+ 
+ 4. Feature scaling  
+Most of the time, the dataset does not remain on the same 
+scale or even not normalized. So, feature scaling is a 
+fundamental data transformation method for coping the 
+dataset to algorithms. We need to scale value of features and 
+provide equal weight to all features in order to obtain the 
+same scale for all data. Moreover, it is possible for scaling 
+to change in different values for different features. There are 
+lots of techniques for feature scaling for example 
+Standardization, Mean Normalization, Min-Max Scaling, 
+Unit Vector and so on. 
+In our research work, we have taken Min-Max Scaling or 
+normalization process as the features are confined within a 
+bounded area. Minmax normalization is a z-series 
+normalization to transform linearly x to x‟ where maxX and 
+minX are the maximum and minimum value for X 
+respectively.  
+ 
+                          (3) 
+ 
+When x=max, then y =1 and x=min, y=1. 
+ 
+The scaling range belongs between 0 and 1(positive value) 
+and -1 to 1(negative) and we have taken the value between 0 
+and 1. 
+ 
+5. Dimension Reducing 
+Dimensionality reduction refers to minimizing random 
+variables by considering the principal set of variables that 
+avoids overfitting. For a large number of dataset, we need to 
+use dimension reduction technique. In our study, we prefer 
+dimension reduction for dimensional graphical visualization. 
+There are a lot of methods for reducing dimension, for 
+instance, LDA, PCA, SVD, NMF, etc. In our system, we 
+have applied Principal Component Analysis (PCA). It is a 
+linear transformation based on the correlation between 
+features in order to identify patterns. High dimensional data 
+are estimated into equal or lower dimensions through 
+maximum variance. We have taken two components of PCA 
+according to their high variance so that we can graphically 
+visualize in Cartesian coordinate system.  
+ 
+C. Training Algorithms 
+The training dataset for T2DM is applied to each algorithm 
+to find out medications and model performance is assessed 
+by obtaining accuracy. 
+ 
+a. Machine Learning Classifier 
+Since we focus on the performance of treatment predictions, 
+we have implemented seven machine learning classifiers 
+such as logistic regression, KNN, SVM, Naive Bayes, 
+decision tree, LDA, random forest tree.   
+Logistic regression is based on the probability model; it is 
+derived from linear regression that mapped the dataset into 
+two categories by considering existing data. At first, features 
+are mapped linearly that are transferred to a sigmoid 
+function layer for prediction. It shows the relationship 
+between the dependent and independent values but output 
+limits the prediction range on [0, 1]. As we need to predict 
+the right treatment of a diabetes person, it is beneficial to 
+use a binary classification problem. 
+Linear Discriminant Analysis (LDA) belongs to a linear 
+classifier to find out the linear correlation between elements 
+in order to support binary and multiclass classification. The 
+chance of inserting a new dataset into every class is detected 
+by LDA. Then, the class that contains the dataset is detected 
+as output. It can calculate the mean function for each class 
+and it is estimated by vectors for finding group variance. 
+Support Vector Machine (SVM) is the most recognized 
+classifier to make decision boundary as hyperplane to keep 
+the widest distance from both sides of points. This 
+hyperplane refers to separating data into two groups in two-
+dimensional space. It performs better with non-linear 
+classification by the kernel function. It is capable of 
+separating and classifying unsupported data.  
+K-nearest neighbours (KNN) works instant learning 
+algorithm and input labeled data that act as training instance. 
+Then, the output produces a group of data. When k=1, 2, 5 
+then it means the class has 1, 2 or 5 neighbours of every data 
+point. For this system, we choose k=5 that means 5 
+neighbours for every data point. We have taken Minkowski 
+distance to provide distance between two points in N-
+dimensional vector space to run data. Suppose, points p1(x1, 
+y1) and p2(x2, y2) illustrates Minkowski distance as, 
+                   
+                  (4) 
+Here, d denotes Minkowski distance between p1 and p2 
+point.  
+ 
+Naive Bayes Classifier is constructed from Bayes theorem, 
+in which features are independent of each other in present 
+class and classification that counts the total number of 
+observations by calculating the probability to create a 
+predictive model in the fastest time. It outperformed with a 
+huge dataset of categorical variables. The main benefits of 
+that it involves limited training data to estimate better 
+results. Naive Bayes theorem probability can be derived 
+from P (T), P(X) and P (X|T). Therefore, 
+ 
+                                    (5) 
+Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.  Restrictions apply. 
+
+at A-(b:.
+17
+noaobmmax(x).
+mmtm(x71
+dp : (x,y) → Ilx -yllp
+Ix - yilP
+i=1The decision tree is a decision-supporting a predictive 
+model based on tree structure by putting logic to interpret 
+features. It provides a conditional control system and marks 
+red or green for died or alive leaves. It has three types of 
+nodes: root node, decision nodes and leaf nodes. The root 
+node is the topmost node among them and data are split into 
+choices to find out the decision‟s result. Decision nodes 
+basically comprise of decision rules to produce the output 
+by considering all information gain and oval shape is used to 
+denote it. The terminal node represents the action that needs 
+to be taken after getting the outcome of all decisions. 
+ 
+Multiple random trees lead to the random forest to calculate 
+elements of molecular structure. A decision tree looks like a 
+tree that is the storehouse of results from the random forest 
+algorithm and bagging is applied to it in order to reduce 
+bias-variance trade-off. It can perform feature selection 
+directly and output represents the mode of all classes. In 
+Random Forest Tree, we took the total number of trees in 
+the forest: 10. 
+ 
+b. Artificial Neural Network 
+An ANN is considered as a human brain due to consisting 
+millions of neurons to communicate with each other. It has 
+three layers; the input layer fed raw data to network, hidden 
+layer is the middle layer based on input, weight and the 
+relationship denoted by activity function. Output layers 
+value is determined by activity, weight and relationship 
+from the second layer. 
+Since we need to find out the probability of each treatment 
+and the objective function is not binary, so we used softmax 
+activation function instead of sigmoid between the hidden 
+layer and output layer. There is no rule of thumb to choose 
+hidden layer in ANN. If our data is linearly separable then 
+we don‟t need any hidden layer. Then the average node 
+between the input and output node is preferable.  
+In our system, we prefer six hidden layers between the input 
+node and the hidden layer and 25 epochs to train a neural 
+network. It has no gradient vanishing problem and uses 
+ReLU activation function to train dataset without 
+pretraining. 
+ 
+c. Validation 
+ 
+The validation is a technique of evaluating the performance 
+of algorithms. It cooperates to evaluate the model and 
+reduce overfitting. Different types of validation method 
+includes 
+Holdout 
+method, 
+K-Fold 
+Cross-Validation, 
+Stratified K-Fold Cross-Validation and Leave-P-Out Cross-
+Validation. We have picked out k-fold validation dataset is 
+divided into k subsets in k times. One k subset act as test set 
+and error is estimated by average k trails. Therefore, k-1 
+subsets produce training set. We prefer k=10 generally 
+which contains 10 folds, repeat one time and stratified 
+sampling as each fold has a similar amount of samples. 
+ 
+IV.EXPERIMENTAL RESULT ANALYSIS 
+ 
+A.  Experimental tool 
+The whole task has been implemented in python 3.6 
+programming language in Anaconda distribution. Python 
+library offers various facilities to implement machine 
+learning and deep learning. The unbeatable library for data 
+representation is pandas that provide huge commands and 
+large data management.  We have used it to read and 
+analyze data in less writing. Afterward, scikit-learn has 
+features for various classification, clustering algorithms to 
+build models. Also, Keras combines the advantages of 
+theano and TensorFlow to train a neural network model. We 
+use to fit and evaluate function to train and assess neural 
+network model respectively bypassing the same input and 
+output, then we apply matplotlib for graphical visualization. 
+B. Model performance  
+For boosting performance, it is always a better idea to 
+increase data size instead of depending on prediction and 
+weak correlations. Also, adding a hidden layer may increase 
+accuracy and speed due to its tendency to make a training 
+dataset overfit. But partially it is dependent on the 
+complexity of the model. Contrarily, increasing the epochs 
+number ameliorate performance though it sometimes 
+overfits training data. It works well for the deep network 
+than shallow network when considering regulation factor. 
+Hereafter, we have added another hidden layer; choose 
+epoch 100 then the Deep ANN accuracy risen up to 95.14% 
+which is superior among all of them. 
+ 
+ 
+Fig.2. Models Performance Comparison. 
+C. Improving Model performance  
+For boosting performance, it is always a better idea to 
+increase data size instead of depending on prediction and 
+weak correlations. Also, adding a hidden layer may increase 
+training accuracy and speed due to its tendency to make 
+training dataset overfit. But partially it is dependent on the 
+complexity of the model. Contrarily, increasing the epochs 
+number ameliorate performance though it sometimes 
+overfits training data. It works well for the deep network 
+than shallow network when considering regulation factor. 
+Hereafter, we added another hidden layer; choose epoch 100 
+then the Deep ANN accuracy of the training and test set is 
+risen up to 96.42% and 95.14% which is superior among all 
+of them.  
+Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.  Restrictions apply. 
+
+ 
+Fig.3. 2-D Graphical Visualization of Test set 
+ 
+D.Final Result 
+After applying feature extraction to the dataset and 
+implementing several types of classification and deep neural 
+network, we found artificial neural network as better 
+performer with best validity and Random forest classifier 
+are preferable among other machine learning classifiers. 
+ 
+ 
+Fig.4. Final Result Comparison 
+ 
+V. CONCLUSIONS  
+Type-2 diabetes can lead to a lot of complications as heart 
+attack, kidney damage, blurred vision, hearing problems and 
+Alzheimer‟s disease. The main problem is lower accuracy of 
+the prediction model, small datasets and inadaptability to 
+various datasets.  In this paper, the medication and treatment 
+are predicted by a comparative study of seven machine 
+learning algorithms and deep neural networks. Artificial 
+neural networks play a vital role in medical science by 
+minimizing classification error that leads to greater 
+accuracy.  Experiment result determines the designed ANN 
+system achieved higher accuracy of 94.7%. It can cooperate 
+with experts to detect T2DM patients at a very early age and 
+provide the best treatment option. 
+In the future, we can enhance the accuracy of early 
+treatment to lessen the suffering of patients. Also, we can 
+implement more classifiers to pick up the leading one for 
+record-breaking performance and extend it to automation 
+analysis. There is a plan to apply this designed system in 
+diabetes or for other diseases. It may increase the 
+performance of prediction of various diseases. Larger 
+dataset leads to the higher training set and it cooperates in 
+advanced accuracy. It is convenient for people to have an 
+application on their smartphones related to T2DM that may 
+have T2DM symptoms, treatment, risk factors, and health 
+management. 
+REFERENCES 
+ 
+[1] 
+https://www.niddk.nih.gov/health-
+information/diabetes/overview/what-is-
+diabetes?fbclid=IwAR36jKI7GXUE4D0PhZ1Wk4zAa49kKXtn3hB7
+OqrYSoAqA925MzkXa_1u_Sk [Accessed: 24 June, 2019] 
+[2] 
+https://www.who.int/health-topics/diabetes [Accessed: 24 June, 2019]  
+[3] 
+Rawal LB, Tapp RJ, Williams ED, Chan C, Yasin S, Oldenburg B. 
+Prevention of type 2 diabetes and its complications in developing 
+countries: a review. Int J Behav Med. 2012; 19:121–133.  
+[4] 
+https://www.diabetes.org.uk/diabetes-the-basics/what-is-type-2-
+diabetes [Accessed: 24 June, 2019]  
+[5] 
+https://en.m.wikipedia.org/wiki/Diabetes?fbclid=IwAR3c20p4V8Np
+MvAwkTZmEK-rXxnBCZ61jhV87-ZnfPMNUJDpm9Easq9dDzA 
+[Accessed: 24 June, 2019]  
+[6] 
+https://idf.org/52-about-diabetes.html [Accessed: 24 June, 2019]  
+[7] 
+E. I. Mohamed, R. Linde, G. Perriello, N. Di Daniele, S. J. Pöppl and 
+A. De Lorenzo. "Predicting type 2 diabetes using an electronic nose-
+based artificial neural network analysis," in Diabetes nutrition & 
+metabolism Vol.15, No.4, (2002). pp. 222-215.  
+[8] 
+R. A. Dunne, Wiley, J., Inc, S. "A Statistical Approach to Neural 
+Networks for Pattern Recognition", New Jersey: John Wiley & Sons 
+Inc; (2007).  
+[9] 
+Ebenezer Obaloluwa Olaniyi and Khashman Adnan..“Onset diabetes 
+diagnosis using artificial neural network”, International Journal of 
+Scientific and Engineering research 5.10 (2014).  
+[10] Nai-Arun, N., Moungmai, R. “Comparison of Classifiers for the Risk 
+of Diabetes Prediction”, Procedia Computer Science vol: 69, pp: 132–
+142, 2015.  
+[11] Deepti Sisodiaa, Dilip Singh Sisodia. “Prediction of Diabetes using 
+Classification 
+Algorithms” 
+International 
+Conference 
+on 
+Computational Intelligence and Data Science, 2018  
+[12] Kowsher, M., Tithi, F. S., Rabeya, T., Afrin, F., & Huda, M. N. 
+(2020). Type 2 Diabetics Treatment and Medication Detection with 
+Machine 
+Learning 
+Classifier 
+Algorithm. In 
+Proceedings 
+of 
+International Joint Conference on Computational Intelligence (pp. 
+519-531). Springer, Singapore.  
+[13] https://www.ncbi.nlm.nih.gov/pubmed/24586457 [Accessed: 24 June, 
+2019]  
+[14] Veena Vijayan V. and Anjali C. Decision support systems for 
+predicting diabetes mellitus –a review. Proceedings of 2015 global 
+conference on communication technologies (GCCT 2015).  
+[15] https://www.researchgate.net/publication/331352518_A_Multi-
+layer_Feed_Forward_Neural_Network_Approach_for_Diagnosing_D
+iabetes  
+[16] https://pdfs.semanticscholar.org/ab93/6e4630720cb7f7ead833222b94
+5dc3801438.pdf  
+[17] Pradhan, M.A., Rahman, A., Acharya, P., Gawade, R., Pateria, A. 
+Design of classifier for Detection of Diabetes using Genetic 
+Programming. In: International Conference on Computer Science and 
+Information Technology, Pattaya, Thailand, pp. 125–130 (2011).  
+[18] Yang Guo, Karlskrona, S Guohua Bai and Yan Hu. Using Bayes 
+Network for Prediction of Type-2 diabetes, IEEE: International 
+Conference on Internet Technology And Secured Transactions, pp: 
+471 - 472, Dec. 2012.  
+[19] https://www.analyticsindiamag.com/5-ways-handle-missing-values-
+machine-learning-datasets/  [Accessed: 24 June, 2019]  
+[20] https://medium.com/@contactsunny/label-encoder-vs-one-hot-
+encoder- [Accessed: 5 August, 2019]  
+ 
+Authorized licensed use limited to: Newcastle University. Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.  Restrictions apply. 
+
+ANN (Test set)
+CI

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+page_content='00 ©2019 IEEE  Prognosis and Treatment Prediction of Type-2  Diabetes Using Deep Neural Network and Machine  Learning Classifiers  Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Kowsher  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' of  Applied Mathematics  Noakhali Science and Technology  University, Noakhali-3814,Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='kowsher@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='com  Mahbuba Yesmin Turaba  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' of Information and  Communication Technology  Comilla University  Comilla, Bangladesh  mahbuba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='yesmin11@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='com  M M Mahabubur Rahman  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' of CSTE  Noakhali Science and Technology  University Noakhali-3814, Bangladesh  toufikrahman098@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='com  Tanvir Sajed  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' of Computing Science  University of Alberta  Edmonton, Canada  tsajed@ualbarta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='ca  Abstract—Type 2 Diabetes is a fast-growing, chronic  metabolic disorder due to imbalanced insulin activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' As lots  of people are suffering from it, access to proper treatment is  necessary to control the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Most patients are unaware of  health complexity, symptoms and risk factors before diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  The motion of this research is a comparative study of seven  machine learning classifiers and an artificial neural network  method to prognosticate the detection and treatment of  diabetes with a high accuracy, in order to identify and treat  diabetes patients at an early age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Our training and test dataset  is an accumulation of 9483 diabetes patients’ information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The  training dataset is large enough to negate overfitting and  provide for highly accurate test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We use  performance measures such as accuracy and precision to find  out the best algorithm deep ANN which outperforms with  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='14% accuracy among all other tested machine learning  classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We hope our high performing model can be used by  hospitals to predict diabetes and drive research into more  accurate prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Keywords—Artificial Neural Network, Type 2 diabetes, Support  Vector Machine, Decision Tree, Naive Bayes, LDA, Random  forest classifier  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  INTRODUCTION  Diabetes Mellitus (DM) is a very common metabolic  disorder that affects millions of people worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It occurs  when the concentration of blood glucose reaches excessive  level due to lack of production of insulin by the pancreas  organ (Type 1 Diabetes) or due to insulin resistance (Type 2  Diabetes) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It has been published that 422 million people  are suffering from diabetes approximately in 2014 and it is  expected to rise to 438 million in 2030[2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Among them,  90% of cases are Type 2 diabetes (T2DM) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It may arise  at an early childhood because of the failure of cells to  respond to insulin appropriately [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' So, patients have to  face excessive tiredness, visual disorders, excessive thirst,  skin infection recurrence, delayed wound healing and  frequent discharge of urine [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It has been pointed out by  Diabetes Research Center that 80 percent of cases of  diabetes can be prevented or delayed if it is detected early  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Also, by controlling blood sugar, it is possible to lessen  the T2DM effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' A healthy diet, physical exercise,  sufficient nutrition for pregnant women, proper medication,  weight at a necessary level are crucial to maintaining a safer  sugar level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  When the diabetes is diagnosed with medical tests, it  shows significantly dangerous symptoms but these methods  do not perform well because of clinical complexity, time-  consuming process and very high expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' However, using  automated machine learning algorithms, a researcher can  predict a disease like diabetes with reduced cost and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In  the field of Artificial Intelligence, classification is  considered a supervised technique that analyses patient data  and classifies whether or not the patient is suffering from a  disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Researchers have created different AI and machine  learning techniques to automate prognosis of various  diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Machine learning techniques studies algorithm and  statistical model that has the capability for accurate  prediction by using implicit programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In medical  science, they take the concept of the human brain as it  contains millions of neurons to complete tasks of the human  body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It is called nonlinear modelling and they are  interconnected like brain cells although the neuron creation  is done by program [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  In this paper, first we have discussed various procedures  and existing works about the prognosis of T2D , though we  emphasized various classification algorithms known as  Logistic Regression, KNN, Decision Tree, Naive Bayes,  SVM, Linear Discriminant Analysis and Random forest  classifier and Artificial Neural Network (ANN) for T2DM  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Our selected model is an Artificial Neural  Network is found to be superior among all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Feedforward neural network contains the signal in one  Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  direction from the input to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It is used in different  medical diagnostic applications such as nephritis disease,  heart disease, myeloid leukemia etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' [ref].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  We have taken a medical dataset from Noakhali Medical  College, Bangladesh, consisting of 9483 samples and 14  symptoms per sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The 80% data and 20% data are  chosen to be training dataset and testing dataset  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Machine learning classification algorithms are  applied to dataset and some elements may be missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Then,  the mean and median method is applied in order to detect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  The contributions of this paper are summarized as We have proposed a prediction model for T2D  using Artificial Neural Network machine learning  classifier We have exerted seven classifier techniques and  ANN on T2D data and provided comparison of  accuracy among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The improvement systems of the model, as well as  accuracy, are mentioned in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  The remaining of the discussion is organized as follows:  Section-II explains related work of various classification  techniques for prediction of diabetes, Section-III describes  the methodology and materials used, Section-IV discusses  evaluated Results and Section-V delineates the conclusion  of the research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  RELATED WORK  In recent years, several studies have been published using  multiple machine learning classifiers, ANN techniques and  various feature extraction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' These have a drastic  change in potential research and some works are discussed  related to T2DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Ebenezer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' used the backpropagation  feature of ANN in order to diagnose diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It finds out  the error by juxtaposing input and output number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Here, the  preceding round error is greater than the present error each  time by means of changing weight to minimize gradient of  errors using a technique known as gradient descent [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Nongyao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' delineated risk prediction by using various  machine learning classification algorithms such as Decision  Tree, Neural Network, Random Forest algorithms, Naïve  Bayes, Logistic Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' All of them followed Bagging  and Boosting approaches to improve robustness except RFA  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Deepti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' proposed a model to identify diabetes at a  premature age by applying Decision Tree, SVM and Naïve  Bayes on Pima Indians Diabetes Database (PIDD) datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  They chose sufficient measures for accuracy including  precision, ROC, F measure, Recall but Naïve Bayes beat  them by acquiring the highest accuracy [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  applied decision tree, logistic regression, neural network,and  rough sets to assess accuracy through various features like  age, right thigh circumference, left thigh circumference,  trunk volume and illustrates thigh circumference as a better  feature than BMI in anthropometrical data [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Al-Rubeaan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' has presented T2DM based on diabetic nephropathy  (DP), then defined high impact risk factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' age and diabetes  duration for microalbuminuria, macroalbuminuria and end-  stage renal disease(ESRD) classifications[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Vijayan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  examines various types of preprocessing techniques which  includes PCA and discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It increases the accuracy of  Naïve Bayes classifier and Decision Tree algorithm but  reduces SVM accuracy [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Micheal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' proposed Multi-Layer Feed Forward  Neural Networks (MLFNN) in order to diagnose diabetes by  considering activation units, learning techniques on Pima  Indian Diabetes (PID) data set and achieved 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='5%  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It performs better than Naïve Bayes, Logistic  Regression (LR) and Random Forest (RF) classifier [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Sadri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' chose data mining algorithms like Naive Bayes,  RBF Network, and J48 to diagnose T2DM for Pima Indians  Diabetes Dataset that has 768 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Each sample has  nine features as the total number of Pregnancy, Plasma  Glucose Concentration, Diastolic Blood Pressure and 2-  Hour Serum Insulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Among them, the Naive Bayes  algorithm is unbeatable and has 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='95% accuracy [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Pradhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' devised a classifier for diabetes detection  using Genetic programming (GP) at low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Simplified  function pool consists of arithmetic operations that are used  in lower validation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Yang Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' applied Naïve  Bayes classifier by using WEKA tool in order to predict  Type2 diabetes and obtained remarkable accuracy [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Unlike these works, we have introduced diabetes‟s  medication detection system using machining learning and  deep ANN that will act like a doctor to choose the right  medication of a patient suffering from diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='MATERIALS AND METHODS  In order to categorize diabetes therapy and drugs system for  patients, the whole workflow is separated into four parts  such as data collection, data preprocessing, training data via  the proposed algorithms, and predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We have exerted  seven machine learning classifiers and deep neural networks  into the pre-processed data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' System Diagram of T2D analysis  Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Data  Processing Data  Training Model  T2Dpatients  ML Classifiers:Logistic  80% Training  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='Missing Value Check  Regression, KNN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='Handling Categorical  Decision Tree, Naive  Bayes, SVM, Linear  20%Testingset  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='Value  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='Features selection  Discriminant Analysis  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='Feature Scaling  and Random Forest  Classifier  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='Dimension Reduction  DeepANN  D=Diet & Lifestyle  I=Insulin  10 Fold Cross  M=Bigreanides  Validation  S=Secretagogues  PredictionThe source of our data came from Noakhali Medical  College, Bangladesh and the data set is separated into two  parts such as training and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The training data are  manipulated to the diagnostic system and 13 factors have  been taken to determine therapy in order to apply machine  learning and multilayer ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The dataset is tested from the  trained machine learning classifiers and artificial neural  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Dataset  As discussed before our data set contains information about  9483 diabetes patients and formatted in comma-separated-  file (CSV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The dimension of the data set is 9483*14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It  preserves 14 different kind of information of a  diabetes  patient such as „Name of patient‟,  “Fasting”, “2 h after  Pressure” “BMI”, “Duration”, “Age”, “Sex”, “Blood  pressure”, “High Cholesterols”, “Heart Diseases”, “Kidney  Diseases”, and , “Medications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The first 13 columns are  considered independent variables and the last one is the  dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It contains kinds of basic medicine  name of diabetes such as Diet and Lifestyle Modification,  Secretagogues, Biguanides, and Insulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  When datasets consist of enough variables, it increases the  accuracy of prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Here, “Fasting” measures blood test  just before taking food, “2 h after glucose load” provides a  blood test after two hours of eating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' “BMI” refers to the  weight and height of patients in kg/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' “Medication”  indicates proper drugs and therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' People who recovered  T2DM at early stage follow some features: age group 30-75  years, diabetes of diagnosis duration is more than half years,  glucose level at fasting plasma is higher than 125 mg/dl,  creation of plasma indicates equal or greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='7 mg/dl,  plasma glucose after two hours is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  When a patient suffers from kidney problems, it may be a  symptom of T2DM as higher sugar level may damage  nephron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Even bleary eyesight is considered as a side effect  for patients as eye‟s retina and the macula is affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Bad  cholesterol may lead to Diabetic dyslipidemia which can  increase heart diseases and atherosclerosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Here, we suggest  treatment for kidney and v---ision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In order to  categorize diabetes therapy and drugs system for patients,  we applied seven machine learning classifiers and eight  deep neural into a data system of Noakhali Medical College,  Bangladesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Training data are manipulated to diagnostic  system and twelve factors have been taken to determine  therapy in order to apply machine learning and multilayer  ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' For the training and testing of the systems, we  divided the data set into 80% training and 20% test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The  training dataset is used to find out the appropriate model and  best hyper-parameters and testing data set contains unseen  data to predict the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Data preprocessing  Data preprocessing involves raw data converting into a  recognizable format from various sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The well-  preprocessed data aids for the best training of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Multi pre-processing training is held in our presented  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Missing Value Check  Usually,  missing  values  may  occur  due  to  data  incompleteness, missing field, programming error, manual  data transfer from a database and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We may ignore  missing values but it causes problems in parameter  calculation and data accuracy for features such as age,  wages and fare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We need to inspect whether a dataset has  any missing value or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' There are many ways to handle  missing values such as delete rows, missing values  prediction, mean, median, mode and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' But the most  prominent policy for missing value replacement is the mean  method and also it is used to exchange the approximate  results in the dataset [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Mean is written in this way in  mathematics,  (1)  Where, denotes the mean and provides the average number  of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Handling Categorical value  Categorical encoding identifies data type and transfers  categorical features into numerical numbers as the majority  of machine learning algorithms could not cope up with label  data directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Then numerical values are fed into the  specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In our data set, there are five categorical  variable names as „Name of patients‟, “Heart Diseases”,  “Kidney Diseases”, “Sex”, and “Medications”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' There are  two popular ways of transforming categorical data into  numerical data such as Integer encoding and one-hot  encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In the label encoder, categorical features are an  integer value and contain a natural order relationship, but  the multiclass relationship will provide different values for  various classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' One hot encoding maps categorical value  into binary vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Firstly, it is obvious to assign binary  value to an integer value of female and male is 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Then converting it to a 2 size of 2 possible integers in a  binary vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Here, a female is encoded as 0 and  represented as [1, 0] in which index 0 has value 1 and vice  versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It chooses this value as a feature to influence model  training [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Features Selection  Feature selection incorporates the identification and  reduction of unnecessary features that have no impact on the  objective function and high impact features are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Our  dataset contains 14 types of elements and we have checked  p-value which is a statistical process for finding out the  probability for the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The features are taken  out whose p-value indicates less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Moreover, multicollinearity refers to determine the high  correlation which exists between two or more independent  features and features that are influential to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It is  called redundancy when two features are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  As we have to handle redundancy, it is essential to choose  some methods such as χ2Test and Correlation Coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  x  KThe Correlation Coefficient can be calculated by numerical  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Assume that A and B are two features and it can be  defined as,  (2)  After performing both p-value and multicollinearity test, we  could come forward with seven features among thirteen  independent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Those are “Fasting”, “2 Hours after  Glucose Load”  “Duration”, “BMI”, “High Cholesterols”,  “Heart Diseases”, and “Kidney Diseases”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Feature scaling  Most of the time, the dataset does not remain on the same  scale or even not normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' So, feature scaling is a  fundamental data transformation method for coping the  dataset to algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We need to scale value of features and  provide equal weight to all features in order to obtain the  same scale for all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Moreover, it is possible for scaling  to change in different values for different features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' There are  lots of techniques for feature scaling for example  Standardization, Mean Normalization, Min-Max Scaling,  Unit Vector and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  In our research work, we have taken Min-Max Scaling or  normalization process as the features are confined within a  bounded area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Minmax normalization is a z-series  normalization to transform linearly x to x‟ where maxX and  minX are the maximum and minimum value for X  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  (3)  When x=max, then y =1 and x=min, y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  The scaling range belongs between 0 and 1(positive value)  and -1 to 1(negative) and we have taken the value between 0  and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Dimension Reducing  Dimensionality reduction refers to minimizing random  variables by considering the principal set of variables that  avoids overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' For a large number of dataset, we need to  use dimension reduction technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In our study, we prefer  dimension reduction for dimensional graphical visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  There are a lot of methods for reducing dimension, for  instance, LDA, PCA, SVD, NMF, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In our system, we  have applied Principal Component Analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It is a  linear transformation based on the correlation between  features in order to identify patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' High dimensional data  are estimated into equal or lower dimensions through  maximum variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We have taken two components of PCA  according to their high variance so that we can graphically  visualize in Cartesian coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Training Algorithms  The training dataset for T2DM is applied to each algorithm  to find out medications and model performance is assessed  by obtaining accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Machine Learning Classifier  Since we focus on the performance of treatment predictions,  we have implemented seven machine learning classifiers  such as logistic regression, KNN, SVM, Naive Bayes,  decision tree, LDA, random forest tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Logistic regression is based on the probability model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' it is  derived from linear regression that mapped the dataset into  two categories by considering existing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' At first, features  are mapped linearly that are transferred to a sigmoid  function layer for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It shows the relationship  between the dependent and independent values but output  limits the prediction range on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' As we need to predict  the right treatment of a diabetes person, it is beneficial to  use a binary classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Linear Discriminant Analysis (LDA) belongs to a linear  classifier to find out the linear correlation between elements  in order to support binary and multiclass classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The  chance of inserting a new dataset into every class is detected  by LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Then, the class that contains the dataset is detected  as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It can calculate the mean function for each class  and it is estimated by vectors for finding group variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Support Vector Machine (SVM) is the most recognized  classifier to make decision boundary as hyperplane to keep  the widest distance from both sides of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' This  hyperplane refers to separating data into two groups in two-  dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It performs better with non-linear  classification by the kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It is capable of  separating and classifying unsupported data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  K-nearest neighbours (KNN) works instant learning  algorithm and input labeled data that act as training instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Then, the output produces a group of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' When k=1, 2, 5  then it means the class has 1, 2 or 5 neighbours of every data  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' For this system, we choose k=5 that means 5  neighbours for every data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We have taken Minkowski  distance to provide distance between two points in N-  dimensional vector space to run data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Suppose, points p1(x1,  y1) and p2(x2, y2) illustrates Minkowski distance as,  (4)  Here, d denotes Minkowski distance between p1 and p2  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Naive Bayes Classifier is constructed from Bayes theorem,  in which features are independent of each other in present  class and classification that counts the total number of  observations by calculating the probability to create a  predictive model in the fastest time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It outperformed with a  huge dataset of categorical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The main benefits of  that it involves limited training data to estimate better  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Naive Bayes theorem probability can be derived  from P (T), P(X) and P (X|T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Therefore,  (5)  Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  at A-(b:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  17  noaobmmax(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  mmtm(x71  dp : (x,y) → Ilx -yllp  Ix - yilP  i=1The decision tree is a decision-supporting a predictive  model based on tree structure by putting logic to interpret  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It provides a conditional control system and marks  red or green for died or alive leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It has three types of  nodes: root node, decision nodes and leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The root  node is the topmost node among them and data are split into  choices to find out the decision‟s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Decision nodes  basically comprise of decision rules to produce the output  by considering all information gain and oval shape is used to  denote it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The terminal node represents the action that needs  to be taken after getting the outcome of all decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Multiple random trees lead to the random forest to calculate  elements of molecular structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' A decision tree looks like a  tree that is the storehouse of results from the random forest  algorithm and bagging is applied to it in order to reduce  bias-variance trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It can perform feature selection  directly and output represents the mode of all classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In  Random Forest Tree, we took the total number of trees in  the forest: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Artificial Neural Network  An ANN is considered as a human brain due to consisting  millions of neurons to communicate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It has  three layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' the input layer fed raw data to network, hidden  layer is the middle layer based on input, weight and the  relationship denoted by activity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Output layers  value is determined by activity, weight and relationship  from the second layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Since we need to find out the probability of each treatment  and the objective function is not binary, so we used softmax  activation function instead of sigmoid between the hidden  layer and output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' There is no rule of thumb to choose  hidden layer in ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' If our data is linearly separable then  we don‟t need any hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Then the average node  between the input and output node is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  In our system, we prefer six hidden layers between the input  node and the hidden layer and 25 epochs to train a neural  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It has no gradient vanishing problem and uses  ReLU activation function to train dataset without  pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Validation  The validation is a technique of evaluating the performance  of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It cooperates to evaluate the model and  reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Different types of validation method  includes  Holdout  method,  K-Fold  Cross-Validation,  Stratified K-Fold Cross-Validation and Leave-P-Out Cross-  Validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We have picked out k-fold validation dataset is  divided into k subsets in k times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' One k subset act as test set  and error is estimated by average k trails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Therefore, k-1  subsets produce training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We prefer k=10 generally  which contains 10 folds, repeat one time and stratified  sampling as each fold has a similar amount of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='EXPERIMENTAL RESULT ANALYSIS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Experimental tool  The whole task has been implemented in python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='6  programming language in Anaconda distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Python  library offers various facilities to implement machine  learning and deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The unbeatable library for data  representation is pandas that provide huge commands and  large data management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We have used it to read and  analyze data in less writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Afterward, scikit-learn has  features for various classification, clustering algorithms to  build models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Also, Keras combines the advantages of  theano and TensorFlow to train a neural network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' We  use to fit and evaluate function to train and assess neural  network model respectively bypassing the same input and  output, then we apply matplotlib for graphical visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Model performance  For boosting performance, it is always a better idea to  increase data size instead of depending on prediction and  weak correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Also, adding a hidden layer may increase  accuracy and speed due to its tendency to make a training  dataset overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' But partially it is dependent on the  complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Contrarily, increasing the epochs  number ameliorate performance though it sometimes  overfits training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It works well for the deep network  than shallow network when considering regulation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Hereafter, we have added another hidden layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' choose  epoch 100 then the Deep ANN accuracy risen up to 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='14%  which is superior among all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Models Performance Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Improving Model performance  For boosting performance, it is always a better idea to  increase data size instead of depending on prediction and  weak correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Also, adding a hidden layer may increase  training accuracy and speed due to its tendency to make  training dataset overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' But partially it is dependent on the  complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Contrarily, increasing the epochs  number ameliorate performance though it sometimes  overfits training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It works well for the deep network  than shallow network when considering regulation factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Hereafter, we added another hidden layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' choose epoch 100  then the Deep ANN accuracy of the training and test set is  risen up to 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='42% and 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='14% which is superior among all  of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' 2-D Graphical Visualization of Test set  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='Final Result  After applying feature extraction to the dataset and  implementing several types of classification and deep neural  network, we found artificial neural network as better  performer with best validity and Random forest classifier  are preferable among other machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Final Result Comparison  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' CONCLUSIONS  Type-2 diabetes can lead to a lot of complications as heart  attack, kidney damage, blurred vision, hearing problems and  Alzheimer‟s disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' The main problem is lower accuracy of  the prediction model, small datasets and inadaptability to  various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In this paper, the medication and treatment  are predicted by a comparative study of seven machine  learning algorithms and deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Artificial  neural networks play a vital role in medical science by  minimizing classification error that leads to greater  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Experiment result determines the designed ANN  system achieved higher accuracy of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It can cooperate  with experts to detect T2DM patients at a very early age and  provide the best treatment option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  In the future, we can enhance the accuracy of early  treatment to lessen the suffering of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Also, we can  implement more classifiers to pick up the leading one for  record-breaking performance and extend it to automation  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' There is a plan to apply this designed system in  diabetes or for other diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It may increase the  performance of prediction of various diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Larger  dataset leads to the higher training set and it cooperates in  advanced accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' It is convenient for people to have an  application on their smartphones related to T2DM that may  have T2DM symptoms, treatment, risk factors, and health  management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  REFERENCES  [1]  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
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+page_content=' Springer, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  [13] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
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+page_content='  [15] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='researchgate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='net/publication/331352518_A_Multi-  layer_Feed_Forward_Neural_Network_Approach_for_Diagnosing_D  iabetes  [16] https://pdfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='semanticscholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='org/ab93/6e4630720cb7f7ead833222b94  5dc3801438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='pdf  [17] Pradhan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=', Rahman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=', Acharya, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=', Gawade, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=', Pateria, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  Design of classifier for Detection of Diabetes using Genetic  Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' In: International Conference on Computer Science and  Information Technology, Pattaya, Thailand, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' 125–130 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  [18] Yang Guo, Karlskrona, S Guohua Bai and Yan Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Using Bayes  Network for Prediction of Type-2 diabetes, IEEE: International  Conference on Internet Technology And Secured Transactions, pp:  471 - 472, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  [19] https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='analyticsindiamag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='com/5-ways-handle-missing-values-  machine-learning-datasets/  [Accessed: 24 June, 2019]  [20] https://medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='com/@contactsunny/label-encoder-vs-one-hot-  encoder- [Accessed: 5 August, 2019]  Authorized licensed use limited to: Newcastle University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Downloaded on May 18,2020 at 05:34:25 UTC from IEEE Xplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content=' Restrictions apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}
+page_content='  ANN (Test set)  CI' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tE1T4oBgHgl3EQfUwM8/content/2301.03093v1.pdf'}

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+arXiv:2301.01444v1  [cond-mat.str-el]  4 Jan 2023
+Magnetic properties of the layered heavy fermion antiferromagnet CePdGa6
+H. Q. Ye,1 T. Le,1 H. Su,1 Y. N. Zhang,1 S. S. Luo,1 M. J. Gutmann,2 H. Q. Yuan,1, 3, 4, 5 and M. Smidman1, 3, ∗
+1Center for Correlated Matter and Department of Physics, Zhejiang University, Hangzhou 310058, China
+2ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot Oxon OX11 0QX, United Kingdom
+3Zhejiang Province Key Laboratory of Quantum Technology and Device,
+Department of Physics, Zhejiang University, Hangzhou 310058, China
+4State Key Laboratory of Silicon Materials, Zhejiang University, Hangzhou 310058, China
+5Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
+(Dated: January 5, 2023)
+We report the magnetic properties of the layered heavy fermion antiferromagnet CePdGa6, and
+their evolution upon tuning with the application of magnetic field and pressure. CePdGa6 orders
+antiferromagnetically below TN = 5.2 K, where there is evidence for heavy fermion behavior from an
+enhanced Sommerfeld coefficient. Our results are best explained by a magnetic ground state of fer-
+romagnetically coupled layers of Ce 4f-moments orientated along the c-axis, with antiferromagnetic
+coupling between layers. At low temperatures we observe two metamagnetic transitions for fields
+applied along the c-axis corresponding to spin-flip transitions, where the lower transition is to a dif-
+ferent magnetic phase with a magnetization one-third of the saturated value. From our analysis of
+the magnetic susceptibility, we propose a CEF level scheme which accounts for the Ising anisotropy
+at low temperatures, and we find that the evolution of the magnetic ground state can be explained
+considering both antiferromagnetic exchange between nearest neighbor and next nearest neighbor
+layers, indicating the influence of long-range interactions. Meanwhile we find little change of TN
+upon applying hydrostatic pressures up to 2.2 GPa, suggesting that significantly higher pressures
+are required to examine for possible quantum critical behaviors.
+I.
+INTRODUCTION
+Heavy fermion compounds are prototypical examples
+of strongly correlated electron systems, and have been
+found to host a range of emergent phenomena including
+unconventional superconductivity, complex magnetic or-
+der and strange metal behavior [1–3]. Ce-based heavy
+fermions contain a Kondo lattice of Ce-ions with an un-
+paired 4f electron, which can both couple to other 4f mo-
+ments via the Ruderman-Kittel-Kasuya-Yosida (RKKY)
+interaction and undergo the Kondo interaction due to
+hybridization with the conduction electrons.
+Here the
+RKKY interaction gives rise to long-range magnetic or-
+der, while the Kondo interaction favors a non-magnetic
+Fermi-liquid ground state with greatly enhanced quasi-
+particle masses. Due to the small energy scales, the rel-
+ative strengths of these competing interactions can often
+be tuned by non-thermal parameters such as pressure,
+magnetic fields and chemical doping [4], and in many
+cases the magnetic ordering can be continuously sup-
+pressed to zero temperature at a quantum critical point
+(QCP).
+A major question for heavy fermion systems is the
+relationship between quantum criticality, and the dome
+of unconventional superconductivity sometimes found to
+encompass the QCP. CeIn3 is a canonical example of this
+phenomenon, which at ambient pressure orders antiferro-
+magnetically below TN = 10.1 K, but exhibits a pressure-
+induced QCP around 2.6 GPa, which is surrounded by
+a superconducting dome with a maximum Tc of 0.2 K
+[5]. The layered CeMIn5 (M= transition metal) com-
+pounds consist of alternating layers of MIn2 and CeIn3
+along the c-axis [6], and among the remarkable proper-
+ties is a significantly enhanced superconducting Tc for
+the M= Rh and Co systems, reaching over 2 K [7, 8],
+giving a strong indication that quasi-two-dimensionality
+is important for promoting heavy fermion superconduc-
+tivity. Meanwhile the Ce2MIn8 compounds correspond
+to a stacked arrangement of two units of CeIn3, and one
+of MIn2 [9], and are expected to have an intermediate
+degree of two dimensionality relative to CeMIn5. Cor-
+respondingly, the superconducting phases have lower Tc
+values of 0.4 and 0.68 K for Ce2CoIn8 [10] and Ce2PdIn8
+[11] at ambient pressure, and a maximum of Tc = 2 K
+at 2.3 GPa for Ce2RhIn8 [12]. On the other hand, these
+different series of related Ce-based heavy fermion sys-
+tems also exhibit different magnetic ground states and
+crystalline electric field (CEF) level schemes [13–17] and
+therefore it is challenging to disentangle the role of these
+factors from that of the reduced dimensionality.
+The
+elucidation of the interplay between these different as-
+pects requires examining additional families of layered
+Ce-based heavy fermion systems for quantum critical be-
+haviors, as well as detailed characterizations of the mag-
+netic ground states and exchange interactions.
+The properties of layered Ce-based heavy fermion gal-
+lides have been less studied than the indium-based sys-
+tems. CeGa6 has a layered tetragonal structure (space
+group P4/nbm), with four Ga-layers between each Ce
+layer [18].
+This compound orders magnetically below
+TN = 1.7 K, and there is evidence for the build-up of
+magnetic correlations at significantly higher tempera-
+tures [19]. A more layered structure is realized in the
+Ce2MGa12 (M= Cu, Ni, Rh, Pd, Ir, Pt) series, where the
+Ce-layers are alternately separated by four Ga-layers, and
+units of MGa6, leading to a larger interlayer separation
+
+2
+of the Ce-atoms [20, 21]. Several members of this series
+show evidence for both antiferromagnetism and heavy
+fermion behavior [20–25], where pressure can readily sup-
+press the antiferromagnetic transitions of Ce2NiGa12 and
+Ce2PdGa12 [26, 27], while evidence for field-induced crit-
+ical fluctuations is revealed in Ce2IrGa12 [25].
+CePdGa6 has a different layered tetragonal structure
+(space group P4/mmm) displayed in Fig. 1(a), consist-
+ing of square layers of Ce-atoms, with each Ce con-
+tained in a CeGa4 prism, separated by PdGa2 layers
+[28]. Correspondingly, there is a distance between Ce-
+layers of 7.92 ˚A, while the nearest neighbor in-plane Ce-
+Ce separation is 4.34 ˚A, compared to respective values
+of 7.54 ˚A
+and 4.65 ˚A in CeRhIn5 [29]. CePdGa6 or-
+ders antiferromagnetically below TN = 5.2 K, and heavy
+fermion behavior is evidenced by an enhanced Sommer-
+feld coefficient [20, 28]. As such, CePdGa6 is a good can-
+didate to look for novel behaviors arising in quasi-two-
+dimensional heavy fermion systems, but there is both a
+lack of detailed characterizations of the magnetic ground
+state, and no reports of the evolution under pressure. In
+addition, most measurements of CePdGa6 are reported
+in Ref. 28, where the results are affected by the inclu-
+sion of an extrinsic antiferromagnetic phase Ce2PdGa12,
+which can be eliminated using a modified crystal growth
+procedure [20].
