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+Mixed moving average field guided learning for spatio-temporal
+data
+Imma Valentina Curato∗ , Orkun Furat† and Bennet Str¨oh ‡
+January 3, 2023
+Abstract
+Influenced mixed moving average fields are a versatile modeling class for spatio-temporal data.
+However, their predictive distribution is not generally accessible. Under this modeling assumption,
+we define a novel theory-guided machine learning approach that employs a generalized Bayesian
+algorithm to make predictions. We employ a Lipschitz predictor, for example, a linear model or
+a feed-forward neural network, and determine a randomized estimator by minimizing a novel PAC
+Bayesian bound for data serially correlated along a spatial and temporal dimension. Performing causal
+future predictions is a highlight of our methodology as its potential application to data with short
+and long-range dependence. We conclude by showing the performance of the learning methodology
+in an example with linear predictors and simulated spatio-temporal data from an STOU process.
+MSC 2020: primary 60E07, 60E15, 60G25, 60G60; secondary 62C10.
+Keywords: stationary models, weak dependence, oracle inequalities, randomized estimators, causal predic-
+tions.
+1
+Introduction
+Modeling spatio-temporal data representing measurements from a continuous physical system introduces
+various methodological challenges. These include finding models that can account for the serial correlation
+typically observed along their spatial and temporal dimensions and simultaneously have good prediction
+performance. Statistical models used nowadays to analyze spatio-temporal data are Gaussian processes
+[5], [23], [53], and [63]; spatio-temporal kriging [19], and [46]; space-time autoregressive moving average
+models [30]; point processes [31], and hierarchical models [19]. An important common denominator of
+statistical modeling is that they enable predictions once the variogram (covariance) structure or the data
+distribution (up to a set of parameters) is carefully chosen in relation to the studied phenomenon and
+practitioners’ experience.
+This paper aims to define a novel theory-guided (or physics-informed) machine learning methodology
+for spatio-temporal data. By this name, go all hybrid procedures that use a stochastic (or deterministic)
+∗Ulm
+University,
+Institute
+of
+Mathematical
+Finance,
+Helmholtzstrae
+18,
+89069
+Ulm,
+Germany.
+E-mail:
+imma.curato@uni-ulm.de.
+†Ulm University, Institute of Stochastics, Helmholtzstrae 18, 89069 Ulm, Germany. E-mail: orkun.furat@uni-ulm.de.
+‡Imperial College, Department of Mathematics, South Kensington Campus, SW7 2AZ London, United Kingdom. E-
+mail: b.stroh@imperial.ac.uk.
+1
+arXiv:2301.00736v1  [stat.ML]  2 Jan 2023
+
+model in synergy with a specific data science one. Such methodologies have started to gain prominence
+in several scientific disciplines such as earth science, quantum chemistry, bio-medical science, climate
+science, and hydrology modeling as, for example, described in [41], [51], [50], and [54]. As in the statistical
+models cited above, we model the spatial-temporal covariance structure of the observed data. However,
+we perform predictions using a generalized Bayesian algorithm. Let us start by introducing the stochastic
+model involved in our methodology.
+We assume throughout to observe data ( ˜Zt(x))(t,x)∈T×L on a regular lattice L ⊂ Rd for d ≥ 1 across
+times T = {1, . . . , N} such that the decomposition
+˜Zt(x) = µt(x) + Zt(x)
+(1)
+holds and no measurement errors are present in the observations. Here, µt(x) is a deterministic function,
+and Zt(x) are considered realizations from a zero mean stationary (influenced) mixed moving average field
+(in brief, MMAF). When the spatial dimension d = 2, a category of data that falls in our assumptions is
+the one of frame images through time (also known as video data or multidimensional raster data). Early
+applications of MMAFs in image modeling can be found in [39].
+An MMAF is defined as
+Zt(x) =
+�
+H
+�
+At(x)
+f(A, x − ξ, t − s) Λ(dA, dξ, ds), (t, x) ∈ R × Rd,
+(2)
+where f is a deterministic function called kernel, H is denoting a non-empty topological space, Λ a L´evy
+basis and At(x) is a so-called ambit set [7], we refer the reader to Section 2 for more details on the
+definition (2). Examples of such models can be found in [7], [11] [47], and [48].
+A significant feature of MMAFs is that they provide a direct way to specify a model for an observed
+physical phenomenon based on a probabilistic understanding of the latter as exemplified by choice of
+the L´evy basis and the kernel function appearing in (2). Choosing an opportune distribution Λ allows
+us to work with Gaussian and non-Gaussian distributed models. Moreover, the random parameter A in
+the kernel function allows us to model short and long-range temporal and spatial dependence. A further
+highlight of these models is that their autocovariance functions can be exponential or power decaying by
+choosing an exponential kernel function f, an opportune distribution of the random parameter A, and
+assuming that the L´evy basis Λ has finite second-order moments. Therefore, such types of autocovariance
+functions can be obtained without the need for further assumptions on the distribution of the random
+field. MMAFs with such properties are, for example, the spatio-temporal Ornstein-Uhlenbeck process (in
+brief, STOU) [47] and its mixed version called the MSTOU process [48]. We also know that the entire
+class of MMAFs is θ-lex weakly dependent. This notion of dependence has first been introduced in [22],
+and it is more general than α∞,v-mixing for random fields as defined in [24] for v ∈ N∪{∞} and α-mixing
+[14] in the particular case of stochastic processes.
+Although the MMAFs are versatile models, only few results in the literature are to be found con-
+cerning their predictive distribution. To our knowledge, the only explicit result concerns Gaussian STOU
+processes defined on cone-shaped ambit sets, see [47, Theorem 13]. We then learn a predictor h ∈ H, for
+H the class of the Lipschitz functions, by determining a randomized estimator ˆρ (i.e., a regular condi-
+tional probability measure) on H using generalized Bayesian learning, see [33] for a review. We call our
+methodology mixed moving average field guided learning. This procedure is applicable, for example, when
+2
+
+using linear models and feed-forward neural networks. The learning task on which we focus is making a
+one-step-ahead prediction of the field Z in a given spatial position x∗.
+The methodology starts by computing the θ-lex coefficients of the underlying MMAF. If the analyzed
+field has finite second-order moments, then such coefficients can be obtained following the calculations in
+Section 2.5. We use this information to select a set of input features from (Zt(x))(t,x)∈T×L and determine
+a training set S. We then prove a PAC Bayesian bound for the sampled data S. We ultimately deter-
+mine a randomized estimator by minimization of the PAC Bayesian bound. The acronym PAC stands
+for Probably Approximately Correct and may be traced back to [66]. A PAC inequality states that with
+an arbitrarily high probability (hence ”probably”), the performance (as provided by a loss function) of
+a learning algorithm is upper-bounded by a term decaying to an optimal value as more data is collected
+(hence ”approximately correct”). PAC-Bayesian bounds have proven over the past two decades to suc-
+cessfully address various learning problems such as classification, sequential or batch learning, and deep
+learning [28]. Indeed, they are a powerful probabilistic tool to derive theoretical generalization guarantees.
+To the best of our knowledge, the PAC Bayesian bounds determined in Section 3.2 are the first results
+in the literature obtained for data serially correlated along a spatial and temporal dimension.
+It is important to emphasize that using a randomized estimator over a classical supervised learning
+methodology has the same advantages as a Bayesian approach, i.e., it allows a deeper understanding of
+the uncertainty of each possible h ∈ H. Moreover, we can enable the analysis of aggregate or ensemble
+predictors ˆh = ˆρ[h]. Despite these similarities, generalized Bayesian learning substantially differs from
+the classical Bayesian learning approach. In the latter, we specify a prior distribution, a statistical model
+connecting output-input pairs called likelihood function, and determine the unique posterior distribution
+by Bayes’ theorem. When using generalized Bayes to determine a randomized estimator, no assumptions
+on the likelihood function are required but just a loss function and a so-called reference distribution. These
+ingredients, together with a PAC Bayesian bound, are employed to determine a randomized predictor,
+which is unique just under a specific set of assumptions, see Section 3.2.
+1.1
+Outline and Contributions
+Between the data-science models used to tackle predictions for spatio-temporal data, we find deep learn-
+ing, see [4], [54], [61] and [62] for a review, and video frame prediction algorithms as in [45] and [68]. Deep
+learning techniques are increasingly popular because they successfully extract spatio-temporal features
+and learn the inner law of an observed spatio-temporal system. However, these models lack interpretabil-
+ity, i.e., it is not possible to disentangle the causal relationship between variables in different spatial-time
+points, and typically no proofs of their generalization (predictive) abilities are available. On the other
+hand, [45] and [68] are methodologies retaining a causal interpretation, see discussion below, but do not
+have proven generalization (predictive) performance.
+Given a model class H, mixed moving average field guided learning selects a predictor h ∈ H that
+has the best generalization performance for the analyzed prediction task. Moreover, the MMAF modeling
+framework has a causal interpretation when using cone-shaped ambit sets. To explain this point, we
+borrow the concept of lightcone from special relativity that describes the possible paths that the light
+can make in space-time leading to a point (t, x) and the ones that lie in its future. In the context of our
+paper, we use their geometry to identify the points in space-time having causal relationships. Let c > 0,
+for a point (t, x), and by using the Euclidean norm to assess the distance between different space-time
+3
+
+points, we define a lightcone as the set
+Alight
+t
+(x) =
+�
+(s, ξ) ∈ R × Rd : ∥x − ξ∥ ≤ c|t − s|
+�
+.
+(3)
+The set Alight with respect to the point (t, x) can be split into two disjoint sets, namely, At(x) and
+At(x)+. The set At(x) is called past lightcone, and its definition corresponds to the one of a cone-shaped
+ambit set
+At(x) :=
+�
+(s, ξ) ∈ R × Rd : s ≤ t and ∥x − ξ∥ ≤ c|t − s|
+�
+.
+(4)
+The set
+At(x)+ = {(s, ξ) ∈ R × Rd : s > t and ∥x − ξ∥ ≤ c|t − s|},
+(5)
+is called instead the future lightcone. By using an influenced MMAF on a cone-shaped ambit set as the
+underlying model, we implicitly assume that the following sets
+l−(t, x) = {Zs(ξ) : (s, ξ) ∈ At(x) \ (t, x)} and l+(t, x) = {Zs(ξ) : (s, ξ) ∈ At(x)+}
+(6)
+are respectively describing the values of the field that have a direct influence on Zt(x) and the future field
+values influenced by Zt(x). We can then uncover the causal relationships described above by estimating
+the constant c from observed data, called throughout the speed of information propagation in the physical
+system under analysis. A similar approach to the modeling of causal relationships can be found in [45],
+[58], and [68]. In [45], and [58], the sets (6) are considered and employed to discover coherent structures, see
+[37] for a formal definition, in spatio-temporal physical systems and to perform video frame prediction,
+respectively. Also, in [68], predictions are performed by embedding spatio-temporal information on a
+Minkowski space-time. Hence, the concept of lightcones enters into play in the definition of their algorithm.
+In machine learning, we typically have two equivalent approaches towards causality: structural causal
+models, which rely on the use of directed acyclical graphs (DAG) [49], and Rubin causal models, which
+rely upon the potential outcomes framework [57]. The concept of causality employed in this paper can be
+inscribed into the latter. In fact, by using MMAFs on cone-shaped ambit sets, the set l+(t, x) describes
+the possible future outcomes that can be observed starting from the spatial position (t, x).
+The paper is structured as follows. In Section 2, we introduce the MMAF framework and define
+STOU and MSTOU processes. In Section 3, we introduce the notations that allow us to bridge the
+MMAF framework (that by definition is continuous in time and space) with a data science one (that
+by definition is discrete in time and space). Important theoretical preliminaries and the input-features
+extraction method can be found in Section 3.1. We then prove PAC Bayesian bounds (also of oracle type)
+in Section 3.2 for Lipschitz predictors, among which we discuss the shape of the bound for linear models
+and feed-forward neural networks. We then focus on linear predictors and show in Section 3.3 how to
+select the best one to be used in a given prediction task. We give in Section 4 an explicit procedure to
+perform one-step ahead casual future predictions in such a framework. In conclusion, we apply our theory-
+guided machine learning methodology to simulated data from an STOU process driven by a Gaussian
+and a NIG-distributed L´evy basis. Appendix A contains further details on the weak dependence measures
+employed in the paper, and a review of the estimation methodologies for STOU and MSTOU processes.
+Appendix B contains detailed proofs of the results presented in the paper.
+4
+
+2
+Mixed moving average fields
+2.1
+Notations
+Throughout the paper, we indicate with N the set of positive integers and R+ the set of non-negative
+real numbers. As usual, we write Lp(Ω) for the space of (equivalence classes of) measurable functions
+f : Ω → R with finite Lp-norm ∥f∥p. When Ω = Rn, ∥x∥1 and ∥x∥ denotes the L1-norm and the Euclidean
+norm, respectively, and we define ∥x∥∞ = maxj=1,...,n |x(j)|, where x(j) represents the component j of
+the vector x.
+To ease the notations in the following sections, unless it is important to keep track of both time and
+space components separately, we often indicate the index set R × Rd with R1+d. A ⊂ B denotes a not
+necessarily proper subset A of a set B, |B| denotes the cardinality of B and dist(A, B) = infi∈A,j∈B∥i −
+j∥∞ indicates the distance of two sets A, B ⊂ R1+d. Let n, k ≥ 1, and F : Rn → Rk, we define ∥F∥∞ =
+supt∈Rd∥F(t)∥. Let Γ = {i1, . . . , iu} ⊂ R1+d for u ∈ N, we define the random vector ZΓ = (Zi1, . . . , Ziu).
+In general, we use bold notations when referring to random elements.
+In the following Lipschitz continuous is understood to mean globally Lipschitz. For u, n ∈ N, G∗
+u
+is the class of bounded functions from Ru to R and Gu is the class of bounded, Lipschitz continuous
+functions from Ru to R with respect to the distance ∥ · ∥1. Moreover, we call L(Ω) the set of all Lipschitz
+functions h on Ω with respect to the distance ∥ · ∥1 and define the Lipschitz constant as
+Lip(h) = sup
+x̸=y
+|h(x) − h(y)|
+∥x − y∥1
+.
+(7)
+Hereafter, we often use the lexicographic order on R1+d. Let the pedex t and s be indicating
+a temporal and spatial coordinate. For distinct elements y = (y1,t, y1,s . . . , yd,s) ∈ R1+d and z =
+(z1,t, z1,s . . . , zd,s) ∈ R1+d we say y <lex z if and only if y1,t < z1,t or yp,s < zp,s for some p ∈ {1, . . . , d}
+and y1,t = z1,t and yq,s = zq,s for q = 1, . . . , p − 1. Moreover, y ≤lex z if y <lex z or y = z holds. Finally,
+let t ∈ R1+d, we define the set V r
+t = Vt ∩ {s ∈ R1+d : ∥t − s∥∞ ≥ r} for r > 0. The definition of the set
+V r
+t is also used when referring to the lexicographic order on Z1+d.
+2.2
+Definition and properties
+Let S = H × R × Rd, where H ⊂ Rq for q ≥ 1, and the Borel σ-algebra of S be denoted by B(S) and let
+Bb(S) contain all its Lebesgue bounded sets.
+Definition 2.1. A family of R-valued random variables Λ = {Λ(B) : B ∈ Bb(S)} is called a L´evy basis
+on (S, Bb(S)) if it is an independently scattered and infinitely divisible random measure. This means that:
+(i) For a sequence of pairwise disjoint elements of Bb(S), say {Bi, i ∈ N}:
+– Λ(�
+n∈N Bn) = �
+n∈N Λ(Bn) almost surely when �
+n∈N Bn ∈ Bb(S)
+– and Λ(Bi) and Λ(Bn) are independent for i ̸= j.
+(ii) Let B ∈ Bb(S). Then, the random variable Λ(B) is infinitely divisible, i.e. for any n ∈ N, there
+exists a law µn such that the law µΛ(B) can be expressed as µΛ(B) = µ∗n
+n , the n-fold convolution of
+µn with itself.
+5
+
+For more details on infinitely divisible distributions, we refer the reader to [59]. In the following, we will
+restrict ourselves to L´evy bases which are homogeneous in space and time and factorizable, i.e., L´evy
+bases with characteristic function
+E
+�
+eiuΛ(B)�
+= eΦ(u)Π(B)
+(8)
+for all u ∈ R and B ∈ Bb(S), where Π = π ×λ1+d is the product measure of the probability measure π on
+H and the Lebesgue measure λ1+d on R×Rd. Note that when using a L´evy basis defined on S = R×Rd,
+Π = λ1+d. Furthermore,
+Φ(u) = iγ u − 1
+2σ2u2 +
+�
+R
+�
+eiux − 1 − iux1[0,1](|x|)
+�
+ν(dx)
+(9)
+is the cumulant transform of an ID distribution with characteristic triplet (γ, σ2, ν), where γ ∈ R, σ2 ≥ 0
+and ν is a L´evy-measure on R, i.e.
+ν({0}) = 0
+and
+�
+R
+�
+1 ∧ x2�
+ν(dx) < ∞.
+The quadruplet (γ, σ2, ν, π) determines the distribution of the L´evy basis entirely, and therefore it
+is called its characteristic quadruplet. An important random variable associated with the L´evy basis, is
+the so-called L´evy seed, which we define as the random variable Λ′ having as cumulant transform (9),
+that is
+E
+�
+eiuΛ′�
+= eΦ(u).
+(10)
+By selecting different L´evy seeds, it is easy to compute the distribution of Λ(B) for B ∈ Bb(S) when
+S = R × Rd. In the following two examples, we compute the L´evy bases used in generating the data sets
+in Section 4.1.