+In this article we report detailed measurements of the
+magnetic properties of single crystals of CePdGa6, in-
+cluding their evolution upon applying magnetic fields and
+hydrostatic pressure. We find that CePdGa6 orders an-
+tiferromagnetically in zero-field, where the Ce-moments
+are orientated along the c-axis and align ferromagneti-
+cally within the ab-plane, but there is antiferromagnetic
+coupling between layers. At low temperatures, two meta-
+magnetic transitions are observed for fields along the c-
+axis, the lower of which corresponds to a spin-flip transi-
+tion to a phase with magnetization one-third of the sat-
+urated value. From our analysis of the magnetic suscep-
+tibility, we propose a CEF level scheme which can ex-
+plain the low temperature Ising anisotropy, and we find
+that from considering interactions between the nearest-
+neighbor and next nearest neighbor Ce-layers, the field
+evolution of the magnetic state can be well accounted for.
+II.
+EXPERIMENTAL DETAILS
+Single crystals of CePdGa6 were grown using a Ga self-
+flux method with a molar ratio of Ce:Pd:Ga of 1:1.5:15
+[20].
+Starting materials of Ce ingot (99.9%), Pd pow-
+der (99.99%) and Ga pieces (99.99%) were loaded into
+an alumina crucible which was sealed in an evacuated
+quartz tube. The tube was heated to 1150 ◦C and held
+at this temperature for two hours, before being rapidly
+cooled to 500 ◦C at a rate of 150 K/h and then cooled
+more slowly to 400
+◦C at 8 K/h.
+After being held
+at 400 ◦C for two weeks, the tube was removed from
+the furnace, and centrifuged to remove excess Ga. The
+2
+0
+1
+FIG. 1.
+(Color online) (a) Crystal structure of CePdGa6
+where the red, blue and green atoms correspond to Ce, Pd
+and Ga, respectively.
+J0 represents magnetic exchange in-
+teractions between nearest neighbor Ce atoms within the ab-
+plane, J1 is between nearest neighboring layers and J2 is
+between next nearest layers.
+An image of a typical single
+crystal of CePdGa6 is also displayed, where each square in
+the background is 2 mm × 2 mm. (b) X-ray diffraction pat-
+tern measured on a single crystal of CePdGa6. The red dashes
+correspond to the positions of the (00l) Bragg peaks, indicat-
+ing that the [001] direction is perpendicular to the large face
+of the plate-like samples.
+obtained crystals are plate-like with typical dimensions
+2 × 1.5 × 0.3 mm3. Note that when slower cooling rates
+of 6 K/h or 4 K/h were used, the resulting crystals were
+significantly smaller. Single crystals of the non-magnetic
+analog LaPdGa6 were also obtained using a similar pro-
+cedure. The composition was confirmed using a cold field
+emission scanning electron microscope (SEM) equipped
+with an energy dispersive x-ray spectrometer. The phase
+of the crystals were checked using both a PANalytical
+X’Pert MRD powder diffractometer using Cu-Kα radi-
+ation, and a Rigaku-Oxford diffraction Xtalab synergy
+single crystal diffractometer equipped with a HyPix hy-
+brid pixel array detector using Mo-Kα radiation. The ob-
+tained lattice parameters from the single crystal diffrac-
+tion data of a = 4.3446(3) ˚A and c = 7.9173(10) ˚A are
+
+pg
+c3
+0
+100
+200
+300
+2
+3
+4
+5
+4
+8
+1.5
+2.0
+ 
+ 
+ 
+ (
+cm)
+T (K)
+T
+N
+ ~ 5.2 K
+ 
+ 
+ 
+ (
+cm)
+T (K)
+FIG. 2.
+(Color online) Temperature dependence of the re-
+sistivity ρ(T ) of CePdGa6 between 1.8 and 300 K. The inset
+displays the low temperature resistivity, where there is a sharp
+anomaly at the antiferromagnetic transition.
+in excellent agreement with previous reports [28]. Mea-
+surements of a crystal using the powder diffractometer
+are displayed in Fig. 1(b), where all the Bragg peaks are
+well-indexed by the (00l) reflections of CePdGa6, demon-
+strating that the c-axis is perpendicular to the large face
+of the crystals.
+Resistivity and specific heat measure-
+ments were performed in applied fields up to 14 T using
+a Quantum Design Physical Property Measurement Sys-
+tem (PPMS-14) down to 1.8 K, and to 0.3 K using a 3He
+insert.
+Resistivity measurements were performed after
+spot welding four Pt wires to the surface, with the exci-
+tation current in the ab-plane. Magnetization measure-
+ments were performed in the range 1.8 - 300 K in applied
+fields up to 5 T using a Quantum Design Magnetic Prop-
+erty Measurement System (MPMS) SQUID magnetome-
+ter.
+Heat capacity measurements under pressure were
+carried out in a piston cylinder cell, using an ac calori-
+metric method.
+III.
+RESULTS
+A.
+Antiferromagnetic transition and CEF
+excitations of CePdGa6
+Figure 2 displays the temperature dependence of
+the resistivity ρ(T ) of CePdGa6 between 1.8 and 300
+K, which has a residual resistivity ratio [RRR =
+ρ(300 K)/ρ(2 K)] = 3.8.
+A broad shoulder is ob-
+served at around 50 K, which likely arises due to both
+the Kondo effect, and as a consequence of CEF excita-
+tions.
+At higher temperatures, quasilinear behavior is
+observed, which could be due to electron-phonon cou-
+pling.
+As shown in the inset, there is an anomaly at
+around TN = 5.2 K, below which ρ(T ) decreases more
+rapidly with decreasing temperature, which corresponds
+C
+m
+/T (J mol
+-1
+ K
+-2
+)
+T (K)
+Rln2
+ 
+ 
+ S
+m
+ (J mol
+-1
+ K
+-1
+)
+ 
+ 
+C
+m
+ (J mol
+-1
+K
+-1
+)
+T (K)
+(b)
+ CePdGa
+6
+ LaPdGa
+6
+ 
+ 
+C (J mol
+-1
+ K
+-1
+)
+T (K)
+(a)
+FIG. 3. (Color online) (a) Magnetic contribution to the spe-
+cific heat Cm at low temperatures, where the red solid line
+shows the results from fitting with Eq. 1. The inset shows
+the total specific heat C of CePdGa6 and the non-magnetic
+analog LaPdGa6. (b) Temperature dependence of Cm/T and
+the magnetic entropy Sm of CePdGa6. The pink dotted line
+displays the low temperature contribution to the specific heat
+calculated from the CEF scheme deduced from the analysis
+of χ(T ).
+to the antiferromagnetic transition reported previously
+[20], while no signature of the spurious transition at
+higher temperatures is detected [28].
+The total spe-
+cific heat of CePdGa6 and nonmagnetic isostructural
+LaPdGa6 are shown in the inset of Fig. 3(a). The tem-
+perature dependence of the magnetic contribution to the
+specific heat Cm was estimated by subtracting the data
+of LaPdGa6, which is shown in Fig. 3(a), while the spe-
+cific heat coefficient Cm/T and the magnetic entropy Sm
+of CePdGa6 are displayed in Fig. 3(b). A pronounced
+λ-like anomaly is observed at TN = 5.2 K, as is typi-
+cal for a second-order magnetic phase transition.
+For
+T > TN, Cm/T increases with decreasing temperature,
+and extrapolates to a relatively large zero temperature
+value of 250 mJ/mol K2. As discussed below, the analysis
+of the magnetic susceptibility χ(T ) suggests the presence
+of a low lying CEF level, which could contribute to Cm/T
+in this temperature range. The dotted line in Fig. 3(b)
+shows the calculated Cm/T for the CEF level scheme de-
+
+8030412JS12
+300
+e0
+0S0
+04
+0
+10
+20
+0.0
+0.1
+0.2
+0
+150
+300
+0
+200
+400
+(b)
+ 
+ 
+ (emu/mol)
+T (K)
+H = 0.1 T
+ H // c
+ H // ab
+(a)
+ 
+H = 0.5 T
+ H // c
+ H // ab 
+1/(
+) (mol/emu)
+T (K)
+ 
+FIG. 4.
+(Color online) (a) Low temperature magnetic sus-
+ceptibility χ(T ) of CePdGa6, with an applied field of µ0H =
+0.1 T both parallel to the c-axis and within the ab-plane. (b)
+Temperature dependence of 1/(χ-χ0) up to 300 K for 0.5 T
+applied along the two field directions, where the dashed and
+solid lines show the results from fitting with the CEF model
+described in the text.
+scribed below, which has a sizeable value in the vicinity
+of the transition. Subtracting the contribution from the
+CEF at TN yields an estimate of γ ∼ 121.4 mJ/mol K2
+associated with the ground state doublet, and such an
+enhanced value could arise both due to heavy fermion
+behavior, as well as the presence of short range magnetic
+correlations, as inferred in CeRhIn5[30, 31].
+The data
+below TN were analyzed using [32]:
+Cm = γT + c∆7/2
+SW
+√
+T exp
+�−∆SW
+T
+�
+×
+�
+1 +
+39T
+20∆SW
++ 51
+32
+�
+T
+∆SW
+�2�
+(1)
+where the first term corresponds to the electronic con-
+tribution and the second term arises due to antiferro-
+magnetic spin-waves. Here the coefficient c is related to
+the spinwave stiffness D via c ∝ D−3, while ∆SW
+is
+the spin-wave gap. The results from fitting the zero-field
+data are displayed in the main panel of Fig. 3(a), where
+γ = 121.4 mJ/mol K2 was fixed, yielding ∆SW = 2.3 K
+and c = 23 mJ/mol K2. The moderate value of ∆SW
+is smaller than TN, unlike the layered heavy fermions
+gallides Ce2PdGa12 and Ce2IrGa12 where ∆SW
+> TN
+[24, 25], likely reflecting the weaker magnetocrystalline
+anisotropy in CePdGa6. The temperature dependence of
+the magnetic entropy Sm of CePdGa6 is also displayed
+in Fig. 3(b), obtained by integrating Cm/T , where Cm/T
+was linearly extrapolated below 0.4 K. At TN, Sm reaches
+0.76R ln 2, which together with the expected sizeable con-
+tribution from the excited CEF level discussed above,
+suggests a reduced entropy corresponding to the ground
+state doublet due to Kondo screening.
+Figure 4(a) displays the temperature dependence of
+the magnetic susceptibility χ(T ) of CePdGa6 at low tem-
+peratures, with an applied field of µ0H = 0.1 T along
+the c-axis and within the ab-plane, which both exhibit
+an anomaly at TN.
+At low temperatures, χ(T ) is sig-
+nificantly larger for fields along the c-axis than in the
+ab-plane, demonstrating that the c-axis is the easy-axis
+of magnetization.
+At TN, there is a peak in χ(T ) for
+H ∥ c, while for H ∥ ab χ(T ) weakly increases below TN,
+indicating that this corresponds to an antiferromagnetic
+transition with moments ordered along the easy c-axis.
+At higher temperatures, the data above 100 K can
+be analyzed using the Curie-Weiss law: χ=χ0+C/(T −
+θCW), where χ0 is a temperature-independent term, C
+is the Curie constant and θCW is the Curie-Weiss tem-
+perature, yielding θc
+CW = −11.7(3) K and an effective
+moment of µc
+eff = 2.35µB/Ce for H ∥ c, as well as
+θab
+CW = −12.9(8) K and µab
+eff = 2.49µB/Ce for H ∥ ab.
+The obtained values of µeff for both directions are close
+to the full value of 2.54 µB for the J = 5
+2 ground state
+multiplet of Ce3+. At lower temperatures, there is a devi-
+ation of χ(T ) from Curie-Weiss behavior, due to the split-
+ting of the ground state multiplet by crystalline-electric
+fields. To analyze the CEF level scheme, we considered
+the following Hamiltonian for a Ce3+ ion in a tetragonal
+CEF [33]
+HCF = B0
+2O0
+2 + B0
+4O0
+4 + B4
+4O4
+4
+(2)
+where Om
+l
+and Bm
+l
+are Stevens operator equivalents and
+parameters, respectively. The B0
+2 parameter can be es-
+timated from the high temperature susceptibility using
+[34]
+B0
+2 = 10kB
+�
+θab
+CW − θc
+CW
+�
+3(2J − 1)(2J + 3) ,
+(3)
+where J =
+5
+2 for the ground state multiplet of Ce3+,
+yielding B0
+2 = -0.01077 meV. χ(T ) along both directions
+was analyzed taking into account the contribution from
+the CEF χi
+CEF, as well as molecular field parameters λi
+using
+χi = χi
+0 +
+χi
+CEF
+1 − λiχi
+CEF
+,
+(4)
+where the superscript i denotes the c-axis or ab-plane.
+With B0
+2 fixed from Eq. 3, values of B0
+4 = -0.0746
+meV and |B4
+4| = 0.496 meV were obtained, together
+with molecular field parameters of λc = -3.55 mol/emu
+and λab = 8.15 mol/emu, χc
+0 = 2.2 × 10−4emu/mol
+and χab
+0
+= −2.3 × 10−3emu/mol, and the fitted re-
+sults are shown in Fig. 4(b).
+These parameters yield
+a CEF scheme with a Γ7 ground state Kramer’s doublet
+��ψ±
+1
+�
+= 0.883
+��± 5
+2
+�
+− 0.469
+��∓ 3
+2
+�
+(for positive B4
+4), and
+excitations to Γ6 and Γ7 levels of ∆1 = 2.8 meV and ∆2 =
+32.1 meV, respectively. At high temperatures, the small
+
+5
+4
+8
+0.5
+1.0
+1.5
+2.0
+ 0    
+ 4
+ 6    
+ 8
+ 10  
+ 12
+4
+8
+(b)
+(a)
+ 
+ 
+C
+P
+/T (J/mol K
+2
+)
+T (K)
+ 0      
+ 1
+ 1.5   
+ 2
+ 2.5   
+ 3
+H // c
+0
+H (T) 
+ 
+ 
+T (K)
+0
+H (T) 
+H // ab
+FIG. 5. (Color online) Temperature dependence of the spe-
+cific heat of CePdGa6 in various applied magnetic fields (a)
+parallel to the c-axis, and (b) within the ab-plane.
+4
+8
+0.1
+0.2
+0.3
+4
+8
+0.04
+0.06
+0.08
+4
+8
+(a)
+ 
+ 
+(emu/mol)
+T (K)
+ 0.1    
+ 0.5
+ 1       
+H // c
+0
+H (T) 
+0
+H (T) 
+H // ab 
+ 
+ 
+ 1   
+ 2
+ 4   
+ 6
+ 8
+(emu/mol)
+T (K)
+(c)
+(b)
+ 
+ 
+T (K)
+ 1.5
+ 2      
+ 3
+H // c
+0
+H (T) 
+FIG. 6. (Color online) Temperature dependence of the mag-
+netic susceptibility χ(T ) of CePdGa6 in different magnetic
+fields parallel to the c-axis for fields (a) below, and (b) above
+1 T. The vertical arrows mark the position of the antiferro-
+magnetic transition. Panel (c) shows χ(T ) for various fields
+applied within the ab-plane, where the dashed line shows the
+evolution of TN with field.
+negative B0
+2 leads to a nearly isotropic χ(T ), while at low
+temperatures, the negative B0
+4 leads to the observed Ising
+anisotropy with an easy c-axis. The predicted moment
+along the c-axis is given by ⟨µz⟩ =
+�
+ψ±
+1 |gJJz| ψ±
+1
+�
+=
+1.4 µB/Ce, which is larger than the value obtained from
+the saturated magnetization. The positive value of λab
+is consistent with ferromagnetic coupling between spins
+within the basal plane, while the smaller negative λc
+is consistent with weaker antiferromagnetic coupling be-
+tween Ce layers.
+B.
+Field dependence of the magnetic properties
+In order to determine the behavior of the magnetic
+ground state in magnetic fields, and to map the field-
+temperature phase diagrams, measurements of the spe-
+H // ab
+H // c
+(b)
+H // c
+T (K)
+ 
+ 
+ 2
+ 3
+ 4
+M (
+/Ce)
+0
+H (T)
+H // c
+(a)
+ 
+ 
+M (
+/Ce)
+0
+H (T)
+ 
+ 
+M (
+/Ce)
+0
+H (T)
+5 K
+3 K
+0.3 K
+ 
+ 
+(
+cm)
+0
+H (T)
+1.8 K
+FIG. 7. (Color online) (a) Isothermal field dependence of the
+magnetization M(H) of CePdGa6 for fields along the c-axis,
+at three temperatures below TN. The lower inset displays the
+low field region of the data in the main panel, demonstrating
+hysteresis about the metamagnetic transition, while the up-
+per inset shows M(H) at 2 K for fields within the ab-plane.
+(b) Field dependence of the resistivity ρ(H) of CePdGa6 at
+several temperatures for fields along the c-axis. The dashed
+lines show the evolution of the two metamagnetic transitions.
+cific heat and magnetization were performed in differ-
+ent applied fields. Figure 5(a) displays the low tempera-
+ture specific heat of CePdGa6 with different fields applied
+along the c-axis. It can be seen that TN is gradually sup-
+pressed with increasing field, and at fields greater than
+2 T, no magnetic transition is observed. Instead, there
+is a broad hump in C/T , which shifts to higher tempera-
+ture with increasing field, corresponding to the Schottky
+anomaly from the splitting of the ground state doublet
+in the applied field. In Fig. 5(b), C/T is displayed for
+fields within the ab-plane, where the antiferromagnetic
+transition is more robust than for fields along the c-axis,
+and the broad Schottky anomaly is only clearly resolved
+in a field of 12 T. The differences in the field dependence
+for the two different field directions is consistent with the
+low temperature Ising anisotropy in CePdGa6, where a
+smaller field along the easy c-axis can bring the system
+to the spin-polarized state.
+The low temperature χ(T ) in different applied fields
+
+744
+00.0
+己.0
+T0
+-0'288ST
+088880.0
+2.00
+V2.0--4806
+2
+4
+6
+8
+0.0
+0.5
+1.0
+ 
+ 
+C
+ac
+/T (a.u.)
+T (K)
+ 0.20GPa    
+ 0.85GPa
+ 1.60GPa    
+ 2.20GPa
+P (GPa)
+FIG. 8. (Color online) Temperature dependence of the ac heat
+capacity of CePdGa6 at various hydrostatic pressures up to
+2.2 GPa. The vertical dashed line shows the position of the
+ambient pressure TN, which remains nearly unchanged with
+pressure.
+are displayed in Fig. 6. For fields along the c-axis dis-
+tinctly different behaviors are observed for different field
+ranges. In a field of 0.1 T, there is a sharp peak at TN,
+corresponding to entering the antiferromagnetic ground
+state. At a larger field of 0.5 T, only a small hump is
+observed at TN, while at low temperatures there is an
+increase in χ(T ), and at higher fields there is broad peak
+which is gradually suppressed with field. Meanwhile for
+fields within the ab-plane up to at least 8 T, there is a
+gradual suppression of TN, in line with the specific heat
+results.
+The isothermal magnetization as a function of field
+along the c-axis at three temperatures below TN is dis-
+played in Fig. 7(a), measured upon both sweeping the
+field up and down.
+In zero-field there is no remanent
+magnetization, consistent with a purely antiferromag-
+netic ground state. At 2 K, there are two metamagnetic
+transitions at Hm1 = 0.4 T and Hm2 = 2.1 T, where
+hysteresis is also observed indicating a first-order nature,
+whereas otherwise the magnetization plateaus, with only
+a weak change of the magnetization with field. This is
+consistent with Hm1 and Hm2 corresponding to spin-flip
+transitions, with the spins remaining orientated along the
+c-axis. For fields above Hm2, no magnetic transition is
+observed in the specific heat, and therefore this likely cor-
+responds to the system reaching the spin polarized state,
+with a saturation magnetization of Ms = 1.1 µB/Ce. On
+the other hand, above Hm1 the magnetization reaches
+a value of 0.35 µB/Ce, corresponding to ≈ Ms/3, in-
+dicating a change of magnetic structure with a ferro-
+magnetic component. While there is little change in the
+field-dependence of the magnetization at 3 K, the curves
+at 4 K are drastically different. Instead of there being
+abrupt step-like metamagnetic transitions, the magneti-
+0
+1
+2
+3
+4
+0
+2
+4
+6
+0
+1
+2
+0.0
+0.5
+1.0
+1.5
+H
+m2
+H
+m1
+H // c
+ 
+ 
+T (K)
+H (T)
+ C (T)
+ 
+ (T)
+ 
+ (T)
+ M (H)
+ 
+ (H)
+ 
+ 
+M (
+/Ce)
+0
+H (T)
+ 2 K
+FIG. 9. (Color online) Temperature-field phase diagram of
+CePdGa6 at ambient pressure for fields along the easy c-axis,
+from measurements of the resistivity, magnetization, and spe-
+cific heat.
+The solid line shows the evolution of TN, while
+the dashed lines show the positions of the low temperature
+metamagnetic transitions.
+The magnetic structures at low
+temperature are also illustrated by the orange arrows, where
+in zero-field there is an antiferromagnetic ground state, while
+upon applying a field the system passes through an interme-
+diate ↑↑↓ phase, before entering the spin polarized state. The
+inset shows the field dependence of the magnetization based
+on mean-field calculations of the magnetic ground state cal-
+culated using the McPhase software package [35], with the
+parameters described in the text.
+zation smoothly increases with field, reaching a very sim-
+ilar saturation value. This suggests that at higher tem-
+peratures, the spins continuously rotate upon increasing
+the applied field, rather than undergoing abrupt spin flip
+transitions. The field dependent magnetization at 2 K
+for fields in the ab-plane is also shown in the inset of
+Fig. 7(a), which smoothly changes with field, with no
+sign of saturation up to at least 5 T, consistent with this
+being the hard direction of magnetization. The metam-
+agnetic transitions are also revealed in the field depen-
+dence of the resistivity ρ(H), as displayed in Fig. 7(b) for
+fields along the c-axis. At 0.3 K, two abrupt anomalies
+are observed corresponding to Hm1 and Hm2, which are
+also detected at 1.8 K and 3 K. Above these transitions,
+there is a decrease of ρ(H), consistent with the reduced
+spin-flip scattering arising from a larger ferromagnetic
+component to the magnetism.
+On the other hand, no
+metamagnetic transitions are detected at 5 K, where in-
+stead there is a broad peak in ρ(H), again consistent
+with a more gradual reorientation of the spins with field
+at higher temperatures.
+
+7
+C.
+Magnetism of CePdGa6 under pressure
+To determine the evolution of the magnetic order un-
+der pressure, the temperature dependence of the ac spe-
+cific heat of CePdGa6 was measured at several differ-
+ent hydrostatic pressures up to 2.2 GPa, which are dis-
+played in Fig. 8.
+It can be seen from the dotted line
+that there is little change of TN with pressure indicat-
+ing the robustness of magnetic order.
+In the case of
+the layered Ce2MGa12 compounds, the TN of Ce2NiGa12
+and Ce2PdGa12 decrease with pressure, and antiferro-
+magnetism is suppressed entirely above 5.5 and 7 GPa,
+respectively [26, 27].
+On the other hand the TN of
+Ce2IrGa12 undergoes a moderate enhancement from 3.1
+to 3.7 K for pressures up to 2.3 GPa, indicating that
+this compound is located on the left side of the Doniach
+phase diagram [25]. In the case of CePdGa6, the robust-
+ness of TN suggests that measurements to higher pres-
+sures are required to situate this compound within the
+framework of the Doniach phase diagram and to exam-
+ine whether there is pressure-induced quantum criticality
+in CePdGa6.
+IV.
+DISCUSSION
+Our measurements of the resistivity, magnetic sus-
+ceptibility and specific heat show that CePdGa6 orders
+antiferromagnetically below TN = 5.2 K, with the mo-
+ments orientated along the c-axis. Figure 9 displays the
+temperature-field phase diagram for magnetic fields ap-
+plied along the c-axis. The phase boundaries obtained
+from different measurements are highly consistent, show-
+ing that TN shifts to lower temperatures with field, before
+abruptly disappearing in a field of 2 T. At low temper-
+atures, there are two step-like metamagnetic transitions
+shown by the dashed lines, where the second transition is
+to the spin polarized state, while the lower transition cor-
+responds to a change of magnetic state to a phase with
+a magnetization of 0.35 µB/Ce, about one-third of the
+saturated value. Such step-like changes in the magne-
+tization suggest that the spins are strongly constrained
+along the c-axis, and therefore there are abrupt spin-
+flip transitions for fields applied along the ordering di-
+rection. On the other hand, at 4 K the magnetization
+changes smoothly with field, reaching the same saturated
+magnetization, indicating that at this temperature the
+spins continuously rotate in the applied field.
+Such a
+change with temperature may be a consequence of only
+a moderate magnetocrystalline anisotropy, as also evi-
+denced by the relatively small value of the spin-wave gap
+∆SW /TN ≈ 0.4, as compared to the other heavy fermion
+gallides Ce2IrGa12 and Ce2PdGa12 which have ∆SW /TN
+of 1.5 and 2.8, respectively [24, 25].
+From the analysis of the magnetic susceptibility includ-
+ing the CEF contribution, the molecular field parameter
+is positive in the ab-plane (λab), while a smaller negative
+value is obtained along the c-axis (λc). Together with the
+fact that only a relatively small field along the c-axis is re-
+quired to reach the spin polarized state, this suggests that
+the antiferromagnetic ground state consists of ferromag-
+netically ordered Ce-layers coupled antiferromagnetically
+along the c-axis. The simplest model for such a system
+would consist of ferromagnetic Heisenberg exchange in-
+teractions between nearest neighbor Ce atoms within the
+ab-plane J0 > 0, and antiferromagnetic exchange inter-
+actions J1 < 0 between nearest neighboring layers, as
+well as a sufficiently strong Ising anisotropy. This yields
+an A-type antiferromagnetic ground state consisting of
+ferromagnetic layers with moments orientated along the
+c-axis, where the moment direction alternates between
+adjacent layers, “↑↓↑↓”. This model however cannot ac-
+count for the field induced phase with one-third magneti-
+zation, since for fields along the c-axis, only a metamag-
+netic transition directly from the ↑↓↑↓ phase to the spin
+polarized state is anticipated.
+In order to realize the intermediate field-induced phase,
+it is necessary to consider an antiferromagnetic exchange
+J2 between next nearest neighboring layers. In this case,
+from considering the classical ground state energies with
+sufficiently strong Ising anisotropy, the same ↑↓↑↓ ground
+state is realized for J1/J2 > 2, while a ↑↑↓↓ state oc-
+curs for J1/J2 < 2 [36]. Upon applying a magnetic field
+along the c-axis, there is a metamagnetic transition at
+a field Hm1 to an ↑↑↓ state with a net magnetization
+one-third of the saturated value, and another at Hm2
+to the spin polarized state, where Hm2/Hm1 is deter-
+mined by J1/J2. We performed mean-field calculations
+of the magnetic ground state and magnetization using
+the McPhase software package [35], which determines
+the most stable magnetic structure at a given temper-
+ature and magnetic field from considering multiple ran-
+dom starting moment configurations.
+These took into
+account the Heisenberg exchange interactions described
+above, as well as the CEF Hamiltonian HCF with our de-
+duced values of the Stevens parameters. As shown in the
+inset of Fig. 9, the observed values of Hm1 = 0.4 T and
+Hm2 = 2.1 T, from the midpoints of the metamagnetic
+transitions at 2 K, are well reproduced from the mean-
+field calculations at 2 K with J1 = −0.023 meV and
+J2 = −0.0085 meV, where for Hm1 < H < Hm2 the ↑↑↓
+ground state has the lowest energy. Keeping these values
+fixed, we find that a nearest neighbor in-plane ferromag-
+netic interaction J0 = 0.034 meV can yield the observed
+value of TN = 5.2 K. Therefore our analysis suggests
+stronger in-plane ferromagnetic interactions, where the
+value of 4J0/(2J1 + 2J2) = 2.16 is close to our fitted
+value of λab/λc = 2.3. Note that here we have assumed
+a ↑↓↑↓ ground state with J1/J2 > 2. Although a ↑↑↓↓
+phase has been reported in CeCoGe3 [37], such a scenario
+is less likely in CePdGa6 due to the larger interlayer dis-
+tances.
+Compared to the layered heavy fermion antiferromag-
+net CeRhIn5, the magnetism in CePdGa6 appears to
+have a much more three dimensional character, whereas
+it is rather two-dimensional in the former, with J1/J0 =
+
+8
+0.13 deduced from inelastic neutron scattering [38]. In
+addition, in CeRhIn5 the easy plane anisotropy and pres-
+ence of in-plane antiferromagnetic interactions give rise
+to spiral magnetic order which is incommensurate along
+the c-axis [13, 14], and these features may be important
+factors for realizing the unconventional quantum critical-
+ity and superconductivity. On the other hand, the TN of
+CePdGa6 is much more robust with pressure, remaining
+almost unchanged at pressures up to 2.2 GPa. Therefore
+an understanding of the relationship between the mag-
+netism and any quantum critical behaviors will require
+measurements at considerably higher pressures.
+In addition, despite the layered arrangement of Ce
+atoms, the local environment of the Ce atoms is rel-
+atively three dimensional, as evidenced by the derived
+CEF parameters being close to that for a cubic sys-
+tem (where B0
+2 = 0 and |B4
+4| = 5|B0
+4|).
+This CEF
+scheme can correctly predict the low-temperature Ising
+anisotropy, but the predicted moment along the c-axis
+is larger than that observed. While such a reduced mo-
+ment compared to that predicted from the CEF level-
+scheme is often observed in heavy fermion antiferromag-
+nets due to screening of the moments by the Kondo effect
+[14, 16, 37, 39, 40], confirming whether such a scenario is
+applicable to CePdGa6 requires a more precise determi-
+nation of the CEF parameters, by measurements such as
+inelastic neutron scattering.
+V.
+CONCLUSION
+In summary, we have characterized the magnetic prop-
+erties of the heavy fermion antiferromagnet CePdGa6,
+and their
+evolution upon the application of mag-
+netic fields and pressure.
+We have constructed the
+temperature-field phase diagram for fields along the c-
+axis, where at low temperatures there are two abrupt
+metamagnetic transitions corresponding to spin-flip tran-
+sitions. From the analysis of the magnetic susceptibility,
+we propose a CEF level scheme for the splitting of the
+ground state J = 5/2 multiplet, indicating that the Ising
+anisotropy at low temperatures is driven by the sizeable
+B0
+4 parameter. Moreover, our results are consistent with
+an antiferromagnetic ground state consisting of ferromag-
+netically coupled Ce-layers, with antiferromagnetic cou-
+pling between layers. We have proposed a model for the
+exchange interactions which can explain the evolution of
+the magnetic ordering with applied magnetic field, which
+has sizeable nearest neighbor and next-nearest neighbor
+layer interactions, indicating the presence of significant
+long-range magnetic interactions. Despite evidence for
+heavy fermion behavior, there is negligible change of TN
+upon applying pressures up 2.2 GPa, and hence measure-
+ments at much higher pressures are necessary to look for
+evidence of quantum criticality.
+VI.
+ACKNOWLEDGMENTS
+We are grateful to Martin Rotter for advice with the
+McPhase software. This work was supported by the Na-
+tional Key R&D Program of China (2017YFA0303100),
+the Key R&D Program of Zhejiang Province, China
+(2021C01002), and the National Natural Science Foun-
+dation of China (12174332, 12034017 and 11974306).
+∗ msmidman@zju.edu.cn
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+

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+arXiv:2301.04192v1  [math.AG]  10 Jan 2023
+QUANTIZATIONS OF LOCAL CALABI–YAU THREEFOLDS
+AND THEIR MODULI OF VECTOR BUNDLES
+E. BALLICO, E. GASPARIM, F. RUBILAR, B. SUZUKI
+Abstract. We describe the geometry of noncommutative deformations of local Calabi–Yau
+threefolds, showing that the choice of Poisson structure strongly influences the geometry of the
+quantum moduli space.
+Contents
+1.
+Introduction
+1
+2.
+Noncommutative deformations
+2
+3.
+Vector bundles on noncommutative deformations
+3
+4.
+Moduli of bundles on noncommutative deformations
+6
+5.
+Quantum moduli of bundles on W1
+8
+6.
+Quantum moduli of bundles on W2
+11
+Appendix A.
+Computations of H1
+15
+References
+16
+1. Introduction
+We discuss moduli of vector bundles on those noncommutative local Calabi–Yau threefolds that
+occur in noncommutative crepant resolutions of the generalised conifolds xy − znwm = 0. Such
+crepant resolutions require lines of type (−1, −1) and (−2, 0), that is, those locally modelled by
+W1 := Tot(OP1(−1) ⊕ OP1(−1))
+or
+W2 := Tot(OP1(−2) ⊕ OP1(0)).
+Their appearance is balanced in a precise sense described in [GKMR] so that no particular
+configuration of such lines is more likely to occur in a crepant resolution than any other.
+Our results show that the structure of the quantum moduli space (Def. 4.3) of vector bundles
+over a noncommutative deformation varies drastically depending on the choice of a Poisson
+structure.
+In the 2-dimensional case, [BG] described the geometry of noncommutative deformations of the
+local surfaces Zk := Tot(OP1(−k)), showing that the quantum moduli space of instantons over
+a noncommutative deformation (Zk, σ) can be viewed as the ´etale space of a constructible sheaf
+over the classical moduli space of instantons on Zk. While in 2 dimensions vector bundles occur
+as mathematical representations of instantons, in the 3-dimensional case vector bundles occur
+as mathematical descriptions of BPS states, with W1 and W2 appearing as building blocks, as
+described in [GKMR, GSTV, OSY].
+1
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+2
+In this work, we describe the geometry of noncommutative deformations W of a Calabi–Yau
+threefold W, showing that the quantum moduli space of vector bundles on together with the
+map taking a vector bundle on W to its classical limit
+Mℏ
+j(W, σ)
+Mj(W)
+has the structure of a constructible sheaf, whose rank and singularity set depend explicitly on
+the choice of noncommutative deformation. In particular, we describe the geometry of noncom-
+mutative deformations of some crepant resolutions. It is at this point yet unclear how these
+compare with Van den Bergh’s noncommutative crepant resolutions [V].
+To each Poisson structure σ on Wk, with k = 1 or k = 2 there corresponds a noncommutative
+deformation (Wk, Aσ) with Aσ = (O[[ℏ]], ⋆σ) where ⋆σ is the star product corresponding to σ. All
+of these Poisson structures were described in [BGKS] in terms of generators over global functions;
+when σ is one of such generators, we refer to it as a basic Poisson structure. There exist Poisson
+structures for which all brackets vanish on the first formal neighbourhood of P1 ⊂ Wk; we call
+them extremal Poisson structures, they behave very differently from the basic ones. Our main
+results are:
+Theorem (5.4,6.4). Let k = 1 or 2. If σ is an extremal Poisson structure on Wk, then the
+quantum moduli space Mℏ
+j(Wk, σ) can be viewed as the ´etale space of a constructible sheaf Ek of
+generic rank 2j − k − 1 over the classical moduli space Mj(Wk) with singular stalks of all ranks
+up to 4j − k − 4.
+If σ′ is another Poisson structure on Wk, then the corresponding sheaf E′
+k is a subsheaf of Ek,
+with the smallest possible sheaf occurring for basic Poisson structures.
+Theorem (5.2,6.2). Let k = 1 or 2. If σ is a basic Poisson structure on Wk, then the quantum
+moduli space Mℏ
+j(Wk, σ) and its classical limit are isomorphic:
+Mℏ
+j(Wk, σ) ≃ Mj(Wk) ≃ P4j−5.
+Therefore, comparing these results, we see that the choice of Poisson structure has a strong
+influence on the geometry of the quantum moduli space.
+2. Noncommutative deformations
+A holomorphic Poisson structure on a complex manifold (or smooth complex algebraic variety)
+X is given by a holomorphic bivector field σ ∈ H0(X, Λ2TX) whose Schouten–Nijenhuis bracket
+[σ, σ] ∈ H0(X, Λ3TX) is zero. The associated Poisson bracket is then given by the pairing ⟨ · , · ⟩
+between vector fields and forms {f, g}σ = ⟨σ, df ∧ dg⟩.
+To obtain a noncommutative deformation of X one must first promote the Poisson structure to
+a
+star product on X, that is, a C[[ℏ]]-bilinear associative product ⋆: OX[[ℏ]] × OX[[ℏ]] → OX[[ℏ]]
+which is of the form f ⋆ g = fg + �∞
+n=1 Bn(f, g) ℏn where the Bn are bidifferential operators.
+The pair (X, ⋆σ) is called a deformation quantization of (X, σ) when the star product on X
+satisfies B1(f, g) = {f, g}σ.
+For a holomorphic Poisson manifold (X, σ) with associated Poisson bracket { · , · }σ, the sheaf
+of formal functions with holomorphic coefficients on the quantization (X, ⋆σ) is
+Aσ := (O[[ℏ]], ⋆σ).
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+3
+We call Wk(σ) = (Wk, Aσ) a noncommutative deformation of Wk, and a vector bundle on a
+noncommutative deformation is by definition a locally free sheaf of Aσ-modules. These vector
+bundles and their moduli are our objects of study here.
+When we work with a fixed Poisson structure, we use the abbreviated notations A, { · , · } and
+⋆. We also use the cut to order n represented as A(n) = O[[ℏ]]/ℏn+1.
+The existence of star products on Poisson manifolds was proven in the seminal papers of Kontse-
+vich [Ko1, Ko2]. For a complex algebraic variety X with structure sheaf OX, if both H1(X, OX)
+and H2(X, OX) vanish, then there is a bijection
+{Poisson deformations of OX}/∼ ↔ {associative deformations of OX}/∼
+where ∼ denotes gauge equivalence [Y, Cor. 11.2]. These cohomological hypothesis are verified
+in the cases of W1 and W2 (but not for W3, see App. A). We now recall the basic properties of
+Poisson structures on Wk or k = 1, 2. All Poisson structures on Wk may be described by giving
+their generators over global functions. This is a consequence of the following result.
+Lemma 2.1. [BGKS, Prop. 1] Let X be a smooth complex threefold and σ a Poisson structure
+on X, then fσ is integrable for all f ∈ O(X).
+Local Calabi–Yau threefolds. For k ≥ 1, we set
+Wk = Tot(OP1(−k) ⊕ OP1(k − 2)).
+The canonical charts for the complex manifold structure of Wk is obtained by gluing the open
+sets
+U = C3
+{z,u1,u2}
+and
+V = C3
+{ξ,v1,v2}
+by the relation
+(ξ, v1, v2) = (z−1, zku1, z−k+2u2).
+All Poisson structures on W1 can be obtained using the following generators [BGKS, Thm. 3.2]
+σ1 = ∂z ∧ ∂u1,
+σ2 = ∂z ∧ ∂u2,
+σ3 = u1∂u1 ∧ ∂u2 − z∂z ∧ ∂u2,
+σ4 = u2∂u1 ∧ ∂u2 + z∂z ∧ ∂u1.
+The W1-Poisson structures σ1, σ2, σ3, σ4 are pairwise isomorphic.
+All Poisson structures on W2 can be obtained using the following generators [BGKS, Lem. 3]
+σ1 = ∂z ∧ ∂u1,
+σ2 = ∂z ∧ ∂u2,
+σ3 = z∂z ∧ ∂u2,
+σ4 = u1∂u1 ∧ ∂u2,
+σ5 = 2zu1∂u1 ∧ ∂u2 − z2∂z ∧ ∂u2.
+The W2-Poisson structures σ2 and σ5 on are isomorphic.
+Moreover, the Poisson structures
+σ1, σ2, σ3, σ4 on W2 are pairwise inequivalent, giving 4 distinct Poisson manifolds.
+3. Vector bundles on noncommutative deformations
+To discuss moduli of vector bundles on noncommutative deformations of Wk, for k = 1 or 2 we
+will consider those bundles that are formally algebraic.
+Definition 3.1. We say that p = � pnℏn ∈ O[[ℏ]] is formally algebraic if pn is a polynomial
+for every n. We say that a vector bundle over (Wk, σ) is formally algebraic if it is isomorphic to
+a vector bundle given by formally algebraic transition functions. In addition, if there exists N
+such that pn = 0 for all n > N, we then say that p is algebraic.
+Lemma 3.2. Let A be a deformation quantization of O. Then an A-module S is acyclic if and
+only if S = S/ℏS is acyclic.
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+4
+Proof. Consider the short exact sequence
+0 −→ S
+ℏ
+−→ S −→ S −→ 0.
+It gives, for j > 0 surjections
+Hj(X, S)
+ℏ
+−→ Hj(X, S) −→ 0.
+This immediately implies that Hj(X, S) = 0 for j > 0. The converse is immediate.
+□
+Notation 3.3. Let Wk be a noncommutative deformation of Wk. Denote by A(j) the line
+bundle over Wk with transition function z−j, hence the pull back of O(j) on P1.
+Proposition 3.4. For k = 1, 2 any line bundle on Wk is isomorphic to A(j) for some j ∈ Z,
+i.e., Pic(Wk) = Z when k = 1, 2.
+Proof. Let f = f0 + �∞
+n=1 �fn ℏn ∈ A∗(U ∩ V ) be the transition function for the line bundle L.
+Then there exist functions a0 ∈ O∗(U) and α0 ∈ O∗(V ) such that α0f0a0 = z−j and viewing
+a0 resp. α0 as elements in A∗(U) resp. A∗(V ) one has α0 ⋆ f ⋆ a0 = z−j + �∞
+n=1 fnℏn for some
+fn ∈ O(U ∩ V ). We may thus assume that the transition function of L is z−j + �∞
+n=1 fnℏn.
+To give an isomorphism L ≃ A(j) it suffices to define functions an ∈ O(U) and αn ∈ O(V )
+satisfying
+�1 + �∞
+n=1 αnℏn� ⋆
+�z−j + �∞
+n=1 fnℏn� ⋆
+�1 + �∞
+n=1 anℏn� = z−j.
+(3.5)
+Collecting terms by powers of ℏ, (3.5) is equivalent to the system of equations
+Sn + z−jan + z−jαn = 0
+n = 1, 2, . . .
+where Sn is a finite sum involving fi, Bi for i ≤ n, but only ai, αi for i < n. The first terms are
+S1 = f1
+S2 = f2 + α1f1 + a1f1 + B1
+�α1, z−j� + B1
+�z−j, a1
+� + α1z−ja1
+S3 = f3 + B2
+�α1, z−j� + B2
+�z−j, a1
+� + B1
+�α2, z−j�
++ B1
+�z−j, a2
+� + B1
+�α1, f1
+� + B1
+�α1, z−ja1
+� + B1
+�z−j, a1
+�
++ α2f1 + α2z−ja1 + α1f2 + α1f1a1 + α1z−ja2 + f2a1 + f1a2
+Since by Lem. A.1 we have H1(Wk, O) = 0 when k = 1, 2, we can solve these equations recur-
+sively, by defining an to cancel out all terms of zjSn having positive powers of z and setting
+αn = zjSn − an.
+□
+Note that this is essentially the same proof as [BG, Prop. 6.7], and it does not work for k ≥ 3,
+in fact Pic(W3) is much larger, see Lem. A.3.
+We now consider vector bundles of higher rank.
+Theorem 3.6. For k = 1, 2, vector bundles over Wk(σ) are filtrable.
+Proof. This is a generalisation of Ballico–Gasparim–K¨oppe [BGK1, Thm. 3.2] to the noncom-
+mutative case. Let E be a sheaf of A-modules. Lem. 3.2 gives that the classical limit E0 = E/ℏE
+is acyclic as a sheaf of A-modules (and equivalently as a sheaf of O-modules) if and only if E is
+acyclic as a sheaf of A-modules.
+Filtrability for a bundle E over Wk, for k = 1, 2 was proved in [K] and is obtained from the
+vanishing of cohomology groups Hi(Wk, E ⊗ SymnN ∗) for i = 1, 2, where N ∗ is the conormal
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+5
+bundle of ℓ ⊂ Wk and n > 0 are integers, the proof proceeds by induction on n.
+In the
+noncommutative case, let S denote the kernel of the projection A(n) → A(n−1). By construction
+we have that S/ℏS = SymnN ∗ and the required vanishing of cohomologies is guaranteed by
+Lem. 3.2.
+□
+The analogous proof does not work for W3, see [K, Rem. 3.13]. It is unknown whether bundles
+on Wk are filtrable when k ≥ 3.
+Remark 3.7. There are also some particular features happening only when k = 1.
+Every
+holomorphic vector bundle on W1 is algebraic [K, Thm. 3.10], and W1 is formally rigid [GKRS,
+Thm. 11]. In contrast, if k > 1, then Wk has as infinite-dimensional family of deformations. In
+particular, a deformation family for W2 can be given by (ξ, v1, v2) =
+�
+z−1, z2u1 + z �
+j>0 tjuj
+2, u2
+�
+[GKRS, Thm. 13] and this family contains infinitely many distinct manifolds [BGS, Thm. 1.13].
+Furthermore, for k > q > 0, Wk can be deformed to Wq [BGS, Thm. 1.28].