+Example 2.2 (Gaussian L´evy basis). Let Λ′ ∼ N(γ, σ2), then its characteristic function is equal to
+exp(iγu − 1
+2σ2u2). Because of (8), we have, in turn, that the characteristic function of Λ(B) is equal to
+exp(iγuλ1+d(B) − 1
+2σ2λ1+d(B)u2). In conclusion, Λ(B) ∼ N(γλ1+d(B), σ2λ1+d(B)) for any B ∈ Bb(S).
+Example 2.3 (Normal Inverse Gaussian L´evy basis). Let K1 denote the modified Bessel function of the
+third order and index 1. Then, for x ∈ R, the NIG distribution is defined as
+f(x : α, β, µ, δ) = αδ(π2(δ2 + (x − µ)2))− 1
+2 exp(δ
+�
+α2 − β2 + β(x − µ))K1(α
+�
+δ2 + (x − µ)2),
+where α, β, µ and δ are parameters such that µ ∈ R, δ > 0 and 0 ≤ |β| < α. Let Λ′ ∼ NIG(α, β, µ, δ),
+then by (8) we have that Λ(B) ∼ NIG(α, β, µλ1+d(B), δλ1+d(B)) for all B ∈ Bb(S).
+We now follow [7], and [10] to formally define ambit sets.
+Definition 2.4. A family of ambit sets (At(x))(t,x)∈R×Rd ⊂ R × Rd satisfies the following properties:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+At(x) = A0(0) + (t, x), (Translation invariant)
+As(x) ⊂ At(x), for s < t
+At(x) ∩ (t, ∞) × Rd = ∅. (Non-anticipative).
+(11)
+6
+
+We consider throughout At(x) to be defined as in (4). We further assume that the random fields
+in the paper are defined on a given complete probability space (Ω, F, P), equipped with the filtration of
+influence (in the sense of Definition 3.8 in [22]) F = (F(t,x))(t,x)∈R×Rd generated by Λ and the family of
+ambit sets (At(x))(t,x)∈R×Rd ⊂ R × Rd, i.e., each F(t,x) is the σ-algebra generated by the set of random
+variables {Λ(B) : B ∈ Bb(H × At(x))}. We call our fields adapted to the filtration of influence F if
+Zt(x) is measurable with respect to the σ-algebra F for each (t, x) ∈ R × Rd. Moreover, we work with
+stationary random fields. Moreover, in the following, we use the term stationary instead of spatio-temporal
+stationary.
+Definition 2.5 (Spatio-temporal stationarity). We say that Zt(x) is spatio-temporal stationary if for ev-
+ery n ∈ N, τ ∈ R, u ∈ Rd, t1, . . . , tn ∈ R and x1, . . . , xn ∈ Rd, the joint distribution of (Zt1(x1), . . . , Ztn(xn))
+is the same as that of (Zt1+τ(x1 + u), . . . , Ztn+τ(xn + u)).
+Definition 2.6 (MMAF). Let Λ = {Λ(B), B ∈ Bb(S)} a L´evy basis, f : H × R × Rd → R a B(S)-
+measurable function and At(x) an ambit set. Then, the stochastic integral (2) is adapted to the filtration F,
+stationary, and its distribution is infinitely divisible. We call the R-valued random field Z an (influenced)
+mixed moving average field and f its kernel function.
+Remark 2.7. On a technical level, we assume all stochastic integrals in this paper to be well defined in
+the sense of Rajput and Rosinski [52]. For more details, including sufficient conditions on the existence
+of the integral as well as the explicit representation of the characteristic triplet of the MMAF’s infinitely
+divisible distribution (which can be directly determined from the characteristic quadruplet of Λ), we refer
+to [22, Section 3.1]. In the latter, there can also be found a multivariate definition of a L´evy basis and
+an MMAF.
+2.3
+Autocovariance Structure
+Moment conditions for MMAFs are typically expressed in function of the characteristic quadruplet of its
+driving L´evy basis and the kernel function f.
+Proposition 2.8. Let Zt(x) be an R-valued MMAF driven by a L´evy basis with characteristic quadruplet
+(γ, σ2, ν, π) with kernel function f : H × R × Rd → R and defined on an ambit set At(x) ⊂ R × Rd.
+(i) If
+�
+|x|>1 |x|ν(dx) < ∞ and f ∈ L1(H × R × Rd, π × λ1+d) ∩ L2(H × R × Rd, π × λ1+d) the first
+moment of Zt(x) is given by
+E[Zt] = E(Λ′)
+�
+H
+�
+At(x)
+f(A, −s, −ξ)ds dξ π(dA),
+where E(Λ′) = γ +
+�
+|x|≥1 x ν(dx).
+7
+
+(ii) If
+�
+R x2 ν(dx) < ∞ and f ∈ L2(H × R × Rd, π × λ1+d), then Zt(x) ∈ L2(Ω) and
+V ar(Zt(x)) = V ar(Λ′)
+�
+H
+�
+R×Rd f(A, −s, −ξ)2ds dξ π(dA),
+Cov(Z0(0), Zt(x)) = V ar(Λ′)
+�
+H
+�
+A0(0)∩At(x)
+f(A, −s, −ξ)f(A, t − s, x − ξ) ds dξ π(dA)
+and
+Corr(Z0(0), Zt(x)) =
+�
+H
+�
+A0(0)∩At(x) f(A, −s, −ξ)f(A, t − s, x − ξ) ds dξ π(dA)
+�
+H
+�
+R×Rd f(A, −s, −ξ)2ds dξ π(dA)
+,
+(12)
+where V ar(Λ′) = σ2 +
+�
+Rd xx′ν(dx).
+(iii) Consider σ2 = 0 and
+�
+|x|≤1 |x| ν(dx) < ∞. If
+�
+|x|>1 |x|ν(dx) < ∞ and f ∈ L1(H ×R×Rd, π×λ1+d),
+the first moment of Zt(x) is given by
+E[Zt(x)] =
+�
+H
+�
+At(x)
+f(A, −s, −ξ)
+�
+γ0 +
+�
+R
+xν(dx)
+�
+ds dξ π(dA),
+where
+γ0 := γ −
+�
+|x|≤1
+x ν(dx).
+(13)
+Proof. Immediate from [59, Section 25] and [22, Theorem 3.3].
+From Proposition 2.8, we can evince that the autocovariance function of an MMAF depends on the
+variance of the L´evy seed Λ′, the kernel function f and the distribution π of the random parameter A.
+Important examples of MMAFs are the spatio-temporal Ornstein-Uhlenbeck field (STOU) and the
+mixed spatio-temporal Ornstein-Uhlenbeck field (MSTOU), whose properties have been thoroughly ana-
+lyzed in [47] and [48], respectively. In Definition 2.9 and 2.11, we give the formal definitions of such fields
+and the explicit expression of their autocovariance functions.
+Definition 2.9. Let Λ = {Λ(B), B ∈ Bb(S)} be a L´evy basis, f : R×Rd → R a B(S)-measurable function
+defined as f(s, ξ) = exp(−As) for A > 0, and At(x) be defined as in (4). Then, the STOU defined as
+Zt(x) :=
+�
+At(x)
+exp(−A(t − s))Λ(ds, dξ)
+(14)
+is adapted to the filtration F, stationary, Markovian and its distribution is ID. Moreover, let u ∈ Rd,
+τ ∈ R, and E[Zt(x)2] ≤ ∞. Then,
+Cov(Zt(x), Zt+τ(x + u)) = V ar(Λ′) exp(−Aτ)
+�
+At(x)∩At+τ (x+u)
+exp(−2A(t − s))ds dξ, and
+(15)
+Corr(Zt(x), Zt+τ(x + u)) =
+exp(−Aτ)
+�
+At(x)∩At+τ (x+u) exp(−2A(t − s)) ds dξ
+�
+At(x) exp(−2A(t − s)) ds dξ
+.
+(16)
+8
+
+Example 2.10. Let d = 1
+ρT (τ) := Corr(Zt(x), Zt+τ(x)) = exp(−A|τ|),
+(17)
+ρS(u) := Corr(Zt(x), Zt(x + u)) = exp
+�
+− A|u|
+c
+�
+,
+(18)
+ρST (τ, u) := Corr(Zt(x), Zt+τ(x + u)) = min
+�
+exp(−A|τ|), exp
+�
+− A|u|
+c
+��
+.
+(19)
+An STOU exhibits exponential temporal autocorrelation (just like the temporal Ornstein-Uhlenbeck
+process) and has a spatial autocorrelation structure determined by the shape of the ambit set. In addition,
+this class of fields admits non-separable autocovariances, which are desirable in practice.
+An MSTOU process is defined by mixing the parameter A in the definition of an STOU process;
+that is, we assume that A is a random variable with support in H = (0, ∞). This modification allows the
+determination of random fields with power-decaying autocovariance functions.
+Definition 2.11. Let Λ = {Λ(B), B ∈ Bb(S)} be a L´evy basis with characteristic quadruplet (γ, σ2, ν, π),
+f : (0, ∞) × R × Rd → R a B(S)-measurable function defined as f(A, s, ξ) = exp(−As), and At(x) be
+defined in (4). Moreover, let l(A) be the density of π with respect to the Lebesgue measure such that
+� ∞
+0
+1
+Ad+1 l(A)dA ≤ ∞.
+Then, the MSTOU defined as
+Zt(x) :=
+� ∞
+0
+�
+At(x)
+exp(−A(t − s))Λ(dA, ds, dξ)
+(20)
+is adapted to the filtration F, stationary and its distribution is ID. Moreover, let u ∈ Rd and τ ∈ R, and
+E[Zt(x)2] ≤ ∞
+Cov(Zt(x), Zt+τ(x + u)) = V ar(Λ′) exp(−Aτ)
+� ∞
+0
+�
+At(x)∩At+τ (x+u)
+exp(−2A(t − s))ds dξ l(A)dA,
+(21)
+Corr(Zt(x), Zt+τ(x + u)) =
+exp(−Aτ)
+� ∞
+0
+�
+At(x)∩At+τ (x+u) exp(−2A(t − s)) ds dξ f(A)dA
+� ∞
+0
+�
+At(x) exp(−2A(t − s)) ds dξ l(A)dA
+.
+(22)
+Example 2.12. Let l(A) =
+βα
+Γ(α)Aα−1 exp(−βA), the Gamma density with shape and rate parameters,
+α > d + 1 and β > 0. For d = 1, u ∈ R and τ ∈ R
+V ar(Zt(x)) =
+V ar(Λ′)cβ2
+2(α − 2)(α − 1)
+(23)
+Cov(Zt(x), Zt+τ(x + u)) =
+V ar(Λ′)cβα
+2(β + max{|τ|, |u|/c})α−2(α − 2)(α − 1),
+(24)
+ρST (τ, u) := Corr(Zt(x), Zt+τ(x + u)) =
+�
+β
+β + max{|τ|, |u|/c}
+�α−2
+.
+(25)
+9
+
+2.4
+Isotropy, short and long range dependence
+Definition 2.13 (Isotropy). Let t ∈ R and x ∈ Rd. A spatio-temporal random field (Zt(x))(t,x)∈R×Rd is
+called isotropic if its spatial covariance:
+Cov(Zt(x), Zt(x + u)) = C(|u|),
+for some positive definite function C.
+STOU and MSTOU processes defined on cone-shaped ambit sets are isotropic random fields.
+Moreover, we consider the following definitions of temporal and spatial short and long-range depen-
+dence in the paper, as given in [48].
+Definition 2.14 (Short and long range dependence). A spatio-temporal random field (Zt(x))(t,x)∈R×Rd
+is said to have temporal short-range dependence if
+� ∞
+0
+Cov(Zt(x), Zt+τ(x)) dτ < ∞,
+and long-range temporal dependence if the integral above is infinite. Similarly, an isotropic random field
+has short-range spatial dependence if
+� ∞
+0
+C(r) dr < ∞,
+where Cov(Zt(x), Zt(x + u)) = C(|u|) and r = |u|. It is said to have long-range spatial dependence if
+the integral is infinite.
+If (Zt(x))(t,x)∈R×Rd is, for example, an STOU process then Z admits temporal and spatial short-
+range dependence. When Z is an MSTOU process, short and long-range dependence models can be
+obtained by carefully modeling the distribution of the random parameter A.
+Example 2.15. Let us consider the model discussed in Example 2.12. Set u = 0, then Z has temporal
+short-range dependence for α > 3, because
+� ∞
+0
+Cov(Zt(x), Zt+τ(x)) dτ =
+cβαV ar(Λ′)
+2(α − 2)(α − 1)
+� ∞
+0
+(β + τ)−(α−2) dτ
+=
+cβ3V ar(Λ′)
+2(α − 1)(α − 2)(α − 3).
+This integral is infinite for 2 < α ≤ 3, and the process has long-range temporal dependence. We obtain
+spatial short- or long-range dependence for the same choice of parameters. In fact, for r = |u| and τ = 0,
+and α > 3
+� ∞
+0
+C(r) dr =
+cβαV ar(Λ′)
+2(α − 2)(α − 1)
+� ∞
+0
+(β + r/c)−(α−2) dτ
+=
+cβ3V ar(Λ′)
+2(α − 1)(α − 2)(α − 3),
+converges, whereas the integral above diverges for 2 < α ≤ 3.
+10
+
+2.5
+Weak dependence coefficients
+MMAFs are θ-lex weakly dependent random fields. For v ∈ N ∪ {∞}}, the latter is a dependence notion
+more general than α∞,v-mixing, see Lemma A.5.
+Definition 2.16. Let Z = (Zt)t∈R1+d be an Rn-valued random field. Then, Z is called θ-lex-weakly
+dependent if
+θlex(r) = sup
+u,v∈N
+θu,v(r) −→
+r→∞ 0,
+where
+θu,v(r) = sup
+�|Cov(F(ZΓ), G(ZΓ′))|
+∥F∥∞vLip(G)
+�
+and F ∈ G∗
+u, G ∈ Gv; Γ = {ti1, . . . , tiu} ⊂ R1+d, Γ′ = {tj1, . . . , tjv} ⊂ R1+d such that |Γ| = u, |Γ′| = v
+and Γ ⊂ V r
+Γ′ = �v
+l=1 V r
+tjl for tjl ∈ Γ′. We call (θlex(r))r∈R+ the θ-lex-coefficients.
+In the MMAF modeling framework, we can show general formulas for the computation of upper
+bounds of the θ-lex coefficients. The latter are given as a function of the characteristic quadruplet of the
+driving L´evy basis Λ and the kernel function f in (2).
+Proposition 2.17. Let Λ be an R-valued L´evy basis with characteristic quadruplet (γ, σ2, ν, π), f :
+H × R1+d → R a B(H × R1+d)-measurable function and Zt(x) be defined as in (2).
+(i) If
+�
+|x|>1 x2ν(dx) < ∞, γ+
+�
+|x|>1 xν(dx) = 0 and f ∈ L2(H ×R1+d, π×λ1+d), then Z is θ-lex-weakly
+dependent and
+θlex(r) ≤ 2
+� �
+H
+� ρ(r)
+−∞
+V ar(Λ′)
+�
+∥ξ∥≤cs
+f(A, −s, −ξ)2dsdξπ(dA)
+� 1
+2 .
+(ii) If
+�
+|x|>1∥x∥2ν(dx) < ∞ and f ∈ L2(H × R1+d, π × λ1+d) ∩ L1(H × R1+d, π × λ1+d), then Z is
+θ-lex-weakly dependent and
+θlex(r) ≤ 2
+� �
+H
+� ρ(r)
+−∞
+V ar(Λ′)
+�
+∥ξ∥≤cs
+f(A, −s)2 dsdξπ(dA)
++
+����
+�
+S
+� ρ(r)
+−∞
+E(Λ′)
+�
+∥ξ∥≤cs
+f(A, −s) dsdξπ(dA)
+����
+2� 1
+2
+.
+(iii) If
+�
+R |x| ν(dx) < ∞, σ2 = 0 and f ∈ L1(H × R1+d, π × λ1+d) with γ0 defined in (13), then Z is
+θ-lex-weakly dependent and
+θlex(r) ≤ 2
+� �
+H
+� ρ(r)
+−∞
+�
+∥ξ∥≤cs
+|f(A, −s)γ0| ds dξπ(dA)
++
+�
+H
+� ρ(r)
+−∞
+�
+∥ξ∥≤cs
+�
+R
+|f(A, −s)x| ν(dx) dsπ(dA)
+�
+.
+11
+
+The results above hold for all r > 0 with
+ρ(r) =
+−r min(1/c, 1)
+�
+(d + 1)(c2 + 1)
+,
+(26)
+V ar(Λ′) = σ2 +
+�
+R x2 ν(dx) and E(Λ′) = γ +
+�
+|x|≥1 xν(dx).
+Given the results of Proposition 2.17, it is then possible to compute upper bounds for the θ-lex-
+coefficients of an MSTOU process.
+Corollary 2.18. Let Z be an MSTOU process as in Definition 2.11 and (γ, σ2, ν, π) be the characteristic
+quadruplet of its driving L´evy basis. Moreover, let the mean reversion parameter A be Gamma(α, β)
+distributed with density l(A) =
+βα
+Γ(α)Aα−1 exp(−βA) where α > d + 1 and β > 0.
+(i) If
+�
+|x|>1 x2 ν(dx) < ∞ and γ +
+�
+|x|>1 xν(dx) = 0, then Z is θ-lex-weakly dependent. Let c ∈ [0, 1],
+then for
+d = 1,
+θlex(h) ≤ 2
+�cV ar(Λ′)βα
+2Γ(α)
+�
+Γ(α − 2)
+(2ψ(h) + β)α−2 + 2ψ(h)Γ(α − 1)
+(2ψ(h) + β)α−1
+�� 1
+2
+,
+and for
+d ≥ 2,
+θlex(h) ≤ 2
+�
+Vd(c)d!V ar(Λ′)βα
+2d+1
+d
+�
+k=0
+(2ψ(h))k
+k!(2ψ(h) + β)α−d−1+k
+Γ(α − d − 1 + k)
+Γ(α)
+� 1
+2
+.
+Let c > 1, then for
+d ∈ N, θlex(h) ≤ 2
+�
+Vd(c)d!V ar(Λ′)βα
+2d+1
+d
+�
+k=0
+�
+2ψ(h)
+c
+�k
+k!
+�
+2ψ(h)
+c
++ β
+�α−d−1+k
+Γ(α − d − 1 + k)
+Γ(α)
+� 1
+2
+.
+The above implies that, in general, θlex(h) = O(h
+(d+1)−α
+2
+).
+(ii) If
+�
+R |x| ν(dx) < ∞, Σ = 0 and γ0 as defined in (13), then Zt(x) is θ-lex-weakly dependent. Let
+c ∈ (0, 1], then for
+d ∈ N,
+θlex(h) ≤ 2Vd(c)d!βαγabs
+d
+�
+k=0
+ψ(h)k
+k!(ψ(h) + β)α−d−1+k
+Γ(α − d − 1 + k)
+Γ(α)
+,
+whereas for c > 1 and
+d ∈ N,
+θlex(h) ≤ 2Vd(c)d!βαγabs
+d
+�
+k=0
+�
+ψ(h)
+c
+�k
+k!
+�
+ψ(h)
+c
++ β
+�α−d−1+k
+Γ(α − d − 1 + k)
+Γ(α)
+,
+where γabs = |γ0| +
+�
+R |x|ν(dx), and Vd(c) denotes the volume of the d-dimensional ball with radius
+c. the above implies that, in general, θlex(h) = O(h(d+1)−α).
+Proof. Proof of this corollary can be obtained by modifying the proof of [22, Section 3.7] in line with the
+calculations performed in Proposition 2.17.
+12
+
+For d = 1, 2, when the MMAF has a kernel function with no spatial component, we can compute a
+more explicit bound for the θ-lex coefficients.
+Proposition 2.19. Let Λ be an R-valued L´evy basis with characteristic triplet (γ, σ2, ν, π) and f : H ×
+R → R a B(H × R)-measurable function not depending on the spatial dimension, i.e.
+Zt(x) =
+�
+H
+�
+At(x)
+f(A, t − s)Λ(dA, ds, dξ),
+(t, x) ∈ R1+d.
+(27)
+(i) For d = 1, if that
+�
+|x|>1 x2ν(dx) < ∞ and γ +
+�
+|x|>1 xν(dx) = 0, then Z is θ-lex weakly dependent
+and
+θlex(r)≤2
+�
+2V ar(Λ′)
+� ∞
+0
+�
+A0(0)∩A0(r min(2,c))
+f(A, −s)2dsdξπ(dA)
+�1/2
+=2
+�
+2Cov(Z0(0), Z0(r min(2, c))),
+where V ar(Λ′) = σ2 +
+�
+R x2 ν(dx).
+(ii) For d = 2, if
+�
+|x|>1 x2ν(dx) < ∞ and γ +
+�
+|x|>1 xν(dx) = 0, then then Z is θ-lex weakly dependent
+and
+θlex(r) ≤ 2
+�
+2Cov
+�
+Z0(0, 0), Z0
+�
+r min
+�
+1, c
+√
+2
+�
+, r min
+�
+1, c
+√
+2
+���
++ 2Cov
+�
+Z0(0, 0), Z0
+�
+r min
+�
+1, c
+√
+2
+�
+, −r min
+�
+1, c
+√
+2
+��� �1/2
+.
+Assumption 2.20. In general, we indicate the bounds of the θ-lex coefficients determined in Proposition
+2.17, Corollary 2.18 or Proposition 2.19 using the sequence (˜θlex(r)))r∈R+ where
+θlex(r) ≤ 2˜θlex(r).
+Example 2.21. Let d = 1 and Z be an STOU as in Definition 2.9. If
+�
+|x|>1 x2 ν(dx) ≤ ∞, γ +
+�
+|x|>1 x ν(dx) = 0, then Z is θ-lex weakly dependent with
+˜θlex(r) =
+� �
+A0(0)∩(A0(ψ)∪A0(−ψ))
+exp(2As)ds dξ
+� 1
+2
+=
+�
+V ar(Λ′)
+�
+A0(0)∩(A0(ψ)∪A0(−ψ))
+exp(2As) ds dξ
+� 1
+2
+≤
+�
+2V ar(Λ′)
+�
+A0(0)∩A0(ψ)
+exp(2As) ds dξ
+� 1
+2
+=
+�
+2V ar(Λ′)
+� − ψ
+2c
+−∞
+� −cs
+ψ+cs
+exp(2As) ds dξ
+� 1
+2 =
+� c
+A2 V ar(Λ′) exp
+�−Aψ
+c
+�� 1
+2
+13
+
+=
+� c
+A2 V ar(Λ′) exp
+�
+− A min(2, c)
+c
+�
+��
+�
+2λ
+r
+�� 1
+2
+=
+�
+2Cov(Z0(0), Z0(r min(2, c))) := ¯α exp(−λr),
+where λ > 0 and ¯α > 0. Because the temporal and spatial autocovariance functions of an STOU are
+exponential, see (15), the model admits spatial and temporal short-range dependence.
+Example 2.22. Let d = 1 and Z be an MSTOU as in Definition 2.11. Moreover, let us define the density
+l(A) as in Example 2.12. If
+�
+|x|>1 x2 ν(dx) ≤ ∞, γ +
+�
+|x|>1 x ν(dx) = 0, then Z is θ-lex weakly dependent
+with
+˜θlex(r) ≤
+� c
+A2 V ar(��)
+� ∞
+0
+exp
+�−Aψ
+c
+�
+π(dA)
+� 1
+2
+=
+�
+V ar(Λ′)cβα
+(β + ψ/c)α−2(α − 2)(α − 1)
+� 1
+2
+=
+� V ar(Λ′)cβα
+(α − 2)(α − 1)
+�
+β + r min(2, c)
+c
+�−(α−2)� 1
+2
+=
+�
+2Cov(Z0(0), Z0(r min(2, c))) := ¯αr−λ,
+where λ = α−2
+2
+and ¯α > 0. As already addressed in Example 2.15, for 2 < α ≤ 3, that is 0 < λ ≤ 1
+2,
+the model admits temporal and spatial long-range dependence whereas for α > 3, that is λ > 1
+2 the model
+admits temporal and spatial short range dependence.
+Statistical inference for STOU and MSTOU processes is reviewed in Appendix A. Such method-
+ologies can be applied to the entire class of influenced mixed moving average fields (as long as certain
+moment conditions are fulfilled). We remind the reader that the parameter c is involved in the definition of
+the ambit set At(x) and therefore of the lightcone (3) which we assume modeling the causal relationships
+between spatial-time points. Other parameters of interest appear in the kernel function or represent the
+variance of the L´evy seed Λ′. We can then determine an estimate of the decay rate of the θ-lex coefficients,
+which is given, for example, by the parameter λ in Examples 2.21 and 2.22.
+In the MMAF framework, we can also find time series models. The latter are θ-weakly dependent,
+a notion of dependence satisfied by causal stochastic processes and defined as follows.
+Definition 2.23. Let Z = (Zt)t∈R be an Rn-valued stochastic process. Then, Z is called θ-weakly
+dependent if
+θ(k) = sup
+u∈N
+θu(k) −→
+k→∞ 0,
+where
+θu(k) = sup
+�|Cov(F(Zi1, . . . , Zi1), G(Zj1))|
+∥F∥∞Lip(G)
+�
+.
+and F ∈ Gu, G ∈ G1; i1 ≤ i2 ≤ . . . ≤ iu ≤ iu + k ≤ j1. We call (θ(K))k∈R+ the θ-coefficients.
+Example 2.24 (Time series case). The supOU process studied in [6] and [11] is an example of a causal
+mixed moving average process. Let the kernel function f(A, s) = e−As1[0,∞)(s), A ∈ R+, s ∈ R and Λ a
+14
+
+L´evy basis on R+ × R with generating quadruple (γ, σ2, ν, π) such that
+�
+|x|>1
+log(|x|) ν(dx) < ∞, and
+�
+R+
+1
+Aπ(dA) < ∞,
+(28)
+then the process
+Zt =
+�
+R+
+� t
+−∞
+e−A(t−s) Λ(dA, ds)
+(29)
+is well defined for each t ∈ R and strictly stationary and called a supOU process where A represents a
+random mean reversion parameter.
+If E(Λ′) = 0 and
+�
+|x|>1 |x|2ν(dx) < ∞, the supOU process is θ-weakly dependent with coefficients
+θZ(r) ≤
+� �
+R+
+� r
+−∞
+e−2Asσ2 ds π(dA)
+� 1
+2 =
+�
+V ar(Λ′)
+�
+R+
+e−2Ar
+2A
+π(dA)
+� 1
+2
+(30)
+= Cov(Z0, Z2r)
+1
+2 ,
+by using Theorem 3.11 [11] and where V ar(Λ′) = σ2 +
+�
+R x2ν(dx).
+If E(Λ′) = µ and
+�
+|x|>1 |x|2ν(dx) < ∞, the supOU process is θ-weakly dependent with coefficients
+θZ(r) ≤
+�
+Cov(Z0, Z2r) +
+4µ2
+V ar(Λ′)2 Cov(Z0, Zr)2� 1
+2 .
+(31)
+If
+�
+R |x|ν(dx) < ∞, σ2 = 0, γ0 = γ −
+�
+|x|≤1 x ν(dx) > 0 and ν(R−) = 0, then the supOU process admits
+θ-coefficients
+θZ(r) ≤ µ
+�
+R+
+e−Ar
+A
+π(dA),
+(32)
+and when in addition
+�
+|x|>1 |x|2ν(dx) < ∞
+θZ(r) ≤
+2µ
+V ar(Λ′)Cov(Z0, Zr).
+(33)
+Note that the necessary and sufficient condition
+�
+R+ 1
+A π(dA) for the supOU process to exist is
+satisfied by many continuous and discrete distributions π, see [64, Section 2.4] for more details. For
+example, a probability measure π being absolutely continuous with density π′ = xhl(x) and regularly
+varying at zero from the right with h > 0, i.e., l is slowly varying at zero, satisfies the above condition. If
+moreover, l(x) is continuous in (0. + ∞) and limx→0+ l(x) > 0 exists, it holds that
+Cov(Z0, Zr) ∼ C
+rh , with a constant C > 0 and r ∈ R+
+where for h ∈ (0, 1) the supOU process exhibits long memory and for h > 1 short memory. In this set-up,
+concrete examples where the covariances are calculated explicitly can be found in [8] and [20].
+Another interesting example of mixed moving average processes is given by the class of trawl pro-
+cesses. A distinctive feature of the class of trawl processes is that it allows one to model its correlation
+structure independently from its marginal distribution, see [9] for further details on their definition. In the
+case of trawl processes, we also have available in the literature likelihood-based methods for estimating
+15
+
+their parameters; see [13] for further details. In general, the generalized method of moments is employed
+to estimate the parameters of an MMA process, see [20].
+3
+Mixed moving average field guided learning
+3.1
+Pre-processing N frames: input-features extraction method
+Our methodology is designed to work for spatio-temporal data when the spatial dimension d ≥ 1. With-
+out loss of generality, we describe the procedure for d = 2, i.e., for frame images data through time
+( ˜Zt(x))(t,x)∈T×L following the decomposition (1). We then represent the regular spatial lattice L as a
+frame made of a finite amount of pixels, i.e., squared-cells representing each of them a unique spatial
+position x ∈ R2, see Figure 1.
+Figure 1: Space-time grid with origin in (t0, x0).
+In several applications, such as satellite imagery, a pixel refers to a spatial cell of several square
+meters. In the paper, a pixel refers to the spatial point x ∈ R2 corresponding to the center of the squared
+cell. We then use the name pixel and spatial position throughout interchangeably. Moreover, we call
+(t0, x0) the origin of the space-time grid, see Figure 1, and ht and hs the time and space discretization
+step in the observed data set. Mixed moving average guided learning has the target to determine a one-time
+ahead prediction of the field Z at a given pixel position x∗, not belonging to the frame boundary.
+Definition 3.1. Let (X, Y )⊤ an input-output vector, H a model class, and a predictor function h ∈ H.
+We define the loss function L as
+L(h(X), Y ) = |Y − h(X)|.
+(34)
+For ϵ > 0 be an accuracy level (specified before learning), we define the truncated absolute loss as
+Lϵ(h(X), Y )) = L(h(X), Y )) ∧ ϵ.
+(35)
+16
+
+Frame
+Pixel
+Origin
+TimeMoreover, we define the generalization error (out-of-sample risk)
+Rϵ(h) = E[Lϵ(h(X), Y )],
+(36)
+and the empirical error (in-sample risk)
+rϵ(h) = 1
+m
+m
+�
+i=1
+Lϵ(h(Xi), Yi).
+(37)
+Remark 3.2 (About the accuracy level ϵ). The boundedness of the loss function Lϵ plays a key role in
+the results of the paper as it allows us to prove Proposition 3.10, which is one of the main technical tools
+used to obtain PAC Bayesian bounds in Section 3.2. Different choices of the parameter ϵ will result in
+different randomized predictors. Therefore ϵ is an hyper-parameter.
+The topic discussed in the remark below holds for a general loss function. However, given the use
+of the truncated absolute loss function in the paper, we focus on this specific example.
+Remark 3.3 (Generalization gap). The function Lϵ is used to measure the discrepancy between a pre-
+dicted output h(X) and the true output Y . Using the in-sample risk, we measure the performance of
+a given predictor h just over an observed sample. Its theoretical counterpart represented by the out-of-
+sample risk gives us the performance of a given predictor h depending on the unknown distribution of the
+data P. The latter represents the measure we want to evaluate when choosing a predictor h. However,
+we do not know the distribution of the data, so typically, we select a predictor h by solely evaluating its
+in-sample risk. We then need a guarantee that a selected predictor will perform well when used on a set of
+out-of-sample observations, i.e., not belonging to the observed data. We can also rephrase the problem as
+finding a predictor h for which the difference between the out-of-sample and in-sample risk Rϵ(h) − rϵ(h)
+is as small as possible. We call the latter generalization gap. The PAC framework, of which in the next
+section we introduce a Bayesian version, aims to find a bound of the generalization gap that holds with
+high probability P, see for example [60] and[67]. Such probability inequalities are also called generalization
+bounds and give a theoretical guarantee on the performance of a predictor on any unseen data.
+We now prove three preliminary results needed to define the input-features extraction method at
+the end of this section. In conclusion, we use these results to define a training data set, which can preserve
+the dependence structure of the data and reduce as much as possible the number of past and neighboring
+space-time points used in learning.
+Preliminary 1: Let us consider a stationary random field (Zt(x))(t,x)∈Z×L, and select a pixel x∗
+in L not belonging to the boundary of the frame. We define
+Xi = L−
+p (t0 + ia, x∗),
+and
+Y i = Zt0+ia(x∗), for i ∈ Z,
+where
+L−
+p (t, x) = (Zi1(ξ1), . . . , Zia(p,c)(ξa(p,c)))⊤, for
+{(is, ξs) : ∥x − ξs∥ ≤ c (t − is) and 0 < t − s ≤ p} := I(t, x), c > 0 and a(p, c) := |I(t, x)|. The
+parameters a > 0 and p > 0 are multiple of ht such that a = atht, p = ptht, at, pt ∈ N and
+at ≥ pt + 1. Moreover, we assume throughout to store in the vectors Xi values of the fields Z in
+17
+
+Figure 2: Time and spatial indices in the definition of the random vector (Xi, Y i)⊤ for c = 1, pt = 2, at =
+3, ht = hs = 1. In particular, the green and red pixels represent the time-spatial indices identifying the
+elements of Z belonging to Xi and Y i, respectively. This sampling scheme cannot be performed starting
+by a pixel x∗ at the boundary of the frame.
+lexicographic order. Then, (Xi, Y i)⊤ are identically distributed for all i ∈ Z. We call ((Xi, Y i)⊤)i∈Z
+a cone-shaped sampling process. An example of such a sampling scheme can be found in Figure 2.
+Preliminary 2: Let us analyze the dependence structure of the stochastic processes (L(h(Xi), Y i))i∈Z
+and (Lϵ(h(Xi), Y i))i∈Z for h a Lipschitz function belonging to the model class H := {h ∈ L(Ra(p,c))}.
+Proposition 3.4. Let ((Xi, Y i)⊤)i∈Z be a cone-shaped sampling process from a stationary and
+θ-lex weakly dependent random field (Zt(x))(t,x)∈R×Rd. Then for all h ∈ H, (L(h(Xi), Yi))i∈Z is a
+θ-weakly dependent process. Moreover, for a, p > 0, k ∈ N, and r = ka − p > 0, it has coefficients
+θ(k) ≤ ˜d(Lip(h)a(p, c) + 1)
+�2
+˜d
+E[|Zt(x) − Z(r)
+t (x)|] + θlex(r)
+�
+,
+(38)
+where ˜d > 0 is a constant independent of r, and Z(r)
+t (x) := Zt(x) ∧ r.