+For each Poisson manifold (Wk, σ), we want to study moduli spaces of vector bundles over
+(Wk, ⋆) where ⋆ is the corresponding star product.
+[K, Prop. 3.1] showed that a rank 2 bundle E on Wk with first Chern class c1(E) = 0 is deter-
+mined by a canonical transition matrix
+�zj
+p
+0
+z−j
+�
+where, using ǫ = 0, 1 we have:
+p =
+2j−2
+�
+s=ǫ
+2j−2−s
+�
+i=1−ǫ
+j−1
+�
+l=i+s−j+1
+pliszlui
+1us
+2
+for
+k = 1,
+(3.8)
+and
+p =
+∞
+�
+s=ǫ
+j−1
+�
+i=1−ǫ
+j−1
+�
+l=2i−j+1
+pliszlui
+1us
+2
+for
+k = 2.
+(3.9)
+Accordingly, for a noncommutative deformation (Wk, σ) we define the notion of canonical tran-
+sition matrix as:
+T =
+�zj
+p
+0
+z−j
+�
+with
+p =
+∞
+�
+n=0
+pnℏn ∈ Ext1(A(j), A(−j)).
+(3.10)
+Where we have that each pn can be given the same canonical form of the classical case, which
+can be seen using:
+Lemma 3.11. Let A be a deformation quantization of OWk with k = 1 or 2. There is an
+injective map of C-vector spaces
+Ext1
+A(A(j), A(−j))
+∞
+�
+n=0
+Ext1
+O(O(j), O(−j))ℏn ≃ Ext1
+O(O(j), O(−j))[[ℏ]]
+p = p0 +
+∞
+�
+n=1
+pnℏn
+(p0, p1ℏ, p2ℏ2, . . . )
+where pi ∈ Ext1(O(j), O(−j)).
+Proof. Ext1
+A(A(j), A(−j)) is the quotient of Ext1
+O(O(j), O(−j))[[ℏ]] by the relations
+qn ≃ qn + � pipn−i.
+□
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+6
+We wish to describe the structure of moduli spaces of vector bundles on Wk. Using the results
+of this section, we may proceed analogously to the classical (commutative) setup, to extract
+moduli spaces out of extension groups of line bundles, by considering extension classes up to
+bundle isomorphism.
+4. Moduli of bundles on noncommutative deformations
+We recall the notion of isomorphism of vector bundles on a noncommutative deformation of Wk.
+Definition 4.1. Let E and E′ be vector bundles over (Wk, σ) defined by transition matrices T
+and T ′ respectively. An isomorphism between E and E′ is given by a pair of matrices AU and
+AV with entries in Aσ(U) and Aσ(V ), respectively, which are invertible with respect to ⋆ and
+such that
+T ′ = AV ⋆ T ⋆ AU.
+Notation 4.2. Denoting by Ext1
+Alg(A(j), A(−j)) the subset of formally algebraic extension
+classes, we denote by Mj(Wk) the quotient
+Mj(Wk) := Ext1
+Alg(A(j), A(−j))/∼
+consisting of those classes of formally algebraic vector bundles (Def. 3.1), whose classical limit is a
+stable vector bundle of charge j. Here ∼ denotes bundle isomorphism as in Def. 4.1 and following
+[BGK2] stability means that the classical limit does not split on the 0-th formal neighbourhood.
+We denote by Mℏn
+j (Wk, σ) the moduli of bundles obtained by imposing the cut-off ℏn+1 = 0,
+that is, the superscript ℏn means quantised to level n.
+Note that Mj(Wk, σ) := Mℏ0
+j (Wk, σ) = Mj(Wk) recovers the classical moduli space obtained
+when ℏ = 0, while Mℏ
+j(Wk, σ) denotes the moduli on the first order quantization, which will be
+the focus of this work. Accordingly:
+Definition 4.3. We call Mj(Wk, σ) the classical moduli space and Mℏ
+j(Wk, σ) the quantum
+moduli space of bundles on Wk.
+Lemma 4.4. [BGS, Thm. 2.7] The classical moduli spaces of vector bundles of rank 2 and
+splitting type j on Wk has dimension 4j − 5.
+Definition 4.5. The splitting type of a vector bundle E on (Wk, σ) is the one of its classical
+limit [BG, Def. 5.2]. Hence, when the classical limit is an SL(2, C) bundle, the splitting type of
+E is the smallest integer j such that E can be written as an extension of A(j) by A(−j).
+We fix a splitting type j and look at rank 2 bundles on the first formal neighbourhood ℓ(1) of
+ℓ ≃ P1 ⊂ W1 together with their extensions up to first order in ℏ. We now calculate isomorphism
+classes. Let p + p′ℏ and q + q′ℏ be two extension classes in Ext1
+A(A(j), A(−j)) which are of
+splitting type j, i.e. in canonical U-coordinates p, p′, q, q′ are multiples of u1, u2.
+According to Def. 4.1 bundles defined by p + p′ℏ and q + q′ℏ are isomorphic, if there exist
+invertible matrices
+�a + a′ℏ
+b + b′ℏ
+c + c′ℏ
+d + d′ℏ
+�
+and
+�α + α′ℏ
+β + β′ℏ
+γ + γ′ℏ
+δ + δ′ℏ
+�
+whose entries are holomorphic on U and V , respectively, such that
+�α + α′ℏ
+β + β′ℏ
+γ + γ′ℏ
+δ + δ′ℏ
+�
+⋆
+�zj
+q + q′ℏ
+0
+z−j
+�
+=
+�zj
+p + p′ℏ
+0
+z−j
+�
+⋆
+�a + a′ℏ
+b + b′ℏ
+c + c′ℏ
+d + d′ℏ
+�
+.
+(4.6)
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+7
+We wish to determine the constraints such an isomorphism imposes on the coefficients of q and
+q′. This is more conveniently rewritten by multiplying by the right-inverse of
+�
+zj q+q′ℏ
+0
+z−j
+�
+, which
+(modulo ℏ2) is
+�z−j
+−q − q′ℏ + 2z−j{zj, q}ℏ
+0
+zj
+�
+.
+We have that the zero section ℓ ≃ P1 is cut out inside Wk by u1 = u2 = 0. Hence, the n-th
+formal neighbourhood of ℓ is by definition ℓ(n) = OW1
+In+1 where I =< u1, u2 >. So, on ℓ(1) we
+have that u2
+1 = u2
+2 = u1u2 = 0 and therefore we may write
+a = a0 + a1
+1u1 + a2
+1u2,
+α = α0 + α1
+1u1 + α2
+1u2,
+etc., where ai
+1, αi
+1, etc. are holomorphic functions of z.
+Following the details of the proof of [G, Prop. 3.3] we assume in (4.6) that a0 = α0, d0 = δ0 are
+constant and b = β = 0. Since we already know that on the classical limit the only equivalence
+on ℓ(1) is projectivization [G, Prop. 3.2], we assume p = q, keeping in mind a projectivization to
+be done in the end. We may also assume that the determinants of the changes of coordinates
+on the classical limit are 1. Accordingly, we rewrite (4.6) as:
+�α + α′ℏ
+β′ℏ
+γ + γ′ℏ
+δ + δ′ℏ
+�
+=
+�zj
+p + p′ℏ
+0
+z−j
+�
+⋆
+�a + a′ℏ
+b′ℏ
+c + c′ℏ
+d + d′ℏ
+�
+⋆
+�z−j
+−p − q′ℏ + 2{zj, p}z−jℏ
+0
+zj
+�
+(4.7)
+where a0 = d0 = α0 = δ0 = 1.
+Since we already know the moduli in the classical limit, we only need to study terms containing
+ℏ, which after multiplying are:
+(1, 1) = a′ + {zja, z−j} + {zj, a}z−j + {pc, z−j} + {p, c}z−j + (pc′ + p′c)z−j
+(1, 2) = {p, d}zj − {a, p}zj − {zj, a}p + {zj, p}a + {pd, zj} + 2z−j{zj, p}pc + z2jb′
+− (pa′ + q′a)zj + (pd′ + p′d)zj − (pc′ + p′c + q′c)p
+(2, 1) = z−2jc′
+(2, 2) = d′ + {z−jd, zj} + {z−j, d}zj − {z−jc, p} − {z−j, c}p − (pc′ + q′c)z−j + 2{zj, p}z−2jc.
+All four terms must be adjusted using the free variables to only contain expressions which are
+holomorphic on V to satisfy (4.7). For example, in the (2, 1) term this condition is satisfied
+precisely when c′ is a section of O(2j). Computing Poisson brackets, we see that the (1, 1) and
+(2, 2) terms can always be made holomorphic on V by appropriate choices of c and d′, leaving
+the coefficients of a′ free. We will need to use these free coefficients for the next step.
+It remains to analyse the (1, 2) term. Because we are working on the first formal neighbourhood
+of ℓ, terms in u2
+1, u1u2, u2
+2 or higher vanish (recall that we assume that p, p′, q′ are multiples of u1
+or u2). Since z2jb′ is there to cancel out any possible terms having power of z greater or equal
+to 2j, we remove it from the expression, keeping in mind that we only need to cancel out the
+coefficients of the monomials ziu1 and ziu2 with i ≤ 2j − 1 in the expression:
+(1, 2) = {p, d+a}zj −{zj, a}p+{zj, p}a+{pd, zj}+2z−j{zj, p}pc+p(d′−a′)zj +(p′−q′)zj. (�)
+To determine the quantum moduli spaces, we must verify what restrictions are imposed on q′ so
+that p′ and q′ define isomorphic bundles. Since this requires computing brackets, the analysis
+must be carried out separately for each noncommutative deformation.
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+8
+5. Quantum moduli of bundles on W1
+The Calabi–Yau threefold we consider in this section is the crepant resolution of the conifold
+singularity xy − zw = 0, that is,
+W1 := Tot(OP1(−1) ⊕ OP1(−1)).
+We will carry out calculations using the canonical coordinates W1 = U ∪ V where U ≃ C3 ≃ V
+with U = {z, u1, u2}, V = {ξ, v1, v2}, and change of coordinates on U ∩ V ≃ C∗ × C × C given
+by
+�ξ = z−1 ,
+v1 = zu1 ,
+v2 = zu2
+� .
+Consequently, global functions on W1 are generated over C by the monomials 1, u1, zu1, u2, zu2.
+For each specific noncommutative deformation (W1, Aσ), we wish to compare the quantum and
+classical moduli spaces of vector bundles, see Def. 4.3.
+This is part of the general quest to
+understand how deformations of a variety affect moduli of bundles on it, and it is worth noting
+that no commutative deformation of W1 is known to exits.
+For a rank 2 bundle E on a noncommutative deformation W1 with a canonical matrix
+�
+zj
+p
+0
+z−j
+�
+as in (3.10) where p = �∞
+n=0 pnℏn, expression (3.8) gives us the general form of the coefficients
+pn. In particular, on the first formal neighbourhood, we have:
+p =
+j−1
+�
+l=−j+2
+pl10zlu1 +
+j−1
+�
+l=−j+2
+pl01zlu2,
+(5.1)
+where p = 0 if j = 1.
+Each noncommutative deformation comes from some Poisson structure which determines the first
+order terms of the corresponding star product, see Sec. 2. The most basic Poisson structures σ
+on W1 are those which generate all others over global functions. We call these generators the
+basic Poisson structures.
+Theorem 5.2. If σ is a basic Poisson structure on W1, then the quantum moduli space Mℏ
+j(Wk, σ)
+and its classical limit are isomorphic:
+Mℏ
+j(W1, σ) ≃ Mj(W1) ≃ P4j−5.
+Proof. We perform the computations using the bracket σ1 = ∂z ∧ ∂u1; the choice of such a
+generator is irrelevant, since all the 4 generators give pairwise isomorphic Poisson manifolds. To
+obtain an isomorphism, we need to cancel out all coefficients of the terms
+z2u1, . . . , z2j−1u1
+and
+z2u2, . . . , z2j−1u2
+appearing in expression �. Calculating σ1 brackets, we have {zj, f} = jzj−1 ∂f
+∂u1
+, and following
+expressions for a and d coming from the classical part
+a = 1 + a1
+1u1 + a2
+1u2,
+d = 1 − a1
+1u1 − a2
+1u2,
+where ai
+1 and di
+1 are functions of z, gives ∂a
+∂u1
+= a1
+1,
+∂d
+∂u1
+= −a1
+1, so that {p, d} − {a, p}zj =
+−2
+�∂p
+∂za1
+1 − ∂a
+∂z
+∂p
+∂u1
+�
+zj. Therefore, expression � becomes
+� = −2
+�∂p
+∂z a1
+1 − ∂a
+∂z
+∂p
+∂u1
+�
+zj + 2j
+� ∂p
+∂u1
+(a1
+1u1 + a2
+1u2 + pc)
+�
+zj−1 + p(d′ − a′)zj + (p′ − q′)zj.
+Now we need to cancel out separately the coefficients of each monomial ziu1 and ziu2 for 2 ≤
+i ≤ 2j − 1, that is, all those terms potentially giving nonholomorphic functions. To determine
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+9
+the classes in the moduli space we need to verify what constraints are imposed on q′. Take for
+instance the monomial ziu1 in (p′d − q′a)zj. Since a′ remains free we can always choose its
+corresponding coefficient in order to cancel out the term in ziu1 in the entire expression of (1, 2).
+Indeed, notice that the expressions p(d′ − a′)zj and (p′d − q′a)zj contain monomials of the same
+orders, all of which may be adjusted to zero by choosing a′. Moreover the first three summands
+in � also contain the same list of monomials, hence may also be absorbed by the appropriate
+choices of coefficients of a, a′ and c.
+Since this process can be independently carried out for each monomial, we then conclude that
+the expression � can be made holomorphic on V for any choice of q′.
+Hence, there are no
+restrictions on q′. Thus, we obtain an equivalence p + p′ℏ ∼ p + q′ℏ for all q′ and the projection
+onto the classical limit (the first coordinate)
+π1 : Mℏ
+j(W1, σ) → Mj(W1)
+taking (p, p′) to p is an isomorphism. The isomorphism type of the moduli space is given in
+[BGS, Lem. 6.2] as P4j−5.
+□
+We now calculate the quantum moduli space for the particular choice of splitting type j = 2 and
+for a different choice of Poisson structure on W1. We use the notation p ∈ Mj(W1) to refer to
+a point in the classical moduli space, that is, a rank 2 bundle is labelled by its extension class.
+Example 5.3 (j = 2 and σ = u1σ1). Here we write
+p = p0zu1 + p1u1 + p2zu2 + p3u2,
+p′ = p′
+0zu1 + p′
+1u1 + p′
+2zu2 + p′
+3u2.
+for the first order part of the extension class, where we have renamed the coefficients to simplify
+notation ( p0 := p110, p1 := p010, p2 := p101, p3 := p001). Lem. 2.1 implies that σ = u1σ1 is also
+a Poisson structure on W1. With this choice, all brackets acquire an extra u1 in comparison
+to the bracket σ1 used in the proof of Thm. 5.2, so that in the first formal neighbourhood the
+(1, 2)-term described in � simplifies to just:
+� = z2p(d′ − a′) + z2(p′ − q′).
+Here a′ = a′
+0 + a′1u1 + a′2u2,
+d′ = d′
+0 − d′1u1 − d′2u2, so that
+d′ − a′ = (d′ − a′)0 + (d′ − a′)1u1 + (d′ − a′)2u2.
+Hence, the total expression of � is
+�
+=
+(p0z3u1 + p1z2u1 + p2z3u2 + p3z2u2)((d′ − a′)0 + (d′ − a′)1u1 + (d′ − a′)2u2))
++(p′
+0 − q′
+0)z3u1 + (p′
+1 − q′
+1)z2u1 + (p′
+2 − q′
+2)z3u2 + (p′
+3 − q′
+3)z2u2,
+where we canceled out all the monomials containing u2
+1, u1u2, and u2
+2, since we work on the first
+formal neighbourhood. We rename (d′ − a′)0(z) = λ0 + λ1z + λ2z2 + . . . to simplify notation,
+and since all terms in (1, 2) having powers of z equal to 4 and higher can be cancelled out by
+the appropriate choice of the z2jb′, it suffices to analyse the expression
+�
+=
+(p0z3u1 + p1z2u1 + p2z3u2 + p3z2u2)(λ0 + λ1z)
++(q′
+0 − p′
+0)z3u1 + (q′
+1 − p′
+1)z2u1 + (q′
+2 − p′
+2)z3u2 + (q′
+3 − p′
+3)z2u2.
+To have an isomorphism q′ ∼ p′, we need to cancel out the coefficients of z3u1, z2u1, z3u2, z2u2
+in � with appropriate choices of λi. Consequently, q′ ∼ p′ if and only if the following equality
+holds for some choice of λ0 and λ1:
+
+
+
+
+q′
+0 − p′
+0
+q′
+1 − p′
+1
+q′
+2 − p′
+2
+q′
+3 − p′
+3
+
+
+
+ = λ0
+
+
+
+
+p0
+p1
+p2
+p3
+
+
+
+ + λ1
+
+
+
+
+p1
+0
+p3
+0
+
+
+
+ .
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+10
+When the vectors v1 = (p0, p1, p2, p3) and v2 = (0, p1, 0, p3) are linearly independent, the point
+q′ belongs to the plane that passes through the point p′ with v1 and v2 as direction vectors.
+Therefore, whenever v1 and v2 are linearly independent vectors, the fibre over p = (p0, p1, p2, p3)
+is a copy of C4 foliated by 2-planes. The leaf containing a point p′ forms the equivalence class
+of p′. Thus, the moduli space over the fibre over p is parametrised by the 2-plane through the
+origin in the direction perpendicular to v1, v2 over the point p, except when p1 = p3 = 0.
+In contrast, the fibre over a point p = (p0, 0, p2, 0) is a copy of C4 foliated by lines in the
+direction of v1 = (p0, 0, p2, 0). In this case, the moduli space over p is parametrised by a copy of
+C3 perpendicular to v1.
+We conclude that Mℏ
+2(W1, σ) → M2(W1) ≃ P3 (where the isomorphism is given by Lem. 4.4) is
+the ´etale space of a constructible sheaf, whose stalks have
+• dimension 2 over the Zariski open set (p1, p3) ̸= (0, 0), and
+• dimension 3 over the P1 cut out by p1 = p3 = 0 in P3.
+The same techniques readily generalise to give a description of the quantum moduli spaces for
+other choices of noncommutative deformations.
+Theorem 5.4. If σ is an extremal Poisson structure on W1, then the quantum moduli space
+Mℏ
+j(W1, σ) can be viewed as the ´etale space of a constructible sheaf of generic rank 2j − 2 over
+the classical moduli space Mj(W1) with singular stalks up to rank 4j − 5.
+Proof. We give the details of the case j = 3, for an extremal Poisson structure, that is, the case
+when all brackets vanish on the first formal neighbourhood. The general case is clear from these
+calculations, just notationally more complicated.
+When j = 3 and σ = u1σ1, expression � becomes:
+� = p(d′ − a′)z3 + (p′d − q′a)z3,
+and we get a system of equations:
+�
+=
+�
+p0z5u1 + p1z4u1 + p2z3u1 + p3z2u1 + p4z5u2 + p5z4u2 + p6z3u2 + p7z2u2
+�
+·(λ0 + λ1z + λ2z2 + λ3z3 + λ4z4) +
++(p′ − q′)0z5u1 + (p′ − q′)1z4u1 + (p′ − q′)2z3u1 + (p′ − q′)3z2u1
++(p′ − q′)4z5u2 + (p′ − q′)5z4u2 + (p′ − q′)6z3u2 + (p′ − q′)7z2u2.
+To have an isomorphism q′ ∼ p′, we need to cancel out the coefficients of z5u1, z4u1, z3u1, z2u1,
+z5u2, z4u2, z3u2, z2u2 in � with appropriate choices of λi. Consequently, q′ ∼ p′ if and only if
+the following equality holds for some choice of λ0, λ1, λ2, λ3:
+
+
+
+
+
+
+
+
+
+
+
+
+
+q′
+0 − p′
+0
+q′
+1 − p′
+1
+q′
+2 − p′
+2
+q′
+3 − p′
+3
+q′
+4 − p′
+4
+q′
+5 − p′
+5
+q′
+6 − p′
+6
+q′
+7 − p′
+7
+
+
+
+
+
+
+
+
+
+
+
+
+
+= λ0
+
+
+
+
+
+
+
+
+
+
+
+
+
+p0
+p1
+p2
+p3
+p4
+p5
+p6
+p7
+
+
+
+
+
+
+
+
+
+
+
+
+
++ λ1
+
+
+
+
+
+
+
+
+
+
+
+
+
+p1
+p2
+p3
+0
+p5
+p6
+p7
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
++ λ2
+
+
+
+
+
+
+
+
+
+
+
+
+
+p2
+p3
+0
+0
+p6
+p7
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
++ λ3
+
+
+
+
+
+
+
+
+
+
+
+
+
+p3
+0
+0
+0
+p7
+0
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+.
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+11
+Consider now the family U of vector spaces over M2(W1) ≃ P7 whose fibre at p is given by
+Up =
+
+
+
+
+
+
+
+
+
+
+
+
+
+p0
+p1
+p2
+p3
+p1
+p2
+p3
+0
+p2
+p3
+0
+0
+p3
+0
+0
+0
+p4
+p5
+p6
+p7
+p5
+p6
+p7
+0
+p6
+p7
+0
+0
+p7
+0
+0
+0
+
+
+
+
+
+
+
+
+
+
+
+
+
+.
+Now, the quantum moduli space is obtained from this family after dividing by the equivalence
+relation ∼ over each point p. Hence
+Mℏ
+2(W1, σ) = U/ ∼ .
+We conclude that Mℏ
+2(W1, σ) → M2(W1) ≃ P7 (where the isomorphism is given by Lem. 4.4) is
+the ´etale space of a constructible sheaf or rank 4, with stalk at p having dimension equal to the
+corank of Up, in this case
+4 ≤ dim Mℏ
+2(W1, σ)p = 8 − rk Up ≤ 7.
+In the general case we have
+2j − 2 ≤ dim Mℏ
+j(W1, σ)p = corank Up =≤ 4j − 5.
+□
+6. Quantum moduli of bundles on W2
+The Calabi–Yau threefold we consider in this section is a crepant resolution of the singularity
+xy − w2 = 0 in C4, that is
+W2 := Tot(OP1(−2) ⊕ OP1) = Z2 × C.
+Similarly to what we did for W1, we will carry out calculations using the canonical coordinates
+W2 = U ∪V where U ≃ C3 ≃ V with U = {z, u1, u2}, V = {ξ, v1, v2}, and change of coordinates
+on U ∩ V ≃ C∗ × C × C given by
+�ξ = z−1 ,
+v1 = z2u1 ,
+v2 = u2
+� .
+Consequently, global holomorphic functions on W2 are generated by 1, u1, zu1, z2u1, u2.
+For each specific noncommutative deformation (W2, Aσ), we wish to compare the quantum and
+classical moduli spaces of vector bundles, see Def. 4.3.
+For a rank 2 bundle E on a noncommutative deformation W2 with a canonical matrix
+�
+zj
+p
+0
+z−j
+�
+as in (3.10) where p = �∞
+n=0 pnℏn, expression (3.8) gives us the general form of the coefficients
+pn. In particular, on the first formal neighbourhood, we have:
+p =
+j−1
+�
+l=−j+3
+pl10zlu1 +
+j−1
+�
+l=−j+1
+pl01zlu2
+(6.1)
+where in case j = 1 we have only p001u2.
+To describe the quantum moduli for Poisson structures on W2, we consider the expression �:
+� = {p, d + a}zj − {zj, a}p + {zj, p}a + {pd, zj} + 2z−j{zj, p}pc + p(d′ − a′)zj + (p′d − q′a)zj,
+where we need to cancel out the coefficients of z3u1, . . . , z2j−1u1
+and
+zu2, . . . , z2j−1u2.
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+12
+Each noncommutative deformation comes from some Poisson structure. The most basic Poisson
+structures σ on W2 are those which generate all others over global functions. We call these
+generators the basic Poisson structures. Now, we compute the quantum moduli of bundles for
+them.
+Remark. We observe that the 4 Poisson manifolds (W2, σi) for i = 1, 2, 3, 4, are pairwise
+nonisomorphic. This can be verified by the table of their degeneracy loci:
+W2 Poisson structures
+bracket
+degeneracy
+σ1
+σ2
+∅
+σ3
+σ4
+∪
+Nevertheless, the 4 quantum moduli spaces defined by these basic Poisson structures turn out to
+be all isomorphic.
+Theorem 6.2. If σ is a basic Poisson structure on W2, then the quantum moduli space Mℏ
+j(Wk, σ)
+and its classical limit are isomorphic:
+Mℏ
+j(W2, σ) ≃ Mj(W2) ≃ P4j−5.
+Proof. We carry out calculations for the basic bracket σ4 = u1∂u1 ∧ ∂u2. It does turn out that
+the result is the same for the the basic brackets. The calculation for σ4 is shorter, since any
+of the brackets having one entry equal to zj vanishes. Because we work on the first formal
+neighbourhood, we also remove the expressions that are quadratic in the ui variables.
+So, the expression � that remains to be analysed simplifies to:
+� = {p, d + a}zj + p(d′ − a′)zj + (p′d − q′a)zj,
+where we must cancel out the coefficients of the monomials z3u1, . . . , z2j−1u1 and zu2, . . . , z2j−1u2.
+On the first formal neighbourhood, we write
+a = 1 + a1(z)u1 + a2(z)u2,
+d = 1 + d1(z)u1 + d2(z)u2,
+and
+a′ = a′
+0(z) + a′
+1(z)u1 + a′
+2(z)u2,
+d′ = d′
+0(z) + d′
+1(z)u1 + d′
+2(z)u2,
+so that the partials are
+∂uia = ai(z)
+∂uid = di(z)
+and
+∂u2a = a2(z)
+∂u2d = d2(z).
+The extension class given in (3.9) becomes p =
+j−1
+�
+l=3−j
+pl10zlu1 +
+j−1
+�
+l=1−j
+pl01zlu2, and computing
+the bracket gives
+{p, d + a}zj =
+
+
+j−1
+�
+l=3−j
+pl10zl
+
+ (d2(z) + a2(z))zju1 +
+
+
+j−1
+�
+l=1−j
+pl01zl
+
+ (d1(z) + a1(z))zju1.
+To work with a simpler notation, we present details of � when j = 2, in which case we can
+express the extension class as
+p = p0zu1 + p1zu2 + p2u2 + p3z−1u2,
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+13
+having renamed the coefficients for simplicity (making p0 := p110, p1 := p101, p2 := p001, p3 :=
+p−101). We will point out the steps for generalising to higher j.
+Assuming j = 2, we have
+{p, d + a}z2 = p0(d2(z) + a2(z))z3u1 + (p1z3 + p2z2 + p3z)(d1(z) + a1(z))u1.
+To obtain equivalence between q′ and p′, we must cancel out coefficients of z3u1, zu2, z2u2, z3u2
+in the expression of �, which becomes
+�
+=
+p0(d2(z) + a2(z))z3u1 + (p1z3 + p2z2 + p3z)(d1(z) + a1(z))u1
++(p0z3u1 + p1z3u2 + p2z2u2 + p3zu2)(d′
+0(z) − a′
+0(z))
++(p′
+0z3u1 + p′
+1z3u2 + p′
+2z2u2 + p′
+3zu2)
+−(q′
+0z3u1 + q′
+1z3u2 + q′
+2z2u2 + q′
+3zu2).
+Since the highest power of z to be considered is 3, we observe that d2(z) + a2(z) may be chosen
+conveniently, we cancel out all terms in z3u1. We may also choose d1(z) + a1(z) = 0, leaving
+�
+=
+(p1z3u2 + p2z2u2 + p3zu2)(d′
+0(z) − a′
+0(z))
++(p′
+1z3u2 + p′
+2z2u2 + p′
+3zu2)
+−(q′
+1z3u2 + q��
+2z2u2 + q′
+3zu2).
+Now we may choose d′
+0 − a′
+0 appropriately to cancel out all terms in u2. We conclude that there
+are no conditions imposed on q′. In other words, here p + p′ℏ is equivalent to p + q′ℏ for any
+choice of q′. Hence, the quantum and classical moduli spaces are isomorphic.
+The generalisation to higher j works out similarly, we can first choose di + ai for i > 0 to cancel
+out the coefficients of u1 and then choose d′
+0 −a′
+0 to take care of the coefficients of u2. So, for all
+j using the bracket σ4 we conclude that the quantum and classical moduli spaces are isomorphic
+Mℏ
+j(W2, σ4) ≃ Mj(W2) ≃ P4j−5
+where the second isomorphism is proven in [K, Prop. 3.24].
+□
+Example 6.3. Now choose any Poisson structure of W2 for which all brackets in � vanish on
+neighbourhood 1, for example σ = u1σ4 = u2
+1∂u1 ∧ ∂u2 works. In such a case, the expression for
+� reduces to:
+� = p(d′ − a′)zj + (p′d − q′a)zj.
+Now, consider the case of j = 2, when we have:
+�
+=
+(p0z3u1 + p1z3u2 + p2z2u2 + p3zu2) + (d′
+0(z) − a′
+0(z))
++(p′
+0z3u1 + p′
+1z3u2 + p′
+2z2u2 + p′
+3zu2)
+−(q′
+0z3u1 + q′
+1z3u2 + q′
+2z2u2 + q′
+3zu2).
+Setting
+d′
+0(z) − a′
+0(z) = λ0 + λ1z + λ2z2,
+we get a system of equations:
+
+
+
+
+q′
+0 − p′
+0
+q′
+1 − p′
+1
+q′
+2 − p′
+2
+q′
+3 − p′
+3
+
+
+
+ =
+
+
+
+
+λ0
+0
+0
+0
+0
+λ0
+λ1
+λ2
+0
+0
+λ0
+λ1
+0
+0
+0
+λ0
+
+
+
+
+
+
+
+
+p0
+p1
+p2
+p3
+
+
+
+ .
+Since we can choose λ1 and λ2 to solve the second and third equations, we see that q′
+1 and q′
+2
+are free. Hence (q′
+0, q′
+1, q′
+2, q′
+3) ∼ λ0(q′
+0, ∗, ∗, q′
+3), and our system of equations reduces to
+�q′
+0 − p′
+0
+q′
+3 − p′
+3
+�
+= λ0
+�p0
+p3
+�
+,
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+14
+which is the parametric equation of a line in the (q′
+0, q′
+3)-plane whenever (p0, p3) ̸= (0, 0). The
+entire question of moduli now reduces to the 2-dimensional case, disregarding p1, p2 coordinates.
+If (p0, p3) ̸= (0, 0), then the equivalence class of q′ in the fibre over the point p is the 1-dimensional
+subspace L directed by the vector (p0, p3) and passing through (q′
+0, q′
+3) in the (p′
+0, p′
+3)-plane.
+If p0 = p3 = 0, then we must have the equality (q′
+0, q′
+3) = (p′
+0, p′
+3). So, its the equivalence class
+consists of a single point.
+Accordingly, the set of equivalence classes over p can be represented either by the line L⊥ by
+the origin perpendicular to L (directed by (−p3, p0) when (p0, p3) ̸= (0, 0) or else by the entire
+(p′
+0, p′
+3)-plane over (0, 0).
+We conclude that Mℏ
+2(W2, σ) → M2(W2) ≃ P3 (where the isomorphism is given by Lem. 4.4) is
+the ´etale space of a constructible sheaf, whose stalks have
+• dimension 1 over the Zariski open set (p0, p3) ̸= (0, 0), and
+• dimension 2 over the P1 cut out by p0 = p3 = 0 in P3.
+In fact, we could express this moduli space as a sheaf given by an extension of OP3(+1) by a
+torsion sheaf.
+Theorem 6.4. If σ is an extremal Poisson structure on W2, then the quantum moduli space
+Mℏ
+j(W2, σ) can be viewed as the ´etale space of a constructible sheaf of generic rank 2j − 3 over
+the classical moduli space Mj(W2) with singular stalks up to rank 4j − 6.
+Proof. Now, for j = 3, we write down the extremal example when the brackets vanish on the
+first formal neighbourhood. The generalisation of the extremal cases to all j becomes clear from
+this example. Where, assuming all brackets vanish on the first formal neighbourhood, we need
+to cancel out the coefficients of z3u1, . . . , z2j−1u1
+and
+zu2, . . . , z2j−1u2 in
+� = p(d′ − a′)zj + (p′d − q′a)zj.
+For j = 3 we have
+p =
+2
+�
+l=0
+pl10zlu1 +
+2
+�
+l=−2
+pl01zlu2,
+which we rewrite as
+p = p0z2u1 + p1zu1 + p2u1 + p3z2u2 + p4z1u2 + p5u2 + p6z–1u2 + p7z−2u2.
+Setting
+d′
+0(z) − a′
+0(z) = λ0 + λ1z + λ2z2 + λ3z3 + λ4z4,
+expression
+� = p(d′ − a′)z3 + (p′d − q′a)z3
+becomes
+�
+=
+�
+p0z5u1 + p1z4u1 + p2z3u1 + p3z5u2 + p4z4u2 + p5z3u2 + p6z2u2 + p7zu2
+�
+·(λ0 + λ1z + λ2z2 + λ3z3 + λ4z4)
++(p′ − q′)0z5u1 + (p′ − q′)1z4u1 + (p′ − q′)2z3u1
++(p′ − q′)3z5u2 + (p′ − q′)4z4u2 + (p′ − q′)5z3u2 + (p′ − q′)6z2u2 + (p′ − q′)7zu2.
+To start with we notice that λ3 and λ4 can always be chosen to solve the equations involving q′
+3
+and q′
+4 so that these 2 coordinates can take any value, that is, there are isomorphisms
+(q′
+0, q′
+1, q′
+2, q′
+3, q′
+4, q′
+5, q′
+6, q′
+7) ∼ (q′
+0, q′
+1, q′
+2, ∗, ∗, q′
+5, q′
+6, q′
+7).
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+15
+Consequently, we may remove q′
+3, q′
+4 and rewrite the reduced system as:
+
+
+
+
+
+
+
+
+
+q′
+0 − p′
+0
+q′
+1 − p′
+1
+q′
+2 − p′
+2
+q′
+5 − p���
+5
+q′
+6 − p′
+6
+q′
+7 − p′
+7
+
+
+
+
+
+
+
+
+
+= λ0
+
+
+
+
+
+
+
+
+
+p0
+p1
+p2
+p5
+p6
+p7
+
+
+
+
+
+
+
+
+
++ λ1
+
+
+
+
+
+
+
+
+
+p1
+p2
+0
+p6
+p7
+0
+
+
+
+
+
+
+
+
+
++ λ2
+
+
+
+
+
+
+
+
+
+p2
+0
+0
+p7
+0
+0
+
+
+
+
+
+
+
+
+
+.
+Here q′ ∼ p′ if and only if the equality holds for some choice of λ0, λ1, λ2. Consider now the
+family U of vector spaces over M2(W2) ≃ P7 whose fibre at p is given by
+Up =
+
+
+
+
+
+
+
+
+
+p0
+p1
+p2
+p1
+p2
+0
+p2
+0
+0
+p5
+p6
+p7
+p6
+p7
+0
+p7
+0
+0
+
+
+
+
+
+
+
+
+
+.
+Now, the quantum moduli space is obtained from this family after dividing by the equivalence
+relation ∼ over each point p. Hence
+Mℏ
+2(W2, σ) = U/ ∼ .
+We conclude that Mℏ
+2(W2, σ) → M2(W2) ≃ P7 (where the isomorphism is given by Lem. 4.4) is
+the ´etale space of a constructible sheaf, with stalk at p having dimension equal to the corank of
+Up, in this case
+3 ≤ dim Mℏ
+2(W2, σ)p = corank Up = 6 − rk Up ≤ 6.
+In the general case we then have
+2j − 3 ≤ dim Mℏ
+j(W2, σ)p = corank Up = 2j − rk Up ≤ 4j − 6.
+□
+Appendix A. Computations of H1
+Lemma A.1. H1(W1, O) = H1(W2, O) = 0.
+Proof. A 1-cocycle τ ∈ O(U ∩ V ) may be written in the form
+τU =
+∞
+�
+l=−∞
+∞
+�
+i=0
+∞
+�
+s=0
+τliszlui
+1us
+2.
+Since terms containing only positive powers of z are holomorphic on the U-chart
+τU ∼
+−1
+�
+l=−∞
+∞
+�
+i=0
+∞
+�
+s=0
+τliszlui
+1us
+2,
+where ∼ denotes cohomological equivalence. Changing to V coordinates we have
+τV =
+−1
+�
+l=−∞
+∞
+�
+i=0
+∞
+�
+s=0
+τlisξ−l+ki+(−k+2)svi
+1vs
+2,
+(A.2)
+where, for k = 1, 2 exponents of ξ are non-negative.
+Thus, τV is holomorphic on V , and
+τ ∼ 0.
+□
+Lemma A.3. H1(W3, O) is infinite dimensional over C.
+
+QUANTIZATION OF CALABI–YAU THREEFOLDS
+16
+Proof. As in the proof of Lem. A.1 we arrive at the expression (A.2) for the 1-cocycle τ on the
+V -chart, which in the case k = 3, gives
+τV ∼
+−1
+�
+l=−∞
+∞
+�
+i=0
+∞
+�
+s=0
+τlisξ−l+3i−svi
+1vs
+2.
+The terms that are not holomorphic on V are all of those satisfying −l + 3i − s < 0.
+We conclude that all terms having s > 3i − l, namely all of
+−1
+�
+l=−∞
+∞
+�
+i=0
+∞
+�
+s=3i−l+1
+τliszlui
+1us
+2
+are nontrivial in first cohomology, so that dim H1(W3, O) = ∞.
+□
+Acknowledgements. E. Ballico is a member of GNSAGA of INdAM (Italy). E. Gasparim
+acknowledges support of Vicerrector´ıa de Investigaci´on y Desarrollo Tecnol´ogico, UCN Chile.
+F. Rubilar acknowledges support of ANID-FAPESP cooperation 2019/13204-0. B. Suzuki was
+supported by Grant 2021/11750-7 S˜ao Paulo Research Foundation - FAPESP.
+References
+[BG]
+S. Barmeier, E. Gasparim, Quantization of local surfaces and rebel instantons, J. Noncommut. Geom.
+16 (2022) 311–351.
+[BGK1]
+E. Ballico, E. Gasparim, T. K¨oppe, Local moduli of holomorphic bundles, J. Pure Appl. Algebra 213
+n.4 (2009) 397–408.
+[BGK2]
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+acteristic. Comm. Algebra 37 n.8 (2009) 2688–2713.
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+threefolds, Rep. Math. Phys. 90 n.3 (2022) 299–324.
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+Rev. Colombiana Mat. 52 n.1 (2018)41–57.
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+alised Conifolds, Journal of Mathematical Physics, 58 (2017) 1–16.
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+294.
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+T. K¨oppe, Moduli of bundles on local surfaces and threefolds, PhD thesis, The University of Edinburgh
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+Ballico - Dept. Mathematics, Univ. of Trento, Povo Italy; ballico@science.unitn.it,
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+

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+Protecting the Texas power grid from tropical cyclones:
+Increasing resilience by protecting critical lines
+Julian Stürmer1,2, Anton Plietzsch1, Thomas Vogt1, Frank Hellmann1, Jürgen Kurths1,3,4,
+Christian Otto1,*, Katja Frieler, Mehrnaz Anvari1,*
+1Potsdam Institute for Climate Impact Research, Telegrafenberg A56, 14473 Potsdam,
+Germany
+2Institute for Theoretical Physics, TU Berlin, 10623 Germany
+3Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany
+4Institute of Physics, Humboldt Universität zu Berlin, 12489 Berlin, Germany
+*Corresponding author(s): anvari@pik-potsdam.de; christian.otto@pik-potsdam.de
+Abstract
+The Texan electric network in the Gulf Coast of the United States is frequently hit by Tropical
+Cyclones (TC) causing widespread power outages, a risk that is expected to substantially
+increase under global warming. Here, we introduce a new approach of combining a
+probabilistic line fragility model with a network model of the Texas grid to simulate the
+temporal evolution of wind-induced failures of transmission lines and the resulting cascading
+power outages from seven major historical hurricanes. The approach allows reproducing
+observed supply failures. In addition, compared to a static approach, it provides a significant
+advantage in identifying critical lines whose failure can trigger large supply shortages. We
+show that protecting only 1% of total lines can reduce the likelihood of the most destructive
+type of outages by a factor of between 5 and 20. The proposed modelling approach could
+represent a tool so far missing to effectively strengthen the power grids against future
+hurricane risks even under limited knowledge.
+Keywords: Electric networks, Extreme weather events (hurricane), Cascading failures
+Introduction
+Modern societies depend heavily on reliable access to electricity. Power outages have the
+potential to disrupt transportation and telecommunication networks, heating and health
+systems, the cooling chain underpinning food delivery and more1–3. Depending on the cause
+of power outages and the amount of physical damages to infrastructures, the recovery of
+the electric network, and the social infrastructures dependent on it, often takes days or even
+months4. Such outages are often driven by extreme weather events. In Norway
+of all
+overhead line failures are caused by extreme weather which involves strong winds, icing and
+lightning strikes5. In February 2021 a winter storm in Texas led to outages that in turn caused
+a breakdown of the gas supply and thus the heating sector6–8. Impacts are particularly
+
+devastating when it comes to tropical cyclones. In the summer months, the Gulf Coast and
+the East Coast of the United States are frequently hit by tropical cyclones (TC) that entail
+widespread outages and costs of billions of dollars. For example, hurricane Ike hitting
+southeast Texas on September 13, 2008 destroyed around 100 towers holding high voltage
+transmission lines and cut off electric power for between 2.8 and 4.5 million customers for
+weeks to months9, 10. On August 29, 2021 hurricane Ida made landfall in Louisiana, and
+destroyed major transmission lines delivering power into New Orleans, causing more than a
+million customers to lose power11.
+Resilience against line failures in power grids is usually discussed in terms of the N-1 (rarely
+also N-2) security of the system, that is, the ability of the system to stay fully functional upon
+the failure of one or two elements12. When a line fails, the power flow automatically
+reroutes through the intact grid. To avoid overloads in the rerouting, relevant lines are
+intentionally taken out of the grid. This secondary failures of lines can trigger a cascade13–19
+of additional failures. N-1 security asserts that single line failures do not trigger such
+cascades. Significant secondary failures do occur in larger events and were, e.g., observed in
+response to the software error leading to the U.S.-Canadian blackout on August 14th, 200320.
+They are typically also induced by the widespread primary damages and line failures caused
+by TCs.
+The N-1 approach to system resilience does not scale to extreme weather events. The tens
+or even hundreds of primary failures during events such as hurricanes can not be fully
+mitigated by an electric network, because N-100 security is not realistic to achieve. N-1
+security is typically studied by simulating the reaction of the system to every possible failure
+scenario. As the number of possible failure scenarios scales exponentially with the number
+of failures, it is computationally infeasible to consider all possible such scenarios in larger
+events. Initialising failure cascade models designed for N-1 studies with many initial failures
+is challenging.
+Here, we present an approach that solves these issues by temporally resolving the potential
+damages induced by hurricanes and a stepwise application of a failure cascade model. This
+approach particularly allows us to identify critical power lines whose protection could most
+effectively reduce the risk of severe widespread power outages. Although the frequency of
+severe hurricanes is expected to increase12–14, such an approach does not exist so far.
+Main text
+Our approach explicitly models the dynamical interplay of an extreme wind event with
+the power grid. It temporally resolves both, the primary wind damages, and the cascades
+and secondary failures that result from them. We will use this approach to study the
+impact of massive TCs on the Texan power grid. Strong hurricanes, such as Harvey that
+
+made landfall on Texas and Louisiana in August 2017, can destroy more than hundreds
+transmission lines in an electric grid (see Fig. 1(a)). These lines do not collapse
+simultaneously, but over the hours or days the TC passage takes. Making use of the
+chronological order of the line destructions, we divide each overall TC scenario into a
+sequence of 5 minute long scenarios. In most of these individual steps, only one line fails.
+We then solve individual scenarios by representing the Texan transmission network in a DC
+power flow approximation with conservative load balancing assumptions (see Methods and
+Supplementary Methods 3 and 4). This approach accounts for the ‘path dependency’ of the
+solution: Everytime a line collapses, secondary failures can occur, but also control
+mechanisms are immediately activated and try to bring back the energy balance to the
+system and, consequently, mitigate the effect of the failure (see Supplementary Methods 4).
+Later primary damages along the TC track then meet a partially destroyed, rebalanced grid.
+Thus, the effect of later failures can be more or, even, less intense. It is the resilience of
+these intermediate, partially destroyed states that ultimately decides whether the impact of
+the TC is amplified by secondary failures.
+Fig. 1: Probability distributions of primary line failures and final power outages (a)
+Probability distribution of the total number of wind-induced line failures
+as generated by
+the probabilistic line fragility model for each of the seven recent hurricanes hitting Texas
+(category in brackets behind the name). TCs are sorted according to the means of the
+distributions which are indicated as solid vertical lines. (b) Probability distribution of the
+associated total power outage
+after TC passage. The inset highlights large cascading
+failures that can also occur for the weaker hurricanes. The dashed vertical lines indicate the
+reported power outages listed in the Supplementary Table 1 and the solid vertical lines
+represent the means. See Methods section for the model parameters used in the
+simulations.