+Remark 3.5 (Locally Lipschitz predictor). Let the predictor h be a locally Lipschitz function such
+that h(0) = 0 and
+|h(x) − h(y)| ≤ ˜c ∥x − y∥1(1 + ∥x∥1 + ∥y∥1) for x, y ∈ Ra(p,c),
+for ˜c > 0. Moreover, let Z be a stationary and θ-weakly dependent random field such that |Z| ≤ C
+a.s. An easy generalization of Proposition 3.4, leads to show that (L(h(Xi), Yi))i∈Z is θ-weakly
+dependent with coefficients
+θ(k) ≤ ˜d(˜c(1 + 2C)a(p, c) + 1)
+�2
+˜d
+E[|Zt(x) − Z(r)
+t
+(x)|] + θlex(r)
+�
+,
+(39)
+where r = ka − p > 0 for a, p > 0 and k ∈ N, ˜d > 0 is a constant independent of r, and
+Z(r)
+t
+(x) := Zt(x) ∧ r.
+MMAFs have a particular definition that allows us to obtain an explicit bound for the coefficients
+18
+
+Timeθlex(r), as proven in Propositions 2.17 and 2.19, and a more refined bound than (38) for the θ-
+coefficients of the cone-shaped sampling process. We consider the following assumptions.
+Assumption 3.6. Let T = {t0, t1, . . . , tN, N ∈ N} and L ⊂ R2, then the data set (Zt(x))(t,x)∈T×L
+is drawn from (Zt(x))(t,x)∈Z×L. Moreover, the latter can be derived from a regular sampling of the
+MMAF Z = (Zt(x))(t,x)∈R×R2, where Z ∈ L2(Ω).
+Proposition 3.7. Let Assumption 2.20 and 3.6 hold. Then (L(h(Xi), Yi))i∈Z is θ-weakly dependent
+for all h ∈ H with coefficients
+θ(k) ≤ 2(Lip(h)a(p, c) + 1)�θlex(r),
+(40)
+where r = ka − p > 0 for a, p > 0 and k ∈ N. Moreover, for linear predictors, i.e., hβ(X) =
+β0 + βT
+1 X, for β := (β0, β1)⊤ ∈ B and B = Ra(p,c)+1, we have that (L(hβ(Xi), Yi))i∈Z is θ-weakly
+dependent for all β ∈ B with coefficients
+θ(k) ≤ 2(∥β1∥1 + 1)�θlex(r).
+(41)
+Lemma A.3 straightforwardly imply that the process Lϵ := (Lϵ(h(X)i, Y i)))i∈Z is θ-weakly depen-
+dent with known θ-coefficients under the assumptions of Proposition 3.4, Remark 3.5 and Proposi-
+tion 3.7.
+Preliminary 3: θ-weakly dependent processes have an essential role in the paper because of the
+property of the process Lϵ. We analyze now an exponential bound for such processes.
+Proposition 3.8. Let (Xi)i∈Z be a Rn valued stationary θ-weakly dependent process, and f : Rn →
+[a, b], for a, b ∈ R such that (f(Xi))i∈Z is itself θ-weakly dependent. Let l = ⌊ m
+k ⌋, for m, k ∈ N,
+such that l ≥ 2 and 0 < s <
+3l
+|b−a|, then
+E
+�
+exp
+� s
+m
+m
+�
+i=1
+(f(Xi) − E[f(Xi)])
+��
+≤ exp
+�
+s2V ar(f(X))
+2l
+�
+1 − s|b−a|
+3l
+�
+�
++ s
+l exp(l − 1 + s|b − a|)θ(k), (42)
+E
+�
+exp
+� s
+m
+m
+�
+i=1
+(E[f(Xi)] − f(Xi))
+��
+≤ exp
+�
+s2V ar(f(X))
+2l
+�
+1 − s|b−a|
+3l
+�
+�
++ s
+l exp(l − 1 + s|b − a|)θ(k). (43)
+Remark 3.9. The assumptions of Proposition 3.8 hold, for example, if f is a Lipschitz function
+or satisfies the assumptions of [21, Section 7]. Moreover, the assumption of Proposition 3.8 hold
+for the function Lϵ(h(·), ·) applied to the process ((Xi, Y i)⊤)i∈Z under the assumptions analyzed in
+the Preliminary 2.
+We can now define a training data set S = {(X1, Y1)⊤, (X2, Y2)⊤, . . . , (Xm, Ym)⊤} as a draw from
+the cone-shaped sampling process defined in the Preliminary 1, namely,
+Xi = L−
+p (t0 + ia, x∗),
+and
+Yi = Zt0+ia(x∗) for i = 1, . . . , m :=
+�N
+at
+�
+,
+(44)
+where
+19
+
+L−
+p (t, x) = (Zi1(ξ1), . . . , Zia(p,c)(ξa(p,c)))⊤, where (is, ξs) ∈ I(t, x) for s = 1, . . . , a(p, c).
+(45)
+We assume that the parameters a, p, and k follow the constraints in Table 1. Moreover, each parameter
+has a precise interpretation, also summarized here. We note that each element of L−
+p (t, x) belongs to
+l−(t, x), where l−(t, x) is defined in (6) and identifies the set of all points in R × Rd that could possibly
+influence the realization Zt(x).
+Parameters
+Constraints
+Interpretation
+a := atht
+pt + 1 ≤ at ≤
+�
+N
+2
+�
+translation vector
+p := ptht
+1 ≤ pt <
+�
+N
+2
+�
+− 1
+past time horizon
+k
+k ∈
+�
+1, . . . ,
+�
+N
+at
+��
+index of the θ-coefficient in Prop. 3.8.
+m := N
+at
+2 < m < N
+number of examples in S
+c
+c > 0
+speed of information propagation
+λ
+λ > 0
+decay rate of the θ-lex coefficients of Z
+Table 1: Parameters involved in pre-processing N observed frames.
+For each observed data set, we know the value of the constants N, ht, and hs. The remaining
+parameters appearing in Table 1 have to be opportunely chosen. We explain in this section how to select
+the parameter at and k by assuming the parameters pt, c, and λ are known. The selection of the parameter
+pt is detailed in Section 3.3 for linear predictors, and c and λ are typically estimated from (Zt(x))(t,x)∈T×L,
+see exemplary estimation methods in Sections A.2 and A.3.
+The parameter at and k are selected to offset the lack of independence of the process Lϵ which is
+θ-weakly dependent, as proven in the Preliminary 2. In Proposition 3.10, and Corollary 3.11 and 3.13,
+we show an important building block in the proof of the PAC Bayesian bounds obtained in the next
+section, which make use of an opportune selection of the parameters at and k. Such a result undergoes
+minor changes when working in the class of linear predictors, feed-forward neural networks, or Lipschitz
+functions. As exemplary, we show this result for linear predictors and discuss how to modify the proof
+for more general model classes. We use the notation ∆(h) = Rϵ(h) − rϵ(h) and ∆′(h) = rϵ(h) − Rϵ(h)
+when referring to a Lipschitz predictor; ∆(β) = Rϵ(β) − rϵ(β) and ∆′(β) = rϵ(β) − Rϵ(β), where we
+call Rϵ(β) and rϵ(β) the generalization and empirical error, in the linear framework; and ∆(hnet,w) =
+Rϵ(hnet,w)−rϵ(hnet,w) and ∆′(hnet,w) = rϵ(hnet,w)−Rϵ(hnet,w), where we call Rϵ(hnet,w) and rϵ(hnet,w)
+the generalization and empirical error, in the case we use a feed-forward neural network.
+Proposition 3.10. Let Assumption 2.20 and 3.6 hold, the parameter pt be known and r = ht(kat − pt).
+Moreover, let us assume that H = {hβ(X) = β0 + βT
+1 X, for β := (β0, β1)⊤ ∈ B} where B ∈ R(a(p,c)+1).
+(i) Let 0 < ϵ < 3, l ≥ 9, β ∈ B and Z be a field with exponential decaying θ-lex coefficients, i.e.,
+�θlex(r) = ¯α exp(−λr)
+for ¯α > 0, λ > 0. If
+atk ≥ (λp − 1) +
+�
+(λp − 1)2 + 8λhtN
+2λht
+,
+(46)
+20
+
+then
+E
+�
+exp
+�√
+l ∆(β)
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2(∥β1∥1 + 1)¯α,
+(47)
+E
+�
+exp
+�√
+l ∆′(β)
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2(∥β1∥1 + 1)¯α.
+(48)
+(ii) Let 0 < ϵ < 3, l ≥ 9, β ∈ B and Z be a field with power decaying θ-lex coefficients
+�θlex(r) = ¯αr−λ
+for ¯α > 0. If
+2N − atk
+atk log(ak − p) ≤ λ,
+(49)
+then the inequalities (47) and (48) hold.
+By substituting the bound (41) with (40) in the proof of Proposition 3.10 we obtain the following
+result.
+Corollary 3.11. Let Assumption 2.20 and 3.6 hold, the parameter pt be known and r = ht(kat − pt).
+Moreover, let us assume that H = {h(X) ∈ L(Ra(p,c))}. Let 0 < ϵ < 3, l ≥ 9, h ∈ H and Z be a field
+with exponential decaying θ-lex coefficients, i.e.,
+�θlex(r) = ¯α exp(−λr)
+for ¯α > 0, λ > 0, or with power decaying θ-lex coefficients
+�θlex(r) = ¯αr−λ
+for ¯α > 0. If condition (46) or (49) hold, then for h ∈ H
+E
+�
+exp
+�√
+l ∆(h)
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2(Lip(h)a(p, c) + 1)¯α,
+(50)
+E
+�
+exp
+�√
+l ∆′(h)
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2(Lip(h)a(p, c) + 1)¯α.
+(51)
+Corollary 3.11 applies to feed-forward neural network predictors as long as we can compute the Lip-
+schitz function of the neural network. There exist numerical methods to compute the Lipschitz constants
+of (deep) neural networks, see [29] and [40], which enable the computation of the inequalities (50) and
+(51) for a given network. We give now an explicit version of the inequality (50) and (51) in the case a
+1-layer neural network, which we define below.
+Definition 3.12. Let σ : R → R an activation function, we define a 1-layer neural network as
+hnet,w(X) = α0 +
+K
+�
+l=1
+αlσ(βT
+l X + γl),
+where w = (α0, α1, . . . , αK, β1, . . . , βK, γ1, . . . , γK) ∈ R2K+1+Ka(p,c) := B′ for K ≥ 1.
+21
+
+Many frequently used activation functions are 1-Lipschitz continuous functions, meaning that their
+Lipschitz constant equals one. Such property is, for example, satisfied by the activation functions ReLU,
+Leaky ReLU, SoftPlus, Tanh, Sigmoid, ArcTan, or Softsign. We can then prove the following result.
+Corollary 3.13. Let Assumption 2.20 and 3.6 hold, the parameter pt be known and r = ht(kat − pt).
+Moreover, let us assume that H = {hnet,w(X) ∈ B′}, where the networks are defined following Definition
+3.12 with a 1-Lipschitz activation function σ. Let 0 < ϵ < 3, l ≥ 9, w ∈ B′ and Z be a field with
+exponential decaying θ-lex coefficients, i.e.,
+�θlex(r) = ¯α exp(−λr)
+for ¯α > 0, λ > 0, or with power decaying θ-lex coefficients
+�θlex(r) = ¯αr−λ
+for ¯α > 0. If condition (46) or (49) hold, then
+E
+�
+exp
+�√
+l ∆(hnet,w)
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2(
+�
+l
+|αl|∥βl∥1 + 1)¯α,
+(52)
+E
+�
+exp
+�√
+l ∆′(hnet,w)
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2(
+�
+l
+|αl|∥βl∥1 + 1)¯α.
+(53)
+Remark 3.14. Similarly, as in Corollary 3.13, using the bounds (38) or (39), Proposition 3.10 can be
+generalized for Lipschitz and locally Lipschitz predictors, respectively, under the assumption that the data
+are generated by a stationary θ-lex weakly dependent random field.
+We then pre-process the data to obtain the largest possible training data set S starting by N
+observed frames.
+Assumption 3.15. We select k∗ being the minimum positive constants that satisfy (46) or (49) and a∗
+t
+as the minimum constant satisfying (46) or (49) such that a∗
+t ≥ pt + 1. Therefore,
+m =
+� N
+a∗
+t
+�
+and l =
+� m
+k∗
+�
+.
+Remark 3.16. Let us assume to work with linear predictors, for C > 1, if we choose
+atk ≥ (λp − 1 − log(1/C)) +
+�
+(λp − 1 − log(1/C))2 + 8λhtN
+2λht
+,
+(54)
+or
+2N − atk(1 + log(1/C))
+atk log(ak − p)
+≤ λ,
+(55)
+then
+E
+�
+exp
+�√
+l ∆(β)
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2
+C (∥β1∥1 + 1)¯α,
+(56)
+E
+�
+exp
+�√
+l ∆′(β)
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2
+C (∥β1∥1 + 1)¯α.
+(57)
+22
+
+For a fixed N, this means that we can obtain tighter bounds than in (47) and (48). However, using such
+results instead of Proposition 3.10 must be evaluated on a case-by-case basis. Typically, we obtain smaller
+training data sets if we have to satisfy the inequalities (54) and (55). Without loss of generality, therefore,
+we consider throughout just the type of bounds analyzed in Proposition 3.10, which results in Assumption
+3.15.
+Remark 3.17. The input-features extraction method discussed in this section, Proposition 3.10, and
+Corollary 3.11 and 3.13 straightforwardly apply to a θ-weakly dependent time series models Z. In such
+case, the parameter c = 0, (Xi, Y i)⊤
+i∈Z is a flat cone-shaped sampling scheme, and straightforwardly it
+is a θ-weakly dependent process with coefficients satisfying the bound (38).
+3.2
+PAC Bayesian bounds for MMAF generated data
+We start with a brief introduction to generalized Bayesian learning.
+Firstly, let π be a reference distribution on the space (H, T ), where T indicates a σ-algebra on the
+space H. The reference distribution gives a structure on the space H, which we can interpret as our belief
+that certain models will perform better than others. The choice of π, therefore, is an indirect way to make
+the size of H come into play; see [16, Section 3] for a detailed discussion on the latter point. Therefore,
+π belongs to M1
++(H), which denotes the set of all probability measures on the measurable set (H, T ).
+Secondly, let S be a training data set with m examples and values in X × Y. Moreover, let us call P
+the distribution of the random vector S. We then aim to determine a posterior distribution, also called a
+randomized estimator, which is the regular conditional probability measure
+ˆρ : (X × Y)m × T → [0, 1],
+satisfying the following properties:
+• for any A ∈ T , the map S → ˆρ(S, A) : (X × Y)m × T → [0, 1] is measurable;
+• for any S ∈ (X × Y)m, the map A → ˆρ(S, A) : T → [0, 1] is a probability measure in M1
++(B).
+From now on, we indicate with π[·], ˆρ[·] the expectations w.r.t. the reference and posterior distribu-
+tions (where the latter is a conditional expectation w.r.t. S), and simply with E[·] the expectation w.r.t.
+the probability distribution P. Moreover, we call ˆρ[Rϵ(h)] and ˆρ[rϵ(h)] the average generalization error
+and the average empirical error, respectively.
+To evaluate the generalization performance of a randomized predictor ˆρ and then have a criterion
+to select such distribution, we determine a PAC bound. The latter is called in this framework a PAC
+Bayesian bound and is used to give with high probability a bound on the (average) generalization gap
+defined as ˆρ[Rϵ(h)] − ˆρ[rϵ(h)]. We then choose a predictor having the best generalization performance in
+the model class H, see also discussion in Remark 3.3, by minimizing the PAC Bayesian bound.
+As exemplary, we first show a PAC Bayesian bound for linear predictors and then discuss how to
+modify the bounds to obtain probabilistic inequalities valid for more general model classes. In general,
+for a given measurable space (H, T ) and for any (ρ, π) ∈ M1
++(H)2, we indicate with
+KL(ρ||π) =
+�
+ρ
+�
+log dρ
+dπ
+�
+if ρ << π
++∞
+otherwise
+23
+
+the Kullback-Leibler divergence. When the model class is given in dependence of a set of parameters, as
+in the case of linear predictors, T indicates the σ-algebra on the parameter space B. We then obtain the
+following result.
+Proposition 3.18 (PAC Bayesian bound). Let 0 < ϵ < 3, Assumption 3.6 hold and let the underlying
+MMAF Z be a random field with exponential or power decaying θ-lex coefficients. Moreover, let l be
+selected following Assumption 3.15 and π be a distribution on B such that π[∥β1∥1] ≤ ∞, then for any ˆρ
+such that ˆρ << π, and δ ∈ (0, 1)
+P
+�
+ˆρ[Rϵ(β)] ≤ ˆρ[rϵ(β)] +
+�
+KL(ˆρ||π) + log
+�1
+δ
+�
++
+3ϵ2
+2(3 − ϵ)
+� 1
+√
+l
++ 1
+√
+l
+log
+�
+π
+�
+1 + 2(∥β1∥1 + 1)¯α
+���
+≥ 1 − δ
+(58)
+and
+P
+�
+ˆρ[rϵ(β)] ≤ ˆρ[Rϵ(β)] +
+�
+KL(ˆρ||π) + log
+�1
+δ
+�
++
+3ϵ2
+2(3 − ϵ)
+� 1
+√
+l
++ 1
+√
+l
+log
+�
+π
+���
+1 + 2(∥β1∥1 + 1)¯α
+���
+≥ 1 − δ.
+(59)
+Example 3.19. The bound in (58) holds for all randomized estimators ˆρ absolutely continuous w.r.t. a
+reference distribution π given a training data set S. Let us assume that the card(B) = M for M ∈ N. We
+choose throughout as reference distribution the uniform distribution π on B and the class of randomized
+predictors ˆρ = δ ˆ
+β, the Dirac mass concentrated on the empirical risk minimizer, i.e., ˆβ := arg infβ rϵ(β).
+For a given S, we have that
+KL(δβ||π) =
+�
+β∈B
+log
+�δ ˆβ{β}
+π{β}
+�
+δ ˆβ{β} = log
+1
+π{ˆβ}
+= log(M).
+(60)
+It is crucial to notice that the bigger the cardinality of the space B is, the more the term log(M) and the
+bound increase.
+Therefore, for a given δ ∈ (0, 1)
+P
+�
+Rϵ(ˆβ) ≤ rϵ(ˆβ) +
+�
+log
+�M
+δ
+�
++
+3ϵ2
+2(3 − ϵ)
+� 1
+√
+l
++ 1
+√
+l
+log
+�
+π[1 + 2(∥β1∥1 + 1)¯α]
+��
+≥ 1 − δ.
+Let us assume to work with an accuracy level ϵ = 1, l = 10000, M = 100, ¯α = 4, δ = 0.05, and
+B := {β : ∥β∥1 ≤ 1}. Then, the generalization gap, as defined in Remark 3.3, is less than 0.12 with
+probability P of at least 95%.
+Remark 3.20. Differently from the i.i.d. setting, the parameter l is not tuned in the bounds obtained in
+Proposition 3.18; see [17] and [65] for a discussion on this topic. The choice of the parameter l in the
+bounds (58) and (59) is a consequence of Assumption 3.15. The value of l is chosen to offset the lack of
+independence in the observed N frames.
+We now determine an oracle inequality for the randomized estimator minimizing the average em-
+pirical error, also known as Gibbs posterior distribution or Gibbs estimator, first in the case of a linear
+predictor and then in the general case of a Lipschitz one.
+24
+
+Proposition 3.21 (Oracle type PAC Bayesian bound). Let 0 < ϵ < 3, Assumption 3.6 hold and let
+the underlying MMAF Z be a random field with exponential or power decaying θ-lex coefficients. Let l
+be selected following Assumption 3.15, π be a distribution on B such that π[∥β1∥1] ≤ ∞, and ¯ρ be a
+regular conditional distribution that is absolutely continuous w.r.t. π and has Radon-Nikodym derivative
+d¯ρ
+dπ =
+exp(−
+√
+lrϵ(β))
+π[exp(−
+√
+lrϵ(β))]. Then, for all δ ∈ (0, 1)
+P
+�
+¯ρ[Rϵ(β)] ≤ inf
+ˆρ
+�
+ˆρ[Rϵ(β)]+
+�
+KL(ˆρ||π)+log
+�1
+δ
+�
++
+3ϵ2
+2(3 − ϵ)
+� 2
+√
+l
++ 2
+√
+l
+log
+�
+π[1+2(∥β1∥1+1)¯α]
+���
+≥ 1−δ. (61)
+By employing Corollary 3.11 instead of Proposition 3.10 in the proof of the oracle inequality, we
+obtain the general result below.
+Corollary 3.22 (Oracle type PAC Bayesian bound for a general Lipschitz predictor). Let 0 < ϵ < 3,
+Assumption 3.6 hold and let the underlying MMAF Z be a random field with exponential or power
+decaying θ-lex coefficients. Let l be selected following Assumption 3.15, π be a distribution on H such that
+π[Lip(h)] ≤ ∞, and ¯ρ be a regular conditional distribution that is absolutely continuous w.r.t. π and has
+Radon-Nikodym derivative d¯ρ
+dπ =
+exp(−
+√
+lrϵ(h))
+π[exp(−
+√
+lrϵ(h))]. Then, for all δ ∈ (0, 1)
+P
+�
+¯ρ[Rϵ(h)] ≤ inf
+ˆρ
+�
+ˆρ[Rϵ(h)]+
+�
+KL(ˆρ||π)+log
+�1
+δ
+�
++
+3ϵ2
+2(3 − ϵ)
+� 2
+√
+l
++ 2
+√
+l
+log
+�
+π[1+2(Lip(h)a(p, c)+1)¯α]
+���
+≥ 1−δ.
+(62)
+Remark 3.23. The PAC Bayesian bound (62) also applies to Lipschitz neural network predictors.
+Better generalization performances are observed, i.e., tighter bounds for the generalization gap, when
+Lip(hnet,w) ≤ 1. In the case of a 1-layer neural network, for which we have an explicit bound of the Lip-
+schitz constant, we can then obtain an oracle inequality by employing Corollary 3.13 instead of Corollary
+3.11 in the proof of the inequality (62). Finally, using the same argument as in Remark 3.14, we can
+obtain an oracle inequality also in the case of locally Lipschitz predictors.
+The rate at which the bounds (61) and (62) converge to zero as m → ∞ is called rate of convergence.
+It allows us to give a measure on how fast the average generalization error of ¯ρ converges to the average
+best theoretical risk inf ˆρ ˆρ[Rϵ(h)], which is obtained by the so-called oracle estimator. The rate depends
+on the choice made for l =
+�
+m
+k
+�
+in the pre-processing step. For k = 1, we obtain the fastest possible rate
+of O(m− 1
+2 ).
+Example 3.24 (Rate of convergence for models with spatial and temporal long range dependence). Let
+us consider a data set with N = 10000 frames drawn from a regular sampling of an MSTOU as defined
+in Example 2.22 with discretization steps ht = hs = 1. Further, let the parameters at = 1000, k = 1, and
+pt = 10 in the pre-processing step of our methodology. From the calculations in Example 2.15, we have
+that the underlying model admits temporal and spatial long-range dependence when its θ-lex coefficients
+have power decay rate 0 < λ ≤ 1
+2. Because of the relationship (49, a Gibbs estimator applies as long as the
+MSTOU admits θ-lex coefficients with λ ≥ 2, 76. Pre-processing the data to include a larger amount of
+observations in an example (X, Y ), i.e., letting pt be a larger parameter, changes the range of applicability
+of the estimator minimally. If, for example, we choose pt = 999 (the largest possible values to choose given
+at = 1000), we obtain that λ ≥ 2, 09.
+To apply the Gibbs estimator to MSTOU with long range dependence, we can choose a k > 1 in
+the pre-processing step. Note that with this choice, however, the rate of convergence of the PAC Bayesian
+25
+
+bound in (61) and (62) becomes then slower than O(m− 1
+2 ). If we can also decide how to sample our
+data along the temporal dimension, we can opt to work with a data set where the temporal discretization
+step ht > 1. So doing, the rate of convergence of the Gibbs estimators can become O(m− 1
+2 ) also in the
+long-range dependence framework. Note that log(ht) appears in (49). Therefore, a careful selection of the
+parameters at, k, and pt has to be done when 0 < λ ≤ 1 such that the sampling frequency is not too low
+and the rate of convergence of the PAC Bayesian bound remains the desired one.
+In the literature, results can be found on the rate of convergence of PAC Bayesian bound just for
+time series models. We review such results in the remark below to highlight the novelty of the results
+obtained in the section.
+Remark 3.25 (Rate of convergence in PAC Bayesian bounds for heavy tailed data). In [1], the au-
+thors determine an oracle inequality for time series with a rate of O(m− 1
+2 ) under the assumption that
+((Xi, Y i)⊤)i∈Z is a stationary and α-mixing process [14] with coefficient (αj)j∈Z such that �
+j∈Z αj < ∞.
+Such bound employs the chi-squared divergence and holds for unbounded losses: it is important to high-
+light that the randomized estimator obtained by minimization of the PAC Bayesian bound is not a Gibbs
+estimator in this framework. An explicit bound for linear predictors can be found in [1, Corollary 2]. This
+result holds under the assumption that π[∥β∥6] ≤ ∞.
+In [3], the authors prove oracle inequalities for a Gibbs estimator for data generated by a stationary
+and bounded θ∞-weakly dependent process– which is an extension of the concept of φ-mixing discussed
+in [55]– or a causal Bernoulli shift process and a Lipschitz predictor. Models that satisfy the dependence
+notion just cited are causal Bernoulli shifts with bounded innovations, uniform φ-mixing sequences, and
+dynamical systems; see [3] for more examples. The oracle inequality is here obtained for an absolute loss
+function and has a rate of O(m− 1
+2 ). An extension of this work for Lipschitz loss functions under φ-mixing
+[38] can be found in [2]. Here, the authors show that an oracle inequality for a Gibbs estimator with the
+optimal rate O(m−1) can be achieved. Note that this rate is considered optimal in the literature when
+using, for example, a squared loss function and independent and identically distributed data [15].
+Interesting results in the literature of PAC Bayesian bounds for heavy-tailed data can also be found
+in [32], and [36]. However, the authors assume independent distribution of the underlying process in their
+works.
+We conclude by giving an oracle inequality for the aggregate Gibbs estimators in the general case
+of a Lipschitz predictor. This result is obtained by using the convexity of the absolute loss function.
+Corollary 3.26 (PAC Bound for the average Gibbs predictor). Let 0 < ϵ < 3, Assumption 3.6 hold and
+let Z be a random field with exponential or power decaying θ-lex coefficients such that |Y − h(X)| < ϵ
+a.s. for each h ∈ L(Ra(p,c)). Let l be selected following Assumption 3.15. Moreover, let π be a distribution
+on H such that π[Lip(h)] ≤ ∞, and ¯ρ a Gibbs posterior distribution. Then, let ¯h = ¯ρ[h], for all δ ∈ (0, 1)
+P
+�
+Rϵ(¯h) ≤ inf
+ˆρ
+�
+ˆρ[Rϵ(h)]+
+�
+KL(ˆρ||π)+log
+�1
+δ
+�
++
+3ϵ2
+2(3 − ϵ)
+� 2
+√
+l
++ 2
+√
+l
+log
+�
+π[1+2(Lip(h)a(p, c)+1)¯α]
+���
+≥ 1−δ.
+3.3
+Modeling selection for the best randomized Gibbs estimator for linear
+predictors
+We discuss in this section the selection of the parameter pt for H = {hβ(X) : β ∈ R(a(p,c)+1)}. For different
+pt, the predictor, we aim to define changes because the cardinality of the input-features vectors Xi changes.
+26
+
+Therefore, choosing this parameter means performing modeling selection. To ease the notations, we will
+refer to pt simply using the symbol p in the section and its related proofs.
+Let the parameter space
+B =
+⌊ N
+2 ⌋−1
+�
+p=1
+Bp
+where the Bp are assumed to be disjoint sets, such that for any β ∈ B, there is only one p such that
+β ∈ Bp. Our modeling selection procedure is designed to select the best predictor in the class below.
+Definition 3.27. For all p = 1, . . . ,
+�
+N
+2
+�
+− 1, and reference distributions πp on M1
++(Bp) such that
+πp[∥β1���1] ≤ ∞, we define the class of Gibbs estimators as the regular conditional probability measures
+which are absolutely continuous w.r.t. πp and have Radon Nikodym derivative
+d¯ρp
+dπp
+=
+exp(−
+√
+lrϵ(β))
+πp[exp(−
+√
+lrϵ(β)]
+.
+(63)
+The proposition below uses Lemma B.6 and B.7 that can be found in Appendix B.
+Proposition 3.28 (Model selection). Let 0 < ϵ < 3, and the assumptions of Proposition 3.10 and 3.18
+hold. Moreover, let
+p∗ = arg inf
+p
+�
+¯ρp
+�
+rϵ(β) + log(2∥β1∥1 + 3)
+√
+l
+�
++ 1
+√
+l
+log
+��N
+2
+���
+.
+(64)
+Then for all δ ∈ (0, 1),
+P
+�
+¯ρp∗[Rϵ(β)] ≤ inf
+p
+�
+¯ρp
+�
+rϵ(β) + log(2∥β1∥1 + 3)
+√
+l
+�
++ 1
+√
+l
+log
+��N
+2
+���
++
+3ϵ2
+2(3 − ϵ)
+√
+l∗ +
+1
+√
+l∗ log ¯α
+δ
+�
+≥ 1 − δ,
+(65)
+where l∗ =
+�
+⌊ N
+a∗ ⌋
+k∗
+�
+, and a∗, k∗ are constants depending on p∗.
+Lastly, for a bounded parameter space B, we can obtain the following oracle inequalities.
+Proposition 3.29 (Oracle type PAC Bayesian bound for the best Gibbs estimator). Let the assumptions
+of Proposition 3.21 hold, and p∗ be defined as in (64). Moreover, let supB ∥β∥ =
+1
+C for C > 0. For all
+δ ∈ (0, 1)
+P
+�
+¯ρp∗[Rϵ(β)] ≤ inf
+p
+�
+inf
+ˆρ∈M1
++(Bp) ˆρ
+�
+Rϵ(β)
+�
++ 2
+√
+l
+2 + 3C
+C
++ 2
+√
+l
+KL(ˆρ||πp) + 1
+√
+l
+log
+��N
+2
+���
++
+3ϵ2
+(3 − ϵ)
+√
+l∗ +
+2
+√
+l∗ log ¯α
+δ
+�
+≥ 1 − δ
+(66)
+Corollary 3.30 (PAC bound for the average best Gibbs estimator). Let the assumptions of Corollary
+3.26 hold, and p∗ be defined as in Proposition 3.28. Moreover, let supB ∥β∥ = 1
+C for C > 0 and ¯β = ¯ρ[hβ].
+27
+
+For all δ ∈ (0, 1),
+P
+�
+Rϵ(¯β) ≤ inf
+p
+�
+inf
+ˆρ∈M1
++(Bp) ˆρ
+�
+Rϵ(β)
+�
++ 2
+√
+l
+2 + 3C
+C
++ 2
+√
+l
+KL(ˆρ||πp) + 1
+√
+l
+log
+��N
+2
+���
++
+3ϵ2
+(3 − ϵ)
+√
+l∗ +
+2
+√
+l∗ log ¯α
+δ
+�
+≥ 1 − δ
+(67)
+Corollary 3.31 (Oracle inequality for a single draw out of the best Gibbs estimator). Let the assumptions
+of Proposition 3.21 hold and p∗ be defined as in (64). Moreover, let supB ∥β∥ =
+1
+C for C > 0 and
+β∗ = arg infB R(β). For all δ ∈ (0, 1) and ˆβ ∼ ¯ρp∗,
+P¯ρp∗
+�
+Rϵ[ˆβ] ≤ Rϵ(β∗) + 1
+C E[|Z|] +
+3ϵ2
+(3 − ϵ)
+√
+l
++ 2
+√
+l
+log
+��N
+2
+��
+1 + 2C + 1
+C
+� ¯α
+δ
+��
+≥ 1 − δ.
+(68)
+Remark 3.32. The modeling selection procedure discussed here selects the best-randomized estimator
+in the Gibbs posterior distributions family in dependence on the parameter p. In the case of non-linear
+models, as the 1-layer neural network in Definition 3.12, a modeling selection procedure has also to select
+the layer’s dimension, i.e., the parameter K. Moreover, for L-layers neural network, the parameter L
+enters the modeling selection procedure. We aim in future research to analyze such a selection strategy
+and determine a feasible procedure for applying the generalized Bayesian algorithms defined in the paper
+to a feed-forward neural network.
+4
+Predicting spatio-temporal data with MMAF guided learning
+and linear predictors
+Let us start by giving in Table 2 an overview of all the parameters appearing in the learning methodology.
+Parameters
+Type
+Interpretation
+ϵ
+Hyperparameters
+Accuracy level in the definition of the loss functions Rϵ(β) and rϵ(β)
+N
+Observed Parameter
+Number of image frames through time composing our data set
+ht
+Observed Parameter
+Discretization step along the temporal dimension
+hs
+Observed Parameter
+Discretization step along the spatial dimension
+λ
+Unknown Parameter
+Decay rate of the θ-lex coefficients of the underlying model
+c
+Unknown Parameter
+Speed of information propagation
+¯α
+Unknown Parameter
+Determines how tight ˜θlex(r) is as bound of θlex(r)
+pt
+Unknown Parameter
+Length of the past information we include in each Xi
+k and at
+Derived Parameters
+Chosen accordingly to Assumption 3.15
+Table 2: Overview of parameters appearing in MMAF guided learning
+If, for example, the underlying MMAF is an STOU or an MSTOU process,i.e., we are assuming that
+the data admits exponential or power-decaying autocorrelation functions, estimation methodologies for
+their parameters can be found in [47] and [48]; see review in Section A.2 and A.3. We highlight that such
+modeling set-up applies to data with short and possibly long-range spatial and temporal dependence. In
+general, we follow the steps below to make one-step-ahead predictions.
+(i) Observed N frames, we first use the entire data set to estimate the parameters λ and c.
+28
+
+Figure 3: Data set with spatial dimension d = 1. The last two frames are indicated with the blue stars,
+whereas the violet circles represent the points in space and time where it is possible to provide predictions
+with MMAF guided learning for pt = c = 1. The prediction in the time-spatial position (5, 3) lies in the
+intersection (between red lines) of the future lightcones of the points (6, 2), (5, 2) and (4, 2) (represented
+with green lines), which belong to L−
+p (5, 3).
+(ii) We fix a pixel position x∗ and we select pt (and as consequence a training data set S following
+assumption 3.15) using the rule (64).
+(iii) We then draw β from the Gibbs estimator determined for the pixel x∗ and the training data set S
+defined in (ii) to determine a prediction at time t = N +1: we can use single draws or averaging a set
+of different draws to this aim. The methodology employs novel-input features given by L−
+p (N+1, x∗).
+Therefore, we can make predictions in a future time point t = N + 1 as long as the set I(N + 1, x∗)
+has cardinality a(p, c).