+Unfortunately, neither detailed information about the topology of the exposed power grid
+nor about the exact power lines destroyed by the considered TC is publically accessible. So
+
+a
+b
+Harvey (4)
+Ike (2)
+Claudette (1)
+Hanna (1)
+Erin (TS)
+Hermine (TS)
+(anod)d
+p(Np)
+Pout
+Laura (4)
+μp
+pout
+0
+20
+40
+60
+80
+100
+120
+140
+0
+10
+20
+30
+40
+Np
+pout [GW]here, we use a synthetic model of the Texan grid introduced by Bircheld et al21 (see
+Supplementary Fig. 2 as well as Methods).
+To represent the TCs impact on the energy supply we combine this grid model with a
+probabilistic line destruction model (see Methods) forced by modelled historical wind fields
+from seven different TCs (see Supplementary Supplementary Methods 2). The probabilistic
+model provides the probability of line failure in terms of wind speeds and allows to generate
+a large sample of temporally resolved realisations of line failure maps. In the default setting
+considered here we assume a homogeneous base failure rate for all transmission lines. This
+is our main adjustable parameter and is tuned to reproduce observed power outages (see
+Fig. 1(b) and Supplementary Methods 5). The TCs are selected to cover several different
+types of trajectories and intensities and particularly include storms that continue to move
+westward after landfall and affect the southern and western parts of Texas such as Hurricane
+Claudette, Tropical Storm Erin, and Hurricane Hanna, contrary to most hurricanes that are
+steered northward by the Coriolis effect before western parts of Texas are reached22.
+Core result
+While the number of primary line failures follows a Poisson binomial distribution, the
+derived distribution of outages is heavily multimodal for all storm tracks with the potential
+of large
+to
+outages (see Fig. 1(b)). These large damages turn out to not
+accumulate gradually over the course of the hurricane but occur suddenly in one or few time
+steps (see Fig. 2(b)). This sudden increase in outages is induced by cascading line failures
+taking the Houston and a weakly connected North-Western section of the grid offline (see
+Fig. 2(d) and Fig. 3).
+Figure 3 shows what damage patterns correspond to the various modes of the outage
+distribution. The disconnection of the North-West occurs due to the non-local effects of
+cascading failures in areas not directly affected by high wind speeds. For example, hurricanes
+Harvey and Hanna never reach this region, but cause a considerable probability of outages
+affected by Harvey (Fig. 3(a)-(c)), but also due to non-local cascades as seen for Hanna (Fig.
+3(h) and (i)). As the most populous city in Texas and a major load centre, the disconnection
+of Houston from the electrical networks causes the disconnection of a huge number of
+consumers from the electrical network and, consequently, the overproduction of generators
+located in the west of Texas, which have key roles to provide the required energy in Houston
+(see Supplementary Methods 5 and Supplementary Fig. 8). Interestingly, the northern part
+of the electric grid is never impacted by outages caused by these three hurricanes. Same
+figures for other hurricanes have been shown in Supplementary Fig. 7.
+
+Figure 2: Simulation of hurricane-induced cascading failures in the Texan electric grid (a)
+The schematic variation of the supplied load in an electric grid before (pre-hurricane), during
+(hurricane phase) and after (restoration phase) a hurricane is loosely based on ERCOT's23.
+The total power outage
+after a hurricane has passed, and the total energy
+(red
+area) that was not supplied are measures for the severity of an outage scenario. (b)
+Summary of all realisations of power outage trajectories simulated for hurricane Claudette
+(see Methods section and Supplementary Methods 5 for specification of the model
+parameters). Trajectories shown in red come in two types, those that aggregate damages
+gradually over time (Type I in the figure) and those that include a large cascade (Type II). The
+distribution of cascade sizes is multimodal and we use an empirical threshold of
+to define large cascades (see Supplementary Fig. 9). (c) and (d) show
+respectively the state of the power grid at the beginning and the end of the hurricane. These
+two states are shown in panel (b), for one realisation of primary line failures. Lines shown in
+black were destroyed by the hurricane or deactivated due to the secondary effects, for the
+other lines the relative line loading is shown, with red lines close to overload. In addition,
+the panel includes the track and a snapshot of the windfields of hurricane Claudette in blue.
+In the Supplement, we also provide a video of the simulation showing how the wind
+damages spread along the passage of hurricane Claudette.
+
+b
+a
+Hurricane
+Restorationphase
+phase
+Supplied load
+60
+Type
+ye
+[GW]
+Eout
+anod
+50
+p
+d
+40
+Time
+60
+70
+80
+90
+100
+110
+t [h]
+c
+d
+1.0
+40
+36
+pout = 0.0 GW
+36 -
+pout = 20.5 GW
+[Pc = 67.i GW
+P: = 46.6 GW
+35
+0.8
+34
+34 
+30
+25
+32 -
+32 
+0.6
+Windspeed [m/s]
+Latitude [°]
+Line Loading
+20
+30 
+30 -
+0.4
+15
+28
+28 -
+10
+ 0.2
+5
+Inactive Parts
+26
+26 -
+- Hurricane Track
+-104
+-102
+-100
+-98
+-96
+-94
+-104
+-102
+-100
+-98 
+-96
+-94
+0.0
+0
+Longitude [°]
+Longitude ["]noC
+15GMFigure 4: Probability of line failure for different parts of the total power outage
+distribution (a-i) Probability that the failure of a given power line is involved in three
+different modes of the power outage distribution. The modes are indicated by the insets and
+the exact range of considered power outages are shown below these insets. The
+probabilities
+are calculated as: number of realisations with a total outage within the
+specified range in each figure where the considered line failed / number of total realisations.
+The rows describe the probabilities for different hurricanes as indicated in the panel. Texan
+electric grid with grid elements colored according to their respective outage probability.
+The probability distributions shown in the insets are identical to the ones shown in Fig. 1(b) .
+
+a
+b
+p
+Cluster of
+pout E[6 GW, 12GW]
+generators
+pout E[23GW, 28GW]
+pout E[34GW, 44GW]
+Dallas
+Austin
+San Antonio
+Generators
+Houston
+Generators
+Generators
+Corpus
+Loads
+Loads
+Loads
+Christi
+Harvey
+Harvey
+Harvey
+d
+pout E[0GW
+pout E[3GW, 12 GW]
+pout E[15GW, 30GW]
+Generators
+Generators
+Generators
+Loads
+Loads
+Loads
+Claudette
+Claudette
+Claudette
+g
+h
+pout E[15GW, 30GW]
+Generators
+Generators
+Generators
+Loads
+Loads
+Loads
+Hanna
+Hanna
+Hanna
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+nodFor all seven hurricanes, the cascades play a major role in the total line failures associated
+with the event (see Fig. 4 ). They are induced by the overload of remaining lines and the
+isolation of grid elements, as well as the failure of islands with unavoidable overproduction.
+Figure 3: The probability of primary damages and secondary failures induced by hurricane
+Harvey In this plot the transmission lines are colored according to their high probability to
+be directly damaged by Harvey (blue lines) or to be deactivated due to the secondary effect
+of the hurricane (red lines). As expected the primary damages are located around the path
+of Harvey. However, secondary failures can occur far away from the hurricane track, which is
+related to the non-local effect of the primary damages in the power grid (see supplementary
+Methods 5). In this plot, grey lines have a higher probability of remaining operational than
+failing due to any reason.
+Our results are not sensitive to the assumption of a homogeneous base failure rate as similar
+characteristics are also derived when assuming randomised base failure rates (see
+Supplementary Table 3). In addition, a temporal resolution of 5 minutes turned out to be
+adequate as time steps where several lines fail are rare. At this resolution it is also
+reasonable to assume that cascades of secondary failures have run their course before
+further lines are destroyed by the hurricane24,25 (for further discussion regarding the
+temporal resolution, see Supplementary Note 2).
+Increasing Resilience
+The fact that large cascades are triggered by the failure of specific lines suggests targeting
+these lines for protection. To identify the critical lines that should be protected we define a
+
+36
+34
+32
+Latitude [°]
+30
+28
+Most probable line status
+Unaffected
+26
+Primary Damage
+Secondary Failure
+-104
+-102
+-100
+-98
+-96
+-94
+Longitude [°]priority index as the probability that the wind-induced damage of this specific line triggers a
+large cascade, that is, a cascade that increase the outage by more than 15 GW, averaged
+over all seven hurricanes (see Fig. 2(b) and Eq. (4) in Methods).
+As a baseline we also consider a conventional, static model (see Methods). The static index
+of a line is the conditional probability of a large outage given that the line is damaged by a
+TC. In both the co-evolution model and the static baseline (see Fig. 5(a) and (b)) the critical
+lines are mostly located around Houston.
+To estimate the reduction in power outages that can be reached by protecting critical lines,
+we order them according to their priority index and evaluate the impact of the TC on the
+system with the first one to twenty lines protected, e.g. by being replaced by underground
+cables. It is worth noting that the co-evolution priority index value for most transmission
+lines is zero. Only
+of them have a value above
+, and only
+lines above
+. By
+protecting these
+lines, large power outages and cascading failures are almost completely
+prevented for smaller storms and dramatically reduced for the larger ones (see Fig. 5 and
+Supplementary Fig.9). For the stronger hurricanes Harvey and Ike, the power outage
+distributions are shifted from the second peak to the first peak with
+(see
+Fig. 5(c)). Protecting the lines one by one shows that the reduction of the largest power
+outages improves smoothly, thus it is effective to protect up to twenty lines (see Fig. 5(c) and
+(d)). While in the original system damage amplification was almost guaranteed, it rarely
+occurs in the reinforced one. In summary
+of total lines reinforced leads to a 5 to 20 time
+reduction of the largest scale outages. The level of protection that can be reached by
+protecting the lines according to the priority index derived from the co-evolution models is
+generally higher than the protection of the same number of lines selected according to the
+priority index derived by the static model (see panel (d) of Fig. 5). The static baseline also
+identifies some of the most critical lines (see Supplementary 4), but additional protections
+stop being effective after the first 6-10 lines (see Fig. 5(c) and (d)). This demonstrates that
+the co-evolution model, with its detailed picture of the partially destroyed states, reveals
+genuinely new and critical information for increasing the resilience of the system.
+It is worth to mention that the results obtained from homogeneous base failure rates are
+similar to the randomised ones (see Supplementary Methods 5 and Supplementary Table 3).
+
+moC
+1OGMFigure 5: Level of risk reduction that can be reached by protecting power lines according to
+the priority index: The co-evolution model against the static model (a)-(b) 20 lines of the
+Texan power grid with the highest priority index (see Eq. Eq. (4)) obtained from the static
+model (orange lines), the co-evolution model (blue lines), and both approaches (green lines).
+The inset (b) shows a close-up view of Houston and Harris County, which contain most of the
+critical lines. As seen in (b) the critical lines obtained from both models are located in the
+same region, however, the co-evolutionary model identifies additional lines whose
+protection has a dramatic effect on increasing resilience. (c) Power outage distributions of
+hurricane Harvey in terms of the number of critical lines protected in both the co-evolution
+(blue) and the static model (orange). The second peak in the power outage distribution is
+strongly reduced as the number of protected lines increases. However, protecting lines
+obtained from the static model does not increase the resilience of the power grid as much as
+occurs in the co-evolution model. (d) Reduction of the large power outages obtained from
+both models. For all three strong hurricanes, i.e. Harvey, Ike and Claudette, the reduction in
+power outages is much greater in co-evolution model than the static one.
+
+b
+a
+31.0
+36 -
+30.5
+34 -
+32 
+30.0
+Latitude [°]
+Latitude [°]
+30
+29.5
+28 
+29.0
+20 most critical lines
+coevolution method
+26 -
+static method
+both methods
+28.5
+-104
+-102
+-100
+-98
+-96
+-94
+-97.0
+96.5
+96.0
+95.5
+95.0
+94.5
+Longitude [°]
+d
+Longitude [°]
+c
+Staticmethod
+1.0
+Coevolutionmethod
+0
+Number of protected lines
+0.8
+0.6
+10
+Method
+0.4
+coevolution
+static
+0.2
+20 
+Harvey
+ike
+Claudette
+0.0
+0
+10
+20
+30
+40
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+p(pout) [GW]
+Number of protected linesConclusion and outlook
+The co-evolution model of the Texan power grid has been introduced as an efficient
+approach to temporally resolve the line failures and secondary grid outages induced by TCs.
+The model can resolve to considerable detail the way secondary failure cascades amplify the
+impact of extreme events. Using this information it can be used to identify critical lines that
+should be protected to effectively increase the system's resilience and prevent the most
+severe outages. Our model goes significantly beyond the state of the art so far represented
+by statistical and economic models that can only capture a static picture of the event and
+the network24–29. We have seen that such static approaches do not easily identify all of the
+critical lines during extended events. Their importance is only revealed by stepwise ‘tracking’
+the destruction of the system and associated power outages and overloads. We expect that
+this co-evolution approach will also be a promising tool to understand and protect other
+grids exposed to spatio-temporally extended extreme events.
+The results of our study are in agreement with a recent TC related risk assessment for
+Texas26. Combining our priority index with additional information about the cost of a
+reinforcement of the considered lines could also enable the identification of the most cost
+efficient way to reduce the probability of power outages above a critical limit to an intended
+value (see Supplementary Methods 6).
+While the model based on wind speeds and historical hurricane tracks already identified
+crucial structures in the grid, the co-evolution approach could naturally be extended to more
+sophisticated models and broader settings. One particularly important goal for future
+research will be to drive the model with potential future storm tracks due to climate
+change27. As the frequency of particularly strong TCs is expected to increase under global
+warming (WGI contribution to the AR6), understanding what lines are critical in the face of
+the weather of the next decades is crucial. Another important avenue of broadening the
+model is to account for TC induced flooding (coastal flooding, pluvial or fluvial flooding) and
+associated destructions. These may follow a different temporal pattern where the adequacy
+of the approach proposed here has to be newly tested. This would also provide a first step
+towards an assessment of genuine compound events in which several stresses for the grid
+coincide.
+Methods
+Electric grid data of Texas
+For the study we used the publicly available electric grid test case ACTIVSg200028, that
+covers the area of the so-called ERCOT Interconnection, which supplies
+percent of the
+electricity demand in Texas29. The test case is synthetic but resembles fundamental
+
+properties of the real grid, such as the spatial distribution of power generation and
+demand21. It encompasses
+buses with geographic locations,
+branches (both
+transmission lines and transformers) and covers four different voltage levels. The test
+case comes with all required electrical parameters ranging from the power injections of
+buses to the power flow capacities of transmission lines and transformers. The flow
+capacities
+play a particularly important role for the simulation of cascading failures
+as they determine the amount of power that can be transported by individual lines and
+transformers without potentially damaging the equipment.
+Historical hurricane data
+Hurricane storm tracks are extracted from the International Best Track Archive for
+Climate Stewardship (IBTrACS)30, 31 as time series of cyclone center coordinates along
+with meteorological variables like maximum sustained wind speeds and minimum
+pressure on a
+h snapshot basis. For this study, a hand-picked selection of seven
+historical storms is used (Supplementary Fig. 1 and 2) to cover several different types of
+trajectories and intensities. Particularly, the selection also includes storms that continue
+to move westward after landfall and affect the southern and western parts of Texas (see
+Hurricane Claudette, Tropical Storm Erin, and Hurricane Hanna in Supplementary Fig. 2
+and the Supplementary Fig. 1), contrary to most hurricanes that are steered northward
+by the Coriolis effect before western parts of Texas are reached22. From the track records,
+we compute time series of wind fields within a radius of
+km from the storm center
+using the Holland model for surface winds, as implemented in the Python-package
+CLIMADA32, 33, at a spatial resolution of
+degrees (approximately
+km) and a
+temporal resolution of
+minutes. The intensities of the considered storms are also
+shown along the respective tracks in Supplementary Fig.1 while other properties of the
+storms are listed in Supplementary Table 1.
+Transmission line fragility model
+To model wind-induced failures of transmission lines, we first differentiate between
+overhead transmission lines and underground cables in the electric grid of Texas.
+Following Birchfield et al., we analyse lines that are shorter than
+km (
+miles)
+and connect a total load of at least
+MW as underground cables21. All other lines are
+assumed to be overhead transmission lines. The latter are then divided into segments of
+length
+m, which corresponds to the average distance between transmission
+towers in Texas34. Our fragility model assigns failure rates to individual line segments
+according to
+
+where,
+denotes the wind force acting on the line segment
+for a given wind
+speed
+and is calculated according to the guidelines published by the American Society
+of Civil Engineers35. The parameter
+represents the inverse of the so-called time to
+failure, which indicates how long a line segment can withstand a wind force equal to the
+breaking force
+. It is used as a free parameter to calibrate the model such that
+historically
+reported
+power
+outages
+are
+reproduced
+in
+our
+simulations
+(see
+Supplementary Methods 5). The full wind force equation as well as the meaning and the
+values of all parameters can be found in Supplementary Methods 2 and Supplementary
+Table 2. In all figures shown in the main text,
+. Using the failure rates
+, we define the probability that a line segment
+fails during the time interval
+as
+This failure probability is inspired by the line fragility model established by Winkler et al.,
+which assumes that the failure probability is proportional to the ratio of the wind force
+and the breaking force36. However, in contrast to their model, we define the failure
+probability
+using a time-dependent failure rate
+that allows us to take the time
+evolution of a field into account. A line is removed from the test case if any of its line
+segments fails during a time interval. It should be noted that multiple lines may be
+destroyed in the same time step, meaning that they are removed from the network
+simultaneously. According to Eq. (2), the probability of simultaneous failures increases
+with time step size
+. A discussion of the role of the time resolution can be found in
+Supplementary Note 2.
+Cascading failure model
+Wind-induced line failures can trigger cascades of overload failures in the branches of
+the electric grid. As cascading failures typically evolve on smaller time scales than the
+temporal resolution
+of the wind field, we can assume a time scale separation. When
+the network topology is changed by a primary damage event, the power flows
+on
+the branches are rerouted using the DC power flow model
+here,
+are the net active power injections at the buses,
+are the bus voltage angles
+and
+are the elements of the nodal susceptance matrix that comprises the network
+topology. More details on the assumptions of the DC power flow model and the software
+used can be found in Supplementary Methods 3. If the new state of the network exhibits
+any overloaded branch (
+), they are deactivated and the process is repeated.
+When the network reaches a state without overloads, the algorithm advances to the
+
+br
+0.002 hnext primary damage event. When a load or generator gets disconnected or the grid is
+split into several parts, the global active power balance (GAPB) has to be restored in each
+network component. Motivated by a primary frequency control in real electric grids, we
+adjust the outputs of generators uniformly, while respecting their output limits defined
+in the data set. Whenever the generator limits do not allow to fully restore the GAPB, we
+either conduct a uniform minimal load shedding or consider the blackout of the whole
+network component in the case of an unavoidable overproduction. The details of the
+algorithm are explained in Supplementary Methods 4.
+Quantification of power outages
+We use the following three different quantities to track the power outages arising in our
+simulations: (i)
+denotes the total supplied load at the end of each time step, i.e.,
+after the cascading algorithm finished, respectively. It is calculated by adding up the
+demands of all connected loads across all islands that exist at the given time. Since our
+co-evolution model assumes that cascading failures happen instantaneously,
+represents a step function for each individual TC scenario as shown in Fig. 2(b). We have
+simulated
+scenarios for each hurricane. (ii) Any cascading failure that actually causes
+a loss of supplied load results in a vertical transition of size
+in
+. One such
+transition is annotated with
+for the highlighted scenario in Fig. 2(b). (iii) All
+cascading failures that are triggered in a given TC scenario lead to a final power outage
+. The interesting statistics of
+are
+shown and discussed in Fig. 1(b) .
+Identification of critical lines
+We identify critical overhead transmission lines by means of a priority index defined for
+each line
+as
+where
+denotes the set of considered hurricanes (seven hurricanes in this study) and
+is the probability of a large cascade being triggered by the wind-induced failure of
+line
+. More specifically, we call cascades large or belonging to type II if their
+associated power outage
+lies above an empirical threshold of
+GW (indicated
+as type II in Fig. 2(b) and Fig. 5(d)). Eq. (4) includes an averaging over all considered
+hurricanes to discern lines that are critical for multiple hurricanes. This allows us to
+propose line reinforcements that increase the resilience not only for a particular
+hurricane. Some properties of the
+most critical lines found in this study are listed in
+
+Dou
+na
+0GM.67.GVSupplementary Table 3. Fig. 5(a) and (b) shows the location of these lines and
+demonstrates that reinforcing them indeed increases the resilience of the electric grid
+substantially. More details of the critical lines and a possibility to incorporate economic
+considerations into our analysis are discussed in Supplementary Methods 6.
+Baseline Method
+Here, we apply the static model as a baseline method. By static model, we mean that all
+primary damages occur simultaneously and, then, the DC power model along with global
+active power balance (see Supplementary Methods 6) are activated once to bring back
+the energy balance in the system and to evaluate the total final power outages
+. As
+discussed in Supplementary Note 2 the final power outage distributions are independent
+of the time resolution of the wind field, however the primary damages leading to large
+outages, i.e.
+to
+, can be completely different ones. To indicate the
+critical lines obtained from the static model, first, we separate all scenarios in which
+. Then, we use Eq. (4) to calculate the priority index of the primary
+damages leading to large cascades. The top
+lines with the highest priority index have
+been listed in Supplementary Table 5. As seen in this table, except for the six lines
+highlighted in red, the other lines are completely different from lines obtained from the
+co-evolution model.
+Code availability
+All code necessary to reproduce the findings in this work is openly available. The
+time-dependent wind fields are computed using the open-source platform CLIMADA32, 33.
+The implementation of the transmission line fragility and the DC power model is
+available from https://gitlab.pik-potsdam.de/stuermer/itcpg.jl.
+Data availability
+The observed TCs from IBTrACS30, 31 are distributed under the permissive WMO open data
+licence
+through
+the
+IBTrACS
+website
+(https://www.ncei.noaa.gov/products/international-best-track-archive)
+and
+can
+be
+directly retrieved through the CLIMADA32, 33 platform. The electrical network data is
+openly available from the Texas
+A&M University’s electric grid test case repository
+(https://electricgrids.engr.tamu.edu/electricgrid-test-cases/activsg2000/).
+Acknowledgements
+This project has received funding from the ConNDyNet2 project under grant no.
+03EF3055F. This research has received funding from the German Academic Scholarship
+Foundation and the German Federal Ministry of Education and Research (BMBF) under
+
+20 GM30 GMnoC
+15GMthe research projects QUIDIC (01LP1907A) and SLICE (FKZ: 01LA1829A), and from the
+CHIPS project, part of AXIS, an ERA-NET initiated by JPI Climate, funded by FORMAS
+(Sweden), DLR/BMBF (Germany, grant no. 01LS1904A), AEI (Spain) and ANR (France)
+with co-funding by the European Union (grant no. 776608).
+Author Contribution
+M. Anvari, F. Hellmann and C. Otto contributed to design and conceive the research. The
+co-evolution model is designed and developed by M. Anvari, J. Stürmer, A. Plietzsch and
+F. Hellmann. All simulations and data analyses of this work have been done by J.
+Stürmer and under supervision of M. Anvari. All hurricane data have been provided by
+T. Vogt during this research. All authors contributed to discussing and interpreting the
+results, and contributed to writing the manuscript.
+Competing Interests
+The authors declare that they have no competing interests.
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+

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+Vision Transformers Are Good Mask Auto-Labelers
+Shiyi Lan1
+Xitong Yang2
+Zhiding Yu1
+Zuxuan Wu3
+Jose M. Alvarez1
+Anima Anandkumar1,4
+1NVIDIA
+2Meta AI, FAIR
+3Fudan University
+4Caltech
+https://github.com/NVlabs/mask-auto-labeler
+Figure 1. Examples of mask pseudo-labels generated by Mask Auto-Labeler on COCO. Only human-annotated bounding boxes are
+used as supervision during training to obtain these results.
+Abstract
+We propose Mask Auto-Labeler (MAL), a high-quality
+Transformer-based mask auto-labeling framework for in-
+stance segmentation using only box annotations. MAL takes
+box-cropped images as inputs and conditionally generates
+their mask pseudo-labels.We show that Vision Transform-
+ers are good mask auto-labelers. Our method significantly
+reduces the gap between auto-labeling and human annota-
+tion regarding mask quality. Instance segmentation models
+trained using the MAL-generated masks can nearly match
+the performance of their fully-supervised counterparts, re-
+taining up to 97.4% performance of fully supervised mod-
+els. The best model achieves 44.1% mAP on COCO in-
+stance segmentation (test-dev 2017), outperforming state-
+of-the-art box-supervised methods by significant margins.
+Qualitative results indicate that masks produced by MAL
+are, in some cases, even better than human annotations.
+1. Introduction
+Computer vision has seen significant progress over the
+last decade. Tasks such as instance segmentation have made
+it possible to localize and segment objects with pixel-level
+accuracy.
+However, these tasks rely heavily on expan-
+sive human mask annotations. For instance, when creat-
+ing the COCO dataset, about 55k worker hours were spent
+on masks, which takes about 79% of the total annotation
+time [1]. Moreover, humans also make mistakes. Human
+annotations are often misaligned with actual object bound-
+aries. On complicated objects, human annotation quality
+tends to drop significantly if there is no quality control. Due
+to the expensive cost and difficulty of quality control, some
+other large-scale detection datasets such as Open Images [2]
+and Objects365 [3], only contain partial or even no instance
+segmentation labels.
+In light of these limitations, there is an increasing in-
+terest in pursuing box-supervised instance segmentation,
+where the goal is to predict object masks from bounding
+box supervision directly. Recent box-supervised instance
+segmentation methods [4–8] have shown promising perfor-
+mance. The emergence of these methods challenges the
+long-held belief that mask annotations are needed to train
+instance segmentation models. However, there is still a non-
+negligible gap between state-of-the-art approaches and their
+fully-supervised oracles.
+Our contributions: To address box-supervised instance
+segmentation, we introduce a two-phase framework consist-
+ing of a mask auto-labeling phase and an instance segmenta-
+tion training phase (see Fig. 2). We propose a Transformer-
+based mask auto-labeling framework, Mask Auto-Labeler
+(MAL), that takes Region-of-interest (RoI) images as inputs
+1
+arXiv:2301.03992v1  [cs.CV]  10 Jan 2023
+
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+0
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+200
+300
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+500
+600Box-supervised 
+Loss
+Cropped
+Regions
+Supervised
+Mask Loss
+Mask   Labels
+MAL
+Generate
+Masks
+Phase 1: Mask Auto-labeling
+Inst Seg
+Image
+Phase 2: Instance Segmentation Training
+Masks
+Figure 2.
+An overview of the two-phase framework of box-
+supervised instance segmentation. For the first phase, we train
+Mask Auto-Labeler using box supervision and conditionally gen-
+erate masks of the cropped regions in training images (top). We
+then train the instance segmentation models using the generated
+masks (bottom).
+and conditionally generates high-quality masks (demon-
+strated in Fig. 1) within the box. Our contributions can be
+summarized as follows:
+• Our two-phase framework presents a versatile design
+compatible with any instance segmentation architecture.
+Unlike existing methods, our framework is simple and
+agnostic to instance segmentation module designs.
+• We show that Vision Transformers (ViTs) used as image
+encoders yield surprisingly strong auto-labeling results.
+We also demonstrate that some specific designs in MAL,
+such as our attention-based decoder, multiple-instance
+learning with box expansion, and class-agnostic training,
+crucial for strong auto-labeling performance. Thanks to
+these components, MAL sometimes even surpasses hu-
+mans in annotation quality.
+• Using MAL-generated masks for training, instance seg-
+mentation models achieve up to 97.4% of their fully
+supervised performance on COCO and LVIS. Our re-
+sult significantly narrows down the gap between box-
+supervised and fully supervised approaches. We also
+demonstrate the outstanding open-vocabulary general-
+ization of MAL by labeling novel categories not seen
+during training.
+Our method outperforms all the existing state-of-the-
+art box-supervised instance segmentation methods by large
+margins. This might be attributed to good representations
+of ViTs and their emerging properties such as meaningful
+grouping [9], where we observe that the attention to objects
+might benefit our task significantly (demonstrated in Fig.
+6). We also hypothesize that our class-agnostic training de-
+sign enables MAL to focus on learning general grouping
+instead of focusing on category information. Our strong re-
+sults pave the way to remove the need for expensive human
+annotation for instance segmentation in real-world settings.
+2. Related work
+2.1. Vision Transformers
+Transformers were initially proposed in natural language
+processing [10].
+Vision Transformers [11] (ViTs) later
+emerged as highly competitive visual recognition models
+that use multi-head self-attention (MHSA) instead of con-
+volutions as the basic building block. These models are re-
+cently marked by their competitive performance in many vi-
+sual recognition tasks [12]. We broadly categorize existing
+ViTs into two classes: plain ViTs, and hierarchical ViTs.
+Standard Vision Transformers. Standard ViTs [11] are
+the first vision transformers. Standard ViTs have the sim-
+plest structures, which consist of a tokenization embedding
+layer followed by a sequence of MHSA layers. However,
+global MHSA layers can be heavy and usually face signif-
+icant optimization issues. To improve their performance,
+many designs and training recipes are proposed to train
+ViTs in data-efficient manners [9,13–19].
+Hierarchical Vision Transformers. Hierarchical Vision
+Transformers [12,20–22] are pyramid-shaped architectures
+that aim to benefit other tasks besides image classification
+with their multi-scale designs. On top of plain ViTs, these
+ViTs [20,21] separate their multi-head self-attention layers
+into hierarchical stages. Between the stages, there are spa-
+tial reduction layers, such as max-pooling layers. These ar-
+chitectures are usually mixed with convolutional layers [23]
+and often adopt efficient self-attention designs to deal with
+long sequence lengths.
+2.2. Instance segmentation
+Instance segmentation is a visual recognition task that
+predicts the bounding boxes and masks of objects.
+Fully supervised instance segmentation. In this setting,
+both bounding boxes and instance-level masks are provided
+as the supervision signals. Early works [24–27] follow a
+two-stage architecture that generates box proposals or seg-
+mentation proposals in the first stage and then produces the
+final segmentation and classification information in the sec-
+ond stage. Later, instance segmentation models are broadly
+divided into two categories: some continue the spirit of
+the two-stage design and extend it to multi-stage architec-
+tures [28, 29].
+Others simplify the architecture and pro-
+pose one-stage instance segmentation, e.g., YOLACT [30],
+SOLO [31, 32], CondInst [33], PolarMask [34, 35]. Re-
+cently, DETR and Deformable DETR [36, 37] show great
+potential of query-based approaches in object detection.
+Then, methods like MaxDeepLab [38], MaskFormer [39],
+2
+
+0
+100
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+400
+0
+100
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+300
+400
+500
+600…
+…
+…
+MHSA
+FFN
+FFN
+MaxPool
+FC
+*
+…
+…
+MHSA
+FFN
+FFN
+*
+Pos. Bags
+Neg. Bags
+Average
+Multiple Instance Learning Loss
++
+-
++ +
++
+++
+-- --
++
+++ +
++
+- -
+-- -
+Self Training
+EMA
+EMA
+Conditional Random Fields Loss
+𝑋!
+Neighbors 𝑋"
+Mean Field Algorithm
+MaxPool
+FC
+𝐸
+𝐷
+𝐷!
+𝐸!
+𝑉!
+𝑉
+𝐾!
+𝐾
+Task   Network
+Teacher   Network
+Figure 3. Overview of MAL architecture. We visualize the architecture of Mask Auto-Labeler. Mask Auto-Labeler takes cropped images
+as inputs. Mask Auto-Labeler consists of two symmetric networks, Task Network and Teacher Network. Each network contains the image
+encoder E(or Et), and the mask decoder D(or Dt). We use the exponential moving average (EMA) to update the weights of the teacher
+network. We apply multiple instance learning (MIL) loss and conditional random fields (CRFs) loss. The CRF loss takes the average mask
+predictions of the teacher network and the task network to make the training more stable and generate refined masks for self-training.
+PanopticSegFormer [40], Mask2Former [41] and Mask
+DINO [42] are introduced along this line and have pushed
+the boundary of instance segmentation. On the other hand,
+the instance segmentation also benefits from more power-
+ful backbone designs, such as Swin Transformers [12, 22],
+ViTDet [43], and ConvNeXt [44].
+Weakly supervised instance segmentation. There are two
+main styles of weakly supervised instance segmentation:
+learning with image-level and box-level labels. The former
+uses image-level class information to perform instance seg-
+mentation [45–49], while the latter uses box-supervision.
+Hsu et al. [4] leverages the tight-box priors. Later, Box-
+Inst [5] proposes to leverage color smoothness to improve
+accuracy. Besides that, DiscoBox [7] proposes to leverage
+both color smoothness and inter-image correspondence for
+the task. Other follow-ups [6,8] also leverage tight-box pri-
+ors and color smoothness priors.
+2.3. Deep learning interpretation
+The interest in a deeper understanding of deep net-
+works has inspired many works to study the interpreta-
+tion of deep neural networks.
+For example, Class Ac-
+tivation Map (CAM) [50] and Grad-CAM [51] visualize
+the emerging localization during image classification train-
+ing of convolutional neural networks (CNNs). This abil-
+ity has also inspired much weakly-supervised localization
+and shows deep connections to general weakly-supervised
+learning, which partly motivates our decoder design in this
+paper. DINO [9] further shows that meaning visual group-
+ing emerges during self-supervised learning with ViTs. In
+addition, FAN [52] shows that such emerging properties in
+ViTs are linked to their robustness.
+3. Method
+Our work differs from previous box-supervised instance
+segmentation frameworks [4–8] that simultaneously learns
+detection and instance segmentation. We leverage a two-
+phase framework as visualized in Fig. 2, which allows us to
+have a network focused on generating mask pseudo-labels
+in phase 1, and another network focused on learning in-
+stance segmentation [24, 28, 41, 43] in phase 2. Our pro-
+posed auto-labeling framework is used in phase 1 to gener-
+ate high-quality mask pseudo-labels.
+We propose this two-phase framework because it brings
+the following benefits:
+• We can relax the learning constraints in phase 1 and
+focus only on mask pseudo-labels. Therefore, in this
+phase, we can take Region-of-interest (RoI) images in-
+stead of untrimmed images as inputs. This change al-
+lows us to use a higher resolution for small objects and
+a strong training technique mentioned in Sec. 3.1, which
+helps improve the mask quality.
+• We can leverage different image encoders and mask de-
+coders in phases 1 and 2 to achieve higher performance.
+We empirically found that phases 1 and 2 favor different
+architectures for the image encoders and mask decoders.
+See the ablation study in Tab. 3 and 4.
+• We can use MAL-generated masks to directly train the
+most fully supervised instance segmentation models in
+phase 2. This makes our approach more flexible than
+previous architecture-specific box-supervised instance
+segmentation approaches [4–8].
+As phase 2 follows the previous standard pipelines,
+which do not need to be re-introduced here, we focus on
+introducing phase 1 (MAL) in the following subsections.
+3
+
+0
+50
+100
+150
+200
+250
+300
+350
+400
+0
+100
+200
+300
+400
+500
+6003.1. RoI input generation
+Most
+box-supervised
+instance
+segmentation
+ap-
+proaches [4–7] are trained using the entire images.
+However, we find that using RoI images might have
+more benefits in box-supervised instance segmentation.
+Moreover, we compare two intuitive sampling strategies
+of RoI images to obtain foreground and background pixels
+and explain the better strategy, box expansion, in detail.
+Benefits of using RoI inputs. There are two advantages of
+using RoI images for inputs. First, using the RoI images
+as inputs is naturally good for handling small objects be-
+cause no matter how small the objects are, the RoI images
+are enlarged to avoid the issues caused by low resolution.
+Secondly, having RoI inputs allows MAL to focus on learn-
+ing segmentation and avoid being distracted from learning
+other complicated tasks, e.g., object detection. RoI sam-
+pling strategy. The sampling strategy should ensure both
+positive and negative pixels are included. We present two
+straightforward sampling strategies:
+• The first strategy is to use bounding boxes to crop the
+images for positive inputs. We crop the images using
+randomly generated boxes containing only background
+pixels for negative inputs. MAL does not generate good
+mask pseudo-labels with cropping strategy. We observe
+that the networks tend to learn the trivial solution (all
+pixels are predicted as either foreground or background).
+• The second is to expand the bounding boxes randomly
+and include background pixels, where negative bags are
+chosen from the expanded rows and columns. We visu-
+alize how we define positive/negative bags in Fig. 3 and
+explain the detail in Sec. 3.3. This detailed design is
+critical to make MAL work as it prevents MAL from
+learning trivial solutions. Without this design, the gen-
+erated masks tend to fill the entire bounding box.
+Box expansion specifics. Given an untrimmed image Iu ∈
+RC×Hu×W u and the bounding box b = (x0, y0, x1, y1) in-
+dicating the x, y coordinates of the top-left corners and the
+bottom-right corners. To obtain background pixels, we ran-
+domly expand the bounding box b to b′ = (xc + βx(x0 −
+xc), yc +β′
+x(y0 −yc), xc +βy(x1 −xc), yc +β′
+y(y1 −yc)),
+where xc = (x0 + x1)/2, yc = (y0 + y1)/2. To gener-
+ate random values of βx, β′
+x, βy, β′
+y, we randomly generate
+θx, θy ∈ [0, θ] for x- and y-direction, where θ is the upper
+bound of box expansion rate. Next, we randomly generate
+βx ∈ [0, θx] and βy ∈ [0, θy]. In the end, we assign β′
+x as
+θx−βx and β′
+y as θy −βy. Finally, we use b′ to crop the im-
+age and obtain trimmed image It. We conduct the ablation
+study for θ in Tab. 5. At last, We resize the trimmed image
+It to the size of C × Hc × W c as the input image Ic.
+Class 
+Tokens
+k
+q
+v
+Transformer
+Layer
+*
+(a)
+(b)
+(c)
+(d)
+Figure 4. (a) The fully connected decoder (b) The fully convolu-
+tional Decoder (c) The attention-based decoder (used in MAL) (d)
+The query-based Decoder.
+3.2. MAL architecture
+MAL can be divided into two symmetric networks: the
+task network and the teacher network. The task network
+consists of an image encoder denoted as E, and a mask de-
+coder denoted as D, demonstrated in Fig. 3. The architec-
+ture of the teacher network is identical to the task network.
+We denote the segmentation output of the task network and
+the teacher network as m, mt ∈ {0, 1}N, respectively.
+Image encoder. We use Standard ViTs [11] as the image
+encoder and drop the classification head of Standard ViTs.
+We compare different image encoders in Sec. 4.4. We also
+try feature pyramid networks on top of Standard ViTs, e.g.,
+FPN [53], but it causes a performance drop. Similar con-
+clusions were also found in ViTDet [43].
+Mask decoder. For the mask decoder D, we use a simple
+attention-based network inspired by YOLACT [30], which
+includes an instance-aware head K and a pixel-wise head
+V , where D(E(I)) = K(E(I)) · V (E(I)), and “ · ” repre-
+sents the inner-product operator.
+For the instance-aware head K, we use a max-pooling
+layer followed by a fully connected layer. The input chan-
+nel dimension of K is equivalent to the output channel di-
+mension of E. The output channel dimension of K is 256.
+For the pixel-wise head V , we use four sequential convo-
+lutional layers. Each is followed by a ReLU layer. Between
+the second and the third convolutional layer, we insert a bi-
+linear interpolation layer to increase the feature resolution
+by 2. The input channel dimension is equivalent to the out-
+put channel dimension of E. We use 256 dimensions for
+hidden channels and output channels. We also compare dif-
+ferent design choices of mask decoders in Sec. 4.5.
+Exponential moving average (EMA) teacher. Instead of
+training the teacher network directly, we leverage exponen-
+tial moving averages (EMA) to update the parameters in the
+teacher network using the parameters in the task network
+similar to MOCO [54]. The goal of using EMA Teacher
+is to eliminate the loss-explosion issues in training since
+optimizing Standard Vision Transformers is usually non-
+trivial [13, 14, 16]. We do not observe any significant per-
+formance drop or improvement on DeiT-small-based MAL
+after removing the teacher network. However, it makes the
+training more stable when we use larger-scale image en-
+coders in MAL, e.g. ViT-MAE-Base [13].
+4
+
+3.3. Losses
+We use Multiple Instance Learning Loss Lmil and Con-
+ditional Random Field Loss Lcrf as the box-supervised loss:
+L = αmilLmil + αcrfLcrf
+(1)
+Multiple Instance Learning Loss. The motivation of the
+Multiple Instance Segmentation is to exploit the priors of
+tight-bounding box annotations.
+After the student network produces the output m, we ap-
+ply the Multiple Instance Learning (MIL) Loss on the out-
+put mask m. We demonstrate the process in Fig. 3.
+We denote mi,j as the mask score at the location i, j in
+the image Ic. We define each pixel as an instance in the
+MIL loss. Inspired by BBTP [4], we treat each row or col-
+umn of pixels as a bag. We determine whether a bag is pos-
+itive or negative based on whether it passes a ground-truth
+box. We define the bags as B, and each bag Bi contains a
+row or column of pixels. Additionally, we define the label
+for each bag g, and each label gi corresponds to a bag Bi.
+Therefore, we use the max pooling as the reduction func-
+tion and dice loss [55]:
+Lmil = 1 −
+2 �
+i gi · max{Bi}2
+�
+i max{Bi}2 + �
+i g2
+i
+(2)
+Conditional Random Field Loss. The goal of CRF loss
+is to refine the mask prediction by imposing the smooth-
+ness priors via energy minimization. Then, we leverage this
+refined mask as pseudo-labels to self-train the mask predic-
+tion in an online-teacher manner. We use the average mask
+prediction ma = 1
+2(m + mt) as the mask prediction to be
+refined for more stable training.
+Next, we define a random field X = {X1, ..., XN},
+where N = Hc × W c is the size of cropped image and
+each Xi represents the label that corresponds to a pixel in
+Ic, therefore we have X ∈ {0, 1}N, meaning the back-
+ground or the foreground. We use l ∈ {0, 1}N to represent
+a labeling of X minimizing the following CRF energy:
+E(l|ma, Xc) = µ(X|ma, Ic) + ψ(X|Ic),
+(3)
+where µ(X|ma, Ic) represents the unary potentials, which
+is used to align Xi and ma
+i since we assume that most of
+the mask predictions are correct. Meanwhile, ψ(X|Ic) rep-
+resents the pairwise potential, which sharpens the refined
+mask. Specifically, we define the pairwise potentials as:
+ψ(X|Ic) =
+�
+i∈{0..N−1},
+j∈N (i)
+ω exp(−|Ic
+i − Ic
+j |2
+2ζ2
+)[Xi ̸= Xj], (4)
+where N(i) represents the set of 8 immediate neighbors to
+Xi as shown in Fig. 3. Then, we use the MeanField al-
+gorithm [7, 56] to efficiently approximate the optimal so-
+lution, denoted as l = MeanField(Ic, ma). We attach
+the derivation and PyTorch code in the supplementary. At
+last, we apply Dice Loss to leverage the refined masks l to
+self-train the models as:
+Lcrf = 1 − 2 �
+i limi
+�
+i l2
+i + m2
+i
+(5)
+4. Experiments
+We evaluate MAL on COCO dataset [1], and LVIS [57].
+The main results on COCO and LVIS are shown in Tab. 1
+and 2. The qualitative results are shown in Fig. 1 and Fig. 5.
+4.1. Datasets
+COCO dataset. contains 80 semantic categories. We fol-
+low the standard partition, which includes train2017 (115K
+images), val2017 (5K images), and test-dev (20k images).
+LVIS dataset. contains 1200+ categories and 164K images.
+We follow the standard partition of training and validation.
+4.2. Implementation Details
+We use 8 NVIDIA Tesla V100s to run the experiments.
+Phase 1 (mask auto-labeling). We use AdamW [58] as
+the network optimizer and set the two momentums as 0.9,
+0.9. We use the cosine and annealing scheduler to adjust the
+learning rate, which is set to 1.5 · 10−6 per image. The MIL
+loss weight αmil, CRF loss weight αcrf, ζ, and ω in CRF
+pairwise potentials are set to 4, 0.5, 0.5, 2, respectively. We
+analyze the sensitivity of the loss weights and CRF hyper-
+parameters in Fig. 8. We use the input resolution of 512 ×
+512, and a batch size of 32 (4 per GPU). For EMA, we use
+a momentum of 0.996. For the task and teacher network,
+we apply random flip data augmentation. On top of that,
+we apply extra random color jittering, random grey-scale
+conversion, and random Gaussian blur for the task network.
+We train MAL for 10 epochs. It takes around 23 hours and
+35 hours to train MAL with Standard ViT-Base [11] on the
+COCO and LVIS datasets, respectively.