+The procedure above can be implemented for any pixel where it is possible to determine a cone-
+shaped training data set S, i.e., for the pixels that do not belong to the frame boundary. Moreover, the
+prediction we obtain in (N +1, x∗) lies in the intersections of the future light cones of each input contained
+in L−
+p (N + 1, x∗), see Figure 3. This means that MMAF guided learning enables us to make predictions
+in spatial-time points that are plausible starting from the set of inputs we observe.
+Similarly to the kriging literature, we need an inference step before being capable of delivering
+spatio-temporal predictions. In this literature, it is often assumed that the estimated parameters used
+in the calculation of kriging weights and kriging variances are the true one, see [18, Chapter 3] and
+[19, Chapter 6] for a discussion on the range of applicability of such estimates. We implicitly make the
+same assumptions when substituting the value of the parameters λ and c in the pre-processing step. It
+remains, however, an interesting open problem to understand the interplay of the estimates’ bias, also of
+the parameter ¯α, on the validity of Proposition (3.10) and the PAC Bayesian bound (58). The biggest
+problem of this analysis relies upon disentangling the effect of the bias of the constant c introduced in
+the pre-processing step which changes the length of the input-features vector X. An optimal selection
+strategy for the hyperparameter ϵ based on the bound (58) is connected to the latter issue.
+29
+
+10
+8
+6
+5
+4
+2
+1
+米
+米
+0
+0
+2
+44.1
+An example with simulated data
+We simulate four data sets from a zero mean STOU process by employing the diamond grid algorithm
+introduced in [47] for a spatial dimension d = 1. The data sets (Zt(x))(t,x)∈T×L is drawn from the
+temporal-spatial interval [0, 1000] × [0, 10], with T = 0, . . . , 20.000 and L = 0, . . . , 200, corresponding to
+a time and spatial discretization step ht = hs = 0.05. We choose as distribution for the L´evy seed Λ′ a
+normal distribution with mean µ = 0 and standard deviation σ = 0.5, and a NIG(α, β, µ, δ) distribution
+with α = 5, β = 0, δ = 0.2 and µ = 0. We use the latter distribution to test the behavior of MMAF
+guided learning for different sets of heavy-tailed data. We also generate data with different mean-reverting
+parameters A and use different seeds in generating the L´evy basis distribution, see summary scheme of
+data’s characteristics in Table 3. The constant c = 1 across all generated data sets.
+Name
+Mean Reverting Parameter
+L´evy seed
+Random generator seed
+GAU1
+A = 1
+Gaussian
+1
+GAU10
+A = 4
+Gaussian
+10
+NIG1
+A = 1
+NIG
+1
+NIG10
+A = 4
+NIG
+10
+Table 3: Overview on simulated data sets with c = 1 and spatial dimension d = 1.
+We start our procedure by step (i), i.e., estimating for each data set the parameter λ and c. We
+use the estimators (77) and (78) presented in Section A.2. The true parameter λ being equal to 1
+2 or
+2 depending on the chosen mean reverting parameter A. The obtained results for each data set are
+given in Table 4. We go to step (ii), and for each pixel position x∗ not belonging to the boundary of
+Name
+A∗
+c∗
+λ∗
+GAU1
+1.014
+1.0003
+0.5068
+GAU10
+4.0145
+1.0011
+2.005
+NIG1
+1, 0024
+0.9992
+0.5016
+NIG10
+3.9682
+1.0000
+1.9841
+Table 4: Estimations of parameters A,c and λ.
+the frame (a total of 199 pixels), we select the parameter pt using the criteria (64) and a value of the
+parameter ϵ = 2.99. For each pixel, we use a multivariate standard normal Gaussian distribution as
+reference distribution and use importance sampling (with Gaussian proposal) to estimate the integral in
+(64). The modeling selection criteria output pt = 1 for each pixel. We then extract from the frames a
+training data set S following assumption 3.15, which we use at step (iii) of our forecasting procedure.
+An acceptance-rejection algorithm with a Gaussian proposal is used to draw β from the best randomized
+Gibbs estimator. We report in Figure 4 the results obtained in each data set and an analysis of the average
+MSE across pixels in Table 5. MMAF guided learning has a low MSE in most of the pixels analyzed.
+Name
+MSE
+GAU1
+0.07042
+GAU10
+0.37028
+NIG1
+0.12365
+NIG10
+0.07102
+Table 5: Average MSE across pixels.
+30
+
+0
+25
+50
+75
+100
+125
+150
+175
+200
+2
+1
+0
+1
+2
+prediction set
+test set
+(a) GAU1
+0
+25
+50
+75
+100
+125
+150
+175
+200
+3
+2
+1
+0
+1
+2
+prediction set
+test set
+(b) GAU10
+0
+25
+50
+75
+100
+125
+150
+175
+200
+1
+0
+1
+2
+3
+4
+prediction set
+test set
+(c) NIG1
+0
+25
+50
+75
+100
+125
+150
+175
+200
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+prediction set
+test set
+(d) NIG10
+Figure 4: One step ahead average prediction on 1.000.000 draws for each pixel position in the data sets.
+31
+
+5
+Conclusions
+We define a novel theory-guided machine learning methodology for spatio-temporal data. The method-
+ology starts by modeling the data using an influenced mixed moving average field. Such a step requires
+the specification of the kernel function, including the presence of random parameters, and that the L´evy
+seed underlying the definition of the model has finite second order moments. To enable one-time ahead
+predictions, we use the underlying model to extract a training data set from the observed data, which
+preserves the dependence structure of the latter and reduces as much as possible the number of past
+and neighboring space-time points used in learning. In the model class of the Lipschitz function, which
+includes linear functions but also neural networks, we show novel PAC Bayesian bounds for MMAF gen-
+erated data. Their minimization leads to the determination of a randomized predictor for our prediction
+task. Finally, in the case of linear models, we test the performance of our learning methodology on a set of
+four simulated data from an STOU process and obtain an average favorable performance in all analyzed
+scenarios.
+Funding
+The fist author was supported by the research grant GZ:CU 512/1-1 of the German Research Foundation
+(DFG).
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+A
+Appendix
+A.1
+Weak dependence notions for casual processes and (influenced) MMAF
+In this section, we discuss more in details the asymptotic dependence notions called θ-weak dependence
+and θ-lex weak dependence. The latter notion has been introduced in [22, Definition 2.1] as an extension to
+the random field case of the notion of θ-weak dependence satisfied by causal stochastic processes [25]. This
+notion of dependence is presented in Definition 2.23. However, the notion of θ-lex weak dependence given
+in Definition 2.16, slightly differs from the one given in [22, Definition 2.1] and represents an extension
+to the random field case of the θ-weak dependence notion defined in [26, Remark 2.1]. Note that the
+definition of θ-weak dependence in [25] and [26, Remark 2.1] differ for the cardinality of the marginal
+distributions on which the function G is computed, namely, G ∈ G1 in the former and G ∈ Gν for ν ∈ N
+in the latter.
+Remark A.1 (Mixingale-type representation of θ-weak dependence). Let L1 = {g : R → R, bounded and
+Lipschitz continuous with Lip(g) ≤ 1}, a stochastic process (Xt)t∈Z, and M = σ{Xt : t < j1, t ∈ Z},
+then it is showed in [26, Proposition 2.3] that
+θ(r) = sup
+g∈L1
+∥E[g(Xj1)|M] − E[g(Xj1)]∥1.
+(69)
+An alternative proof of this result can be also found in [25, Lemma 1].
+36
+
+Let us now analyze the relationship between θ-weak dependence, α-mixing, and φ-mixing. Most
+of the PAC Bayesian literature for stationary and heavy tailed data employs the following two mixing
+conditions, see Remark 3.25, namely α-mixing and φ-mixing. The results in the Lemma below give us a
+proof that the θ-weak dependence is more general than α-mixing and φ-mixing and therefore describes
+the dependence structure of a bigger class of models.
+Let M and V be two sub-sigma algebras of F. First of all, the strong mixing coefficient [56] is
+defined as
+α(M, V) = sup{|P(M)P(V ) − P(M ∩ V )|, M ∈ M, V ∈ V}.
+A stochastic process X is said to be α-mixing if
+α(r) = α(σ{Xs, s ≤ 0}, σ{Xs, s ≥ r})
+converges to zero as r → ∞. The φ-mixing coefficient has been introduced in [38] and defined as
+φ(M, V) = sup{|P(V |M) − P(V )|, M ∈ M, V ∈ V, P(M) > 0}.
+A stochastic process X is said to be φ-mixing if
+φ(r) = φ(σ{Xs, s ≤ 0}, σ{Xs, s ≥ r})
+converges to zero as r → ∞.
+Lemma A.2. Let (Xt)t∈Z be a stationary real-valued stochastic process such that E[|X0|q] ≤ ∞ for
+some q > 1. Then,
+(a) θ(r) ≤ 2
+2q−1
+q α(r)
+q−1
+q ∥X0∥q ≤ 2
+q−1
+q φ(r)
+q−1
+q ∥X0∥q, and
+(b) θ-weak dependence is a more general dependence notion than α-mixing and φ-mixing.
+Proof. The proof of the first inequality at point (a) is proven in [22, Proposition 2.5] using the represen-
+tation of the θ-coefficients (69). The proof of the second inequality follows from a classical result in [14,
+Proposition 3.11]. In [22, Proposition 2.7], it is defined a stochastic process which is θ-weak dependent
+but neither α-mixing or φ-mixing.
+As seen in Definition 2.16 by using the lexicographic order in R1+d, an opportune extension of
+θ-weak dependence valid for random fields can be defined.
+The definition of θ-lex coefficients for G ∈ G1 is given in [22, Definition 2.1]. The latter can be
+represented as θv
+lex(r) := supu∈N{θu,v(r)} for v = 1. Therefore, an alternative way to define the θ-lex
+coefficients in Definition 2.16 is obviously
+θlex(r) = sup
+v∈N
+θv
+lex(r), v ∈ N for all r ∈ R+.
+(70)
+The following Lemma has important applications in the following sections.
+Lemma A.3. Let Z be a θ-lex weakly dependent random field, then ZM
+i
+= Zi ∨ (−M) ∧ M is θ-lex
+weakly dependent.
+37
+
+Proof. Let u, v ∈ N, M > 0, F ∈ G∗
+u, G ∈ Gv, and i1, i2, . . . , iu ∈ V r
+Γ′ where Γ′ = {j1, . . . , jv}. Let
+F M(Zi1, . . . , Ziu) = F(ZM
+i1 , . . . , ZM
+iu ), and GM(Zj1, . . . , Zjv) = G(ZM
+j1 , . . . , ZM
+jv ). We have that F M is
+a bounded function on (Rn)u and GM is a bounded and Lipschitz function on (Rn)v (with the same
+Lipschitz coefficients as the function G): let (Z1, . . . , Zv) and ( ˜Z1, . . . , ˜Zv),
+|GM(Z1, . . . , Zv) − GM( ˜Z1, . . . , ˜Zv)| ≤ Lip(G)
+v
+�
+i=1
+|ZM
+i
+− ˜Z
+M
+i |
+≤ Lip(G)
+v
+�
+i=1
+|Zi − ˜Zi|.
+Then, it holds that
+|Cov(F(ZM
+i1 , . . . , ZM
+iu ), G(ZM
+j1 , . . . , ZM
+jv ))| ≤ ∥F∥∞vLip(G)θlex(r),
+where θlex(r) are the θ-coefficients of the field Z, and so the field ZM
+t (x) is θ-lex weakly dependent.
+Note that the above result also holds for X a θ-weakly dependent process. Therefore, the truncated
+XM
+t
+= Xt ∨ (−M) ∧ M is a θ-weakly dependent process.
+The notion of θ-lex weak dependence also admits a mixingale-type representation.
+Remark A.4. Let L1 = {g : R → R, bounded and Lipschitz continuous with Lip(g) ≤ 1} and Γ′ =
+{j1, . . . , jv} for {j1, . . . jv} ∈ Z1+d. For a random field (Zt)t∈Z1+d, by readily applying [22, Lemma 5.1]
+on the σ-algebra M = σ{Zt : t ∈ V r
+Γ′ ⊂ Z1+d}, the following result can be easily proved:
+θlex(r) = sup
+g∈L1
+sup
+v∈N
+∥E[g(Zj1, . . . , Zjv)|M] − E[g(Zj1, . . . , Zjv)]∥1.
+(71)
+We now use the representation of the θ-lex coefficients (71) to understand its relationships to α∞,v-
+mixing and φ∞,v-mixing for v ∈ N∪{∞}. These notions are defined in [24] and are strong mixing notions
+used in the study of stationary random fields.
+In general, for u, v ∈ N ∪ {∞}, given coefficients
+αu,v(r) = sup{α(σ(ZΓ), σ(ZΓ′)), Γ, Γ′ ∈ R1+d, |Γ| ≤ u, |Γ′| ≤ v, dist(Γ, Γ′) ≥ r},
+and
+φu,v(r) = sup{φ(σ(ZΓ), σ(ZΓ′)), Γ, Γ′ ∈ R1+d, |Γ| ≤ u, |Γ′| ≤ v, dist(Γ, Γ′) ≥ r}.
+a random field Z is said to be αu,v-mixing or φu,v-mixing if the coefficients (αu,v(r))r∈R+ or (φu,v(r))r∈R+
+converge to zero as r → ∞. We then have the following result.
+Lemma A.5. Let (Zt)t∈Z1+d be a stationary real-valued random field such that E[|Z0|q] ≤ ∞ for some
+q > 1. Then, for v ∈ N ∪ {∞},
+(a) θlex(r) ≤ 2
+2q−1
+q α∞,v(r)
+q−1
+q ∥Z0∥q ≤ 2
+q−1
+q φ∞,v(r)
+q−1
+q ∥Z0∥q, and
+(b) it holds that θ-lex weak dependence is more general than α∞,v-mixing and α-mixing in the special
+case of stochastic processes. Moreover, θ-lex weak dependence is more general than φ∞,v-mixing.
+38
+
+Proof. From the proof of [22, Proposition 2.5], we have that
+θ1
+lex(r) ≤ 2
+2q−1
+q α∞,1(r)
+q−1
+q ∥Z0∥q.
+Because of (70) and [14, Proposition 3.11], we have that
+θlex(r) ≤ 2
+2q−1
+q α∞,1(r)
+q−1
+q ∥Z0∥q ≤ 2
+q−1
+q φ∞,1(r)
+q−1
+q ∥Z0∥q.
+Equally,
+θlex(r) ≤ 2
+2q−1
+q α∞,v(r)
+q−1
+q ∥Z0∥q ≤ 2
+q−1
+q φ∞,v(r)
+q−1
+q ∥Z0∥q.
+The proof of the point (b) follows directly by [22, Proposition 2.7]. In fact θlex(r) = θ1,∞(r) following
+the notations of [26, Definition 2.3] and the process used in the proof of the Proposition is θ-lex weakly
+dependent but neither α∞,v, α or φ∞,v-mixing.
+A.2
+Inference on STOU processes
+Let us start by explaining the available estimation methodologies for the parameter vector θ0 = {A, c, V ar(L′)}
+under the STOU modeling assumption when the spatial dimension d = 1. Throughout, we refer to the
+notations and calculations in Example 2.21.
+We have two ways of estimating the parameter vector θ0 in such a scenario. The first one is presented
+in [47]. Here, the parameters A and c are first estimated using normalized spatial and temporal variograms
+defined as
+γS(u) := E((Zt(x) − Zt(x − u))2)
+V ar(Zt(x))
+= 2(1 − ρS(u)) = 2
+�
+1 − exp
+�
+− Au
+c
+��
+,
+(72)
+and
+γT (τ) := E((Zt(x) − Zt−τ(x))2)
+V ar(Zt(x))
+= 2(1 − ρT (τ)) = 2(1 − exp(−Aτ)),
+(73)
+where ρS and ρT are defined in Example 2.10. Note that normalized variograms are used to separate the
+estimation of the parameters A and c from the parameter V ar(Λ′). Let N(u) be the set containing all the
+pairs of indices at mutual spatial distance u for u > 0 and the same observation time. Let N(τ) be the
+set containing all the pairs of indices where the observation times are at a distance τ > 0 and have the
+same spatial position. |N(u)| and |N(τ)| give the number of the obtained pairs, respectively. Moreover,
+let ˆk2 be the empirical variance which is defined as
+ˆk2 =
+1
+(D − 1)
+D
+�
+i=1
+Z2
+ti(xi),
+(74)
+where D denotes the sample size. The empirical normalized spatial and temporal variograms are then
+39
+
+defined as follows:
+ˆγS(u) =
+1
+|N(u)|
+�
+i,j∈N(u)
+(Zti(xi) − Ztj(xj))2
+ˆk2
+(75)
+ˆγT (τ) =
+1
+|N(τ)|
+�
+i,j∈N(τ)
+(Zti(xi) − Ztj(xj))2
+ˆk2
+.
+(76)
+By matching the empirical and the theoretical forms of the normalized variograms, we can estimate A
+and c by employing the estimators
+A∗ = −τ −1 log
+�
+1 − ˆγT (τ)
+2
+�
+, and c∗ = −
+A∗u
+log
+�
+1 − ˆγS(u)
+2
+�.
+(77)
+Alternatively, we can use a least square methodology to estimate the parameters A and c, i.e. (75) and
+(76) are computed at several lags, and a least-squares estimation is used to fit the computed values to the
+theoretical curves. The authors in [47] use the methodology discussed in [43] to achieve the last target. We
+refer the reader also to [18, Chapter 2] for further discussions and examples of possible variogram model
+fitting. The parameter V ar(Λ′) can be estimated by matching the second-order cumulant of the STOU
+with its empirical counterpart. The consistency of this estimation procedure is proven in [47, Theorem
+12].