+Phase 2 (Training instance segmentation models). We
+select a couple of high-performance fully supervised in-
+stance segmentation models, which are ConvNeXts [44]
+with Cascade R-CNN [28], Swin Transformers [12] with
+Mask2Former [41], ResNets [59] and ResNeXts [60] with
+SOLOv2 [31]. MAL works extremely well with these ar-
+chitectures, which demonstrates the great power of Mask
+Auto-Labeler from the perspective of accuracy and gener-
+alization. We leverage the codebase in MMDetection [61]
+for phase 2. Again, we only replace the GT masks with
+MAL-generated mask pseudo-labels to adjust all these fully
+supervised models to box-supervised learning.
+4.3. Instance segmentation results
+Retention Rate. We argue that the sole mAP of instance
+segmentation is not fair enough to evaluate box-supervised
+instance segmentation since the performance gain can be
+5
+
+Method
+Labeler Backbone
+InstSeg Backbone
+InstSeg Model
+Sup
+(%)Mask APval
+(%)Mask APtest
+(%)Ret.val
+(%)Ret.test
+Mask R-CNN∗ [24]
+-
+ResNet-101
+Mask R-CNN
+Mask
+38.6
+38.8
+-
+-
+Mask R-CNN∗ [24]
+-
+ResNeXt-101
+Mask R-CNN
+Mask
+39.5
+39.9
+-
+-
+CondInst [33]
+-
+ResNet-101
+CondInst
+Mask
+38.6
+39.1
+-
+-
+SOLOv2 [31]
+-
+ResNet-50
+SOLOv2
+Mask
+37.5
+38.4
+-
+-
+SOLOv2 [31]
+-
+ResNet-101-DCN
+SOLOv2
+Mask
+41.7
+41.8
+-
+-
+SOLOv2 [31]
+-
+ResNeXt-101-DCN
+SOLOv2
+Mask
+42.4
+42.7
+-
+-
+ConvNeXt [44]
+-
+ConvNeXt-Small [44]
+Cascade R-CNN
+Mask
+44.8
+45.5
+-
+-
+ConvNeXt [44]
+-
+ConvNeXt-Base [44]
+Cascade R-CNN
+Mask
+45.4
+46.1
+-
+-
+Mask2Former [41]
+-
+Swin-Small
+Mask2Former
+Mask
+46.1
+47.0
+-
+-
+BBTP† [4]
+-
+ResNet-101
+Mask R-CNN
+Box
+-
+21.1
+-
+59.1
+BoxInst [5]
+-
+ResNet-101
+CondInst
+Box
+33.0
+33.2
+85.5
+84.9
+BoxLevelSet [6]
+-
+ResNet-101-DCN
+SOLOv2
+Box
+35.0
+35.4
+83.9
+83.5
+DiscoBox [7]
+-
+ResNet-50
+SOLOv2
+Box
+30.7
+32.0
+81.9
+83.3
+DiscoBox [7]
+-
+ResNet-101-DCN
+SOLOv2
+Box
+35.3
+35.8
+84.7
+85.9
+DiscoBox [7]
+-
+ResNeXt-101-DCN
+SOLOv2
+Box
+37.3
+37.9
+88.0
+88.8
+BoxTeacher [8]
+-
+Swin-Base
+CondInst
+Box
+-
+40.0
+-
+-
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ResNet-50
+SOLOv2
+Box
+35.0
+35.7
+93.3
+93.0
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ResNet-101-DCN
+SOLOv2
+Box
+38.2
+38.7
+91.6
+92.6
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ResNeXt-101-DCN
+SOLOv2
+Box
+38.9
+39.1
+91.7
+91.6
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ConvNeXt-Small [44]
+Cascade R-CNN
+Box
+42.3
+43.0
+94.4
+94.5
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ConvNeXt-Base [44]
+Cascade R-CNN
+Box
+42.9
+43.3
+94.5
+93.9
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+Swin-Small [12]
+Mask2Former [41]
+Box
+43.3
+44.1
+93.9
+93.8
+Table 1. Main results on COCO. Ret means the retention rate of box-supervised mask AP
+supervised mask AP . MAL with SOLOv2/ResNeXt-101 outperforms
+DiscoBox with SOLOv2/ResNeXt-101 by 1.6% on val2017 and 1.3% on test-dev. Our best model (Mask2former/Swin-Small) achieves
+43.3% AP on val and 44.1% AP on test-dev.
+achieved by improving box quality unrelated to segmenta-
+tion quality. However, the retention rate can better reflect
+the real mask quality because the fully supervised counter-
+parts also get boosted by the better box results.
+Results on COCO. In table 1, we show that various mod-
+ern instance segmentation models can achieve up to 94.5%
+performance with the pseudo-labels of the fully supervised
+oracles. Our best results are 43.3% mAP on COCO test-dev
+and 44.1% mAP on COCO val, achieved by using MAL
+(Standard ViT-Base [11] pretrained with MAE) for phase
+1, and using Mask2Former (Swin-Small) [12,41] for phase
+2. There is no significant retention drop when we use the
+mask pseudo-labels to train more powerful instance seg-
+mentation models. On the contrary, the higher retention
+rates on COCO are achieved by the heavier instance seg-
+mentation models, e.g., Cascade R-CNN with ConvNeXts
+and Mask2Former with Swin-Small. However, other meth-
+ods have significantly lower retention rates compared with
+MAL. The experiment results quantitatively imply that the
+mask quality outperforms other methods by a large margin.
+Results on LVIS. In table 2, we also observe that all in-
+stance segmentation models work very well with the mask
+pseudo-labels generated by MAL (Ret. = 93% ˜ 98%). We
+visualize part of the results in figure 5. We also evaluate
+the open-vocabulary ability of MAL by training MAL on
+COCO dataset but generating mask pseudo-labels on LVIS,
+and thus training instance segmentation models using these
+mask pseudo-labels.
+4.4. Image encoder variation
+To support our claim that Vision Transformers are good
+auto-labelers, we compare three popular networks as the im-
+age encoders of MAL: Standard Vision Transformers [11,
+13,16], Swin Transformer [12], ConvNeXts [44] in Tab. 4.
+First,
+we compare the fully supervised pretrained
+weights of these three models. We choose the official fully
+supervised pre-trained weights of ConvNeXts and Swin
+Transformers. For Standard Vision Transformers, we adopt
+a popular fully supervised approach, DeiT [16]. We ob-
+serve that fully supervised Standard Vision Transformers
+(DeiT) as image encoders of Mask Auto-Labeler are better
+than Swin Transformers and ConvNeXts even though the
+imaganet-1k performance of Swin Transformers and Con-
+vNeXts is higher than that of DeiT. We argue that the suc-
+cess of Standard Vision Transformers might be owed to the
+self-emerging properties of Standard ViTs [9, 11] (visual-
+ized in Fig. 6), and the larger-receptive field brought by
+global multi-head self-attention layers.
+Second, the mask pseudo-labels can be further improved
+by Mask AutoEncoder (MAE) pretraining [13]. The poten-
+tial reason might be that MAE pretraining enhances Stan-
+dard ViTs via learning pixel-level information, which is
+very important for dense-prediction tasks like segmentation.
+4.5. Mask decoder variation
+We compare four different modern designs of mask de-
+coders: the fully connected Decoder [62], the fully convolu-
+tional decoder [24,63], the attention-based decoder [30,31],
+and the query-based decoder [41] in Tab.
+3. We visual-
+ize different designs of mask decoders in Figure 4. For the
+fully connected Decoder, we use two fully connected layers
+with a hidden dimension of 2048 and then output a confi-
+dence map for each pixel. We reshape this output vector as
+the 2D confidence map. We introduce the attention-based
+decoder in Sec 3.2. For the fully convolutional Decoder,
+We adopt the pixel-wise head V in the attention-based De-
+6
+
+Figure 5. Qualitative results of mask pseudo-labels generated by Mask Auto-Labeler on LVIS v1.
+Method
+Autolabeler Backbone
+InstSeg Backbone
+InstSeg Model
+Training Data
+Sup
+(%)Mask APval
+(%)Ret.val
+Mask R-CNN [24]
+-
+ResNet-50-DCN
+Mask R-CNN [24]
+-
+Mask
+21.7
+-
+Mask R-CNN [24]
+-
+ResNet-101-DCN
+Mask R-CNN [24]
+-
+Mask
+23.6
+-
+Mask R-CNN [24]
+-
+ResNeXt-101-32x4d-FPN
+Mask R-CNN [24]
+-
+Mask
+25.5
+-
+Mask R-CNN [24]
+-
+ResNeXt-101-64x4d-FPN
+Mask R-CNN [24]
+-
+Mask
+25.8
+-
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ResNet-50-DCN
+Mask R-CNN [24]
+LVIS v1
+Box
+20.7
+95.4
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ResNet-101-DCN
+Mask R-CNN [24]
+LVIS v1
+Box
+23.0
+97.4
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ResNeXt-101-32x4d-FPN
+Mask R-CNN [24]
+LVIS v1
+Box
+23.7
+92.9
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ResNeXt-101-64x4d-FPN
+Mask R-CNN [24]
+LVIS v1
+Box
+24.5
+95.0
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ResNeXt-101-32x4d-FPN
+Mask R-CNN [24]
+COCO
+Box
+23.3
+91.8
+Mask Auto-Labeler
+ViT-MAE-Base [13]
+ResNeXt-101-64x4d-FPN
+Mask R-CNN [24]
+COCO
+Box
+24.2
+93.8
+Table 2. Main results on LVIS v1. Training data means the dataset we use for training MAL. We also finetune it on COCO and then generate
+pseudo-labels of LVIS v1. Compared with trained on LVIS v1 directly, MAL finetuned on COCO only caused around 0.35% mAP drop
+on the final results, which indicates the great potential of the open-set ability of MAL. Ret means the retention rate of box-supervised mask AP
+supervised mask AP .
+Mask decoder
+(%)Mask APval
+(%)Ret.val
+Fully connected decoder
+35.5
+79.2
+Fully convolutional decoder
+36.1
+80.5
+Attention-based decoder
+42.3
+94.4
+Query-based decoder
+-
+-
+Table 3. Ablation study of box expansion. We use Standard ViT-
+MAE-Base as the image encoder of MAL in phase 1 and Cascade
+RCNN with ConvNext-Small as the instance segmentation models
+in phase 2. The numbers are reported in % Mask mAP. Among
+different designs, the attention-based decoder performs the best.
+We can not obtain reasonable results with Query-based Decoder.
+coder. For the query-based decoder, we follow the design
+in Mask2Former [41]. We spend much effort exploring the
+query-based Decoder on MAL since it performs extremely
+well on fully supervised instance segmentation. However,
+the results are surprisingly unsatisfactory. We suspect the
+slightly heavier layers might cause optimization issues un-
+der the box-supervised losses.
+Experiments show that box-supervised instance segmen-
+tation favors the attention-based decoder. However, state-
+of-the-art instance segmentation and object detection meth-
+ods often adopt the fully convolutional decoder [15, 43]
+or the query-based decoder [41]. Our proposed two-phase
+framework resolves this dilemma and allows the networks
+to enjoy the merits of both the attention-based Decoder and
+the non-attention-based Decoders.
+4.6. Clustering analysis
+As the results are shown in Tab. 4, we wonder why the
+Standard ViTs outperform other modern image encoders in
+auto-labeling. As the comparison of classification ability
+Backbone
+IN-1k Acc@1
+Mask APval
+Ret.val
+ConvNeXt-Base [44]
+83.8
+39.6
+88.4
+Swin-Base [12]
+83.5
+40.2
+89.7
+ViT-DeiT-Small [64]
+79.9
+40.8
+91.0
+ViT-DeiT-Base [64]
+81.8
+41.1
+91.7
+ViT-MAE-Base [13]
+83.6
+42.3
+94.4
+ViT-MAE-Large [13]
+85.9
+42.3
+94.4
+Table 4.
+Ablation study of different backbones.
+All models
+are pre-trained on ImageNet-1k.
+ConvNeXt and Swin Trans-
+former outperform DeiT on image classification, but standard ViT-
+Small [16] (ViT-DeiT-Small) outperforms ConvNeXt-base and
+Swin-Base on mask Auto-labeling.
+Standard ViT-Base (ViT-
+MAE-Base) and Standard ViT-Large (ViT-MAE-Large) pretrained
+via MAE achieve the best performance on mask Auto-labeling.
+does not seem to reflect the actual ability of auto-labeling,
+we try to use the ability clustering to evaluate the image en-
+coders because foreground(FG)/background(BG) segmen-
+tation is very similar to the binary clustering problem.
+Specifically, we extract the feature map output by the last
+layers of Swin Transformers [12], ConvNeXts [44], Stan-
+dard ViTs [11]. Then, we use the GT mask to divide the
+feature vectors into the FG and BG feature sets. By evalu-
+ating the average distance from the FG/BG feature vectors
+to their clustering centers, we can reveal the ability of the
+networks to distinguish FG and BG pixels empirically.
+Formally, we define the feature vector of token i gener-
+ated by backbone E as f E
+i . We define the FG/BG clustering
+centers f ′
+1, f ′
+0 as the mean of the FG/BG feature vectors.
+Then, we use the following metric as the clustering score:
+S = 1
+N
+N
+�
+i
+( f E
+i
+|f E
+i | −
+f ′
+γ(i)
+|f ′
+γ(i)|)2,
+(6)
+7
+
+50
+100
+150
+200
+250
+300
+350
+400
+0
+100
+200
+300
+400
+500
+6000
+100
+200
+300
+400
+500
+600
+0
+100
+200
+300
+40050
+100
+150
+200
+250
+300
+350
+400
+0
+100
+200
+300
+400
+500
+600100
+200
+300
+400
+500
+600
+0
+100
+200
+300
+400
+500
+600100
+200
+300
+400
+0
+100
+200
+300
+400
+500
+6000
+50
+100
+150
+200
+250
+300
+350
+400
+0
+100
+200
+300
+400
+500
+600Figure 6. Attention visualization of two RoI images produced by MAL. In each image group, the left-most image is the original image.
+We visualize the attention map output by the 4th, 8th, 12th MHSA layers of the Standard ViTs in MAL.
+(a) MAL-generated Masks are sharper and more boundary-sticky
+(b) Occlusion Issues
+MAL Mask 
+Pseudo-labels 
+Ground-truth
+Masks
+Figure 7. The lateral comparison between MAL-generated pseudo-labels (top) and GT masks (bottom) on COCO val2017. On the left, we
+observe that MAL-generated pseudo-labels are sharper and more boundary-sticky than GT masks in some cases. On the right, we observe
+that in highly occluded situations, human-annotated masks are still better.
+25
+30
+35
+40
+0.1 0.25 0.5 0.75 1
+𝛼!"#
+32
+34
+36
+1
+2
+4
+8
+16
+𝛼$%&
+30
+32
+34
+36
+0.5
+1
+2
+4
+6
+𝜔
+27
+30
+33
+36
+0.1 0.25 0.5 0.75 1
+𝜁
+Figure 8. Sensitivity analysis of loss weights and CRF hyper-
+parameters. We use ViT-Base [11] pretrained via MAE [13] as the
+image encoder for the first phase and SOLOv2 (ResNet-50) for the
+second phase. The x-axis and y-axis indicate the hyper-parameter
+values and the (%)mask AP, respectively.
+θ
+Mask APval
+Ret.val
+0.6
+41.3
+92.2
+0.8
+41.7
+93.1
+1.0
+42.2
+94.2
+1.2
+42.3
+94.4
+1.4
+42.0
+93.8
+1.6
+41.8
+93.3
+Table 5. Ablation on box ex-
+pansion ratio. We use Standard
+ViT-Base pretrained via MAE
+(ViT-MAE-Base) and Cascade
+R-CNN (ConvNeXt-Small) for
+phase 1 and 2.
+Backbone
+Score (↓)
+ConvNeXt-Base [44]
+0.459
+Swin-Base [12]
+0.425
+ViT-DeiT-Small [64]
+0.431
+ViT-DeiT-Base [64]
+0.398
+ViT-MAE-Base [13]
+0.324
+ViT-MAE-Large [13]
+0.301
+Table 6.
+Clustering scores
+for different image encoders.
+The smaller clustering scores
+imply a better ability to dis-
+tinguish foreground and back-
+ground features.
+where if pixel i is FG, γ(i) = 1, otherwise γ(i) = 0.
+We show the clustering evaluation on the COCO val
+2017 in Tab. 6. The results align our conclusion that Stan-
+dard Vision Transformers are better at mask auto-labeling.
+4.7. MAL masks v.s. GT masks
+We show the apples to apples qualitative comparison in
+Fig. 7 and make the following observations. First, MAL-
+generated mask pseudo-labels are considerably sharper and
+boundary-sticky than human-annotated ones since humans
+have difficulties in aligning with the true boundaries. Sec-
+ond, severe occlusion also presents a challenging issue.
+5. Conclusion
+In this work, we propose a novel two-phase frame-
+work for box-supervised instance segmentation and a
+novel Transformer-based architecture, Mask Auto-Labeler
+(MAL), to generate high-quality mask pseudo-labels in
+phase 1. We reveal that Standard Vision Transformers are
+good mask auto-labelers. Moreover, we find that random
+using box-expansion RoI inputs, the attention-based De-
+coder, and class-agnostic training are crucial to the strong
+mask auto-labeling performance. Moreover, thanks to the
+two-phase framework design and MAL, we can adjust al-
+most all kinds of fully supervised instance segmentation
+models to box-supervised learning with little performance
+drop, which shows the great generalization of MAL.
+Limitations. Although great improvement has been made
+by our approaches in mask auto-labeling, we still observe
+many failure cases in the occlusion situation, where human
+annotations are much better than MAL-generated masks.
+Additionally, we meet saturation problems when scaling the
+model from Standard ViT-Base to Standard ViT-Large. We
+leave those problems in the future work.
+Broader impacts. Our proposed Transformer-based mask
+auto-labeler and the two-phase architecture serve as a stan-
+dard paradigm for high-quality box-supervised instance
+segmentation. If follow-up work can find and fix the issues
+under our proposed paradigm, there is great potential that
+expansive human-annotated masks are no longer needed for
+instance segmentation in the future.
+8
+
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+
+A. Appendix
+A.1. Additional details of CRF
+In the main paper, we define the energy terms of CRF but
+skip the details on how we use the Mean Field algorithm to
+minimize the energy. Here, we provide more details on how
+we use the Mean Field algorithm [56].
+We define l = {l1, ..., lN} as the label being inferred,
+where N = H × W is the size of the input image and xi
+is the label of the i-th pixel in I. We also assume that the
+network predicts a mask m = {m1, ..., mN} is where mi
+is the unary mask score of the i-th pixel in I. The pseudo-
+code to obtain l using mean field is attached in Alg. 1:
+Algorithm 1 Mean field algorithm for CRFs.
+1: procedure MEANFIELD(m, I)
+2:
+Ki,j ←− ω exp(− |Ii−Ij|
+2ζ2
+)
+3:
+▷ Initialize the Gaussian kernels
+4:
+l ← m
+▷ Initialize l using m
+5:
+while not converge do
+▷ Iterate until convergence
+6:
+for i ← 1 to |l| do
+7:
+ˆli ← li
+8:
+for j ∈ N(i) do
+9:
+ˆli ← ˆli + Kj ∗ lj
+10:
+▷ Message passing
+11:
+end for
+12:
+end for
+13:
+l ← ϕ(ˆl)
+▷ ϕ is a clamp function
+14:
+end while
+15:
+return λ(l)
+▷ λ is a threshold function
+16: end procedure
+A.2. Additional implementation details
+We use the same hyper-parameters on all benchmarks
+for all image encoders (Standard ViTs [11, 13, 16], Swin
+Transformers [12], and ConvNeXts [44]) and mask de-
+coders (fully connected decoder, fully convolutional de-
+coder, attention-based decoder, ), including batch size, opti-
+mization hyper-parameters. We observe a performance drop
+when we add parametric layers or multi-scale lateral/skip
+connections [43, 53] between the image encoder (Standard
+ViTs, Swin Transformers, ConvNeXts) and the mask de-
+coder (attention-based decoder). We insert a couple of the
+bi-linear interpolation layers to resize the feature map be-
+tween the image encoder and the mask decoder and resize
+the segmentation score map. Specifically, we resize the fea-
+ture map produced by the image encoder to 1/16 (small),
+1/8 (medium), 1/4 (large) size of the raw input according
+to the size of the objects. We divide the objects into three
+scales regarding to the area of their bound boxes. We use
+the area ranges of [0, 322), [322, 962), [962, ∞) to cover
+small, medium, and large objects, respectively. We resize
+the mask prediction map to 512 × 512 to reach the original
+resolution of the input images.
+Moreover, we also try three naive ways to add classifica-
+tion loss, but it does not work well with MAL. First, we add
+another fully connected layer as the classification decoder,
+which takes the feature map of the first fully connected layer
+of the instance-aware head K. With this design, the classi-
+fication causes a significant performance drop. Secondly,
+we use two extra fully connected layers or the original clas-
+sification decoder of standard ViTs as the classification de-
+coder, which directly takes the feature map of the image
+encoder. However, the classification loss does not provide
+performance improvement or loss in this scenario.
+A.3. Benefits for object detection
+The supervised object detection models benefit from the
+extra mask supervision [24], which improves detection re-
+sults.
+Specifically, we follow the settings in Mask R-
+CNN [24]. First, we use RoI Align, the box branch, and
+the box supervision without mask supervision. Second, we
+add the mask branch and ground-truth mask supervision on
+top of the first baseline. The second baseline is the original
+Mask R-CNN. Thirdly, we replace the ground-truth masks
+with the mask pseudo-labels generated by MAL on top of
+the second baseline. It turns out that using MAL-generated
+mask pseudo-labels for mask supervision brings in an im-
+provement similar to ground-truth masks on detection. We
+show the results in Tab. 7.
+A.4. Additional qualitative results
+We also visualize the prediction results produced by
+the instance segmentation models trained with ground-truth
+masks and mask pseudo-labels in Fig. 9. In most cases, we
+argue that humans cannot tell which results are produced by
+the models supervised by human-annotated labels.
+12
+
+InstSeg Backbone
+Dataset
+Mask Labels
+(%)AP
+(%)AP50
+(%)AP75
+(%)APS
+(%)APM
+(%)APL
+ResNet-50-DCN [59]
+LVIS v1
+None
+22.0
+36.4
+22.9
+16.8
+29.1
+33.4
+ResNet-50-DCN [59]
+LVIS v1
+GT mask
+22.5
+36.9
+23.8
+16.8
+29.7
+35.0
+ResNet-50-DCN [59]
+LVIS v1
+MAL mask
+22.6
+37.2
+23.8
+17.3
+29.8
+34.6
+ResNet-101-DCN [59]
+LVIS v1
+None
+24.4
+39.5
+26.1
+17.9
+32.2
+36.7
+ResNet-101-DCN [59]
+LVIS v1
+GT mask
+24.6
+39.7
+26.1
+18.3
+32.1
+38.3
+ResNet-101-DCN [59]
+LVIS v1
+MAL mask
+25.1
+40.0
+26.7
+18.4
+32.5
+37.8
+ResNeXt-101-32x4d-FPN [53,59]
+LVIS v1
+None
+25.5
+41.0
+27.1
+18.8
+33.7
+38.0
+ResNeXt-101-32x4d-FPN [53,59]
+LVIS v1
+GT mask
+26.7
+42.1
+28.6
+19.7
+34.7
+39.4
+ResNeXt-101-32x4d-FPN [53,59]
+LVIS v1
+MAL mask
+26.3
+41.5
+28.3
+19.5
+34.5
+39.6
+ResNeXt-101-64x4d-FPN [53,59]
+LVIS v1
+None
+26.6
+42.0
+28.3
+19.8
+34.7
+39.9
+ResNeXt-101-64x4d-FPN [53,59]
+LVIS v1
+GT mask
+27.2
+42.8
+29.2
+20.2
+35.7
+41.0
+ResNeXt-101-64x4d-FPN [53,59]
+LVIS v1
+MAL mask
+27.2
+42.7
+29.1
+19.8
+35.9
+40.7
+ConvNeXt-Small [44]
+COCO
+None
+51.5
+70.6
+56.1
+34.8
+55.2
+66.9
+ConvNeXt-Small [44]
+COCO
+GT mask
+51.8
+70.6
+56.3
+34.5
+55.9
+66.6
+ConvNeXt-Small [44]
+COCO
+MAL mask
+51.7
+70.5
+56.2
+35.2
+55.7
+66.8
+Table 7. Results of detection by adding different mask supervision. The models are evaluated on COCO val2017 and LVIS v1. By adding
+mask supervision using ground-truth masks or mask pseudo-labels, we can get around 1% improvement on different AP metrics on LVIS
+v1. On COCO val2017, the detection performance also benefits from mask pseudo-labels. Although the improvement is less than COCO’s,
+the improvement is consistent over different random seeds.
+Mask2former
+(Swin-S)
+trained with
+GT Mask
+Mask2former
+(Swin-S)
+trained with
+MAL Mask
+Mask2former
+(Swin-S)
+trained with
+GT Mask
+Mask2former
+(Swin-S)
+trained with
+MAL Mask
+Figure 9. The qualitative comparison between Mask2Former trained with GT mask and Mask2Former trained with MAL-generated mask
+pseudo-labels. Note that we use ViT-MAE-Base as the image encoder of MAL and Swin-Small as the backbone of the Mask2Former.
+13
+
+Y
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+couchl0.30
+couchjo.99dog/0.97
+ngel08
+benchjo.47dog/0.98
+ngej0.93
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+tv0.38
+WORKPLACE
+keyboardj0.98
+mnuse|0.34
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+mouse|0.89
+keyboardj0.99
+mouse0.9gtv/0.99
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+WORKPLACE
+keyboardj0.99
+660/
+mouse|0.95
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+mouse/0.99baseballbat/0.98
+personj0.97
+personj0.98
+oersonbaseball batj0.98
+personjo.98
+personjo.99chairl0.80
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+cellphone|0.72
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ANE5T4oBgHgl3EQfSg9u/content/tmp_files/2301.05529v1.pdf.txt

@@ -0,0 +1,2921 @@
+UNIFORM GLOBAL STABILITY OF SWITCHED NONLINEAR
+SYSTEMS IN THE KOOPMAN OPERATOR FRAMEWORK∗
+CHRISTIAN MUGISHO ZAGABE† AND ALEXANDRE MAUROY ‡
+Abstract.
+In this paper, we provide a novel solution to an open problem on the global uniform stability
+of switched nonlinear systems. Our results are based on the Koopman operator approach and, to
+our knowledge, this is the first theoretical contribution to an open problem within that framework.
+By focusing on the adjoint of the Koopman generator in the Hardy space on the polydisk, we
+define equivalent linear (but infinite-dimensional) switched systems and we construct a common
+Lyapunov functional for those systems, under a solvability condition of the Lie algebra generated by
+the linearized vector fields. A common Lyapunov function for the original switched nonlinear systems
+is derived from the Lyapunov functional by exploiting the reproducing kernel property of the Hardy
+space. The Lyapunov function is shown to converge in a bounded region of the state space, which
+proves global uniform stability of specific switched nonlinear systems on bounded invariant sets.
+Key words.
+Koopman operator, Hardy space on the polydisk, Switched systems, Uniform
+stability, Common Lyapunov function.
+AMS subject classifications. 47B32, 47B33, 47D06, 70K20, 93C10, 93D05.
+1. Introduction. Switched systems are hybrid-type models encountered in ap-
+plications where the dynamics abruptly jump from one behavior to another. They
+are typically described by a family of subsystems that alternate according to a given
+commutation law. Stability properties of switched systems have been the focus of
+intense research effort (see e.g. [32] for a review). In this context, a natural question
+is whether a switched system with an equilibrium point is uniformly stable, that is,
+stable for any commutation law. It turned out that the uniform stability problem
+is counter-intuitive and challenging. In the linear case, it is well-known that stable
+subsystems may induce an unstable switched system. However, uniform stability is
+guaranteed if the matrices associated with the subsystems are stable and commute
+pairwise [24], a result which is extended in [15] to subsystems described by stable
+matrices generating a solvable Lie algebra. This latter result can be explained by the
+well-known equivalence between solvable Lie algebra of matrices and the existence
+of a common invariant flag for those matrices, which allows to construct a common
+Lyapunov function for the subsystems [34].
+In the case of switched nonlinear systems, an open problem was posed in [13] on
+the relevance of Lie-algebraic conditions of vector fields for global uniform stability.
+Partial solutions have been proposed in this context. It was proven in [17] that uniform
+stability holds if the vector fields are individually stable and commute, in which case
+a common Lyapunov function can be constructed [30, 33]. Uniform stability was also
+shown for a pair of vector fields generating a third-order nilpotent Lie algebra [29]
+and for particular r-order nilpotent Lie algebras [18]. However, no result has been
+obtained, which solely relies on the more general solvability property of Lie algebras
+of the subsystems vector fields.
+In this paper, we provide a partial solution to the problem introduced in [13] by
+∗Submitted to the editors
+†Department of Mathematics and Namur Research Institute for Complex Systems (naXys), Uni-
+versity of Namur (christian.mugisho@unamur.be),
+‡Department of Mathematics and Namur Research Institute for Complex Systems (naXys), Uni-
+versity of Namur (alexandre.mauroy@unamur.be)
+1
+This manuscript is for review purposes only.
+arXiv:2301.05529v1  [math.DS]  13 Jan 2023
+
+2
+C. M. ZAGABE AND A. MAUROY
+proving global uniform stability results for switched nonlinear systems under a gen-
+eral solvability property of Lie algebras. To do so, we rely on the Koopman operator
+framework [3, 21]: we depart from the classical pointwise description of dynami-
+cal systems and consider instead the evolution of observable functions (here in the
+Hardy space of holomorphic functions defined on the complex polydisk). Through this
+approach, equivalent infinite-dimensional dynamics are generated by linear Koopman
+generators, so that nonlinear systems are represented by Koopman linear systems that
+are amenable to global stability analysis [19]. In particular, building on preliminary
+results obtained in [34], we construct a common Lyapunov functional for switched
+Koopman linear systems. A key point is to focus on the adjoint of the Koopman
+generators and notice that these operators have a common invariant maximal flag if
+the linear parts of the subsystems generate a solvable Lie algebra, a condition that is
+milder than the original assumption proposed in [13]. Finally, we derive a common
+Lyapunov function for the original switched nonlinear system and prove its conver-
+gence under specific algebraic conditions on the vector field. This allows us to obtain
+a bounded invariant region where the switched nonlinear system is globally uniformly
+asymptotically stable. To our knowledge, this is the first time that a novel solution
+to an open theoretical problem is obtained within the Koopman operator framework.
+The rest of the paper is organized as follows.
+In Section 2, we present some
+preliminary notions on uniform stability of switched nonlinear systems and give a
+general introduction to the Koopman operator framework, as well as some specific
+properties in the Hardy space on the polydisk. In Section 3, we state and prove our
+main result. We recast the open problem given in [13] in terms of the existence of
+an invariant maximal flag and we provide a constructive proof for the existence of
+a common Lyapunov function. Additional corollaries are also given, which focus on
+specific classes of vector fields. Our main results are illustrated with two examples in
+Section 4. Finally, concluding remarks and perspectives are given in Section 5.
+Notations. We will use the following notation throughout the manuscript. For
+multi-index notations α = (α1, ..., αn) ∈ Nn, we define |α| = α1 + · · · + αn and
+zα = zα1
+1 · · · zαn
+n . The complex conjugate and real part of a complex number a are
+denoted by ¯a and ℜ(a), respectively. The transpose-conjugate of a matrix (or vector)
+A is denoted by A†. The Jacobian matrix of the vector field F at x is given by JF(x).
+The complex polydisk centered at 0 and of radius ρ is defined by
+Dn(0, ρ) = {z ∈ Cn : |z1| < ρ, · · · , |zn| < ρ} .
+In particular, Dn denotes the unit polydisk (i.e. with ρ = 1) and ∂Dn is its boundary.
+Finally, the floor of a real number is denoted by ⌊x⌋.
+2. Preliminaries. In this section, we introduce preliminary notions and results
+on the stability theory for switched systems and on the Koopman operator framework.
+2.1. Stability of switched systems. We focus on the uniform asymptotic sta-
+bility property of switched systems and on the existence of a common Lyapunov
+function. Some existing results that connect these two main concepts are presented
+in both linear and nonlinear cases.
+Definition 2.1 (Switched system). A switched system ˙x = F (σ)(x) is a (finite)
+set of subsystems
+(2.1)
+�
+˙x = F (i)(x), x ∈ X ⊂ Rn�m
+i=1
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+3
+associated with a commutation law σ : R+ → {1, · · · , m} indicating which subsystem
+is activated at a given time.
+In this paper, we make the following standing assumption.
+Assumption 1. The commutation law σ is a piecewise constant function with a
+finite number of discontinuities on every bounded time interval (see e.g. [12]).
+2.1.1. Uniform stability. According to [16], stability analysis of switched sys-
+tems revolves around three important problems:
+• decide whether an equilibrium is stable under the action of the switched
+system for any commutation law σ, in which case the equilibrium is said to
+be uniformly stable,
+• identify the commutation laws for which the equilibrium is stable, and
+• construct the commutation law for which the equilibrium is stable.
+In this paper we focus on the first problem related to uniform stability.
+Definition 2.2 (Uniform stability). Assume that F (i)(xe) = 0 for all i = 1, . . . , m.
+The equilibrium xe is
+• uniformly asymptotically stable (UAS) if ∀ϵ > 0, ∃δ > 0 such that
+∥x(0) − xe∥ ≤ δ ⇒ ∥x(t) − xe∥ ≤ ϵ, ∀t > 0, ∀σ
+and
+∥x(0) − xe∥ ≤ δ ⇒ lim
+t→∞ x(t) = xe, ∀σ,
+• globally uniformly asymptotically stable (GUAS) on D ⊆ Rn if it is UAS
+and
+x(0) ∈ D ⇒ lim
+t→∞ x(t) = xe, ∀σ,
+• globally uniformly exponentially stable (GUES) on D ⊆ Rn if ∃β, λ > 0 such
+that
+x(0) ∈ D ⇒ ∥x(t) − xe∥ ≤ β∥x(0) − xe∥e−λt, ∀t > 0, ∀σ.
+This definition implies that the subsystems share a common equilibrium. More-
+over, a necessary condition is that this equilibrium is asymptotically stable with re-
+spect to the dynamics of all individual subsystems. However, this condition is not
+sufficient, since the switched system might be unstable for a specific switching law.
+A sufficient condition for uniform asymptotic stability is the existence of a common
+Lyapunov function (CLF).
+Definition 2.3 (Common Lyapunov function [12]). A positive C1- function V :
+D ⊆ Rn → R is a common Lyapunov function on D ⊆ Rn for the family of subsystems
+(2.1) if
+∇V · F (i)(x) < 0
+∀x ∈ D \ {xe},
+∀i = 1, . . . , m.
+For switched systems with a finite number of subsystems, a converse Lyapunov result
+also holds ([12], [17]).
+Theorem 2.4 ([17]). Suppose that D ⊆ Rn is compact and forward-invariant
+with respect to the flow induced by the subsystems (2.1). The switched system (2.1)
+is GUAS on D if and only if all subsystems share a CLF on D.
+This manuscript is for review purposes only.
+
+4
+C. M. ZAGABE AND A. MAUROY
+A corollary of this result provides a necessary condition for GUAS, which is based on
+convex combinations of vector fields.
+Corollary 2.5 ([12]). If the equilibrium of the switched system (2.1) is GUAS,
+then it is a globally asymptotically stable equilibrium for the dynamics
+˙x = αF (i)(x) + (1 − α)F (j)(x),
+for all i, j ∈ {1, · · · , m} and for all α ∈ [0, 1].
+2.1.2. Lie-algebraic conditions in the linear case. In the case of switched
+linear systems { ˙x = A(i)x, A(i) ∈ Cn×n}m
+i=1, several results related to uniform stability
+have been proved (see [32] for a review). We focus here on specific results based on
+Lie-algebraic conditions.
+Let g = span
+�
+A(i)�
+Lie denote the Lie algebra generated by the matrices A(i),
+with i = 1, · · · , m, and equipped with the Lie bracket [A(i), A(j)] = A(i)A(j)−A(j)A(i).
+Definition 2.6 (Solvable Lie algebra). A Lie algebra g equipped with the Lie
+bracket [., .] is said to be solvable if there exists k ∈ N such that gk = 0, where
+{gj}j∈N∗ is a descendant sequence of ideals defined by
+�
+g1 := g
+gj+1 :=
+�
+gj, gj�
+.
+A general Lie-algebraic criterion for uniform exponential (asymptotic) stability of
+switched linear systems is given in the following theorem.
+Theorem 2.7 ([15]).
+If all matrices A(i), i = 1, · · · , m, are stable (i.e. with
+eigenvalues λ(i)
+j
+such that ℜ
+�
+λ(i)
+j
+�
+< 0) and if the Lie algebra g is solvable, then the
+switched linear system { ˙x = A(i)x}m
+i=1 is GUES.
+As shown in [23, 31], this result follows from the simultaneous triangularization of
+the matrices A(i), which is a well-known property of solvable Lie algebras (see Lie’s
+theorem A.5 in Appendix A). This property is in fact equivalent to the existence of a
+common invariant flag for complex matrices [6].
+Definition 2.8 (Invariant flag). An invariant maximal flag of the set of matrices
+{A(i)}m
+i=1 is a set of subspaces {Sj}n
+j=1 ⊆ Cn such that (i) A(i)Sj ⊂ Sj for all i, j, (ii)
+dim(Sj) = j for all j, and (iii) Sj ⊂ Sj+1 for all j < n.
+The subspaces Sj can be described through an orthonormal basis (v1, · · · , vn), so
+that Sj = span {v1, · · · , vj}. Note that the vector v1 is a common eigenvector of the
+matrices A(i). This basis can be used to construct a CLF.
+Proposition 2.9 ([34]).
+Let
+(2.2)
+�
+˙x = A(i) x, A(i) ∈ Cn×n, x ∈ Cn�m
+i=1
+be a switched linear system. Suppose that all matrices A(i) are stable and admit a
+common invariant maximal flag
+{0} ⊂ S1 ⊂ · · · ⊂ Sn = Cn,
+Sj = span{v1, . . . , vj}.
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+5
+Then there exist ϵj > 0, j = 1, . . . , n, such that
+(2.3)
+V (x) =
+n
+�
+j=1
+ϵj|v†
+jx|2
+is a CLF for (2.2).
+The values ϵj must satisfy the condition
+(2.4)
+ϵj >
+max
+i∈{1,...,m}
+k∈{1,...,j−1}
+ϵk
+(n − 1)2
+4
+���v†
+kA(i)vj
+���
+2
+���ℜ
+�
+λ(i)
+j
+����
+���ℜ
+�
+λ(i)
+k
+����
+where λ(i)
+j
+are the eigenvalues of A(i).
+They can be obtained iteratively from an
+arbitrary value ϵ1 > 0. The geometric approach followed in [34] provides a constructive
+way to obtain a CLF, a result that we will leverage in an infinite-dimensional setting
+for switched nonlinear systems.
+2.1.3. Lie-algebraic condition in the nonlinear case. In the context of
+switched nonlinear systems, one has to consider the Lie algebra of vector fields
+(2.5)
+gF = span
+�
+F (i), i = 1, . . . , m
+�
+Lie
+equipped with the Lie bracket
+(2.6)
+[F (i), F (j)](x) = JF (j)(x) F (i)(x) − JF (i)(x) F (j)(x).
+It has been conjectured in [13] that Lie-algebraic conditions on (2.5) could be
+used to characterize uniform stability.
+This problem has been solved partially in
+[29] for third-order nilpotent Lie algebras and in [18] for particular r-order nilpotent
+Lie algebras. Another step toward more general Lie-algebraic conditions based on
+solvability has been made in [34], a preliminary result that relies on the so-called
+Koopman operator framework. However, the results obtained in [34] are restricted
+to specific switched nonlinear systems that can be represented as finite-dimensional
+linear ones.
+In this paper, we build on this preliminary work, further exploiting
+the Koopman operator framework to obtain general conditions that characterize the
+GUAS property of switched nonlinear systems.
+2.2. Koopman operator approach to dynamical systems. In this section,
+we present the Koopman operator framework, which is key to extend the result of
+Proposition 2.9 to switched nonlinear systems. We introduce the Koopman semigroup
+along with its Koopman generator, cast the framework in the context of Lie groups,
+and describe the finite-dimensional approximation of the operator.
+2.2.1. Koopman operator. Consider a continuous-time dynamical system
+(2.7)
+˙x = F(x),
+x ∈ X ⊂ Rn,
+F ∈ C1
+which generates a flow ϕt : X → X, with t ∈ R+. The Koopman operator is defined
+on a (Banach) space F and acts on observables, i.e. functions f : X → R, f ∈ F.
+Definition 2.10 (Koopman semigroup [11]). The semigroup of Koopman opera-
+tors (in short, Koopman semigroup) is the family of linear operators (Ut)t≥0 defined
+by
+Ut : F → F,
+Utf = f ◦ ϕt.
+This manuscript is for review purposes only.
+
+6
+C. M. ZAGABE AND A. MAUROY
+We can also define the associated Koopman generator.
+Definition 2.11 (Koopman generator [11]). The Koopman generator associated
+with the vector field (2.7) is the linear operator
+(2.8)
+LF : D(LF ) → F,
+LF f := F · ∇f
+with the domain D(LF ) = {f ∈ F : F · ∇f ∈ F}.
+As shown below (see Lemma 2.13), the Koopman semigroup and the Koopman gen-
+erators are directly related. When the Koopman semigroup is strongly continuous
+[7], i.e.
+lim
+t→0+ ∥Utf − f∥F = 0, the Koopman generator is the infinitesimal generator
+LF f := lim
+t→0+(Utf −f)/t of the Koopman semigroup. Since the Koopman operator Ut
+and the generator LF are both linear, we can describe the dynamics of an observable
+f on F through the linear abstract ordinary differential equation
+(2.9)
+˙f = LF f.
+We can also briefly discuss the spectral properties of the Koopman operator.
+Definition 2.12 (Koopman eigenfunction and eigenvalue [3, 21]). An eigenfunc-
+tion of the Koopman operator is an observable φλ ∈ F \ {0} such that
+LF φλ = λφλ.
+The value λ ∈ C is the associated Koopman eigenvalue.
+Under the strong continuity property, the Koopman eigenfunction also satisfies
+Utφλ = eλtφλ,
+∀t ≥ 0.
+For a linear system ˙x = Ax, with x ∈ Rn, we denote an eigenvalue of A by ˜λj
+and its associated left eigenvector by wj. Then ˜λj is a Koopman eigenvalue and the
+associated Koopman eigenfunction is given by φ˜λj(x) = w†
+jx [22]. For a nonlinear
+system of the form (2.7) which admits a stable equilibrium xe, the eigenvalues of
+JF(xe) are typically Koopman eigenvalues and the associated eigenfunctions are the
+so-called principal Koopman eigenfunctions (see Remark 2.14 below).
+2.2.2. Koopman operator in the Hardy space H2(Dn). From this point
+on, we define the Koopman operator in the Hardy space on the polydisk (see e.g.
+[25, 26, 28] for more details). This choice is well-suited to the case of analytic vector
+fields that admit a stable hyperbolic equilibrium, where it allows to exploit convenient
+spectral properties of the operator.
+Let D be the open unit disk in C, ∂D its boundary, and Dn the unit polydisk in
+Cn. The Hardy space of holomorphic functions on Dn is the space
+H2(Dn) =
+�
+f : Dn → C, holomorphic : ∥f∥2 = lim
+r→1−
+�
+(∂D)n |f (rω) |2dmn(ω) < ∞
+�
+,
+where mn is the normalized Lebesgue measure on (∂D)n. It is equipped with an inner
+product defined by
+⟨f, g⟩ =
+�
+(∂D)n f (ω) ¯g (ω) dmn(ω),
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+7
+so that the set {zα : α ∈ Nn} is the standard orthonormal basis of monomials on
+H2(Dn). The monomials will be denoted by ek(z) = zα(k), where the map α : N → Nn,
+k �→ α(k) refers to the lexicographic order, i.e. ek1 < ek2 if |α(k1)| < |α(k2)|, or if
+|α(k1)| = |α(k2)| and αj(k1) > αj(k2) for the smallest j such that αj(k1) ̸= αj(k2).
+For f and g in H2(Dn), with f =
+�
+k∈N
+fkek and g =
+�
+k∈N
+gkek, the isomorphism
+�
+k∈N
+fkek �→ (fk)k≥0
+between H2(Dn) and the l2-space allows to rewrite the norm and the inner product
+as
+∥f∥2 =
+�
+k∈N
+|fk|2
+and
+⟨f, g⟩ =
+�
+k∈N
+fk ¯gk.
+We also note that H2(Dn) is a reproducing kernel Hilbert space (RKHS) with the
+Cauchy kernel ([25, Chapter 1])
+(2.10)
+k (z, ξ) =
+n
+�
+i=1
+1
+1 − ¯ξizi
+, z, ξ ∈ Dn.
+It follows that one can define the evaluation functional f(z) = ⟨f, kz⟩ with kz(ω) =
+k (z, ω).
+If the vector field F is analytic, we can consider its analytic continuation on Dn.
+Moreover, if it generates a holomorphic flow that is invariant in Dn, we can define
+the Koopman semigroup on H2(Dn), which is also known as the composition operator
+with symbol ϕt. The required assumptions are summarized as follows.
+Assumption 2. The components Fl, l = 1, · · · , n, of the vector field F belong to
+the Hardy space H2(Dn). Moreover, F generates a flow which is holomorphic and
+maps Dn to Dn (forward invariance).
+It is shown in [4] that the flow ϕt is holomorphic on Dn if and only if the vector field
+components have a specific form (see Proposition (A.1) in the Appendix, and the works
+[4, 5]). Note also that this property holds if the dynamics possess a globally stable
+hyperbolic equilibrium (Assumption 3 below) in the case of non-resonant eigenvalues
+(see Remark 2.14).
+Now, we recall some important properties that we will use to prove our results.
+Lemma 2.13. Consider a function f ∈ H2(Dn) and an evaluation functional kz,
+with z ∈ Dn. Then,
+1. LF zα ∈ H2(Dn) and the domain D (LF ) is dense in H2(Dn),
+2. U ∗
+t kz = kϕt(z),
+3.
+d
+dt ⟨U ∗
+t kz, f⟩ = ⟨L∗
+F U ∗
+t kz, f⟩.
+Proof.
+1. For all z ∈ Dn, we have
+LF zα = F(z) · ∇zk =
+n
+�
+l=1
+Fl(z) αl z(��1,...,αl−1,αl−1,αl+1,...,αn).
+Since ∥fzα∥ = ∥f∥ for all f ∈ H2(Dn) and for all α ∈ Nn, it follows from
+Assumption 2 that ∥LF eα∥ =
+�����
+n
+�
+l=1
+αlFl
+����� ≤
+n
+�
+l=1
+|αl| ∥Fl∥ < ∞. Moreover
+D (LF ) is dense in H2(Dn) since the monomials zα form a complete basis.
+This manuscript is for review purposes only.
+
+8
+C. M. ZAGABE AND A. MAUROY
+2. For all f ∈ H2(Dn), we have
+⟨U ∗
+t kz, f⟩ = ⟨kz, Utf⟩ = (Utf) (z)
+and
+�
+kϕt(z), f
+�
+= f (ϕt(z)) = (Utf) (z),
+so that
+U ∗
+t kz = kϕt(z).
+3. For all z ∈ Dn and all f ∈ D(LF ),
+d
+dt ⟨U ∗
+t kz, f⟩ = d
+dt
+�
+kϕt(z), f
+�
+= d
+dtf ◦ ϕt(z)
+= F (ϕt(z)) .∇f (ϕt(z))
+=
+�
+kϕt(z), LF f
+�
+= ⟨L∗
+F U ∗
+t kz, f⟩ .
+The result follows for all f since D(LF ) is dense in H2(Dn).
+In the previous lemma, the second property is a well-known property of the composi-
+tion operator on a RKHS. The third property is also known in the context of strongly
+continuous semigroup theory (see [7]).
+Finally, we make the following additional standing assumption.
+Assumption 3. The vector field F admits on Dn a unique hyperbolic stable equi-
+librium at 0 (without loss of generality), i.e. F(0) = 0 and the eigenvalues ˜λj of the
+Jacobian matrix JF(0) satisfy ℜ{˜λj} < 0.
+Remark 2.14 (Holomorphic flow and spectral properties). If Assumption 3 holds
+and if the eigenvalues ˜λj are non-resonant1, then the Poincar´e linearization theorem [2]
+implies that the flow ϕt is topologically conjugated to the linear flow ˜ϕt(z) = eJF (0)tz,
+i.e. there exists a bi-holomorphic map h such that ϕt = h−1 ◦ ˜ϕt ◦ h. In this case, the
+flow ϕt is clearly holomorphic. Moreover, the components of h are associated with
+holomorphic Koopman eigenfunctions φ˜λj ∈ H2(Dn) associated with the eigenvalues
+˜λj [9, 20]. These eigenfunctions are called principal eigenfunctions. Also, it can easily
+be shown that, for all α ∈ Nn,
+n
+�
+j=1
+αj˜λj is a Koopman eigenvalue associated with the
+eigenfunction φα1
+˜λ1 · · · φαn
+˜λn.
+2.2.3. Koopman infinite matrix. Since H2(Dn) is isomorphic to l2, the Koop-
+man generator can be represented by the Koopman infinite matrix
+(2.11)
+¯LF =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+⟨LF e0, e0⟩
+⟨LF e0, e1⟩
+⟨LF e0, e2⟩
+⟨LF e0, e3⟩
+· · ·
+⟨LF e1, e0⟩
+⟨LF e1, e1⟩
+⟨LF e1, e2⟩
+⟨LF e1, e3⟩
+· · ·
+⟨LF e2, e0⟩
+⟨LF e2, e1⟩
+⟨LF e2, e2⟩
+⟨LF e2, e3⟩
+· · ·
+⟨LF e3, e0⟩
+⟨LF e3, e1⟩
+⟨LF e3, e2⟩
+⟨LF e3, e3⟩
+· · ·
+⟨LF e4, e0⟩
+⟨LF e4, e1⟩
+⟨LF e4, e2⟩
+⟨LF e4, e3⟩
+· · ·
+...
+...
+...
+...
+· · ·
+�
+�
+�
+�
+�
+�
+�
+�
+�
+,
+1The eigenvalues ˜λj are non-resonant if
+n
+�
+j=1
+αj ˜λj = 0 with α ∈ Zn implies that α = 0.
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+9
+where the kth row contains the components of LF ek in the basis of monomials. For
+f =
+�
+k∈N
+fkek, we also have that
+⟨LF f, ej⟩ =
+�
+k∈N
+fk⟨LF ek, ej⟩.
+Remark 2.15. We note that, since LF e0 = 0, the first row and column of ¯LF
+contains only zero entries. By removing the first row and column, one obtains the
+representation of the restriction of the Koopman generator to the subspace of functions
+f that satisfy f(0) = 0. This subspace is spanned by the basis (ek)k≥1. Note that
+kz − k0 belongs to this subspace, since (2.10) implies that kz(0) − k0(0) = 0.
+For Fl(z) =
+�
+|β|≥1
+al,βzβ, the action of the Koopman operator on a basis element
+is given by
+LF zα =
+n
+�
+l=1
+Fl(z) αl z(α1,...,αl−1,αl−1,αl+1,...,αn)
+=
+n
+�
+l=1
+�
+|β|≥1
+al,βzβ αl z(α1,...,αl−1,αl−1,αl+1,...,αn)
+=
+n
+�
+l=1
+αl
+�
+�
+�
+(β1,...,βn)∈Nn
+al,(β1,...,βn)z(β1+α1,...,βl+αl−1,βl+1+αl+1,...,βn+αn)
+�
+� .
+By setting γ1 = β1 + α1, . . . , γl = βl + αl − 1, . . . , γn = βn + αn we obtain
+LF zα =
+n
+�
+l=1
+αl
+�
+|γ|≥|α|
+al,(γ1−α1,...,γl−αl+1,...,γn−αn)z(γ1,γ2,...,γn)
+=
+n
+�
+l=1
+αl
+�
+|γ|≥|α|
+al,(γ−α)lzγ,
+(2.12)
+where we denote
+(2.13)
+(γ − α)l = (γ1 − α1, · · · , γl − αl + 1, · · · , γn − αn) .
+It follows that the entries of (2.11) are given by
+(2.14)
+⟨LF ek, ej⟩ =
+�
+�
+�
+�
+�
+n
+�
+l=1
+αl(k) al,(α(j)−α(k))l
+if |α(j)| ≥ |α(k)|
+0
+if |α(j)| < |α(k)|.
+Remark 2.16. For the linear part of the vector field F, where |α(j)| = 1, j =
+1, · · · , n, it is clear that α(j) is the canonical basis vector of Cn, i.e. αi(j) = δij, and we
+have that a(l)
+α(j) = [JF(0)]lj. Also, if |α(j)| = |α(k)|, we have that (α(j)−α(k))l = α(r)
+for some r ≤ n (i.e. |α(r)| = 1), with αr(j) = αr(k) + 1, αl(j) = αl(k) − 1, and
+This manuscript is for review purposes only.
+
+10
+C. M. ZAGABE AND A. MAUROY
+αi(j) = αi(k) for all i /∈ {l, r}. Then, it follows from (2.14) that
+(2.15)
+⟨LF ek, ej⟩ =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+n
+�
+l=1
+αl(j) [JF(0)]ll
+if j = k
+αl(k) [JF(0)]lr
+if α(j) = (α1(k), · · · , αl(k) − 1, · · · , αr(k) + 1, · · · , αn(k)),
+0
+otherwise .
+2.2.4. Switched Koopman systems and Lie-algebraic conditions. In the
+case of a switched nonlinear system (2.1), the Koopman operator description yields
+a switched linear infinite-dimensional system (in short, switched Koopman system) of
+the form
+(2.16)
+�
+˙f = LF (i)f, f ∈ D
+�m
+i=1
+with D = ∩m
+i=1D(LF (i)). Similarly, the Lie algebra gF spanned by F (i) (see (2.5)) is
+replaced by gL = span {LF (i), i = 1, . . . , m}Lie, equipped with the Lie bracket
+[LF (i), LF (j)] = LF (i)LF (j) − LF (j)LF (i) .
+In particular, we have the well-known relationship
+(2.17)
+[LF (i), LF (j)] = L[F (i),F (j)]
+so that the two algebras gF and gL are isomorphic.
+It follows that Lie-algebraic
+conditions in gF can be recast into Lie-algebraic criteria in gL, a framework where
+we can expect to obtain new results on switched systems that are reminiscent to the
+linear case. In particular, since the solvability property of gF is equivalent to the
+solvability property of gL, we will investigate whether this latter condition implies
+the existence of a common Lyapunov functional for the switched Koopman system
+(2.16).
+3. Main result. This section presents our main result. We first use an illus-
+trative example to show that Lie’s theorem A.5 cannot be used for nonlinear vector
+fields, in contrast to the linear case (see Proposition 2.9). We then relax the algebraic
+conditions suggested in [13] in order to obtain a triangular form in the Koopman
+matrix representation (2.11), a property which is equivalent to the existence of an in-
+variant flag for the adjoint operator L∗
+F . We finally prove uniform stability of switched
+nonlinear systems under these conditions.
+3.1. A first remark on the existence of the common invariant flag. The
+following example shows that Lie’s theorem does not hold for infinite-dimensional
+switched Koopman systems.
+Example 1. Consider the two vector fields
+F (1)(x1, x2) = (−αx1, −αx2)
+and
+F (2)(x1, x2) = (−βx1+γ
+�
+x2
+1 − x2
+2
+�
+, −βx2+2γx1x2),
+where α, β and γ are real parameters. These two vector fields generate the Lie alge-
+bra g = span
+�
+F (1), F (2), F (3)�
+Lie with F (3)(x1, x2) = (αγ(x2
+1 − x2
+2), 2αγx1x2) since
+[F (1), F (2)] = F (3), [F (1), F (3)] = αF (3) and [F (2), F (3)] = βF (3).
+Moreover, one
+has g1 = [g, g] = span
+�
+F (3)�
+Lie and g2 = [g1, g1] = 0, which implies that g is a
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+11
+solvable Lie algebra. However, the Koopman generators LF (1) and LF (2) associated
+with the two vector fields do not share a common eigenfunction, and therefore can-
+not have a common invariant flag. Indeed, the principal eigenfunctions of LF (1) are
+φ˜λ(1)
+1 (x1, x2) = x1 and φ˜λ(1)
+2 (x1, x2) = x2, while those of LF (2) are given by
+φ˜λ(2)
+1 (x1, x2) = βγ
+�
+βx1 − γ
+�
+x2
+1 − x2
+2
+��
+(β − γx1)2 + γ2x2
+2
+and
+φ˜λ(2)
+2 (x1, x2) =
+β2γx2
+(β − γx1)2 + γ2x2
+2
+.
+We conclude that Lie’s theorem A.5 does not hold setting for the above example,
+so that we cannot directly extend Proposition 2.9 to this case. The two Koopman
+generators are not simultaneous triangularizable and do not have a common invariant
+flag (see [10] for more details about simultaneous triangularization of operators and
+its connection to the existence of an invariant infinite maximal flag). However, it
+can be easily seen that the Koopman infinite matrices (2.11) related to the vector
+fields F (1) and F (2) are both upper triangular, and therefore admit a common infinite
+invariant maximal flag. In fact, this implies that the adjoint operators L∗
+F (i) have a
+common invariant flag. For this reason, we will depart from the solvability condition
+on vector fields (i.e. on Koopman generators), and we will deal with simultaneous
+triangularization of adjoints of Koopman generators. The following result provides a
+sufficient condition on the vector fields for the simultaneous triangularization of ad-
+joints of Koopman generators, which appears to be less restrictive than the solvability
+condition.
+Lemma 3.1. Let F be an analytic vector field on Dn such that the Jacobian matrix
+JF(0) is upper triangular. Then the Koopman matrix (2.11) is upper triangular, i.e.
+⟨LF ek, ej⟩ = 0 for all k > j. Moreover, the adjoint L∗
+F of the Koopman generator
+admits an infinite invariant maximal flag generated by the monomials ek, i.e. Sk =
+span{e1, . . . , ek}.
+Proof. It follows from (2.14) that ⟨LF ek, ej⟩ = 0 if |α(k)| > |α(j)| (i.e. the Koop-
+man matrix (2.11) is always upper triangular by matrix blocks related to monomials of
+the same total degree). In the case |α(k)| = |α(j)| with k > j, the lexicographic order
+implies that one can have α(j) = (α1(k), · · · , αl(k)−1, · · · , αr(k)+1, · · · , αn(k)) only
+with r < l. Since [JF(0)]lr = 0 for all l > r, it follows from (2.15) that ⟨LF ek, ej⟩ = 0
+when k > j. Finally, it is clear that L∗
+F ej ∈ span{e1, . . . , ej} since ⟨ek, L∗
+F ej⟩ = 0 for
+all k > j.
+Remark 3.2. When the Jacobian matrix is upper triangular, it is well-known that
+[JF(0)]jj = ˜λj. In this case, it follows from (2.15) that the diagonal entries of the
+(upper triangular) Koopman matrix are given by
+(3.1)
+⟨LF ej, ej⟩ =
+n
+�
+l=1
+αl(j)˜λl.
+Since these values are the Koopman eigenvalues in the case of non-resonant eigenvalues
+˜λj (see Remark 2.14), we will denote λj = ⟨LF ej, ej⟩ by a slight abuse of notation.
+Corollary 3.3. Let
+�
+F (i)�m
+i=1 be a switched nonlinear system on Dn and sup-
+pose that the Lie algebra of matrices span
+�
+JF (i)(0)
+�
+Lie is solvable. Then there ex-
+ists a change of variables z �→ �z = P −1z on Cn such that the adjoint operators
+L∗
+�
+F (i) of the Koopman generators (with �F (i)(�z) = P −1F (i)(P �z)) admit a common in-
+This manuscript is for review purposes only.
+
+12
+C. M. ZAGABE AND A. MAUROY
+finite invariant maximal flag. Moreover,
+�
+�F (i)�m
+i=1 is a switched nonlinear system on
+Dn �
+0, ∥P −1∥∞
+�
+.
+Proof. Since span
+�
+JF (i)(0)
+�
+Lie is solvable, Lie’s theorem A.5 implies that the
+matrices JF (i)(0) are simultaneously triangularizable, i.e. there exists a matrix P such
+that JF (i)(0) = PT (i)P −1 for all i, where T (i) is upper triangular. Let set F (i)(z) =
+JF (i)(0)z + ˜F (i)(z) to separate the linear and the nonlinear parts of the dynamics. In
+the new coordinates �z = P −1z, we obtain the dynamics �F (i)(�z) = P −1JF (i)(0)P �z +
+P −1 ˜Fi(P �z) = T (i)�z + �˜F i(�z). It follows from Lemma 3.1 that monomials �ek, with
+�ek(�z) = (�z)α(k), generate a common invariant maximal flag for L∗
+�
+F (i). In addition, for
+all z ∈ Dn and all j = 1, · · · , n, we have
+|�zj| ≤ ∥�z∥∞ =
+��P −1z
+��
+∞ ≤
+��P −1��
+∞ ∥z∥∞ < ∥P −1∥∞.
+Remark 3.4. It is clear that the change of coordinates z �→ �z = P −1z is defined
+up to a multiplicative constant. Without loss of generality, we will consider in the
+sequel that ∥P −1∥∞ = 1, so that
+�
+�F (i)�m
+i=1 is a switched nonlinear system on the
+unit polydisk Dn.
+Instead of a nilpotency or solvability condition on the vector fields F (i), we only
+require a milder solvability condition on the Jacobian matrices JF (i)(xe) to guarantee
+the triangular form of the Koopman matrix (2.11). It is noticeable that this local
+condition is much less restrictive than the global solvability condition mentioned in
+the original open problem [13]. Also, it was shown in [1] that the triangular form of the
+vector fields (and therefore of the Jacobian matrices) is not sufficient to guarantee the
+GUAS property of a switched nonlinear system on Rn. In the next section, however,
+we use the solvability condition on the Jacobian matrices to prove the GUAS property
+in a bounded invariant region of the state space. This result is consistent with the
+local stability result derived in [14].
+3.2. A common Lyapunov function for switched nonlinear systems. We
+now aim to show that, for some positive sequence (ϵk)∞
+k=1, the series
+(3.2)
+V(f) =
+∞
+�
+k=1
+ϵk |⟨f, ek⟩|2
+is a Lyapunov functional for the switched Koopman system (2.16). Before starting
+our main result, we need a few lemmas.
+Lemma 3.5. Let ˙z = F(z) be a vector field on the polydisk Dn which generates a
+flow ϕt. Suppose that there exist a sequence of positive numbers (ϵk)k≥1 and ρ ∈]0, 1]
+such that Dn(0, ρ) is forward invariant with respect to ϕt and such that the series
+�
+k≥1
+|α(k)|ϵkρ2|α(k)|
+is convergent. Then, the series
+(3.3)
+V(kz − k0) =
+∞
+�
+k=1
+ϵk |⟨kz, ek⟩|2
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+13
+and
+(3.4)
+∞
+�
+k=1
+ϵk
+d
+dt |⟨U ∗
+t (kz − k0), ek⟩|2
+are absolutely and uniformly convergent on Dn(0, ρ) for all t > 0.
+Proof. For the first series, we have
+ϵk |⟨kz − k0, ek⟩|2 = ϵk |⟨kz, ek⟩|2 = ϵk
+���zα(k)���
+2
+< ϵkρ2|α(k)| ≤ |α(k)|ϵkρ2|α(k)|
+for all z ∈ Dn(0, ρ) and all k ≥ 1. For the second series, we have
+ϵk
+����
+d
+dt |⟨U ∗
+t (kz − k0), ek⟩|2
+���� = ϵk
+����
+d
+dt
+���(ϕt(z))α(k) − (ϕt(0))α(k)���
+2����
+= 2ϵk
+����ℜ
+� d
+dt (ϕt(z))α(k) ·
+�
+ϕt(z)
+�α(k)�����
+≤ 2ϵk
+����
+d
+dt (ϕt(z))α(k)
+���� ·
+����
+�
+ϕt(z)
+�α(k)����
+< 2ϵkρ|α(k)|
+����
+d
+dt (ϕt(z))α(k)
+����
+≤ 2ϵkρ|α(k)|
+n
+�
+s=1
+αs(k)
+���F (s) (ϕt(z))
+���
+��(ϕt(z))s
+��αs(k)−1
+n
+�
+l=1,l̸=s
+��(ϕt(z))l
+��αl(k)
+< 2ϵkρ2|α(k)|−1
+n
+�
+s=1
+αs(k)
+���F (s) (ϕt(z))
+���
+for all z ∈ Dn(0, ρ), t > 0 and k ≥ 1. By using the maximum modulus principle for
+bounded domains A.2 with the holomorphic function F (s) ◦ ϕt, we can denote
+M =
+max
+z∈∂Dn,s=1,··· ,n |(F (s) ◦ ϕt)(z)|
+and we obtain
+ϵk
+����
+d
+dt |⟨U ∗
+t (kz − k0), ek⟩|2
+���� < 2M|α(k)|ϵkρ2|α(k)|−1.
+Finally, absolute and uniform convergence of both series follow from the Weierstrass
+test (A.4).
+Lemma 3.6. Let ˙z = F(z) be a nonlinear system on Dn with an upper triangular
+Jacobian matrix JF(0) and let ˙f = LF f be its corresponding Koopman system on
+D (LF ) ⊂ H2(Dn).
+If the series (3.3) and (3.4) are absolutely and uniformly convergent in Dn for all
+t > 0, then, for all double sequences of positive real numbers (bjk)j≥1,k≥1 such that
+(3.5)
+∞
+�
+k=1
+bjk ≤ 1,
+This manuscript is for review purposes only.
+
+14
+C. M. ZAGABE AND A. MAUROY
+one has
+d
+dtV (U ∗
+t (kz − k0)) ≤2
+∞
+�
+j=1
+bjjϵj |cj|2 ℜ (λj)
++ 2
+∞
+�
+j=2
+j−1
+�
+k=1
+�
+bjkϵj |cj|2 ℜ (λj) + bkjϵk |ck|2 ℜ (λk) + ϵkℜ (cj¯ck ⟨ej, LF ek⟩)
+�
+,
+(3.6)
+with cj = ⟨U ∗
+t (kz − k0), ej⟩ and λj = ⟨LF ej, ej⟩.
+Proof. Suppose that z ∈ Dn is such that the series (3.3) and (3.4) are absolutely
+and uniformly convergent. Then, by using Lemma 2.13 (3), we obtain
+d
+dtϵk |⟨U ∗
+t (kz − k0), ek⟩|2 = 2ϵkℜ
+�� d
+dtU ∗
+t (kz − k0), ek
+�
+⟨U ∗
+t (kz − k0), ek⟩
+�
+= 2ϵkℜ
+�
+⟨L∗
+F U ∗
+t (kz − k0), ek⟩ ⟨U ∗
+t (kz − k0), ek⟩
+�
+= 2ϵk
+∞
+�
+j=1
+ℜ (cj¯ck ⟨ej, LF ek⟩)
+where we used the decomposition U ∗
+t (kz − k0) =
+∞
+�
+j=1
+cjej. Since (3.4) is absolutely
+and uniformly convergent, term by term derivation yields
+d
+dtV (U ∗
+t (kz − k0)) =
+∞
+�
+k=1
+d
+dtϵk |⟨U ∗
+t (kz − k0), ek⟩|2
+= 2
+∞
+�
+k=1
+∞
+�
+j=1
+ϵkℜ (cj¯ck ⟨ej, LF ek⟩)
+= 2
+∞
+�
+j=1
+ϵj |cj|2 ℜ (λj) + 2
+∞
+�
+j=2
+j−1
+�
+k=1
+ϵkℜ (cj¯ck ⟨ej, LF ek⟩)
+where we used the triangular form of ¯LF (which follows from Lemma 3.1 since JF(0)
+is triangular) and λj = ⟨LF ej, ej⟩.
+Using (3.5), we have
+d
+dtV (U ∗
+t (kz − k0)) ≤ 2
+∞
+�
+j=1
+�
+�
+j
+�
+k=1
+bjk +
+∞
+�
+k=j+1
+bjk
+�
+� ϵj |cj|2 ℜ (λj) + 2
+∞
+�
+j=2
+j−1
+�
+k=1
+ϵkℜ (cj¯ck ⟨ej, LF ek⟩)
+= 2
+∞
+�
+j=1
+bjjϵj |cj|2 ℜ (λj) + 2
+∞
+�
+j=2
+j−1
+�
+k=1
+bjkϵj |cj|2 ℜ (λj)
++ 2
+∞
+�
+j=1
+∞
+�
+k=j+1
+bjkϵj |cj|2 ℜ (λj) + 2
+∞
+�
+j=2
+j−1
+�
+k=1
+ϵkℜ (cj¯ck ⟨ej, LF ek⟩)
+= 2
+∞
+�
+j=1
+bjjϵj |cj|2 ℜ (λj)
++ 2
+∞
+�
+j=2
+j−1
+�
+k=1
+�
+bjkϵj |cj|2 ℜ (λj) + bkjϵk |ck|2 ℜ (λk) + ϵkℜ (cj¯ck ⟨ej, LF ek⟩)
+�
+.
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+15
+Under Assumption 3, it follows from (3.1) that ℜ{λj} < 0 for all j. Therefore, the
+time derivative (3.6) of the Lyapunov functional is negative if negative terms related
+to the diagonal entries ⟨LF ej, ej⟩ = λj and ⟨LF ek, ek⟩ = λk compensate (possibly
+positive) cross-terms related to ⟨ej, LF ek⟩. We note that a term associated with a
+diagonal entry will be used to compensate an infinity of cross-terms (associated with
+entries in the corresponding row and column of the Koopman matrix), and the values
+bjk play the role of weights in the compensation process.
+We are now in position to state our main result.
+Theorem 3.7. Let
+(3.7)
+�
+˙z = F (i)(z)
+�m
+i=1
+be a switched nonlinear system on Dn and assume that
+• all subsystems of (3.7) have a common hyperbolic equilibrium ze = 0 that is
+globally asymptotically stable on the polydisk Dn,
+• the Lie algebra span
+�
+JF (i)(0)
+�
+Lie is solvable (and therefore there exists a
+matrix P such that P −1JF (i)(0)P are upper triangular),
+• there exists ρ ∈]0, 1] such that Dn (0, ρ) is forward invariant with respect to
+the flows ϕt
+i of F (i).
+Consider double sequences of positive real numbers
+�
+b(i)
+jk
+�
+j≥1,k≥1, with i = 1, . . . , m,
+such that b(i)
+jk b(i)
+kj > 0 if ⟨L �
+F (i)�ek, �ej⟩ ̸= 0 (where �ej(�z) = �zα(j) are monomials in the
+new coordinates �z = P −1z) and such that
+∞
+�
+k=1
+b(i)
+jk ≤ 1, and define the double sequence
+(3.8)
+�
+�
+�Q(i)
+jk
+def
+=
+�
+�
+�
+�
+�
+���
+L �
+F (i)�ek, �ej
+���2
+4
+��ℜ
+��
+L �
+F (i)�ej, �ej
+���� ��ℜ
+��
+L �
+F (i)�ek, �ek
+����
+1
+b(i)
+jk b(i)
+kj
+if
+�
+L �
+F (i)�ek, �ej
+�
+̸= 0
+0
+otherwise
+�
+�
+�
+j≥2,1≤k≤j−1
+.
+If the series
+(3.9)
++∞
+�
+k=1
+|α(k)| ϵk ρ2|α(k)|
+is convergent with
+(3.10)
+ϵj >
+max
+i=1,··· ,m
+k=1,...,j−1
+ϵk Q(i)
+jk ,
+then the switched system (3.7) is GUAS on Dn(0, ρ). Moreover the series
+V (z) =
+∞
+�
+k=1
+ϵk
+���
+�
+P −1z
+�α(k)���
+2
+is a common global Lyapunov function on Dn(0, ρ).
+This manuscript is for review purposes only.
+
+16
+C. M. ZAGABE AND A. MAUROY
+Proof. Consider the switched system
+(3.11)
+�
+˙�z = �F (i)(�z)
+�m
+i=1
+defined on Dn. By Corollary 3.3, the monomials �ek(�z) = (�z)α(k) generate a common
+infinite invariant maximal flag for ¯L �
+F (i). We first show that the candidate Lyapunov
+functional �V(f) =
+∞
+�
+k=1
+ϵk |⟨f, �ek⟩|2 satisfies
+d
+dt
+�V
+�
+(�U (i)
+t )∗ (k�z − k0)
+�
+< 0
+for all i = 1, · · · , m, where �U (i)
+t
+denotes the Koopman semigroup associated with the
+subsystem ˙�z = �F (i)(�z). Lemma 3.5 with (3.9) implies that the series (3.3) and (3.4)
+are absolutely convergent on Dn (0, ρ). Then, it follows from Lemma 3.6 that
+d
+dt
+�V
+�
+(�U (i)
+t )∗ (k�z − k0)
+�
+≤2
+∞
+�
+j=1
+b(i)
+jj ϵj
+���c(i)
+j
+���
+2
+ℜ
+�
+λ(i)
+j
+�
++ 2
+∞
+�
+j=2
+j−1
+�
+k=1
+b(i)
+jk ϵj
+���c(i)
+j
+���
+2
+ℜ
+�
+λ(i)
+j
+�
++ b(i)
+kj ϵk
+���c(i)
+k
+���
+2
+ℜ
+�
+λ(i)
+k
+�
++ ϵkℜ
+�
+c(i)
+j ¯c(i)
+k
+�
+�ej, L �
+F (i)�ek
+��
+where c(i)
+j
+=
+�
+(�U (i)
+t )∗ (k�z − k0) , �ej
+�
+and λ(i)
+j
+=
+�
+L �
+F (i)�ej, �ej
+�
+. Since ℜ{λ(i)
+j } < 0 (see
+(3.1)), one has to find a sequence of positive numbers (ϵj)j≥1 such that
+b(i)
+jk ϵj
+���c(i)
+j
+���
+2 ���ℜ
+�
+λ(i)
+j
+���� + b(i)
+kj ϵk
+���c(i)
+k
+���
+2 ���ℜ
+�
+λ(i)
+k
+���� > ϵk
+���ℜ
+�
+c(i)
+j ¯c(i)
+k
+�
+�ej, L �
+F (i)�ek
+�����
+for all i = 1, · · · , m and for all j, k with j > k such that
+(3.12)
+�
+�ej, L �
+F (i)�ek
+�
+̸= 0.
+By using the inequality
+���ℜ
+�
+c(i)
+j ¯c(i)
+k
+�
+�ej, L �
+F (i)�ek
+����� ≤
+���c(i)
+j
+���
+���c(i)
+k
+���
+���
+�ej, L �
+F (i)�ek
+��� ,
+one has to satisfy
+b(i)
+jk ϵj
+���c(i)
+j
+���
+2 ���ℜ
+�
+λ(i)
+j
+���� + b(i)
+kj ϵk
+���c(i)
+k
+���
+2 ���ℜ
+�
+λ(i)
+k
+���� > ϵk
+���c(i)
+j
+���
+���c(i)
+k
+���
+���
+L �
+F (i)�ek, �ej
+���
+or equivalently
+(3.13)
+ϵj > ϵk
+�
+�−
+b(i)
+kj
+b(i)
+jk
+���ℜ
+�
+λ(i)
+k
+����
+���ℜ
+�
+λ(i)
+j
+����
+�����
+c(i)
+k
+c(i)
+j
+�����
+2
++
+���
+L �
+F (i)�ek, �ej
+���
+b(i)
+jk
+���ℜ
+�
+λ(i)
+j
+����
+�����
+c(i)
+k
+c(i)
+j
+�����
+�
+� def
+= ϵk h
+������
+c(i)
+k
+c(i)
+j
+�����
+�
+.
+It is easy to see that the real quadratic function h has the maximal value
+Q(i)
+jk =
+���
+L �
+F (i)�ek, �ej
+���2
+4
+���ℜ
+�
+λ(i)
+j
+����
+���ℜ
+�
+λ(i)
+k
+����
+1
+b(i)
+jk b(i)
+kj
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+17
+so that (3.13) is satisfied if we choose iteratively ϵj according to (3.10). It follows that
+we have
+d
+dt
+�V
+�
+(�U (i)
+t )∗ (k�z − k0)
+�
+< 2
+∞
+�
+j=1
+b(i)
+jj ϵj
+���c(i)
+j
+���
+2
+ℜ
+�
+λ(i)
+j
+�
+< − min
+j
+�
+b(i)
+jj
+���ℜ
+�
+λ(i)
+j
+����
+� ∞
+�
+j=1
+ϵj
+���c(i)
+j
+���
+2
+= − min
+j
+�
+b(i)
+jj
+���ℜ
+�
+λ(i)
+j
+����
+�
+�V
+�
+(�U (i)
+t )∗ (k�z − k0)
+�
+.
+With the evaluation functional k�z, we can define
+�V : Dn (0, ρ) → R+,
+�V (�z) = �V (k�z − k0)
+and, using Lemma 2.13, we verify that
+�V
+�
+�ϕ(i)
+t (�z)
+�
+= �V
+�
+k�ϕ(i)
+t
+(�z) − k0
+�
+= �V
+�
+k�ϕ(i)
+t
+(�z) − k�ϕ(i)
+t
+(0)
+�
+= �V
+�
+(�U (i)
+t )∗ (k�z − k0)
+�
+< �V (k�z − k0) = �V (�z).
+In addition, if we define V = �V ◦ P −1 : Dn (0, ρ) → R+, we have
+V
+�
+ϕ(i)
+t (z)
+�
+= �V
+�
+P −1ϕ(i)
+t (P �z)
+�
+= �V
+�
+�ϕ(i)
+t (�z
+�
+< �V (�z) = V (P �z) = V (z).
+Therefore, we have the CLF
+(3.14)
+V (z) =
+∞
+�
+k=1
+ϵk |⟨k�z − k0, �ek⟩|2 =
+∞
+�
+k=1
+ϵk |⟨k�z, �ek⟩|2 =
+∞
+�
+k=1
+ϵk
+���
+�
+P −1z
+�α(k)���
+2
+for the switched nonlinear system (3.7). Finally, since Dn (0, ρ) is forward invariant
+with respect to ϕ(i)
+t , the switched system (3.7) is GUAS on Dn(0, ρ).
+Note that, if the assumptions of Theorem 3.7 are satisfied but the polydisk
+Dn(0, ρ) is not forward invariant with respect to the flow generated by the subsys-
+tems, then the switched system is GUAS in the largest sublevel set of the Lyapunov
+function that is contained in Dn(0, ρ).
+The condition on the boundedness of the double sequence (3.8) could be inter-
+preted as the dominance of diagonal entries of the matrix ¯L �
+F (i) (i.e., the Koopman
+eigenvalues (3.1)) with respect to the other entries. Moreover, the number of non-
+zero cross-terms (3.12) to be compensated affects the way we define the sequence of
+weights b(i)
+jk and therefore the sequence ϵj in (3.10). If the double sequence (3.8) has
+an upper bound Q < 1, one can set ϵk = Q for all k. However, such case rarely
+appears. Instead, if Q > 1, one might have ϵk = O(Qk) and it is clear that (3.9)
+diverges for all ρ since k − 2|α(k)| → ∞ as k → ∞ (except in the case n = 1 where
+|α(k)| = k, see also Remark 3.10 below). In the following, we will consider specific
+vector fields such that the series (3.9) converges for a proper choice of sequence b(i)
+jk ,
+so that Theorem 3.7 can be used.
+For polynomial vector fields of the form F (i)
+l
+(z) =
+r
+�
+k=1
+a(i)
+l,kzα(k), we denote by K(i)
+the number of nonzero terms (without counting the monomial zl in F (i)
+l
+), i.e.
+(3.15)
+K(i) =
+m
+�
+l=1
+#
+�
+k ̸= l : a(i)
+l,k ̸= 0
+�
+This manuscript is for review purposes only.
+
+18
+C. M. ZAGABE AND A. MAUROY
+where # is the cardinal of a set. In this case, we have the following result.
+Corollary 3.8. Let
+(3.16)
+�
+˙z = F (i)(z)
+�m
+i=1
+be a switched nonlinear system on Dn, where F (i) are polynomial vector fields. Assume
+that
+• all subsystems of (3.16) have a common hyperbolic equilibrium ze = 0 that is
+globally asymptotically stable on Dn,
+• the Lie algebra span
+�
+JF (i)(0)
+�
+Lie is solvable (and therefore there exists a
+matrix P such that PJF (i)(0)P −1 are upper triangular),
+• the unit polydisk Dn is forward invariant with respect to the flows ϕ(i)
+t
+gener-
+ated by F (i).
+If
+(3.17)
+max
+i=1,··· ,m lim sup
+j∈N
+max
+k=1,...,j−1
+( �K(i))2 ���
+L �
+F (i)�ek, �ej
+���2
+��ℜ
+��
+L �
+F (i)�ej, �ej
+���� ��ℜ
+��
+L �
+F (i)�ek, �ek
+���� < 1,
+where �K(i) is the number of nonzero terms of �F (i)(�z) = P −1F (i)(P �z) (see (3.15)) and
+where �ej(�z) = �zα(j) are the monomials in the new coordinates �z = P −1z, then (3.16)
+is GUAS on Dn.
+Proof. The result follows from Theorem 3.7 with the sequence
+(3.18)
+�
+�
+�
+�
+�
+�
+�
+b(i)
+jj = (1 − ξ)
+b(i)
+jk =
+ξ
+2K(i)
+if j ̸= k with
+�
+L �
+F (i)�ek, �ej
+�
+̸= 0 or
+�
+L �
+F (i)�ej, �ek
+�
+̸= 0
+b(i)
+jk = 0,
+if j ̸= k with
+�
+L �
+F (i)�ek, �ej
+�
+= 0 or
+�
+L �
+F (i)�ej, �ek
+�
+= 0,
+with ξ ∈]0, 1[.
+It is clear from (2.14) that, for a fixed j and for all k ∈ N \ {j},
+there are at most K(i) nonzero values ⟨L �
+F (i)�ek, �ej⟩ and at most K(i) nonzero values
+⟨L �
+F (i)�ej, �ek⟩, so that the sequence (3.18) satisfies
+∞
+�
+k=1
+b(i)
+jk ≤ 1. The elements Q(i)
+jk of
+the double sequence (3.8) are given by
+(3.19)
+Q(i)
+jk =
+( �K(i))2 ���
+L �
+F (i)�ek, �ej
+���2
+ξ2 ��ℜ
+��
+L �
+F (i)�ej, �ej
+���� ��ℜ
+��
+L �
+F (i)�ek, �ek
+����.
+The condition (3.17) implies that
+max
+i=1,··· ,m lim sup
+j∈N
+max
+k=1,...,j−1 Q(i)
+jk
+def
+= Q < 1 for some
+ξ ∈]0, 1[, so that (3.10) is satisfied with
+(3.20)
+ϵj ∼ max
+k∈Kj {ϵk Q}
+for j ≫ 1, with Kj = {k ∈ {1, . . . , j − 1} :
+�
+L �
+F (i)�ek, �ej
+�
+̸= 0 for some i ∈ {1, . . . , m}}.
+The sequence (3.20) yields ϵj = O(Qj) for j ≫ 1. It follows that (3.9) is convergent
+for any ρ ≤ 1 and Theorem 3.7 implies that the switched system (3.16) is GUAS on
+Dn.
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+19
+Another result is obtained when a diagonal dominance property is assumed for
+the Jacobian matrices JF (i)(0).
+Corollary 3.9. Let
+(3.21)
+�
+˙z = F (i)(z)
+�m
+i=1
+be a switched nonlinear system on Dn, with F (i)
+l
+(z) =
++∞
+�
+k=1
+a(i)
+l,kzα(k) and
++∞
+�
+k=1
+|a(i)
+l,k| < ∞
+for all i = {1, . . . , m} and l ∈ {1, . . . , n}. Assume that
+• all subsystems of (3.21) have a common hyperbolic equilibrium ze = 0 that is
+globally asymptotically stable on Dn,
+• the Lie algebra span
+�
+JF (i)(0)
+�
+Lie is solvable (and therefore there exists a
+matrix P such that PJF (i)(0)P −1 are upper triangular),
+• there exists ρ ∈]0, 1] such that Dn (0, ρ) is forward invariant with respect to
+the flows ϕ(i)
+t
+generated by F (i).
+If there exist ξ ∈]0, 1[ and κ ∈]0, 1[ with ξ + κ < 1 such that, for all q, r ∈
+{1, · · · , n} with q < r (when n > 1),
+(3.22)
+���J �F (i)(0)]qr
+���
+2
+<
+�
+2ξ
+n2 − n
+�2 ���ℜ([J �F (i)(0)]rr)
+���
+���ℜ([J �F (i)(0)]qq)
+��� ,
+(3.23)
+���[J �F (i)(0)]qr
+��� <
+2ξ
+n2 − n
+���ℜ([J �F (i)(0)]qq)
+���
+and
+(3.24)
+max
+i=1,··· ,m lim sup
+j∈N
+max
+k=1,...,j−1
+⟨L �
+F (i) �ek,�ej⟩̸=0
+�∞
+l=1
+���
+L �
+F (i)�el, �ej
+��� �∞
+l=1
+���
+L �
+F (i)�ek, �el
+���
+κ2 ��ℜ
+��
+L �
+F (i)�ej, �ej
+���� ��ℜ
+��
+L �
+F (i)�ek, �ek
+���� < 1
+ρ2 ,
+where �ej(�z) = �zα(j) are monomials in the new coordinates �z = P −1z, then (3.21) is
+GUAS on Dn(0, ρ).
+Proof. We will denote by D
+def
+= (n2 − n)/2 the number of upper off-diagonal
+entries of the Jacobian matrices J �F (i)(0). The result follows from Theorem 3.7 with
+the sequence
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+b(i)
+jj = (1 − ξ − κ)
+b(i)
+jk =
+ξ
+2D
+if j ̸= k with |α(j)| = |α(k)|, and if
+�
+L �
+F (i)�ek, �ej
+�
+̸= 0 or
+�
+L �
+F (i)�ej, �ek
+�
+̸= 0
+b(i)
+jk = 0
+if |α(j)| = |α(k)|,
+�
+L �
+F (i)�ek, �ej
+�
+= 0 and
+�
+L �
+F (i)�ej, �ek
+�
+= 0
+b(i)
+jk = κ
+2
+���
+L �
+F (i)�ek, �ej
+���
+�∞
+l=1
+���
+L �
+F (i)�el, �ej
+���
+if |α(k)| < |α(j)|
+b(i)
+jk = κ
+2
+���
+L �
+F (i)�ej, �ek
+���
+�∞
+l=1
+���
+L �
+F (i)�ej, �el
+���
+if |α(k)| > |α(j)|
+with ξ ∈]0, 1[ and κ ∈]0, 1[. It follows from (2.15) in Remark 2.16 and the fact that
+the Jacobian matrices J �F (i)(0) are upper triangular that, for a fixed j and all k ̸= j
+This manuscript is for review purposes only.
+
+20
+C. M. ZAGABE AND A. MAUROY
+with |α(k)| = |α(j)|, there are at most D nonzero values
+�
+L �
+F (i)�ek, �ej
+�
+and at most D
+nonzero values
+�
+L �
+F (i)�ej, �ek
+�
+. Therefore, the sequence b(i)
+jk satisfies
+∞
+�
+k=1
+b(i)
+jk < (1 − ξ − κ) + ξ + κ
+2
+�j
+k=1
+���
+L �
+F (i)�ek, �ej
+���
+�∞
+l=1
+���
+L �
+F (i)�el, �ej
+��� + κ
+2
+�∞
+k=j+1
+���
+L �
+F (i)�ej, �ek
+���
+�∞
+l=1
+���
+L �
+F (i)�ej, �el
+���
+< 1.
+The elements Q(i)
+jk of the double sequence (3.8) are given by
+(3.25)
+Q(i)
+jk =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+D2 ���
+L �
+F (i)�ek, �ej
+���2
+ξ2 ��ℜ
+��
+L �
+F (i)�ej, �ej
+���� ��ℜ
+��
+L �
+F (i)�ek, �ek
+����
+if |α(j)| = |α(k)|
+�∞
+l=1
+���
+L �
+F (i)�el, �ej
+��� �∞
+l=1
+���
+L �
+F (i)�ek, �el
+���
+κ2 ��ℜ
+��
+L �
+F (i)�ej, �ej
+���� ��ℜ
+��
+L �
+F (i)�ek, �ek
+����
+if |α(k)| ̸= |α(j)| and
+�
+L �
+F (i)�ek, �ej
+�
+̸= 0
+0
+otherwise.
+We note that
+∞
+�
+l=1
+���
+L �
+F (i)�el, �ej
+��� and
+∞
+�
+l=1
+���
+L �
+F (i)�ek, �el
+��� are finite according to the as-
+sumption.
+Next, we show that the conditions (3.22) and (3.23) imply that Q(i)
+jk < 1 if |α(j)| =
+|α(k)|. Indeed, it follows from (2.15) and (3.25) that this latter inequality is equivalent
+to
+α2
+q(k)|[J �F (i)(0)]qr|2 < ξ2
+D2
+�����
+n
+�
+l=1
+αl(j)ℜ([J �F (i)(0)]ll)
+�����
+�����
+n
+�
+l=1
+αl(k)ℜ([J �F (i)(0)]ll)
+�����
+for all j > k such that α(j) = (α1(k), · · · , αq(k)−1, · · · , αr(k)+1, · · · , αn(k)) for some
+q < r. Since the diagonal entries of the (upper-triangular) Jacobian matrices J �F (i)(0)
+are the eigenvalues and therefore have negative real parts, the most restrictive case is
+obtained with αl(k) = 0 for all l ̸= q, which yields
+α2
+q(k)|[J �F (i)(0)]qr|2 < ξ2
+D2
+���(αq(k) − 1)ℜ([J �F (i)(0)]qq) + ℜ([J �F (i)(0)]rr)
+���
+���αq(k)ℜ([J �F (i)(0)]qq)
+��� .