+A second possible methodology for estimating the vector θ0 employs a generalized method of moment
+estimator (GMM), as in [48]. It is essential to notice that by using such an estimator, we cannot separate
+the parameter V ar(Λ′) from the estimation of the parameters A and c. Instead, all moment conditions
+must be combined into one optimization criterion, and all the estimations must be found simultaneously.
+Consistency and asymptotic normality of the GMM estimator are discussed in [48] and [22], respectively.
+It is important to notice that the parameter λ can be inferred by knowing the parameter A and c
+alone (this also holds for d ≥ 2). We can then plug in the estimations (77) (or the ones obtained by least
+squares or GMM) and obtain the estimator
+λ∗ = A∗ min(2, c∗)
+2c∗
+,
+(78)
+of the decay rate of the θ-lex coefficients of an STOU with spatial dimension d = 1. This estimator is
+consistent because of [47, Theorem 12] and the continuous mapping theorem. Furthermore, by using the
+estimation of the parameter V ar(Λ′), we can also obtain a consistent estimator for ¯α.
+For d ≥ 2, a least square methodology is still applicable for estimating the variogram’s parameters.
+The estimator used in [47] is a normalized version of the least-square estimator for spatial variogram’s
+parameters discussed in [43], which also applies for d > 1. This method, paired with a method of moments
+(matching the second order cumulant of the field Z with its empirical counterparts), allows estimating
+the parameter V ar(Λ′). The GMM methodology discussed in [48] also continues to apply for d ≥ 2.
+However, when the spatial dimension is increasing, the shape of the normalized variograms and the field’s
+moments become more complex, and higher computational effort is required to navigate through the high
+dimensional surface of the optimization criterion behinds least-squares or GMM estimators.
+40
+
+A.3
+Inference for MSTOU processes
+When estimating the parameter vector θ1 = {α, β, c, V ar(L′)} under an MSTOU modeling assumption–
+see, for example, the shape of the coefficients in Example 2.22– it is evident that the shape of the
+autocorrelation function, and therefore of the normalized temporal and spatial variograms, become more
+complex for increasing d. As already addressed in the previous sections, when estimating the parameters
+(α, β, c) alone, we can use the least-squares type estimator discussed in [43]. Moreover, by pairing the
+latter with a method of moments or using a GMM estimator, we can estimate the vector θ1.
+B
+Appendix
+B.1
+Proofs of Section 2.5
+In [22, Proposition 3.11], it is given a general methodology to show that an MMAF Z is θ-lex weakly
+dependent. Given that the definition of θ-lex-weakly dependence used in the paper slightly differs from
+the one given in [22], the proof of Proposition 2.17 and 2.19 differ from the one of [22, Proposition 3.11].
+Before giving a detailed account of these proofs, let us state some notations used in both. Let r > 0,
+{(tj1, xj1), . . . (tjv, xjv)} = Γ′ ⊂ R1+d and {(ti1, xi1), . . . , (tiu, xiu)} = Γ ⊂ V r
+Γ′ such that |Γ| = u and
+|Γ′| = v for (u, v) ∈ N × N. We call the truncated (influenced) MMAF the vector
+Z(ψ)
+Γ′ =
+�
+Z(ψ)
+tj1 (xj1), . . . , Z(ψ)
+tjv (xjv)
+�⊤
+,
+where ψ := ψ(r) > 0. In particular, for all a ∈ {1, . . . , u} and a b ∈ {1, . . . , v} , ψ has to be chosen such
+that it exists a set Bψ
+tjb (xjb) with the following properties.
+• |Bψ
+tjb (xjb)| → ∞ as r → ∞ for all b, and
+• Iia = H × Atia (xia) and Ijb = H × Bψ
+tjb (xjb) are disjoint sets or intersect on a set H × O, where
+O ∈ R1+d and dim(O) < d + 1, for all a and b
+Let us now assume that it is possible to construct the sets Bψ
+tjb (xjb). Then, since π×λ1+d(H×O) = 0
+and by the definition of a L´evy basis, it follows that
+Ztia (xia) =
+�
+H
+�
+Ati(xi)
+f(A, ti − s, xi − ξ)Λ(dA, ds, dξ) and
+Z(ψ)
+tjb (xjb) =
+�
+H
+�
+Bψ
+tjb (xjb)
+f(A, tj − s, xj − ξ)Λ(dA, ds, dξ),
+and ZΓ and Z(ψ)
+Γ′
+are independent. Hence, for F ∈ G∗
+u and G ∈ Gv, F(ZΓ) and G(Z(ψ)
+Γ′ ) are also
+41
+
+independent. Now
+|Cov(F(ZΓ), G(ZΓ′))|
+≤ |Cov(F(ZΓ), G(Z(ψ)
+Γ′ ))| + |Cov(F(ZΓ), G(ZΓ′) − G(Z(ψ)
+Γ′ ))|
+= |E[(G(ZΓ′) − G(Z(ψ)
+Γ′ ))F(ZΓ)] − E[G(ZΓ′) − G(Z(ψ)
+Γ′ )]E[F(ZΓ)]|
+≤ 2∥F∥∞E[|G(ZΓ′) − G(Z(ψ)
+Γ′ )|] ≤ 2Lip(G)∥F∥∞
+v
+�
+l=1
+E[|Ztjl (xjl) − Z(ψ)
+tjl (xjl)|] =
+= 2Lip(G)∥F∥∞vE[|Ztj1 (xj1) − Z(ψ)
+tj1 (xj1)|],
+(79)
+because an (influenced) MMAF is a stationary random field. To show that a field satisfy Definition 2.16, is
+then enough to prove that E[|Ztj1 (xj1)−Z(ψ)
+tj1 (xj1)|] in the above inequality converges to zero as r → ∞.
+The proofs of Proposition 2.17 and 2.19 differ in the definition of the sequence ψ and the sets Bψ
+tjb (xjb).
+Proof of Proposition 2.17. In this proof, we assume that Bψ
+tjb (xjb) = At(x)\V ψ
+(t,x), where ψ = ψ(r) :=
+r
+√
+(d+1)(c2+1).
+(i) Using the translation invariance of At(x) and V (ψ)
+(t,x) we obtain
+E[|Ztj1 (xj1) − Z(ψ)
+tj1 (xj1)|] ≤
+��
+H
+�
+A0(0)∩V ψ
+0
+V ar(Λ′)f(A, −s, −ξ)2dξdsπ(dA)
+� 1
+2
+=
+��
+H
+�
+−r min(1/c,1)
+√
+(d+1)(c2+1)
+−∞
+V ar(Λ′)
+�
+∥ξ∥≤cs
+f(A, −s, −ξ)2 dξdsπ(dA)
+� 1
+2
+,
+where we have used Proposition 2.8-(ii) to bound the L1-distance from above. Overall, we obtain
+θlex(r) ≤ 2
+��
+H
+�
+−r min(1/c,1)
+√
+(d+1)(c2+1)
+−∞
+V ar(Λ′)
+�
+∥ξ∥≤cs
+f(A, −s, −ξ)2dξdsπ(dA)
+� 1
+2
+,
+which converges to zero as r tends to infinity by applying the dominated convergence theorem.
+(ii) By applying Proposition 2.8-(i) and (ii), we obtain
+E[|Ztj1 (xj1) − Z(ψ)
+tj1 (xj1)|]
+≤
+� �
+H
+�
+−r min(1/c,1)
+√
+(d+1)(c2+1)
+−∞
+V ar(Λ′)
+�
+∥ξ∥≤cs
+f(A, −s, −ξ)2dξdsπ(dA)
++
+��
+H
+�
+−r min(1/c,1)
+√
+(d+1)(c2+1)
+−∞
+E(Λ′)
+�
+∥ξ∥≤cs
+f(A, −s, −ξ)dξdsπ(dA)
+�2 � 1
+2
+.
+Finally, we proceed similarly to proof (i) and obtain the desired bound.
+42
+
+(iii) We apply now Proposition 2.8-(iii). Then,
+E[|Ztj1 (xj1) − Z(ψ)
+tj1 (xj1)|]
+≤
+� �
+S
+�
+−r min(1/c,1)
+√
+(d+1)(c2+1)
+−∞
+�
+∥ξ∥≤cs
+|f(A, −s, −ξ)γ0|dξdsπ(dA)
++
+�
+H
+�
+−r min(1/c,1)
+√
+(d+1)(c2+1)
+−∞
+�
+∥ξ∥≤cs
+�
+R
+|f(A, −s, −ξ)y|ν(dy)dξdsπ(dA)
+�
+.
+The bound for the θ-lex-coefficients is obtained following the proof line in (i).
+Proof of Proposition 2.19.
+(i) W.l.o.g., let us determine the truncated set when (tjb, xjb) = (0, 0). We
+use, to this end, two auxiliary ambit sets translated by a value ψ > 0 along the spatial axis, namely,
+the cones A0(ψ) and A0(−ψ), as illustrated in Figure 5-(a),(c) for c ≤ 1 and in Figure 5-(b),(d)
+for c > 1. Then, we set the truncated integration set to Bψ
+0 (0) = A0(0)\(A0(ψ) ∪ A0(−ψ)). Since
+(tia, xia) ∈ V r
+(0,0), it is sufficient to choose ψ such that the integration set of Z(ψ)
+0
+(0) is a subset of
+(V r
+(0,0))c. To this end, the three intersecting points ( −ψ
+2c , −ψ
+2 ), ( −ψ
+c , 0) and ( −ψ
+2c , ψ
+2 ) have to be inside
+the set (V r
+(0,0))c, as illustrated in Figure 5-(e) for c ≤ 1 and in Figure 5-(f) for c > 1. Clearly, this
+leads to the conditions ψ ≤ rc, ψ ≤ 2r and ψ ≤ 2rc, which are satisfied for ψ = r min(2, c). Hence,
+by using Proposition 2.8-(ii), we have that
+θlex(r) ≤ 2V ar(Λ′)1/2
+�� ∞
+0
+�
+A0(0)∩(A0(ψ)∪A0(−ψ))
+f(A, −s)2dsdξπ(dλ)
+�1/2
+= 2V ar(Λ′)1/2
+� � ∞
+0
+�
+A0(0)∩A0(ψ)
+f(A, −s)2dsdξπ(dλ) +
+� ∞
+0
+�
+A0(0)∩A0(−ψ)
+f(A, −s)2dsdξπ(dλ)
+−
+� ∞
+0
+�
+A0(0)∩A0(ψ)∩A0(−ψ)
+f(A, −s)2dsdξπ(dλ)
+�1/2
+≤ 2
+�
+2Cov(Z0(0), Z0(r min(2, c))).
+which converges to zero as r → ∞ for the dominated convergence theorem.
+(ii) In this proof, we indicate the spatial components and write xia = (yia, zia) ∈ R × R and xjb =
+(yjb, zjb) ∈ R×R for a �� {1, . . . , u} and b ∈ {1, . . . , v}. W.l.o.g., let us then determine the truncated
+set when (tjb, yjb, zjb) = (0, 0, 0). To this end, we use four additional ambit sets that are translated
+by a value ψ > 0 along both spatial axis, namely, the cones A0(ψ, ψ), A0(ψ, −ψ), A0(−ψ, ψ) and
+A0(−ψ, −ψ), as illustrated in Figure 7-(c) for c ≤ 1 and in Figure 7-(d) for c > 1). Then, we set the
+truncated integration set to Bψ
+0 (0) = A0(0, 0)\(A0(ψ, ψ) ∪ A0(ψ, −ψ) ∪ A0(−ψ, ψ) ∪ A0(−ψ, −ψ)).
+Since (tia, yia, zia) ∈ V r
+(0,0,0), it is sufficient to choose ψ such that the integration set of Z(ψ)
+0
+(0, 0)
+is a subset of (V r
+(0,0,0))c, i.e.
+sup
+b∈Bψ
+0 (0)
+∥b∥∞ ≤ r.
+(80)
+43
+
+(a) Integration set A0(0) for c = 1/
+√
+2 together with the
+complement of V r
+(0,0) for r = 3.
+(b) Integration set A0(0) for c = 2
+√
+2 together with the
+complement of V r
+(0,0) for r = 3.
+(c) Integration set A0(0) together with A0(ψ) and A0(−ψ)
+for ψ = r min(2, c).
+(d) Integration set A0(0) together with A0(ψ) and A0(−ψ)
+for ψ = r min(2, c).
+(e) Integration set of Z(ψ)
+0
+(0) together with the complement
+of V r
+(0,0).
+(f) Integration set of Z(ψ)
+0
+(0) together with the complement
+of V r
+(0,0).
+Figure 5: Integration set and truncated integration set of an MMAF Zt(x), exemplary for (t, x) = (0, 0).
+44
+
+6
+ = rmin(2,c) = 3/V2
+Ao(2)
+4
+(0, 2b)
+2
+0
+-2
+(0,一)
+-4
+Ao(2b)
+-6
+1
+-6
+-5
+-4 
+-3 
+-2 
+-1
+0
+1(0)K
+Ao(cb)
+6
+4
+2
+ob = rmin(2, c)
+=6
+-2
+-4
+Ao(-)
+(0, -山
+-6
+-6
+-5
+-4
+-3
+-2
+-1
+0Ao(O)(Ao() U Ao()
+6
+2c
+2
+0
+-2
+-4
+2c
+2
+-6
+-6
+-5
+-4
+-3
+-2
+-1
+0
+16
+D
+2c
+2
+4
+2
+Ao(O)(Ao()
+UAo(-))
+-2
+-4
+C
+2c
+-6
+2
+-6
+-5
+-4
+-3
+-2
+-1
+06
+c= 1/V2
+4
+r=3
+Ao(0)
+2
+(0,0)
+0
+((00)A)
+-2
+-4
+-6
+-6
+-5
+-4
+-3
+-2
+-1
+0
+16
+Ao(0)
+4
+3
+2
+(0,0)
+0
+-2
+-4
+(V(0,0))c
+-6
+-6
+-5
+-4
+-3
+-2
+0In the following we prove that the choice ψ = r min(1, c/
+√
+2) is sufficient for (80) to hold. We
+investigate cross sections of the truncated integration set Bψ
+0 (0) along the time axis. For a fixed
+time point t, we call this cross-section Bt and, similarly, we denote the cross-section of an ambit
+set by At. Note that the cross sections of our ambit sets along the time axis are circles with radius
+|ct| (see also Figure 6).
+t ∈
+�
+−ψ
+√
+2c, 0
+�
+: As the distance between the center of the circle At
+0(0, 0) and the centers of the circles At
+0(ψ, ψ),
+At
+0(−ψ, ψ), At
+0(ψ, −ψ), At
+0(−ψ, −ψ) is
+√
+2ψ, respectively, the set At
+0(0, 0) (which is a circle with
+radius |ct|) is disjoint from every of the additional ambit sets’ cross-sections at t and hence
+Bt = At
+0(0, 0) (see Figure 6-(a)). Clearly, we obtain
+sup
+t∈
+�
+−ψ/(
+√
+2c),0
+� sup
+b∈Bt∥b∥ =
+sup
+t∈
+�
+−ψ/(
+√
+2c),0
+� max(c|t|, |t|) = max
+� ψ
+√
+2,
+ψ
+√
+2c
+�
+.
+(81)
+t ∈
+�
+−ψ
+c , −ψ
+√
+2c
+�
+: For such t the set At
+0(0, 0) intersects with every additional ambit sets’ cross-section (see Figure
+6-(b)). However, as the additional ambit sets’ cross-sections do not intersect with each other,
+the point p1(t) = (t, c|t|, 0) ∈ Bt on the boundary of At
+0(0, 0) (see the red point in Figure 6-(b))
+is not excluded from Bt by any additional ambit set. Note that symmetry makes it sufficient
+to look at p1(t). Hence, we obtain
+sup
+t∈
+�
+−ψ/c,−ψ/(
+√
+2c)
+� sup
+b∈Bt∥b∥ =
+sup
+t∈
+�
+−ψ/c,−ψ/(
+√
+2c)
+�∥p1(t)∥ = max
+�
+ψ, ψ
+c
+�
+.
+(82)
+t ∈
+�
+−
+√
+2ψ
+c
+, −ψ
+c
+�
+: For such t the set At
+0(0, 0) intersects with every additional ambit sets’ cross-section. Such
+intersection additionally restrict At
+0(0, 0) (see Figure 6-(c)). Straightforward calculations show
+that the point where the boundaries of At
+0(−ψ, ψ) and At
+0(−ψ, ψ) as well as the set At
+0(0, 0)
+intersect, say p2(t), is given by (t, 0, ψ −
+�
+(ct)2 − ψ2) (see red point in Figure 6-(c)). Note
+that symmetry makes it sufficient to look at p2(t). We obtain
+sup
+t∈
+�
+−
+√
+2ψ/c,−ψ/c
+� sup
+b∈Bt∥b∥ =
+sup
+t∈
+�
+−
+√
+2ψ/c,−ψ/c
+�∥p2(t)∥ = max
+�
+ψ,
+√
+2ψ
+c
+�
+.
+(83)
+t ≤ −
+√
+2ψ
+c
+: In the following, we show that for such t, the set At(0, 0) is entirely included in the union of the
+additional ambit sets’ cross sections. Clearly, this is true if the upper point where the bound-
+aries of At
+0(−ψ, ψ) and At
+0(−ψ, ψ) intersect, say p3(t), is outside of At
+0(0, 0) (see the red point
+in Figure 6-(d)). Note that symmetry makes it sufficient to look at p3(t). As straighforward
+calculations show that p3(t) = (t, 0, ψ +
+�
+(ct)2 − ψ2), this is true if ψ +
+�
+(ct)2 − ψ2 ≥ c|t|,
+or equivalently (ψ +
+�
+(ct)2 − ψ2)2 ≥ (ct)2. Moreover, we have
+ψ2 + 2ψ
+�
+(ct)2 − ψ2 + (ct)2 − ψ2 ≥ (ct)2 ⇐⇒ ψ ≥ 0.