+When αq(k) = 1, this inequality is equivalent to (3.22). When αq(k) > 1, we can
+rewrite
+(αq(k) − 1)|[J �F (i)(0)]qr|2 + |[J �F (i)(0)]qr|2
+< ξ2
+D2
+�
+(αq(k) − 1)
+���ℜ([J �F (i)(0)]qq)
+���
+2
++
+���ℜ([J �F (i)(0)]rr)
+���
+���ℜ([J �F (i)(0)]qq)
+���
+�
+.
+Using (3.22), we have that the above inequality is satisfied if
+(αq(k) − 1)|[J �F (i)(0)]qr|2 < ξ2
+D2 (αq(k) − 1)
+���ℜ([J �F (i)(0)]qq)
+���
+2
+,
+which is equivalent to (3.23).
+While Q(i)
+jk < 1 for |α(j)| = |α(k)|, it is easy to see that Q(i)
+jk > 1 for |α(j)| > |α(k)|.
+The condition (3.24) therefore implies that
+max
+i=1,··· ,m lim sup
+j∈N
+max
+k=1,...,j−1 Q(i)
+jk
+def
+= Q < 1/ρ2
+and (3.10) is satisfied with
+(3.26)
+ϵj ∼ max
+k∈Kj {ϵk Q}
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+21
+for j ≫ 1, with
+Kj = {k ∈ 1, . . . , j − 1 :
+�
+L �
+F (i)�ek, �ej
+�
+̸= 0 for some i ∈ {1, . . . , m} and |α(k)| < |α(j)|}.
+Hence, the sequence (3.26) yields ϵj = O(Q|α(j)|). It follows that (3.9) is convergent
+and Theorem 3.7 implies that the switched system (3.21) is GUAS on Dn(0, ρ).
+For the particular case where the Jacobian matrices JF (i)(0) are simultaneously
+diagonalizable (i.e. they are diagonalizable and they commute), the diagonal dom-
+inance conditions (3.22) and (3.23) are trivially satisfied. We should mention that
+the Lie-algebraic property of commutation is only needed for the Jacobian matrices
+JF (i)(0), an assumption which contrasts with the commutation property imposed on
+vector fields in [17], [30] and [33].
+Remark 3.10. In the case n = 1, we recover the trivial GUAS property of switched
+systems from Corollary 3.9. Indeed, consider the vector fields F (i)(z) =
+∞
+�
+k=1
+a(i)
+k zk
+on D, with
+∞
+�
+k=1
+|a(i)
+k | < ∞, and assume that the subsystems have a globally stable
+equilibrium at the origin. The Lie-algebra generated by the scalars JF (i)(0) is trivially
+solvable and D(0, ρ) is forward invariant for all ρ. Moreover, the conditions (3.22) and
+(3.23) are trivially satisfied. Then Corollary 3.9 implies that the switched system is
+GUAS on D(0, ρ) for ρ ∈]0, 1[ which satisfies (3.24). It follows from (2.14) that
+∞
+�
+l=1
+|⟨LF (i)el, ej⟩| =
+j
+�
+l=1
+l|a(i)
+j−l+1| =
+j
+�
+l=1
+(j − l + 1)|a(i)
+l |,
+∞
+�
+l=1
+|⟨LF (i)ek, el⟩| = k
+∞
+�
+l=1
+|a(i)
+l |,
+and |ℜ (⟨ej, LF (i)ej⟩)| = j
+���ℜ(a(i)
+1 )
+���. With κ arbitrarily close to 1 (since ξ can be taken
+arbitrarily small in (3.22) and (3.23)), condition (3.24) is rewritten as
+max
+i=1,··· ,m lim sup
+j∈N
+�j
+l=1(j − l + 1)|a(i)
+l | �∞
+l=1 |a(i)
+l |
+j
+���ℜ(a(i)
+1 )
+���
+2
+< 1
+ρ2
+and, using (j − l + 1)|a(i)
+l | ≤ j|a(i)
+l | for all l, we obtain
+ρ <
+min
+i=1,··· ,m
+�∞
+l=1 |a(i)
+l |
+���ℜ(a(i)
+1 )
+���
+.
+4. Examples. This section presents two examples that illustrate our results. We
+will focus on specific cases that satisfy the assumptions of Corollaries 3.8 and 3.9 and,
+without loss of generality, we will directly consider Jacobian matrices in triangular
+form.
+4.1. Example 1: polynomial vector fields. Similarly to Example 1, we con-
+sider the vector fields on the bidisk D2
+(4.1)
+F (1)(z1, z2) =
+�
+−az1
+−az2
+and F (2)(z1, z2) =
+�
+�
+�
+−az1 + b
+�
+z2
+1 − z1z2
+2
+�
+−az2 + b
+2z1z2,
+This manuscript is for review purposes only.
+
+22
+C. M. ZAGABE AND A. MAUROY
+where b > 0 and a > 3b. For all ρ < 1, the bidisk D2(0, ρ) is invariant with respect
+to the flows of F (i). Indeed, for all z ∈ ∂D2(0, ρ) (i.e. |zl| = ρ for some l), one has to
+verify that ℜ
+�
+F (i)
+l
+(z) ¯zl
+�
+< 0. We have
+• |zl| = ρ ⇒ ℜ
+�
+F (1)
+l
+(z)¯zl
+�
+= −aρ2 < 0,
+• |z1| = ρ ⇒ ℜ
+�
+F (2)
+1
+(z)¯z1
+�
+= −aρ2 + bρ2ℜ
+�
+z1 − z2
+2
+�
+< 0,
+• |z2| = ρ ⇒ ℜ
+�
+F (2)
+2
+(z)¯z2
+�
+= ρ2
+�
+−a + b
+2ℜ (z1)
+�
+< 0.
+It is clear that vector field F (1) generates a holomorphic flow on D2.
+The same
+property holds for F (2) since the conditions of Proposition A.1 are satisfied with
+h1 (z′
+1) = h2 (z′
+2) = 0 (i.e. ℜ{G1(z)} = ℜ{−a + b
+�
+z1 − z2
+2
+�
+} < 0 and ℜ{G2(z)} =
+ℜ{−a + b
+2z1} < 0). The unique global stable equilibrium of the subsystems in the
+bidisk D2 is the origin. According to (2.14), the entries of the Koopman matrices
+¯LF (1) and ¯LF (2) are given by
+⟨LF (1)ek, ej⟩ =
+�
+−a |α(j)|
+if k = j
+0
+otherwise
+and
+⟨LF (2)ek, ej⟩ =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−a |α(j)|
+if k = j
+b
+�
+α1(j) + α2(j) − 2
+2
+�
+if α1(k) = α1(j) − 1 ≥ 0 and α2(k) = α2(j)
+−b α1(j)
+if α1(k) = α1(j) and α2(k) = α2(j) − 2 ≥ 0
+0
+otherwise .
+Since ⟨LF (1)ek, ej⟩=0 for all k ̸= j, we have Q(1)
+jk = 0 for all k, j in (3.17). Moreover,
+K(2) = 3 and the condition (3.17) can be rewritten as
+Q = lim sup
+j∈N
+|α(j)|>1
+max
+�
+�
+�
+�
+�
+9b2
+a2
+�
+α1(j) + α2(j)−2
+2
+�2
+|α(j)| (|α(j)| − 1) , 9b2
+a2
+(α1(j))2
+|α(j)| (|α(j)| − 2)
+�
+�
+�
+�
+�
+= 9b2
+a2 < 1
+and is satisfied since a > 3b. Hence, it follows from Corollary 3.8 that the switched
+system (4.1) is GUAS in D2. Note that, in this case, a CLF is given by
+V (z) =
+∞
+�
+k=1
+Q−k ���zα(k)���
+2
+.
+4.2. Example 2: analytic vector fields. The following example is taken from
+[1] and [12]. Consider the switched system defined by the vector fields
+F (1)(x1, x2) =
+�
+−x1 + 1
+µ sin2(x1)x2
+1x2, −x2
+�
+F (2)(x1, x2) =
+�
+−x1 + 1
+µ cos2(x1)x2
+1x2, −x2
+�
+,
+where µ ≥ 12/5. Both subsystems are globally asymptotically stable, but the switched
+system is not GUAS in R2, as shown in [1] by using the fact that the convex com-
+bination F =
+�
+F (1) + F (2)�
+/2 of the two subsystems is not globally asymptotically
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+23
+stable in R2 (see Corollary 2.5). Yet, our result allows to infer the GUAS property
+in a specific region of the invariant real square ] − 1, 1[2. To do so, we complexify
+the dynamics and define the vector fields on the bidisk D2. For all ρ < 1, the bidisk
+D2(0, ρ) is invariant with respect to the flows of F (i). Indeed, we have
+• |z1| = ρ ⇒ ℜ
+�
+F (1)
+1
+(z)¯z1
+�
+= ρ2
+�
+−1 + 1
+µℜ
+�
+z1z2 sin2(z1)
+��
+< 0 since
+1
+µ
+��ℜ
+�
+z1z2 sin2(z1)
+��� ≤ ρ2 ��sin2(z1)
+��
+µ
+< 1
+where we used max
+z1∈D
+��sin2(z1)
+�� < 12/5 ≤ µ,
+• |z2| = ρ ⇒ ℜ
+�
+F (1)
+2
+(z)¯z2
+�
+= −ρ2 < 0.
+The same result follows for F (2) (with max
+z1∈D
+��cos2(z1)
+�� < 12/5 ≤ µ). The vector field
+F (1) generates a holomorphic flow on D2 since the conditions of Proposition A.1 are
+satisfied with h1 (z′
+1) = h2 (z′
+2) = 0 (i.e. ℜ{G1(z)} = ℜ{−1 + 1
+µ sin2(z1)z1z2} < 0 and
+ℜ{G2(z)} = −1 < 0. The same result holds for F (2). The Taylor expansion of the
+vector fields yields
+F (1)(z) =
+�
+−z1 + 1
+µ
+∞
+�
+p=1
+(−1)p+122p−1
+(2p)!
+z2p+2
+1
+z2, −z2
+�
+F (2)(z) =
+�
+−z1 + 1
+µz2
+1z2 + 1
+µ
+∞
+�
+p=1
+(−1)p22p−1
+(2p)!
+z2p+2
+1
+z2, −z2
+�
+.
+According to (2.14), the entries of the Koopman matrices ¯LF (1) and ¯LF (2) are given
+by
+⟨LF (1)ek, ej⟩ =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− |α(k)|
+if k = j
+α1(k)
+µ
+(−1)p+122p−1
+(2p)!
+if α1(k) = α1(j) − 1 − 2p ≥ 0 and α2(k) = α2(j) − 1 ≥ 0
+0
+otherwise
+and
+⟨LF (2)ek, ej⟩ =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− |α(k)|
+if k = j
+α1(k)
+µ
+if α1(k) = α1(j) − 1 ≥ 0 and α2(k) = α2(j) − 1 ≥ 0
+α1(k)
+µ
+(−1)p22p−1
+(2p)!
+if α1(k) = α1(j) − 1 − 2p ≥ 0 and α2(k) = α2(j) − 1 ≥ 0
+0
+otherwise
+This manuscript is for review purposes only.
+
+24
+C. M. ZAGABE AND A. MAUROY
+where p = 1, · · · ,
+�α1(j) − 1
+2
+�
+. This implies that we have
+(4.2)
+∞
+�
+l=1
+|⟨LF (1)el, ej⟩| = |α(j)| + 1
+µ
+⌊ α1(j)−1
+2
+⌋
+�
+l=1
+(α1(j) − 1 − 2l)22l−1
+(2l)!
+∞
+�
+l=1
+|⟨LF (2)el, ej⟩| = |α(j)| + α1(j) − 1
+µ
++ 1
+µ
+⌊ α1(j)−1
+2
+⌋
+�
+l=1
+(α1(j) − 1 − 2l)22l−1
+(2l)!
+∞
+�
+l=1
+|⟨LF (1)ek, el⟩| = |α(k)| + α1(k)
+µ
+∞
+�
+p=1
+22p−1
+(2p)! = |α(k)| + α1(k)
+2µ
+(cosh(2) − 1)
+∞
+�
+l=1
+|⟨LF (2)ek, el⟩| = |α(k)| + α1(k)
+µ
+�
+1 +
+∞
+�
+p=1
+22p−1
+(2p)!
+�
+= |α(k)| + α1(k)
+2µ
+(cosh(2) + 1) .
+Since the Jacobian matrices JF (i)(0) are diagonal, the conditions (3.22) and (3.23)
+are trivially satisfied (with ξ arbitrarily small). Moreover, we observe from (4.2) that
+∞
+�
+l=1
+|⟨LF (1)el, ej⟩| ≤
+∞
+�
+l=1
+|⟨LF (2)el, ej⟩| and
+∞
+�
+l=1
+|⟨LF (1)ek, el⟩| ≤
+∞
+�
+l=1
+|⟨LF (2)ek, el⟩| for all
+k, j. It follows that, with κ arbitrarily close to 1, condition (3.24) can be rewritten as
+lim sup
+j∈N
+max
+k=1,...,j−1
+�∞
+l=1 |⟨LF (2)el, ej⟩| �∞
+l=1 |⟨LF (2)ek, el⟩|
+|ℜ (⟨LF (2)ej, ej⟩)| |ℜ (⟨LF (2)ek, ek⟩)|
+< 1
+ρ2 ,
+which is verified for ρ =
+�
+1 + 1
+2µ (cosh(2) + 1)
+�−1
+. Indeed, from (4.2), we have
+∞
+�
+l=1
+|⟨LF (2)el, ej⟩| < |α(j)| + α1(j)
+µ
+�
+1 +
+∞
+�
+l=1
+22l−1
+(2l)!
+�
+= |α(j)| + α1(j)
+2µ
+(cosh(2) + 1)
+≤ |α(j)|
+�
+1 + 1
+2µ (cosh(2) + 1)
+�
+.
+and
+∞
+�
+l=1
+|⟨LF (2)ek, el⟩| ≤ |α(k)|
+�
+1 + 1
+2µ (cosh(2) + 1)
+�
+.
+It follows from Corollary 3.9 that the switched system is GUAS on D2(0, ρ). See
+Figure 1 for the different values of ρ depending on µ. Note that a CLF is given by
+V (z) =
+∞
+�
+k=1
+Q−2|α(k)| ���zα(k)���
+2
+where Q = 1/ρ.
+5. Conclusion and perspectives. This paper provides new advances on the
+uniform stability problem for switched nonlinear systems satisfying Lie-algebraic solv-
+ability conditions. First, we have shown that the solvability condition on nonlinear
+This manuscript is for review purposes only.
+
+UNIFORM STABILITY OF SWITCHED NONLINEAR SYSTEMS
+25
+Fig. 1. The switched system is shown to be GUAS on a polydisk of radius ρ that depends on
+the parameter µ.
+vector fields does not guarantee the existence of a common invariant flag and, instead,
+we have imposed the solvability condition only on the linear part of the vector fields.
+Then we have constructed a common Lyapunov functional for an equivalent infinite-
+dimensional switched linear system obtained with the adjoint of the Koopman genera-
+tor on the Hardy space of the polydisk. Finally we have derived a common Lyapunov
+function via evaluation functionals to prove that specific switched nonlinear systems
+are uniformly globally asymptotically stable on invariant sets. Our results heavily
+rely on the Koopman operator framework, which appears to be a valid tool to tackle
+theoretical questions from a novel angle.
+We envision several perspectives for future research. Our results apply to specific
+types of switched nonlinear systems within the frame of Lie-algebraic solvability con-
+ditions. They could be extended to more general dynamics, including dynamics that
+possess a limit cycle or a general attractor. In the same line, the Koopman operator-
+based techniques developed in this paper could be applied to other types of stability
+than uniform stability. More importantly, the obtained stability results are limited
+to bounded invariant sets, mainly due to the convergence properties of the Lyapunov
+functions and the very definition of the Hardy space on the polydisk. We envision
+that these results could possibly be adapted to infer global stability in Rn. Finally,
+our results are not restricted to switched systems and have direct implications in the
+global stability properties of nonlinear dynamical systems, which will be investigated
+in a future publication.
+Appendix A. General theorems.
+We recall here some general results that are used in the proofs of our results.
+Proposition A.1 ([4]).
+Let F : Dn → Cn be holomorphic.
+Then F is an
+infinitesimal generator on Dn if and only if, for all l = 1, · · · , n and for all z ∈ Dn,
+Fl(z) = Gl(z) (zl − hl (z′
+l))
+where z′
+l = (z1, · · · , zl−1, zl+1, · · · zn), hl : Dn−1 → D is holomorphic, Gl : Dn → C is
+holomorphic, and ℜ ((1 − hl (z′
+l) ¯zl) Gl(z)) ≤ 0.
+Theorem A.2 (Maximum Modulus Principle for bounded domains [27]).
+Let
+Dn ⊂ Cn be a bounded domain and f : Dn → C be a continuous function, whose
+restriction to Dn is holomorphic. Then |f| attains a maximum on the boundary ∂Dn.
+Theorem A.3 (Abel’s multidimensional lemma [27] p.36).
+Let
+�
+α∈Nn
+aαzα be
+This manuscript is for review purposes only.
+
+1
+0.9
+0.8
+0.7
+0.6
+0.5
+0
+10
+20
+30
+40
+50
+μ26
+C. M. ZAGABE AND A. MAUROY
+a power series. If there exist r ∈ Cn such that sup
+α∈Nn |aαrα| < ∞, then the series
+�
+α∈Nn
+aαzα is normally convergent for all z ∈ Cn such that |z1| < |r1|, · · · , |zn| < |rn|.
+Theorem A.4 (Weierstrass’s M-test).
+Let
++∞
+�
+k=1
+fn(z) be a series of functions on
+a domain Dn of Cn. If there exists a sequence of real numbers Mk such that
+• Mk > 0 for all k,
+• the numerical series
++∞
+�
+k=1
+Mk is convergent and
+• ∀k, ∀z ∈ Dn, |fk(z)| ≤ Mk.
+Then the series
++∞
+�
+k=1
+fn(z) is absolutely and uniformly convergent on Dn.
+Theorem A.5 (Lie’s theorem [8] p.49).
+Let X be a nonzero n-complex vector
+space, and g be a solvable Lie subalgebra of the Lie algebra of n × n complex matrices.
+Then X has a basis (v1, . . . , vn) with respect to which every element of g has an upper
+triangular form.
+REFERENCES
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+[32] R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King, Stability criteria for switched
+and hybrid systems, SIAM review, 49 (2007), pp. 545–592.
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+systems, Systems & control letters, 54 (2005), pp. 405–416.
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+This manuscript is for review purposes only.
+

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+Augmenting data-driven models for energy systems
+through feature engineering: A Python framework
+for feature engineering
+Sandra Wilfling
+Abstract—Data-driven modeling is an approach in energy systems
+modeling that has been gaining popularity. In data-driven mod-
+eling, machine learning methods such as linear regression, neural
+networks or decision-tree based methods are being applied.
+While these methods do not require domain knowledge, they are
+sensitive to data quality. Therefore, improving data quality in a
+dataset is beneficial for creating machine learning-based models.
+The improvement of data quality can be implemented through
+preprocessing methods. A selected type of preprocessing is feature
+engineering, which focuses on evaluating and improving the
+quality of certain features inside the dataset. Feature engineering
+methods include methods such as feature creation, feature ex-
+pansion, or feature selection. In this work, a Python framework
+containing different feature engineering methods is presented.
+This framework contains different methods for feature creation,
+expansion and selection; in addition, methods for transforming
+or filtering data are implemented. The implementation of the
+framework is based on the Python library scikit-learn. The
+framework is demonstrated on a case study of a use case
+from energy demand prediction. A data-driven model is created
+including selected feature engineering methods. The results show
+an improvement in prediction accuracy through the engineered
+features.
+Keywords: Energy Systems Modeling, Data-driven Modeling,
+Feature Engineering, Python, Frameworks
+I. INTRODUCTION
+Modeling and simulation is an crucial step in the design and
+optimization of energy systems. While traditional modeling
+methods rely on system parameters, a recent approach focuses
+on creating data-driven models based on measurement data
+from an underlying system. In data-driven modeling, models
+are not created based on system parameters, but on existing
+measurement data. These models are based on machine learn-
+ing (ML) methods [1]. While the area of machine learning
+includes a wide range of methods such as clustering algorithms
+or classifiers, the focus in data-driven modeling is set to
+regression analysis for prediction and forecasting [2]. In re-
+gression analysis, methods such as linear regression, decision-
+tree based regression, or neural networks are being applied
+[3]. While some of these methods, such as linear regression,
+can be classified as white-box ML methods, others, such as
+neural networks, are classified as black-box ML methods due
+to their lack of comprehensibility [4]. While white-box ML
+methods give more insight about their internal structure than
+black box ML methods, their architecture is simpler, making
+it more difficult to model complex dependencies, for instance
+non-linearities [5]. To capture such dependencies using white-
+box ML models, information about the dependencies can be
+passed to the model through the dataset. This step is called
+feature engineering. The main purpose of feature engineering
+is to augment the existing dataset [6]. This can be done through
+adding new information, or expanding or reducing the existing
+feature set. In addition, the quality of a single feature can be
+improved, for instance through transformation or filtering [7].
+The area of feature engineering covers a wide number of
+methods, such as feature expansion [8] or feature selection
+[9]. The term feature creation covers the creation of features
+to add new information. Methods of feature creation include
+encodings of time-based features, such as cyclic features [10],
+or categorical encoding [11]. Similarly, feature expansion
+is the method of creating new features based on existing
+features. Feature expansion covers classical methods such as
+polynomial expansion [8] or spline interpolation [12].
+In contrast to feature creation and expansion, feature selection
+aims to reduce the size of the feature set. While large feature
+sets may contain more information than smaller feature sets,
+there may be redundancy in the data, as well as sparsity [13] or
+multicollinearity [14]. To reduce the sparsity or multicollinear-
+ity, as well as to remove redundant features, feature selection
+mechanisms are applied. While methods such as Principal
+Component Analysis (PCA) [15] aim to reduce the feature
+set through transformation, feature selection methods discard
+features based on certain criteria [16]. Feature selection can
+be implemented for instance through sequential methods, such
+as forward or backward selection [17], or through correlation
+criteria [9]. Correlation criteria include measures based on
+the Pearson Correlation Coefficient, as well as entropy-based
+criteria [16]. The feature selection is then implemented through
+a threshold-based selection. Threshold-based feature selection
+analyzes features based on the selected criterion, and discards
+features below a certain threshold.
+Mainly, the methods of feature engineering are applied during
+the first steps of creating a data-driven model, creating an en-
+gineered dataset. This engineered dataset is then used to train
+the model [18]. However, feature engineering methods can
+also be used in combination with model selection procedures,
+such as grid search [19]. Feature engineering methods are
+widely used in applications from the energy domain, such as
+in prediction for building energy demand [20] or photovoltaic
+power prediction [18].
+arXiv:2301.01720v1  [cs.LG]  4 Jan 2023
+
+A. Main Contribution
+In the creation of data-driven models, a significant factor is
+the quality of the underlying dataset. To improve the dataset
+quality, feature engineering methods can be applied.
+The main contribution of this work is a Python framework for
+feature engineering that can be used for data-driven model
+creation. The framework implements different methods for
+feature creation, feature expansion, feature selection or trans-
+formation. The feature engineering framework is implemented
+in Python based on the scikit-learn framework and can be
+imported as a Python package. The functionality of the frame-
+work is demonstrated on a case study of an energy demand
+prediction use case. The results of the case study show an
+improvement prediction accuracy through the applied feature
+engineering steps.
+II. METHOD
+The presented framework implements various feature engi-
+neering methods in Python based on the research in [21] and
+on the interfaces defined by scikit-learn. The methods are im-
+plemented using either scikit-learn’s TransformerMixin or
+SelectorMixin interface. The framework implements meth-
+ods for feature expansion, feature creation, feature selection,
+as well as transformation and filtering operations.
+A. Feature Creation and Expansion
+In the framework, different methods for feature creation and
+expansion are implemented. These methods create new fea-
+tures from time values or from expansion of existing features.
+To create new features, the implemented framework supports
+categorical encoding and cyclic encoding of time-based values.
+Cyclic Features Cyclic features can be used to model time
+values through cyclic functions [10]. Cyclic features were
+implemented in [21], as well as in [22] and [23]. In the
+implementation of the framework, sinusoidal signals xsin, xcos
+with a selected frequency f can be created based on a sample
+series n:
+xsin[n] = sin(2πfn)
+(1)
+xcos[n] = cos(2πfn)
+(2)
+The implementation offers the creation of features with a zero-
+order hold function for a certain time period, for instance TS =
+1 day for a signal with a time period of T = 1 week.
+Categorical Features Categorical encoding creates a repre-
+sentation of discrete numerical values through a number of
+features with boolean values [11], [21]. In this implementation,
+for a number of categorical features x0,....,N for a feature x
+with discrete possible values v0,....,N, a single feature xi is
+defined as:
+xi =
+�
+1
+x = vi
+0
+else
+(3)
+The framework offers categorical encoding for time-based
+values. In addition, a division factor is implemented to create
+an encoding of a downsampled version of the time values.
+Feature Expansion For feature expansion, the framework im-
+plements wrappers for scikit-learn’s PolynomialFeatures and
+SplineTransformer classes. The method of polynomial expan-
+sion was applied in [21]. The parameters for the expansion
+methods are passed through the wrapper.
+Time-based Features The framework implements a method of
+dynamic timeseries unrolling to create features xn−1, xn−2,
+... xn−N from an existing feature x. The method of dynamic
+timeseries unrolling is based on the research in [24], [25], and
+[22]. While [25] and [24] use dynamic timeseries unrolling for
+both input and target features of a model, allowing the creation
+of auto-recursive models, this implementation only supports
+dynamic timeseries unrolling for the input features, similar
+to the method used in [22]. In this implementation, dynamic
+timeseries unrolling is implemented through filter operations
+from the scipy.signal library. The dynamic features are created
+through the convolution of the signal x with a Kronecker delta
+for i = 1...N:
+xdyn,i[n] = x[n] ∗ δ[n − i]
+(4)
+This operation creates delayed signals xdyn,1, ..., xdyn,N. In
+our implementation, for the samples in the delayed signals,
+for which no values are available, zero values are used.
+B. Feature Selection
+In the framework, several threshold-based feature selection
+methods are implemented. These methods analyze the input
+and target features based on a certain criterion, and then
+discard features with a low value of the criterion. A widely
+used criterion is the Pearson Correlation Coefficient, which
+is used to detect linear correlations between features [18].
+The Pearson Correlation Coefficient calculates the correlation
+between two features for samples x0,....,N, y0,...,N with mean
+values ¯x and ¯y:
+rx,y =
+�N
+i=0(xi − ¯x)(yi − ¯y)
+��N
+i=0(xi − ¯x)2 �N
+i=0(yi − ¯y)2
+(5)
+While the Pearson correlation identifies linear correlations,
+non-linear dependencies are not detected. To detect non-linear
+dependencies, criteria such as Maximum Information Coeffi-
+cient (MIC) [26], ennemi [27], dCor [28] or the Randomized
+Dependence Coefficient (RDC) [29] can be used.
+The framework provides classes for the criteria Pearson Corre-
+lation Coefficient, F-statistic based on the Pearson Correlation
+Coefficient, as well as thresholds based on the MIC, ennemi
+and RDC.
+C. Transformation and Filtering Operations
+To transform features, the framework implements the Box-
+cox transformation as well as the square root and inverse
+transformation. In addition, the framework provides filtering
+operations, which were applied in timeseries prediction for
+instance in [7]. Discrete-time based filters can be implemented
+in Python through the functions implemented in scipy.signal.
+The scipy.signal library offers functions for calculating the
+coefficients for different types of digital filters. A digital filter
+
+of order N can be defined through the transfer function H(z)
+in a direct form:
+H(z) =
+�N
+i=0 bizi
+�N
+i=0 aizi
+(6)
+The filter coefficients ai and bi define the behavior of the
+filter. The scipy.signal library offers functions to compute the
+filter coefficients for filter types such as the Butterworth or
+Chebyshev filter [30]. While scipy.signal offers the compu-
+tation of analog and digital filter coefficients, the framework
+implementation focuses on digital filter implementations. The
+framework implements the Butterworth and Chebyshev fil-
+ter as scikit-learn TransformerMixin classes. In addition, an
+envelope detection filter was implemented for demodulation
+of modulated signals. This filter was implemented using the
+pandas rolling average function. For all filters, offset com-
+pensation before and after applying the filter operation and a
+mask for handling NaN values were implemented. The direct
+form filter classes of the framework offer a simple option for
+extension. Different architectures can be implemented by re-
+defining the implemented method for coefficient calculation.
+This allows to create filters with different Finite Impulse
+Response (FIR) or Infinite Impulse Response (IIR) structures.
+D. Composite Transformers
+In feature engineering, it is often the case that only a se-
+lected subset of features should be transformed. To offer the
+possibility to transform only selected features, a composite
+transformer wrapper was implemented. This wrapper offers to
+either automatically replace features through their transformed
+versions, or add transformed features separately to the dataset.
+E. Implementation
+The framework offers compatibility with the sklearn.Pipeline
+implementation,
+making
+it
+possible
+to
+use
+objects
+as
+part of a ML pipeline. The parameters of each objects
+can be adapted through grid search, for instance using
+sklearn.model_selection.GridSearchCV. In addition, every cre-
+ated object can be stored to and loaded from a Pickle file using
+the save_pkl or load_pkl method.
+While the filtering, feature expansion and feature cre-
+ation methods support operations on a numpy.ndarray or
+pd.Dataframe or pd.Series object, the feature creation methods
+require a pd.Dataframe or pd.Series object with a DateTimeIn-
+dex or TimedeltaIndex to create samples based on a certain
+date.
+III. CASE STUDY
+The framework is demonstrated on a use case from prediction
+for energy systems modeling. For this purpose, a mixed office-
+campus building is selected. A prediction model should be
+trained based on existing measurement data. The data-driven
+model is created using a workflow based on the implemented
+methods.
+A. Application
+In this case study, the energy demand of a mixed office-campus
+building should be evaluated. The data was provided from the
+research in [24]. The energy demand of a building is subject
+to various factors. Main factors that influence building energy
+demand are thermal characteristics and Heating, Ventilation,
+Air Conditioning and Cooling (HVAC) system behavior [31].
+Additionally, building energy demand may be dependent on
+occupancy [3] or subject to seasonal trends [10]. Many of these
+factors show non-linear behavior, which makes it difficult to
+address them through a purely linear model. Therefore, feature
+engineering was used to model additional factors.
+B. Data-driven Model
+For the selected application, a data-driven model of the build-
+ing energy demand should be created. To demonstrate the
+effect of feature engineering, two models were trained based
+on the existing measurement data: a basic regression model
+and a regression model with engineered features.
+Measurement Data The energy demand was measured during
+a period from 05/2019 to 03/2020, with a sampling time of
+1h [24]. The measurement data includes features based on
+weather data, such as temperature, as well as occupancy data,
+such as registrations. The rest of the features are time-based,
+such as daytime or weekday.
+TABLE I
+FEATURE SET FOR ENERGY CONSUMPTION PREDICTION
+Feature Name
+Unit
+Description
+temperature
+°C
+Outdoor Temperature
+daytime
+h
+Daytime
+weekday
+d
+Weekday from 0 to 6
+holiday
+Public holiday
+daylight
+day or night
+registrations
+registrations for lectures
+Consumption
+kWh
+Energy Consumption
+Model Architecture For the energy demand, a linear regression
+model should be trained. The linear regression architecture was
+selected due to its simplicity and comprehensibility as a white-
+box ML model. Non-linear behavior of the underlying system
+should be incorporated through feature engineering.
+Feature Engineering To model the non-linear behavior of
+the energy demand, categorical features and cyclical features
+were used in combination with Butterworth Filtering, dynamic
+timeseries unrolling and feature selection through the Pearson
+Correlation Coefficient. An overview of the implemented
+workflow is depicted in Figure 1.
+Training Parameters For the model training, a train-test split
+of 0.8 was selected together with a 5-fold cross-validation.
+For the model with engineered features, the parameters for
+the steps timeseries unrolling and feature selection were deter-
+mined through a grid search based on the metrics Coefficient
+of Determination (R2), mean squared error (MSE) and Mean
+Absolute Percentage Error (MAPE).
+
+Basic Feature 
+Set
+Extended 
+Feature Set
+Engineered 
+Feature Set
+Feature Selection – 
+Pearson Correlation
+Fig. 1. Implemented Workflow.
+C. Experimental Results
+The two models were trained on the measurement data and
+compared in terms of performance metrics. Additionally, anal-
+yses of the predicted values through timeseries analysis and
+prediction error plots were performed.
+Performance Metrics To evaluate the performance of the
+model, the metrics R2, Coefficient of Variation of the Root
+Mean Square Error (CV-RMSE) and MAPE were used [21].
+Table II gives an overview of the metrics.
+TABLE II
+PERFORMANCE METRICS
+Model
+R2
+CV-RMSE
+MAPE
+Basic Regression
+0.548
+0.267
+22.764 %
+Engineered Features
+0.638
+0.201
+17.493%
+From the performance metrics, an improvement in prediction
+accuracy for the linear regression model through the engi-
+neered features could be observed.
+Timeseries Analysis The improvement in prediction accuracy
+could also be observed from the timeseries analysis depicted
+in Figure 2.
+2020-01-05
+2020-01-06
+2020-01-07
+2020-01-08
+2020-01-09
+2020-01-10
+2020-01-11
+2020-01-12
+2020-01-13
+2020-01-14
+2020-01-15
+2020-01-16
+2020-01-17
+2020-01-18
+2020-01-19
+2020-01-20
+2020-01-21
+2020-01-22
+2020-01-23
+2020-01-24
+2020-01-25
+20
+40
+60
+80
+Time [Days]
+Consumption [kWh]
+Measurement value
+Basic Regression
+Engineered Features
+Fig. 2. Timeseries Analysis for period of 25 days from test set.
+The timeseries analysis showed that the cyclic behavior of
+the day-night changes in the energy demand could be more
+accurately replicated by the model with engineered features.
+Additionally, the prediction using engineered features shows a
+higher accuracy in replicating low energy demand values than
+the basic regression. This effect can be observed in Figure 3.
+2020-01-15
+2020-01-16
+2020-01-17
+2020-01-18
+2020-01-19
+2020-01-20
+20
+40
+60
+Time [Days]
+Consumption [kWh]
+Measurement value
+Basic Regression
+Engineered Features
+Fig. 3. Timeseries Analysis for period of five days from test set.
+For both models, the residual error was analyzed through
+prediction error plots (Figure 4). The prediction error plots
+show that the residual error is decreased for the model with
+engineered features. In addition, the homogenity of the error
+distribution is improved through the applied feature engineer-
+ing methods.
+20
+30
+40
+50
+60
+70
+80
+20
+30
+40
+50
+60
+70
+80
+True Value [kWh]
+Predicted Value [kWh]
+Basic Regression
+Optimal Prediction
+20
+25
+30
+35
+40
+45
+50
+55
+20
+25
+30
+35
+40
+45
+50
+55
+True Value [kWh]
+Predicted Value [kWh]
+Engineered Features
+Optimal Prediction
+Fig. 4. Prediction Error Plots for Energy Consumption
+Since performance metrics, timeseries analysis and prediction
+error plots show an improvement in accuracy, the feature engi-
+neering steps are suggested to be beneficial for the prediction
+model.
+IV. RELATED WORK
+In the creation of data-driven models in Python, many frame-
+works have been implemented. One of the most well-known
+Python ML frameworks is the scikit-learn framework, which
+provides methods such as data preprocessing, feature engineer-
+ing, clustering, and implementations of various ML models.
+The scikit-learn framework offers interfaces which can be used
+to implement additional methods. Due to the popularity of
+scikit-learn, various frameworks extending scikit-learn have
+been implemented. For instance, the imblearn framework [32]
+focuses on extending scikit-learn’s functionality to processing
+imbalanced datasets. In addition, the imblearn framework
+offers different resampling methods. The mlxtend framework
+[33] offers feature extraction methods such as PCA, or fea-
+ture selection methods such as sequential feature selection.
+Additionally, different evaluation and utility functions are
+implemented. In contrast, libraries such as statsmodels [34]
+provide their own interface for their regression models. The
+
+statsmodels framework provides models based on stochastic
+and statistical methods, such as the Weighted Least Squares
+(WLS). In the area of feature engineering, different Python
+packages have been created. The feature-engine [35] library
+contains a large collection of feature engineering methods,
+which are implemented based on scikit-learn. The featuretools
+framework [36] allows the synthesis of features from relational
+databases. offers functionality for feature encoding, as well as
+different transformations or aggregate functions. Additionally,
+this framework offers transformations, feature encoding, ag-
+gregate functions, as well as coordinate transformations.
+V. CONCLUSION
+This paper presents a Python framework for feature engineer-
+ing that provides different methods through a standardized
+interface. The framework is based on the scikit-learn package
+and offers different methods. The framework offers classic
+feature engineering methods such feature expansion, as well
+as as feature creation, feature selection or transformation and
+filter operations. The framework is implemented as a Python
+package and can be included in different projects. Through
+the specifically defined interfaces of the framework, additional
+methods can be added with low effort. Finally, we demonstrate
+the framework on a case study of energy demand prediction,
+using a workflow created from a subset of the implemented
+methods for data-driven model creation.
+A. Future Work
+The current version of the framework gives many options
+for extensions. For instance, additional feature engineering
+methods can be added using the provided interfaces of the
+framework. In addition, combinations of the implemented
+feature engineering methods can be used for prediction in
+different use cases.
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+

CNAzT4oBgHgl3EQfwP6m/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf,len=500
+page_content='Augmenting data-driven models for energy systems  through feature engineering: A Python framework  for feature engineering  Sandra Wilfling  Abstract—Data-driven modeling is an approach in energy systems  modeling that has been gaining popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In data-driven mod-  eling, machine learning methods such as linear regression, neural  networks or decision-tree based methods are being applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  While these methods do not require domain knowledge, they are  sensitive to data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Therefore, improving data quality in a  dataset is beneficial for creating machine learning-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  The improvement of data quality can be implemented through  preprocessing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' A selected type of preprocessing is feature  engineering, which focuses on evaluating and improving the  quality of certain features inside the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Feature engineering  methods include methods such as feature creation, feature ex-  pansion, or feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In this work, a Python framework  containing different feature engineering methods is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  This framework contains different methods for feature creation,  expansion and selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' in addition, methods for transforming  or filtering data are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The implementation of the  framework is based on the Python library scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The  framework is demonstrated on a case study of a use case  from energy demand prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' A data-driven model is created  including selected feature engineering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The results show  an improvement in prediction accuracy through the engineered  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Keywords: Energy Systems Modeling, Data-driven Modeling,  Feature Engineering, Python, Frameworks  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' INTRODUCTION  Modeling and simulation is an crucial step in the design and  optimization of energy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' While traditional modeling  methods rely on system parameters, a recent approach focuses  on creating data-driven models based on measurement data  from an underlying system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In data-driven modeling, models  are not created based on system parameters, but on existing  measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' These models are based on machine learn-  ing (ML) methods [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' While the area of machine learning  includes a wide range of methods such as clustering algorithms  or classifiers, the focus in data-driven modeling is set to  regression analysis for prediction and forecasting [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In re-  gression analysis, methods such as linear regression, decision-  tree based regression, or neural networks are being applied  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' While some of these methods, such as linear regression,  can be classified as white-box ML methods, others, such as  neural networks, are classified as black-box ML methods due  to their lack of comprehensibility [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' While white-box ML  methods give more insight about their internal structure than  black box ML methods, their architecture is simpler, making  it more difficult to model complex dependencies, for instance  non-linearities [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' To capture such dependencies using white-  box ML models, information about the dependencies can be  passed to the model through the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' This step is called  feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The main purpose of feature engineering  is to augment the existing dataset [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' This can be done through  adding new information, or expanding or reducing the existing  feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In addition, the quality of a single feature can be  improved, for instance through transformation or filtering [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  The area of feature engineering covers a wide number of  methods, such as feature expansion [8] or feature selection  [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The term feature creation covers the creation of features  to add new information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Methods of feature creation include  encodings of time-based features, such as cyclic features [10],  or categorical encoding [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Similarly, feature expansion  is the method of creating new features based on existing  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Feature expansion covers classical methods such as  polynomial expansion [8] or spline interpolation [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  In contrast to feature creation and expansion, feature selection  aims to reduce the size of the feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' While large feature  sets may contain more information than smaller feature sets,  there may be redundancy in the data, as well as sparsity [13] or  multicollinearity [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' To reduce the sparsity or multicollinear-  ity, as well as to remove redundant features, feature selection  mechanisms are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' While methods such as Principal  Component Analysis (PCA) [15] aim to reduce the feature  set through transformation, feature selection methods discard  features based on certain criteria [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Feature selection can  be implemented for instance through sequential methods, such  as forward or backward selection [17], or through correlation  criteria [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Correlation criteria include measures based on  the Pearson Correlation Coefficient, as well as entropy-based  criteria [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The feature selection is then implemented through  a threshold-based selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Threshold-based feature selection  analyzes features based on the selected criterion, and discards  features below a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Mainly, the methods of feature engineering are applied during  the first steps of creating a data-driven model, creating an en-  gineered dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' This engineered dataset is then used to train  the model [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' However, feature engineering methods can  also be used in combination with model selection procedures,  such as grid search [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Feature engineering methods are  widely used in applications from the energy domain, such as  in prediction for building energy demand [20] or photovoltaic  power prediction [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='01720v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='LG]  4 Jan 2023  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Main Contribution  In the creation of data-driven models, a significant factor is  the quality of the underlying dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' To improve the dataset  quality, feature engineering methods can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  The main contribution of this work is a Python framework for  feature engineering that can be used for data-driven model  creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The framework implements different methods for  feature creation, feature expansion, feature selection or trans-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The feature engineering framework is implemented  in Python based on the scikit-learn framework and can be  imported as a Python package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The functionality of the frame-  work is demonstrated on a case study of an energy demand  prediction use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The results of the case study show an  improvement prediction accuracy through the applied feature  engineering steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' METHOD  The presented framework implements various feature engi-  neering methods in Python based on the research in [21] and  on the interfaces defined by scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The methods are im-  plemented using either scikit-learn’s TransformerMixin or  SelectorMixin interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The framework implements meth-  ods for feature expansion, feature creation, feature selection,  as well as transformation and filtering operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Feature Creation and Expansion  In the framework, different methods for feature creation and  expansion are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' These methods create new fea-  tures from time values or from expansion of existing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  To create new features, the implemented framework supports  categorical encoding and cyclic encoding of time-based values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Cyclic Features Cyclic features can be used to model time  values through cyclic functions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Cyclic features were  implemented in [21], as well as in [22] and [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In the  implementation of the framework, sinusoidal signals xsin, xcos  with a selected frequency f can be created based on a sample  series n:  xsin[n] = sin(2πfn)  (1)  xcos[n] = cos(2πfn)  (2)  The implementation offers the creation of features with a zero-  order hold function for a certain time period, for instance TS =  1 day for a signal with a time period of T = 1 week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Categorical Features Categorical encoding creates a repre-  sentation of discrete numerical values through a number of  features with boolean values [11], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In this implementation,  for a number of categorical features x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='.,N for a feature x  with discrete possible values v0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='.,N, a single feature xi is  defined as:  xi =  �  1  x = vi  0  else  (3)  The framework offers categorical encoding for time-based  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In addition, a division factor is implemented to create  an encoding of a downsampled version of the time values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Feature Expansion For feature expansion, the framework im-  plements wrappers for scikit-learn’s PolynomialFeatures and  SplineTransformer classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The method of polynomial expan-  sion was applied in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The parameters for the expansion  methods are passed through the wrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Time-based Features The framework implements a method of  dynamic timeseries unrolling to create features xn−1, xn−2,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' xn−N from an existing feature x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The method of dynamic  timeseries unrolling is based on the research in [24], [25], and  [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' While [25] and [24] use dynamic timeseries unrolling for  both input and target features of a model, allowing the creation  of auto-recursive models, this implementation only supports  dynamic timeseries unrolling for the input features, similar  to the method used in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In this implementation, dynamic  timeseries unrolling is implemented through filter operations  from the scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='signal library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The dynamic features are created  through the convolution of the signal x with a Kronecker delta  for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='N:  xdyn,i[n] = x[n] ∗ δ[n − i]  (4)  This operation creates delayed signals xdyn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=', xdyn,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In  our implementation, for the samples in the delayed signals,  for which no values are available, zero values are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Feature Selection  In the framework, several threshold-based feature selection  methods are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' These methods analyze the input  and target features based on a certain criterion, and then  discard features with a low value of the criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' A widely  used criterion is the Pearson Correlation Coefficient, which  is used to detect linear correlations between features [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  The Pearson Correlation Coefficient calculates the correlation  between two features for samples x0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='.,N, y0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=',N with mean  values ¯x and ¯y:  rx,y =  �N  i=0(xi − ¯x)(yi − ¯y)  ��N  i=0(xi − ¯x)2 �N  i=0(yi − ¯y)2  (5)  While the Pearson correlation identifies linear correlations,  non-linear dependencies are not detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' To detect non-linear  dependencies, criteria such as Maximum Information Coeffi-  cient (MIC) [26], ennemi [27], dCor [28] or the Randomized  Dependence Coefficient (RDC) [29] can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  The framework provides classes for the criteria Pearson Corre-  lation Coefficient, F-statistic based on the Pearson Correlation  Coefficient, as well as thresholds based on the MIC, ennemi  and RDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Transformation and Filtering Operations  To transform features, the framework implements the Box-  cox transformation as well as the square root and inverse  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In addition, the framework provides filtering  operations, which were applied in timeseries prediction for  instance in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Discrete-time based filters can be implemented  in Python through the functions implemented in scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  The scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='signal library offers functions for calculating the  coefficients for different types of digital filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' A digital filter  of order N can be defined through the transfer function H(z)  in a direct form:  H(z) =  �N  i=0 bizi  �N  i=0 aizi  (6)  The filter coefficients ai and bi define the behavior of the  filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='signal library offers functions to compute the  filter coefficients for filter types such as the Butterworth or  Chebyshev filter [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' While scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='signal offers the compu-  tation of analog and digital filter coefficients, the framework  implementation focuses on digital filter implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The  framework implements the Butterworth and Chebyshev fil-  ter as scikit-learn TransformerMixin classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In addition, an  envelope detection filter was implemented for demodulation  of modulated signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' This filter was implemented using the  pandas rolling average function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' For all filters, offset com-  pensation before and after applying the filter operation and a  mask for handling NaN values were implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The direct  form filter classes of the framework offer a simple option for  extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Different architectures can be implemented by re-  defining the implemented method for coefficient calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  This allows to create filters with different Finite Impulse  Response (FIR) or Infinite Impulse Response (IIR) structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Composite Transformers  In feature engineering, it is often the case that only a se-  lected subset of features should be transformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' To offer the  possibility to transform only selected features, a composite  transformer wrapper was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' This wrapper offers to  either automatically replace features through their transformed  versions, or add transformed features separately to the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Implementation  The framework offers compatibility with the sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='Pipeline  implementation,  making  it  possible  to  use  objects  as  part of a ML pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The parameters of each objects  can be adapted through grid search, for instance using  sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='model_selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='GridSearchCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In addition, every cre-  ated object can be stored to and loaded from a Pickle file using  the save_pkl or load_pkl method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  While the filtering, feature expansion and feature cre-  ation methods support operations on a numpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='ndarray or  pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='Dataframe or pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='Series object, the feature creation methods  require a pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='Dataframe or pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='Series object with a DateTimeIn-  dex or TimedeltaIndex to create samples based on a certain  date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' CASE STUDY  The framework is demonstrated on a use case from prediction  for energy systems modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' For this purpose, a mixed office-  campus building is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' A prediction model should be  trained based on existing measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The data-driven  model is created using a workflow based on the implemented  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Application  In this case study, the energy demand of a mixed office-campus  building should be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The data was provided from the  research in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The energy demand of a building is subject  to various factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Main factors that influence building energy  demand are thermal characteristics and Heating, Ventilation,  Air Conditioning and Cooling (HVAC) system behavior [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Additionally, building energy demand may be dependent on  occupancy [3] or subject to seasonal trends [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Many of these  factors show non-linear behavior, which makes it difficult to  address them through a purely linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Therefore, feature  engineering was used to model additional factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Data-driven Model  For the selected application, a data-driven model of the build-  ing energy demand should be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' To demonstrate the  effect of feature engineering, two models were trained based  on the existing measurement data: a basic regression model  and a regression model with engineered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Measurement Data The energy demand was measured during  a period from 05/2019 to 03/2020, with a sampling time of  1h [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The measurement data includes features based on  weather data, such as temperature, as well as occupancy data,  such as registrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The rest of the features are time-based,  such as daytime or weekday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  TABLE I  FEATURE SET FOR ENERGY CONSUMPTION PREDICTION  Feature Name  Unit  Description  temperature  °C  Outdoor Temperature  daytime  h  Daytime  weekday  d  Weekday from 0 to 6  holiday  Public holiday  daylight  day or night  registrations  registrations for lectures  Consumption  kWh  Energy Consumption  Model Architecture For the energy demand, a linear regression  model should be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The linear regression architecture was  selected due to its simplicity and comprehensibility as a white-  box ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Non-linear behavior of the underlying system  should be incorporated through feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Feature Engineering To model the non-linear behavior of  the energy demand, categorical features and cyclical features  were used in combination with Butterworth Filtering, dynamic  timeseries unrolling and feature selection through the Pearson  Correlation Coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' An overview of the implemented  workflow is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Training Parameters For the model training, a train-test split  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='8 was selected together with a 5-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  For the model with engineered features, the parameters for  the steps timeseries unrolling and feature selection were deter-  mined through a grid search based on the metrics Coefficient  of Determination (R2), mean squared error (MSE) and Mean  Absolute Percentage Error (MAPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Basic Feature  Set  Extended  Feature Set  Engineered  Feature Set  Feature Selection –  Pearson Correlation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Implemented Workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Experimental Results  The two models were trained on the measurement data and  compared in terms of performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Additionally, anal-  yses of the predicted values through timeseries analysis and  prediction error plots were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Performance Metrics To evaluate the performance of the  model, the metrics R2, Coefficient of Variation of the Root  Mean Square Error (CV-RMSE) and MAPE were used [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Table II gives an overview of the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  TABLE II  PERFORMANCE METRICS  Model  R2  CV-RMSE  MAPE  Basic Regression  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='548  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='267  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='764 %  Engineered Features  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='638  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='201  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='493%  From the performance metrics, an improvement in prediction  accuracy for the linear regression model through the engi-  neered features could be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Timeseries Analysis The improvement in prediction accuracy  could also be observed from the timeseries analysis depicted  in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  2020-01-05  2020-01-06  2020-01-07  2020-01-08  2020-01-09  2020-01-10  2020-01-11  2020-01-12  2020-01-13  2020-01-14  2020-01-15  2020-01-16  2020-01-17  2020-01-18  2020-01-19  2020-01-20  2020-01-21  2020-01-22  2020-01-23  2020-01-24  2020-01-25  20  40  60  80  Time [Days]  Consumption [kWh]  Measurement value  Basic Regression  Engineered Features  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Timeseries Analysis for period of 25 days from test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  The timeseries analysis showed that the cyclic behavior of  the day-night changes in the energy demand could be more  accurately replicated by the model with engineered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Additionally, the prediction using engineered features shows a  higher accuracy in replicating low energy demand values than  the basic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' This effect can be observed in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  2020-01-15  2020-01-16  2020-01-17  2020-01-18  2020-01-19  2020-01-20  20  40  60  Time [Days]  Consumption [kWh]  Measurement value  Basic Regression  Engineered Features  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Timeseries Analysis for period of five days from test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  For both models, the residual error was analyzed through  prediction error plots (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The prediction error plots  show that the residual error is decreased for the model with  engineered features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In addition, the homogenity of the error  distribution is improved through the applied feature engineer-  ing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  20  30  40  50  60  70  80  20  30  40  50  60  70  80  True Value [kWh]  Predicted Value [kWh]  Basic Regression  Optimal Prediction  20  25  30  35  40  45  50  55  20  25  30  35  40  45  50  55  True Value [kWh]  Predicted Value [kWh]  Engineered Features  Optimal Prediction  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Prediction Error Plots for Energy Consumption  Since performance metrics, timeseries analysis and prediction  error plots show an improvement in accuracy, the feature engi-  neering steps are suggested to be beneficial for the prediction  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' RELATED WORK  In the creation of data-driven models in Python, many frame-  works have been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' One of the most well-known  Python ML frameworks is the scikit-learn framework, which  provides methods such as data preprocessing, feature engineer-  ing, clustering, and implementations of various ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  The scikit-learn framework offers interfaces which can be used  to implement additional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Due to the popularity of  scikit-learn, various frameworks extending scikit-learn have  been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' For instance, the imblearn framework [32]  focuses on extending scikit-learn’s functionality to processing  imbalanced datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In addition, the imblearn framework  offers different resampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The mlxtend framework  [33] offers feature extraction methods such as PCA, or fea-  ture selection methods such as sequential feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  Additionally, different evaluation and utility functions are  implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In contrast, libraries such as statsmodels [34]  provide their own interface for their regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The  statsmodels framework provides models based on stochastic  and statistical methods, such as the Weighted Least Squares  (WLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In the area of feature engineering, different Python  packages have been created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The feature-engine [35] library  contains a large collection of feature engineering methods,  which are implemented based on scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The featuretools  framework [36] allows the synthesis of features from relational  databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' offers functionality for feature encoding, as well as  different transformations or aggregate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Additionally,  this framework offers transformations, feature encoding, ag-  gregate functions, as well as coordinate transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' CONCLUSION  This paper presents a Python framework for feature engineer-  ing that provides different methods through a standardized  interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The framework is based on the scikit-learn package  and offers different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The framework offers classic  feature engineering methods such feature expansion, as well  as as feature creation, feature selection or transformation and  filter operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' The framework is implemented as a Python  package and can be included in different projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Through  the specifically defined interfaces of the framework, additional  methods can be added with low effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Finally, we demonstrate  the framework on a case study of energy demand prediction,  using a workflow created from a subset of the implemented  methods for data-driven model creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' Future Work  The current version of the framework gives many options  for extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' For instance, additional feature engineering  methods can be added using the provided interfaces of the  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
+page_content=' In addition, combinations of the implemented  feature engineering methods can be used for prediction in  different use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAzT4oBgHgl3EQfwP6m/content/2301.01720v1.pdf'}
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+arXiv:2301.11802v1  [cs.LG]  27 Jan 2023
+Decentralized Online Bandit Optimization on Directed Graphs with Regret
+Bounds
+Johan ¨Ostman 1 Ather Gattami 1 Daniel Gillblad 1
+Abstract
+We consider a decentralized multiplayer game,
+played over T rounds, with a leader-follower hi-
+erarchy described by a directed acyclic graph.
+For each round, the graph structure dictates the
+order of the players and how players observe
+the actions of one another. By the end of each
+round, all players receive a joint bandit-reward
+based on their joint action that is used to update
+the player strategies towards the goal of minimiz-
+ing the joint pseudo-regret. We present a learn-
+ing algorithm inspired by the single-player multi-
+armed bandit problem and show that it achieves
+sub-linear joint pseudo-regret in the number of
+rounds for both adversarial and stochastic ban-
+dit rewards. Furthermore, we quantify the cost
+incurred due to the decentralized nature of our
+problem compared to the centralized setting.
+1. Introduction
+Decentralized multi-agent online learning concerns agents
+that, simultaneously, learn to behave over time in order
+to achieve their goals.
+Compared to the single-agent
+setup, novel challenges are present as agents may not
+share the same objectives, the environment becomes non-
+stationary, and information asymmetry may exist between
+agents (Yang & Wang, 2020).
+Traditionally, the multi-
+agent problem has been addressed by either relying on
+a central controller to coordinate the agents’ actions or
+to let the agents learn independently.
+However, access
+to a central controller may not be realistic and indepen-
+dent learning suffers from convergence issues (Zhang et al.,
+2019). To circumvent these issues, a common approach
+is to drop the central coordinator and allow informa-
+tion exchange between agents (Zhang et al., 2018; 2019;
+Cesa-Bianchi et al., 2021).
+Decision-making that involves multiple agents is often
+1AI Sweden, Gothenburg, Sweden. Correspondence to: Johan
+¨Ostman <johan.ostman@ai.se>.
+modeled as a game and studied under the lens of game
+theory to describe the learning outcomes.1
+Herein, we
+consider games with a leader-follower structure in which
+players act consecutively. For two players, such games
+are known as Stackelberg games (Hicks, 1935). Stackel-
+berg games have been used to model diverse learning situ-
+ations such as airport security (Balcan et al., 2015), poach-
+ing (Sessa et al., 2020), tax planning (Zheng et al., 2020),
+and generative adversarial networks (Moghadam et al.,
+2021).
+In a Stackelberg game, one is typically con-
+cerned with finding the Stackelberg equilibrium, some-
+times called Stackelberg-Nash equilibrium, in which the
+leader uses a mixed strategy and the follower is best-
+responding. A Stackelberg equilibrium may be obtained by
+solving a bi-level optimization problem if the reward func-
+tions are known (Sch¨afer et al., 2020; Aussel & Svensson,
+2020) or, otherwise, it may be learnt via online learn-
+ing techniques (Bai et al., 2021; Zhong et al., 2021), e.g.,
+no-regret algorithms (Shalev-Shwartz, 2012; Deng et al.,
+2019; Goktas et al., 2022).
+No-regret algorithms have emerged from the single-player
+multi-armed bandit problem as a means to alleviate
+the exploitation-exploration trade-off (Bubeck & Slivkins,
+2012). An algorithm is called no-regret if the difference be-
+tween the cumulative rewards of the learnt strategy and the
+single best action in hindsight is sublinear in the number
+of rounds (Shalev-Shwartz, 2012). In the multi-armed ban-
+dit problem, rewards may be adversarial (based on random-
+ness and previous actions), oblivious adversarial (random),
+or stochastic (independent and identically distributed) over
+time (Auer et al., 2002). Different assumptions on the ban-
+dit rewards yield different algorithms and regret bounds.
+Indeed, algorithms tailored for one kind of rewards are
+sub-optimal for others, e.g., the EXP3 algorithm due
+to Auer et al. (2002) yields the optimal scaling for adversar-
+ial rewards but not for stochastic rewards. For this reason,
+best-of-two-worlds algorithms, able to optimally handle
+both the stochastic and adversarial rewards, have recently
+been pursued and resulted in algorithms with close to op-
+timal performance in both settings (Auer & Chiang, 2016;
+1The convention is to use agents in learning applications and
+players in game theoretic applications, we shall use the game-
+theoretic nomenclature in the remainder of the paper.
+
+Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds
+Wei & Luo, 2018; Zimmert & Seldin, 2021). Extensions to
+multiplayer multi-armed bandit problems have been pro-
+posed in which players attempt to maximize the sum of
+rewards by pulling an arm each, see, e.g., (Kalathil et al.,
+2014; Bubeck et al., 2021).
+No-regret algorithms are a common element also when an-
+alyzing multiplayer games.
+For example, in continuous
+two-player Stackelberg games, the leader strategy, based
+on a no-regret algorithm, converges to the Stackelberg equi-
+librium if the follower is best-responding (Goktas et al.,
+2022). In contrast, if also the follower adopts a no-regret al-
+gorithm, the regret dynamics is not guaranteed to converge
+to a Stackelberg equilibrium point (Goktas et al., 2022,
+Ex. 3.2).
+In (Deng et al., 2019), it was shown for two-
+player Stackelberg games that a follower playing a, so-
+called, mean-based no-regret algorithm, enables the leader
+to achieve a reward strictly larger than the reward achieved
+at the Stackelberg equilibrium.
+This result does, how-
+ever, not generalize to n-player games as demonstrated
+by D’Andrea (2022). Apart from studying the Stackelberg
+equilibrium, several papers have analyzed the regret. For
+example, Sessa et al. (2020) presented upper-bounds on the
+regret of a leader, employing a no-regret algorithm, playing
+against an adversarial follower with an unknown response
+function. Furthermore, Stackelberg games with states were
+introduced by Lauffer et al. (2022) along with an algorithm
+that was shown to achieve no-regret.
+As the follower in a Stackelberg game observes the leader’s
+action, there is information exchange. A generalization
+to multiple players has been studied in a series of pa-
+pers (Cesa-Bianchi et al., 2016; 2020; 2021). In this line
+of work, players with a common action space form an ar-
+bitrary graph and are randomly activated in each round.
+Active players share information with their neighbors by
+broadcasting their observed loss, previously received neigh-
+bor losses, and their current strategy. The goal of the play-
+ers is to minimize the network regret, defined with respect
+to the cumulative losses observed by active players over
+the rounds. The players, however, update their strategies
+according to their individually observed loss. Although we
+consider players connected on a graph, our work differs
+significantly from (Cesa-Bianchi et al., 2016; 2020; 2021),
+e.g., we allow only actions to be observed between players
+and the players update their strategies based on a common
+bandit reward rather than an individual reward.
+Contributions:
+We introduce the joint pseudo-regret,
+defined with respect to the cumulative reward where
+all the players observe the same bandit-reward in each
+round. We provide an online learning-algorithmfor general
+consecutive-play games that relies on no-regret algorithms
+developed for the single-player multi-armed bandit prob-
+lem. The main novelty of our contribution resides in the
+joint analysis of players with coupled rewards where we
+derive upper bounds on the joint pseudo-regret and prove
+our algorithm to be no-regret in the adversarial setting. Fur-
+thermore, we quantify the penalty incurred by our decen-
+tralized setting in relation to the centralized setting.
+2. Problem formulation
+In this section, we formalize the consecutive structure of
+the game and introduce the joint pseudo-regret that will
+be used as a performance metric throughout. We consider
+a decentralized setting where, in each round of the game,
+players pick actions consecutively. The consecutive nature
+of the game allows players to observe preceding players’
+actions and may be modeled by a DAG. For example, in
+Fig. 1, a seven-player game is illustrated in which player 1
+initiates the game and her action is observed by players 2, 5,
+and 6. The observations available to the remaining players
+follow analogously. Note that for a two-player consecutive
+game, the DAG models a Stackelberg game.
+We let G = (V, E) denote a DAG where V denotes the ver-
+tices and E denotes the edges. For our setting, V constitutes
+the n different players and E = {(j, i) : j → i, j ∈ V, i ∈
+V} describes the observation structure where j → i indi-
+cates that player i observes the action of player j. Accord-
+ingly, a given player i ∈ V observes the actions of its direct
+parents, i.e., players j ∈ Ei = {k : (k, i) ∈ E}. Further-
+more, each player i ∈ V is associated with a discrete action
+space Ai of size Ai. We denote by πi(t), the mixed strat-
+egy of player i over the action space Ai in round t ∈ [T ]
+such that πi(t) = a with probability pi,a for a ∈ Ai. In
+the special case when pi,a = 1 for some a ∈ Ai, the strat-
+egy is referred to as pure. Let AB denote the joint action
+space of players in a set B given by the Cartesian product
+AB = �
+i∈B Ai. If a player i has no parents, i.e., Ei = ∅,
+we use the convention |AEi| = 1.
+We consider a collaborative setting with bandit rewards
+given by a mapping rt : AV → [0, 1] in each round t ∈ [T ].
+The bandit rewards are assumed to be adversarial. Let C de-
+note a set of cliques in the DAG (Koller & Friedman, 2009,
+Def. 2.13) and let Nk ∈ C for k ∈ [|C|] denote the players
+in the kth clique in C with joint action space ANk such that
+Nk ∩ Nj = ∅ for j ̸= k. For a joint action a(t) ∈ AV,
+we consider bandit rewards given by a linear combination
+of the clique-rewards as
+rt(a(t)) =
+|C|
+�
+k=1
+βkrk
+t (P k(a(t))),
+(1)
+where rk
+t
+:
+ANk
+→
+[0, 1], βk
+≥
+0 is the weight
+of the kth clique reward such that �|C|
+k=1 βk
+=
+1,
+and P k(a(t)) denotes the joint action of the players in
+Nk. As an example, Fig. 2 highlights the cliques C =
+
+Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds
+1
+2
+3
+4
+5
+6
+7
+Figure 1. A game with seven players.
+1
+2
+3
+4
+5
+6
+7
+Figure 2. Colored cliques comprising the bandit reward.
+{{2, 3, 4}, {1, 5}, {6}, {7}} and we have, e.g., N1
+=
+{2, 3, 4}, and P 1(a(t)) = (a2(t), a3(t), a4(t)). Note that
+each player influences only a single term in the reward (1).
+In each round t ∈ [T ], the game proceeds as follows for
+player i ∈ V:
+1) the player is idle until the actions of all parents in Ei
+have been observed,
+2) the player picks an action ai(t) ∈ Ai according to its
+strategy πi(t),
+3) once all the n players in V have chosen an action,
+the player observes the bandit reward rt(a(t)) and up-
+dates its strategy.
+The goal of the game is to find policies {πi(t)}n
+i=1 that de-
+pend on past actions and rewards in order to minimize the
+joint pseudo-regret R(T ) which is defined similarly to the
+pseudo regret (Shalev-Shwartz, 2012, Ch. 4.2) as
+R(T ) = r(a⋆) − E
+� T
+�
+t=1
+rt(a(t))
+�
+,
+(2)
+where
+r(a⋆) = max
+a∈AV E
+� T
+�
+t=1
+rt(a)
+�
+,
+and the expectations are taken with respect to the rewards
+and the player actions.2 Note that r(a⋆) corresponds to the
+largest expected reward obtainable if all players use pure
+strategies. Hence, the pseudo-regret in (2) quantifies the
+difference between the expected reward accumulated by the
+learnt strategies and the reward-maximizing pure strategies
+in hindsight.
+Our problem formulation pertains to a plethora of appli-
+cations.
+Examples include resource allocation in cog-
+nitive radio networks where available frequencies are
+obtained via channel sensing (Janatian et al., 2015) and
+semi-autonomous vehicles with adaptive cruise control,
+i.e., vehicles ahead are observed before an action is de-
+cided (Marsden et al., 2001).
+Also recently, the impor-
+tance of coupled rewards and partner awareness through
+implicit communications, e.g., by observation, has been
+highlighted in human-robot and human-AI collaborative
+settings (Bıyık et al., 2022). Furthermore, our formulation
+is applicable in simple scenarios within to reinforcement
+learning (Ibarz et al., 2021).
+As will be shown in the next section, any no-regret al-
+gorithm can be used as a building block for the games
+considered herein to guarantee a sub-linear pseudo-regret
+in the number of rounds T .
+As our goal is to study
+the joint pseudo-regret (2) for adversarial, we start from
+a state-of-the-art algorithm for the adversarial multi-
+armed bandit problem. In particular, we will utilize the
+TSALLIS-INF algorithm that guarantees a pseudo-regret
+with the optimal scaling in the adversarial single-player set-
+ting (Zimmert & Seldin, 2021).
+3. Analysis of the joint pseudo-regret
+Our analysis of the joint pseudo-regret builds upon learning
+algorithms for the single-player multi-armed bandit prob-
+lem. First, let us build intuition on how to use a multi-
+armed bandit algorithm in the DAG-based game described
+in Section 2. Consider a 2-player Stackelberg game where
+the players choose actions from A1 and A2, respectively,
+and where player 2 observes the actions of player 1. For
+simplicity, we let player 1 use a mixed strategy whereas
+player 2 is limited to a pure strategy. Furthermore, con-
+sider the rewards to be a priori known by the players and let
+T = 1 for which the Stackelberg game may be viewed as a
+bi-level optimization problem (Aussel & Svensson, 2020).
+In this setting, the action of player 1 imposes a Nash game
+on player 2 whom attempts to play optimally given the ob-
+servation. Hence, player 2 has A1 pure strategies, one for
+each of the A1 actions of player 1.
+We may generalize this idea to the DAG-based multiplayer
+game with unknown bandit-rewards and T ≥ 1 to achieve
+2This is called pseudo-regret as r(a⋆) is obtained by a maxi-
+mization outside of the expectation.
+
+Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds
+no-regret. Indeed, a player i ∈ V may run |AEi| different
+multi-armed bandit algorithms, one for each of the joint
+actions of its parents.
+Algorithm 1 illustrates this idea
+in conjunction with the TSALLIS-INF update rule intro-
+duced by Zimmert & Seldin (2021), which is given in Al-
+gorithm 2 for completeness.3 In particular, for the 2-player
+Stackelberg game, the leader runs a single multi-armed ban-
+dit algorithm whereas the follower runs A1 learning algo-
+rithms. For simplicity, Algorithm 1 assumes that player i
+knows the size of the joint action space of its parents, i.e.,
+|AEi|. Dropping this assumption is straightforward: simply
+keep track of the observed joint actions and initiate a new
+multi-armed bandit learner upon a unique observation.
+Algorithm 1 Learning algorithm of player i ∈ V
+1: Input: for ease of notation, let the actions in AEi be
+labeled as 1, 2, . . ., |AEi|
+2: initialize cumulative loss Lk ← 0 ∈ RAi for k ∈
+[|AEi|]
+3: initialize fixed-point xk ← 0 for k ∈ [|AEi|]
+4: initialize counter nk ← 0 for k ∈ [|AEi|]
+5: for t = 1, 2, . . ., T do
+6:
+observe the joint action j ∈ [|AEi|] of the preceding
+players
+7:
+increase counter nj ← nj + 1
+8:
+obtain
+new
+strategy
+and
+new
+fixed-point
+(πi(t), xj) ← TSALLIS-INF(nj, Lj, xj)
+9:
+play action ai(t) ∼ πi(t)
+10:
+observe the joint bandit-reward rt(a(t))
+11:
+update the cumulative loss for all k ∈ [Ai] as
+Lj,k ← Lj,k + 1{ai(t) = k}(1 − rt(a(t)))/pk
+12: end for
+Algorithm 2 Strategy update for player i ∈ V
+1: Input: time step t, cumulative rewards L ∈ RAi
++ , pre-
+vious fixed point x Output strategy πi(t), fixed point
+x
+2: set learning rate η ← 2
+�
+1/t
+3: repeat
+4:
+pj ← 4(η(Lj − x))−2 for all j ∈ [Ai]
+5:
+x ← x −
+��Ai
+j=1 pj − 1
+�
+/
+�
+η �Ai
+j=1 p3/2
+j
+�
+6: until convergence
+7: update strategy πi(t) ← (p1, . . . , pAi)
+Next, we go on to analyze the joint pseudo-regret of Algo-
+rithm 1. First, we present a result on the pseudo-regret for
+the single-player multi-armed bandit problem that will be
+used throughout.
+3The original TSALLIS-INF Algorithm is given in terms of
+losses. To use rewards, one may simply use the relationship l =
+1 − r.
+Theorem 3.1 (Pseudo-regret of TSALLIS-INF). Consider
+a single-player multi-armed bandit problem with A1 arms,
+played over T rounds. Let the player operate according to
+Algorithm 1. Then, the pseudo-regret satisfies
+R(T ) ≤ 4
+�
+A1T + 1.
+Proof. For a single player, E1 = ∅ and we have |AE1| =
+1 by convention. Hence, our setting becomes equivalent
+to that of Zimmert & Seldin (2021, Th 1) and the result
+follows thereof.
+Next, we consider a two-player Stackelberg game with
+joint bandit-rewards defined over a two-player clique. We
+have the following upper bound on the joint pseudo-regret.
+Theorem 3.2 (Joint pseudo-regret over cliques of size 2).
+Consider a 2-player Stackelberg game with bandit-rewards,
+given by (1), defined over a single clique containing both
+players. Furthermore, let each of the players follow Algo-
+rithm 1. Then, the joint pseudo-regret satisfies
+R(T ) ≤ 4
+�
+A1A2T + 4
+�
+A1T + A1 + 1.
+Proof. Without loss of generality, let player 2 observe the
+actions of player 1. Let a1(t) ∈ A1 and a2(t) ∈ A2 denote
+the actions of player 1 and player 2, respectively, at time t ∈
+[T ] and let a⋆
+1 and a⋆
+2(a1) denote the reward-maximizing
+pure strategies of the players in hindsight, i.e.,
+a⋆
+1 = arg max
+a1∈A1 E
+� T
+�
+t=1
+rt(a1, a⋆
+2(a1))
+�
+,
+(3)
+a⋆
+2(a1) = arg max
+a2∈A2 E
+� T
+�
+t=1
+rt(a1, a2)
+�
+.
+(4)
+Note that the optimal joint decision in hindsight is given by
+(a⋆
+1, a⋆
+2(a⋆
+1)). The joint pseudo-regret is given by
+R(T ) =
+T
+�
+t=1
+E [rt(a⋆
+1, a⋆
+2(a⋆
+1)) − rt(a⋆
+1, a2(t))]
++ E [rt(a⋆
+1, a2(t)) − rt(a1(t), a2(t))]
+≤
+T
+�
+t=1
+max
+at∈A1 E [rt(at, a⋆
+2(at)) − rt(at, a2(t))]
++ E
+� T
+�
+t=1
+rt(a⋆
+1, a2(t)) − rt(a1(t), a2(t))
+�
+.
+(5)
+Next, let
+a+
+1 (t) = arg max
+at∈A1 E [rt(at, a⋆
+2(at)) − rt(at, a2(t))]
+and let Ta = {t : a+
+1 (t) = a}, for a ∈ A1, denote all the
+rounds that player 1 chose action a and introduce Ta = |Ta|.
+
+Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds
+Then, the first term in (5) is upper-bounded as
+T
+�
+t=1
+max
+at∈A1 E [rt(at, a⋆
+2(at)) − rt(at, a2(t))]
+=
+�
+a∈A1
+�
+t∈Ta
+E [rt(a, a⋆
+2(a)) − rt(a, a2(t))]
+≤
+�
+a∈A1
+4
+�
+A2Ta + 1
+(6)
+≤
+max
+�
+a Ta=T
+�
+a∈A1
+4
+�
+A2Ta + 1
+= 4
+�
+A1A2T + A1
+(7)
+where (6) follows from Theorem 3.1 and because player 2
+follows Algorithm 1. Note that the actions in Ta may not be
+consecutive. However, as we consider adversarial rewards,
+Theorem 3.1 is still applicable.
+Next, we consider the second term in (5). Note that, accord-
+ing to (4), a⋆
+1 is obtained from the optimal pure strategies
+in hindsight of both the players. Let
+a◦
+1 = arg max
+a1∈A1
+T
+�
+t=1
+E [rt(a1, a2(t))]
+and note that
+E
+� T
+�
+t=1
+rt(a⋆
+1, a2(t))
+�
+≤ E
+� T
+�
+t=1
+rt(a◦
+1, a2(t))
+�
+.
+By adding and subtracting rt(a◦
+1, a2(t)) to the second term
+in (5), we get
+E
+� T
+�
+t=1
+rt(a⋆
+1, a2(t)) − rt(a1(t), a2(t))
+�
+≤ E
+� T
+�
+t=1
+rt(a◦
+1, a2(t)) − rt(a1(t), a2(t))
+�
+≤ 4
+�
+A1T + 1
+(8)
+where the last equality follows from Theorem 3.1. The re-
+sult follows from (7) and (8).
+From Theorem 3.2, we note that the joint pseudo-regret
+scales with the size of the joint action space as R(T ) =
+O(√A1A2T). This is expected as a centralized version
+of the cooperative Stackelberg game may be viewed as
+a single-player multi-armed bandit problem with A1A2
+arms where, according to Theorem 3.1, the pseudo-regret
+is upper-bounded by 4√A1A2T + 1.
+Hence, from
+Theorem 3.2, we observe a penalty of 4√A1T + A1
+due to the decentralized nature of our setup.
+More-
+over, in the single-player setting, Algorithm 2 was shown
+in Zimmert & Seldin (2021) to achieve the same scaling as
+the lower bound in Cesa-Bianchi & Lugosi (2006, Th. 6.1).
+Hence, Algorithm 1 achieves the optimal scaling. Next, we
+extend Theorem 3.2 to cliques of size larger than two.
+Theorem 3.3 (Joint pseudo-regret over a clique of arbitrary
+size). Consider a DAG-based game with bandit rewards
+given by (1), defined over a single clique containing m
+players. Let each of the players operate according to Al-
+gorithm 1. Then, the joint pseudo-regret satisfies
+R(T ) ≤ 4
+√
+T
+m
+�
+i=1
+i�
+k=1
+�
+Ak +
+m−1
+�
+i=1
+i�
+k=1
+Ak + 1.
+Proof. Let Rub(T, m) denote an upper bound on the joint
+pseudo-regret when the bandit-reward is defined over a
+clique containing m players. From Theorem 3.1 and Theo-
+rem 3.2, we have that
+Rub(T, 1) = 4
+�
+A1T + 1
+Rub(T, 2) = 4
+�
+A1T + 4
+�
+A1A2T + A1 + 1,
+respectively. Therefore, we form an induction hypothesis
+as
+Rub(T, m) = 4
+√
+T
+m
+�
+i=1
+i�
+k=1
+�
+Ak +
+m−1
+�
+i=1
+i�
+k=1
+Ak + 1. (9)
+Assume that (9) is true for a clique containing m − 1
+players and add an additional player, assigned player in-
+dex 1, whose actions are observable to the original m − 1
+players.
+The m players now form a clique C of size
+m.
+Let a(t) ∈ AC denote the joint action of all the
+players in the clique at time t ∈ [T ] and let a−i(t) =
+(a1(t), . . . , ai−1(t), ai+1(t), . . . , am(t)) ∈ AC\i denote
+the joint action excluding the action of player i. Further-
+more, let
+a⋆
+1 = arg max
+a1∈A1 E
+� T
+�
+t=1
+rt(a1, a⋆
+−1(a1))
+�
+a⋆
+−1(a1) = arg max
+a∈AC\1 E
+� T
+�
+t=1
+rt(a1, a)
+�
+denote the optimal actions in hindsight of player 1 and the
+optimal joint action of the original m − 1 players given the
+action of player 1, respectively. The optimal joint action in
+hindsight is given as a⋆ = (a⋆
+1, a⋆
+−1(a⋆
+1)). Following the
+
+Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds
+steps in the proof of Theorem 3.2 verbatim, we obtain
+R(T ) =
+T
+�
+t=1
+E [rt(a⋆) − rt(a⋆
+1, a−1(t))]
++ E [rt(a⋆
+1, a−1(t)) − rt(a(t))]
+≤
+T
+�
+t=1
+max
+a1 E
+�
+rt(a1, a⋆
+−1(a1)) − rt(a1, a−1(t))
+�
++
+T
+�
+t=1
+E [rt(a⋆
+1, a−1(t)) − rt(a1(t), a−1(t))]
+≤
+�
+a∈A1
+�
+t∈Ta
+E
+�
+rt(a, a⋆
+−1(a)) − rt(a, a−1(t))
+�
++
+T
+�
+t=1
+E [rt(a◦
+1, a−1(t)) − rt(a1(t), a−1(t))]
+≤
+�
+a∈A1
+Rub(Ta, m − 1) + 4
+�
+A1T + 1
+≤ A1Rub(T/A1, m − 1) + 4
+�
+A1T + 1
+(10)
+where Ta, Ta, and a◦
+n are defined analogously as in the
+proof of Theorem 3.2. By using the induction hypothe-
+sis (9) in (10) and by accounting for the original m − 1
+players being indexed from 2 to m, we obtain
+R(T ) ≤ A1
+�
+4
+�
+T/A1
+m
+�
+i=2
+i�
+k=2
+�
+Ak +
+m−1
+�
+i=2
+i�
+k=2
+Ak + 1
+�
++ 4
+�
+A1T + 1
+= Rub(T, m)
+which is what we wanted to show.
+As in the two-player game,
+the joint pseudo-regret
+of
+Algorithm 1
+achieves the
+optimal scaling,
+i.e.,
+R(T )
+=
+O(
+√
+T �m
+k=1
+√Ak), but exhibits a penalty
+due to the decentralized setting which is equal to
+4
+√
+T �m−1
+i=1
+�i
+k=1
+√Ak + �m−2
+i=1
+�i
+k=1 Ak.
+Up until this point, we have considered the pseudo-regret
+when the bandit-reward (1) is defined over a single clique.
+The next theorem leverages the previous results to provide
+an upper bound on the joint pseudo-regret when the bandit-
+reward is defined over an arbitrary number of independent
+cliques in the DAG.
+Theorem 3.4 (Joint pseudo-regret in DAG-based games).
+Consider a DAG-based game with bandit rewards given as
+in (1) and let C contain a collection of independent cliques
+associated with the DAG. Let each player operate accord-
+ing to Algorithm 1. Then, the joint pseudo-regret satisfies
+R(T ) = O
+��
+T max
+k∈[|C|] |ANk|
+�
+where ANk denotes the joint action-space of the players in
+the kth clique Nk ∈ C.
+Proof. Let Nk ∈ C denote the players belonging to the kth
+clique in C with joint action space ANk. The structure of (1)
+allows us to express the joint pseudo-regret as
+R(T ) = E
+� T
+�
+t=1
+rt(a⋆) − rt(a(t))
+�
+≤
+|C|
+�
+k=1
+βkE
+� T
+�
+t=1
+rk
+t (a⋆
+k) − rk
+t (P k(a(t)))
+�
+(11)
+where
+a⋆ = arg max
+a∈AV E
+� T
+�
+t=1
+rt(a)
+�
+,
+a⋆
+k = arg max
+a∈ANk
+E
+� T
+�
+t=1
+rk
+t (a)
+�
+,
+and the inequality follows since E
+��T
+t=1 rk
+t (P k(a⋆))
+�
+≤
+E
+��T
+t=1 rk
+t (a⋆
+k)
+�
+. Now, for each clique Nk ∈ C, let the
+player indices in Nk be ordered according to the order of
+player observations within the clique.
+As Theorem 3.3
+holds for any Nk ∈ C, we may, with a slight abuse of nota-
+tion, bound the joint pseudo-regret of each clique as
+R(T ) ≤
+|C|
+�
+k=1
+βkRub(T, Nk) ≤ max
+k∈[|C|] βkRub (T, Nk)
+where Rub(T, Nk) follows from Theorem 3.3 as
+Rub(T, Nk) = 4
+√
+T
+�
+i∈Nk
+�
+j≤i,j∈Nk
+�
+Aj
++
+�
+i∈N −
+k
+�
+j≤i,j∈N −
+k
+Aj + 1
+where N −
+k excludes the last element in Nk. The result fol-
+lows as Rub(T, Nk) = O(
+�
+T |ANk|) where the βk has
+been consumed in the prefactor.
+4. Numerical results
+The experimental setup in this section is inspired by the
+socio-economic simulation in (Zheng et al., 2020).4
+We
+consider a simple taxation game where one player acts as
+a socio-economic planner and the remaining M players act
+4The source code of our experiments is available on
+https://anonymous.4open.science/r/bandit_optimization_dag-242C/.
+
+Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds
+as workers that earn an income by performing actions, e.g.,
+constructing houses. The socio-economic planner divides
+the possible incomes into N brackets where [βi−1, βi] de-
+notes the ith bracket with β0 = 0 and βN = ∞. In each
+round t ∈ [T ], the socio-economic planner picks an action
+ap(t) = (ap,1(t), . . . , ap,N(t)) that determines the taxation
+rate where ap,i(t) ∈ Ri denotes the the marginal taxation
+rate in income bracket i and Ri is a finite set. We use the
+discrete set Ap = �N
+i=1 Ri of size Ap to denote the action
+space of the planner.
+In each round, the workers observe the taxation policy
+ap(t) ∈ Ap and choose their actions consecutively, see
+Fig. 3. Worker j ∈ [M] takes actions aj(t) ∈ Aj where
+Aj is a finite set. A chosen action aj(t) ∈ Aj translates
+into a tuple (xj(t), ˜lj(t)) consisting of a gross income and
+a marginal default labor cost, respectively. Furthermore,
+each worker has a skill level sj that serves as a divisor of
+the default labor, resulting in an effective marginal labor
+lj(t) = ˜lj(t)/sj. Hence, given a common action, high-
+skilled workers exhibit less labor than low-skilled workers.
+The gross income xj(t) of worker j in round t is taxed ac-
+cording to ap(t) as
+ξ(xj(t)) =
+N
+�
+i=1
+ap,i(t)(βi − βi−1)1{xj(t) > βi}
++ (xj(t) − βi−1)1{xj(t) ∈ [βi−1, βi]}
+where ap,i(t) is the taxation rate of the ith income bracket
+and ξ(xj(t)) denotes the collected tax. Hence, worker j’s
+cumulative net income zj(t) and cumulative labor ℓj(t) in
+round t are given as
+zj(t) =
+t
+�
+u=1
+xj(u) − ξ(xj(u)),
+ℓj(t) =
+t
+�
+u=1
+lj(u).
+In round t, the utility of worker j depends on the cumula-
+tive net income and the cumulative labor as
+rj
+t (zj(t), ℓj(t)) = (zj(t))1−η − 1
+1 − η
+− ℓj(t)
+(12)
+where η > 0 determines the non-linear impact of income.
+An example of the utility function in (12) is shown in Fig. 4
+for η = 0.3, income xj(t) = 10, and a default marginal la-
+bor ˜lj(t) = 1 at different skill levels. It can be seen that the
+utility initially increases with income until a point at which
+the cumulative labor outweighs the benefits of income and
+the worker gets burnt out.
+We consider bandit-rewards defined with respect to the
+Socio-economic planner
+1
+2
+3
+workers
+Figure 3. Socio-economic setup with 4 players among which 3 are
+designated workers.
+100
+101
+102
+103
+104
+0
+200
+400
+600
+800
+1,000
+houses built
+worker utility
+s = 1
+s = 2
+s = 3
+Figure 4. Example of utility functions for different skill levels
+when xj(t) = 10 and ˜ℓj(t) = 1.
+worker utilities and the total collected tax as
+rt(ap(t), a1(t), . . . , aM(t)) =
+1
+(M + 1)
+
+
+M
+�
+j=1
+wrj
+t (zj(t), ℓj(t)) + wp
+M
+�
+j=1
+ξ(xj(t))
+
+
+(13)
+where the weights trade off worker utility for the col-
+lected tax and satisfy Mw + wp
+=
+M + 1.
+The
+individual rewards are all normalized to [0, 1], hence,
+rt(ap(t), a1(t), . . . , aM(t)) ∈ [0, 1].
+For the numerical experiment, we consider N = 2 in-
+come brackets where the boundaries of the income brackets
+are {0, 14, ∞} and the socio-economic planner chooses a
+marginal taxation rate from R = {0.1, 0.3, 0.5} in each
+income bracket, hence, Ap = 9. We consider M = 3 work-
+ers with the same action set A of size 3. Consequently,
+the joint action space is of size 243. Furthermore, we let
+the skill level of the workers coincide with the worker in-
+dex, i.e., sj = j for j ∈ [M].
+Simply, workers able
+to observe others have higher skill. The worker actions
+translate to a gross marginal income and a marginal la-
+bor as aj(t) → (xj(t), lj(t)) where xj(t) = 5aj(t) and
+
+Decentralized Online Bandit Optimization on Directed Graphs with Regret Bounds
+lj(t) = aj(t)/sj for aj(t) ∈ {1, 2, 3}. Finally, we set
+η = 0.3 and let w = 1/M and wp = M to model a sit-
+uation where the collected tax is preferred over workers’
+individual utility.
+The joint pseudo-regret of the socio-economic simulation is
+illustrated in Fig. 5 and Fig. 6 (different scales) along with
+the upper bound in Theorem 3.4. We collect 100 realiza-
+tions of the experiment and, along with the pseudo-regret
+R(T ), two standard deviations are also presented. It can
+be seen that the players initially explore the action space
+and are able to eventually converge on an optimal strategy
+from a pseudo-regret perspective. The upper bound in the
+figures is admittedly loose and does not exhibit the same
+asymptotic decay as the simulation due to different con-
+stants in the scaling law, see Fig. 6. However, it remains
+valuable as it provides an asymptotic no-regret guarantee
+for the learning algorithm.
+100
+101
+102
+103
+104
+105
+106
+0.1
+0.2
+0.3
+0.4
+0.5
+T
+Regret
+Upper bound
+R(T )
+Figure 5. Pseudo regret vs the upper bound in Theorem 3.4 (linear
+scale).
+5. Conclusion
+We have studied multiplayer games with joint bandit-
+rewards where players execute actions consecutively and
+observe the actions of the preceding players.
+We intro-
+duced the notion of joint pseudo-regret and presented an
+algorithm that is guaranteed to achieve no-regret for adver-
+sarial bandit rewards. A bottleneck of many multi-agent
+algorithms is that the complexity scales with the joint ac-
+tion space (Jin et al., 2021) and our algorithm is no ex-
+ception. An interesting venue of further study is to find
+algorithms that have more benign scaling properties, see
+e.g., (Jin et al., 2021; Daskalakis et al., 2021).
+Further-
+more, recent results on correlated multi-armed bandits have
+demonstrated that multi-armed bandits with many arms
+may become significantly more feasible if one is able to
+101
+102
+103
+104
+105
+106
+10−3
+10−2
+10−1
+100
+101
+102
+T
+Regret
+Upper bound
+R(T )
+Figure 6. Pseudo regret vs the upper bound in Theorem 3.4 (log-
+scale).
+exploit dependencies among arms (Gupta et al., 2021). It
+would be interesting to explore how the scaling of our algo-
+rithm is affected by modelling and exploiting dependencies
+among players.
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