+(84)
+In view of condition (80) we combine (81), (82) and (83) and set ψ = r min(1, c/
+√
+2), which also
+satisfies (84).
+45
+
+In addition to the cross sectional views from Figure 6, we give a full three-dimensional view of the
+set Bψ
+0 (0) for c ≤ 1 in Figure 7-(e) and for c > 1 in Figure 7-(f) that highlight the points on the
+boundary of Bψ
+0 (0) with maximal ∞-norm for ψ = r min(1, c/
+√
+2).
+Therefore, because of Proposition 2.8-(ii), we can conclude that
+θlex(r) ≤ 2V ar(Λ′)1/2
+�� ∞
+0
+�
+A0(0,0)∩(A0(ψ,ψ)∪A0(−ψ,ψ)∪A0(ψ,−ψ)∪A0(−ψ,−ψ))
+f(A, −s)2dsdξπ(dλ)
+�1/2
+= 2V ar(Λ′)1/2
+� � ∞
+0
+�
+A0(0,0)∩A0(ψ,ψ)
+f(A, −s)2dsdξπ(dλ)
++
+� ∞
+0
+�
+A0(0,0)∩A0(−ψ,ψ)
+f(A, −s)2dsdξπ(dλ)
++
+� ∞
+0
+�
+A0(0,0)∩A0(ψ,−ψ)
+f(A, −s)2dsdξπ(dλ)
++
+� ∞
+0
+�
+A0(0,0)∩A0(−ψ,−ψ)
+f(A, −s)2dsdξπ(dλ)
+−
+� ∞
+0
+�
+A0(0)∩A0(ψ,−ψ)∩A0(−ψ,−ψ)
+f(A, −s)2dsdξπ(dλ)
+−
+� ∞
+0
+�
+A0(0)∩A0(−ψ,ψ)∩(A0(ψ,−ψ)∪A0(−ψ,−ψ))
+f(A, −s)2dsdξπ(dλ)
+−
+� ∞
+0
+�
+A0(0)∩A0(ψ,ψ)∩(A0(−ψ,ψ)∪A0(ψ,−ψ)∪A0(−ψ,−ψ))
+f(A, −s)2dsdξπ(dλ)
+�1/2
+≤ 2
+�
+2Cov(Z0(0, 0), Z0(ψ, ψ)) + 2Cov(Z0(0, 0), Z0(ψ, −ψ)).
+which converges to zero as r → ∞ for the dominated convergence theorem.
+B.2
+Proofs of Section 3
+Proof of Proposition 3.4. We drop the bold notations indicating random fields and stochastic processes
+in the following. Let h ∈ H, we call Li = L(h(Xi), Yi) for i ∈ Z, Z(M)
+t
+(x) := Zt(x) ∧ M for M > 1, and
+L(M) = (L(h(X(M)
+i
+), Y (M)
+i
+))i∈Z where
+X(M)
+i
+= L−(M)
+p
+(t0 + ia, x∗),
+and
+Y (M)
+i
+= Z(M)
+t0+ia(x∗), for i ∈ Z,
+where
+L−(M)
+p
+(t, x) = {Z(M)
+s
+(ξ) : (s, ξ) ∈ Z × L, ∥x − ξ∥ ≤ c (t − s) and t − s ≤ p}.
+For u ∈ N, i1 ≤ i2 ≤ . . . ≤ iu ≤ iu + k = j with k ∈ N, let us consider the marginal of the field
+�
+(Xi1, Yi1), . . . , (Xiu, Yiu), (Xj, Yj)
+�
+,
+(85)
+46
+
+(a) Cross sections of the auxiliary
+ambit sets and At
+0(0, 0) for t = −1
+and c = 2.
+(b) Cross sections of the auxiliary
+ambit sets and At
+0(0, 0) for t = −5/4
+and c = 2.
+(c) Cross sections of the auxiliary
+ambit sets and At
+0(0, 0) for t = −7/4
+and c = 2.
+(d) Cross sections of the auxiliary
+ambit sets and At
+0(0, 0) for t = −2.2
+and c = 2.
+Figure 6: Cross sections of the auxiliary ambit sets and At
+0(0, 0) at different time points for ψ = 3.
+and let us define
+Γ = {(ti, xi) ∈ Z1+d: Zti(xi) ∈ L−
+p (t0 + isa, x∗) or (ti, xi) = (t0 + isa, x∗) for s = 1, . . . , u},
+and
+Γ′ = {(ti, xi) ∈ Z1+d: Zti(xi) ∈ L−
+p (t0 + ja, x∗) or (ti, xi) = (t0 + isa, x∗)}.
+Then r = dist(Γ, Γ′). In particular Γ ⊂ V r
+Γ′, and r = (j − iu)a − p. For F ∈ G∗
+u and G ∈ G1, then
+|Cov(F(Li1, . . . , Liu), G(Lj))|
+(86)
+≤|Cov(F(Li1, . . . , Liu), G(Lj) − G(L(M)
+j
+))|
+(87)
++ |Cov(F(Li1, . . . , Liu), G(L(M)
+j
+))|.
+(88)
+The summand (87) is less than or equal to
+2∥F∥∞Lip(G)E[|Lj − L(M)
+j
+|] ≤ 2∥F∥∞Lip(G)(E[|Yj − Y (M)
+j
+|] + E[|h(Xj) − h(X(M)
+j
+)|])
+≤ 2∥F∥∞Lip(G)(Lip(h)a(p, c) + 1)E[|Zt1(x1) − Z(M)
+t1
+(x1)|]
+by stationarity of the field Z, and because L and h are Lipschitz functions. Moreover, the function G(L(M)
+j
+)
+47
+
+A(, )
+At(, b)
+6
+4
+2
+A(0, 0)
+2
+0
+-2
+-4
+t=-1
+-6
+A(,)
+A(,-)
+-8
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+yA(, )
+At(,)
+6
+4
+2
+Pi(t)
+—A(O, 0)
+-2 
+-4 
+t = -5/4
+-6
+A(，)
+A(,)
+-8 
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+yA(, )
+At(, b)
+p2(t)
+6
+4
+2
+A(0, 0)
+2
+0
+-2
+-4
+t = -7/4
+9-
+A(,)
+A(,一)
+-8
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+y8 
+P3(t)
+6
+A(,)
+A(,)
+4
+2
+At (O, 0
+0
+-2 
+-4 
+A(,)
+A(,)
+-6
+t = -2.2
+-8
+-8
+-6
+-4
+-2
+0
+2
+4
+6
+8
+y(a) Integration set A0(0, 0) for c = 1/
+√
+2 together with the
+complement of V r
+(0,0,0) for r = 3.
+(b) Integration set A0(0, 0) for c = 2 together with the
+complement of V r
+(0,0,0) for r = 3.
+(c) Integration
+set
+A0(0, 0)
+together
+with
+A0(ψ, ψ),
+A0(−ψ, ψ),
+A0(ψ, −ψ)
+and
+A0(−ψ, −ψ)
+for
+ψ
+=
+r min(1, c/
+√
+2).
+(d) Integration
+set
+A0(0, 0)
+together
+with
+A0(ψ, ψ),
+A0(−ψ, ψ),
+A0(ψ, −ψ)
+and
+A0(−ψ, −ψ)
+for
+ψ
+=
+r min(1, c/
+√
+2).
+(e) B is the integration set of Z(ψ)
+0
+(0, 0). In addition, we
+illustrate A0(−ψ, −ψ) for c = 1/
+√
+2 together with the com-
+plement of V r
+(0,0,0) for r = 3.
+(f) B is the integration set of Z(ψ)
+0
+(0, 0). In addition, we
+illustrate A0(−ψ, −ψ) for c = 2 together with the comple-
+ment of V r
+(0,0,0) for r = 3.
+Figure 7: Integration set and truncated integration set of an MMAF Zt(y, z) for d = 2.
+48
+
+8-
+9
+(0, 0,0)
+4
+r=3
+2
+At(0, 0)
+2
+0
+-2
+-4
+(000A)
+9-
+-8
+c = 1/V2
+5
+0
+0
+-1
+-5
+-2
+-3
+y
+-4
+t8
+Ao(0, 0)
+6
+(0, 0,0)
+4
+r=3
+0
+2
+-2
+-4 
+(V(o,0,0)
+9-
+-8
+5
+C
+0
+0
+-1
+-5
+-2
+-3
+y
+-4
+t8
+(0,-,)
+9
+(0, b, )
+=rmin
+4
+3/2
+2
+0
+-2
+-4
+9-
+-8
+(0, 4,-)
+5
+0,-山二山)
+0
+0
+-1
+-5
+-2
+-3
+y
+-4
+t(0,一, )
+(0,,
+8
+6
+b
+= rmin
+4
+3
+0
+2
+-2
+-4
+9-
+-8
+0一一山
+5
+(0,,-山)
+0
+0
+-1
+-5
+-2
+-3
+y
+-4
+t,0,4
+0,0
+4
+3
+B
+1
+2
+.0
+-1
+-2
+-3
+Ao(一,一山
+4
+-4
+(V0,0,0))℃
+2
+0
+-3
+-2
+-2.5
+-2
+-1.5
+-1
+0.5
+-4
+0
+0.5
+y
+t于,0,4
+b.
+0
+(V(0.0,0)°
+4
+0
+3
+B
+1
+2
+.0
+-1
+-2
+-3
+Ao(-,一山)
+-4
+2
+0
+-3
+-2.5
+-2
+-2
+-1.5
+-1
+-0.5
+-4
+0
+0.5
+y
+tbelongs to Ga(p,c)+1. Let (X, Y ), (X′, Y ′) ∈ Ra(p,c)+1, then
+|G(L(h(X(M)), Y (M))) − G(L(h(X′(M)), Y ′(M)))| ≤ Lip(G)|L(h(X(M)), Y (M)) − L(h(X′(M)), Y ′(M))|
+≤ Lip(G)(|h(X(M))) − h(X′(M))| + |Y (M) − Y ′(M)|
+≤ Lip(G)(Lip(h) + 1)(∥X − X′∥1 + |Y − Y ′|),
+and Lip(G(L(M)
+j
+)) ≤ Lip(G)(Lip(h) + 1).
+Because Z is a θ-lex weakly dependent random field, (88) is less than or equal to
+˜d∥F∥∞Lip(G)(Lip(h) + 1)θlex(r).
+We choose now M = r and obtain that (86) is less than or equal than
+∥F∥∞Lip(G) ˜d(Lip(h)a(p, c) + 1)
+�2
+˜d
+E[|Zt1(x1) − Z(r)
+t1 (x1)|] + θlex(r)
+�
+.
+The quantity above converges to zero as r → ∞. Therefore, (Li)i∈Z is a θ-weakly dependent process.
+Proof of Proposition 3.7. We drop the bold notations indicating random fields and stochastic processes in
+the following. We call Li = L(hβ(Xi), Yi) for i ∈ Z, as defined in Proposition 2.17, and β ∈ B. Moreover
+we will use the notation Z(ψ)
+t
+(x) to indicate a truncated of the field Z and L(ψ) = (L(hβ(X(ψ)
+i
+), Y (ψ)
+i
+))i∈Z
+where
+X(ψ)
+i
+= L−(ψ)
+p
+(t0 + ia, x∗),
+and
+Y (ψ)
+i
+= Z(ψ)
+t0+ia(x∗), for i ∈ Z,
+where
+L−(ψ)
+p
+(t, x) = {Z(ψ)
+s
+(ξ) : (s, ξ) ∈ Z × L, ∥x − ξ∥ ≤ c (t − s) and t − s ≤ p}.
+For u ∈ N, i1 ≤ i2 ≤ . . . ≤ iu ≤ iu + k = j with k ∈ N, let us consider the marginal of the field
+�
+(Xi1, Yi1), . . . , (Xiu, Yiu), (Xj, Yj)
+�
+,
+(89)
+and let us define
+Γ = {(ti, xi) ∈ Z1+d: Zti(xi) ∈ L−
+p (t0 + isa, x∗) or (ti, xi) = (t0 + isa, x∗) for s = 1, . . . , u},
+and
+Γ′ = {(ti, xi) ∈ Z1+d: Zti(xi) ∈ L−
+p (t0 + ja, x∗) or (ti, xi) = (t0 + isa, x∗)}.
+Then r = dist(Γ, Γ′). In particular Γ ⊂ V r
+Γ′, and r = (j − iu)a − p. For F ∈ G∗
+u and G ∈ G1, then
+|Cov(F(Li1, . . . , Liu), G(Lj))|
+≤|Cov(F(Li1, . . . , Liu), G(Lj) − G(L(ψ)
+j
+))|
+(90)
++ |Cov(F(Li1, . . . , Liu), G(L(ψ)
+j
+))|.
+(91)
+The summand (91) is equal to zero because Γ ⊂ V r
+Γ′, see proof of Proposition 2.19 for more details about
+49
+
+this part of the proof. We can then bound (90) from above by
+2∥F∥∞Lip(G)E[|Lj − L(ψ)
+j
+|] ≤ 2∥F∥∞Lip(G)(E[|Yj − Y (ψ)
+j
+|] + E[|h(Xj) − h(X(ψ)
+j
+)|])
+(92)
+≤ 2∥F∥∞Lip(G)(Lip(h)a(p, c) + 1)E[|Zt1(x1) − Z(ψ)
+t1 (x1)|]
+(93)
+where (92) holds because L is a function with Lipschitz constant equal to one, and (93) holds given that
+h ∈ L(Ra(p,c)).
+In this case, we work with linear functions,
+E[|hβ(Xj) − hβ(X(ψ)
+j
+)∥] = E
+����
+a(p,c)
+�
+l=1
+β1,l(Ztl(xl) − Z(ψ)
+tl (xl))
+���
+�
+.
+(94)
+By stationarity of the field Z, we have that (94) is smaller or equal than ∥β1∥1E[|Zt1(x1) − Z(ψ)
+t1 (x1)|].
+On overall, we have that (90) is smaller or equal than
+2∥F∥∞Lip(G)(Lip(h)a(p, c) + 1)E[|Zt1(x1) − Z(ψ)
+t1 (x1)|] for h a Lipschitz function,
+or it is smaller or equal than
+2∥F∥∞Lip(G)(∥β1∥1 + 1)E[|Zt1(x1) − Z(ψ)
+t1 (x1)|] for hβ a linear function.
+Because of the properties of the truncated field Z(ψ)
+t1 (x1), we have that the above bounds converge to
+zero as r → ∞. Therefore, L is a θ-weakly dependent process.
+The proof of Proposition 3.8 is based on a blocks technique used in the papers [34] and [44]. Such
+results are based on the use of several lemmas. To ease the complete understanding of the proof of
+Proposition 3.8, we prove these Lemmas below, given that they undergo several modifications in our
+framework. Let us start by partitioning a set {1, 2, . . . , m} into k blocks. Each block will contain l = ⌊ m
+k ⌋
+terms. Let h = m − k l < k denote the remainder when we divide m by k. We now construct k blocks
+such that the number of elements in the jth-block is defined by
+¯lj =
+�
+l + 1
+if j = 1, 2, . . . , h
+l
+if j = h + 1, . . . , k
+.
+Let (U i)i∈Z a stationary process, and V m = �m
+i=1 U i, for j = 1, . . . , k we define the j-th block as
+V j,m = U j + U j+k + . . . U j+(¯lj−1) k =
+¯lj
+�
+i=1
+U j+(i−1) k
+such that
+V m =
+k
+�
+j=1
+V j,m =
+k
+�
+j=1
+¯lj
+�
+i=1
+U j+(i−1) k.
+For j = 1, 2, . . . , k, let us define pj =
+¯lj
+m. It follows that �k
+j=1 pj = 1
+m
+�k
+j=1 ¯lj = 1.
+50
+
+Lemma B.1. For all s ∈ R
+E
+�
+exp
+�
+sV m
+m
+��
+≤
+k
+�
+j=1
+pjE
+�
+exp
+�
+sV j,m
+¯lj
+��
+The proof of the result above is due to Hoeffding [35].
+Lemma B.2. Let the assumptions of Proposition 3.8 hold and define the process (U i)i∈Z such that
+U i := f(Xi) − E[f(Xi)]. For all j = 1, 2, . . . , k, l ≥ 2 and 0 < s <
+3l
+|b−a|
+M¯lj(s) =
+���E
+� ¯lj
+�
+i=1
+exp
+�s U j+(i−1) k
+¯lj
+���
+−
+¯lj
+�
+i=1
+E
+�
+exp
+�s U j+(i−1) k
+¯lj
+����� ≤ exp
+�
+l − 1 + s|b − a|
+�
+θ(k)s
+l
+(95)
+The same result holds when defining the process (U i)i∈Z for U i = E[f(Xi)] − f(Xi).
+Proof. Let us first discuss the case when U i = f(Xi) − E[f(Xi)], we have that the process U has mean
+zero and |U i| ≤ |b − a|. Let us define Fj = σ(U i, i ≤ j).
+M¯lj :=
+�����E
+� ¯lj
+�
+i=1
+exp
+�sU j+(i−1)k
+¯lj
+��
+−
+¯lj
+�
+i=1
+E
+�
+exp
+�sU j+(i−1)k
+¯lj
+�������
+=
+�����E
+� ¯lj−1
+�
+i=1
+exp
+�sU j+(i−1)k
+¯lj
+�
+E
+�
+exp
+�sU j+(¯lj−1)k
+¯lj
+����Fj+(¯lj−2)k
+��
+−
+¯lj
+�
+i=1
+E
+�
+exp
+�sU j+(i−1)k
+¯lj
+�������
+≤
+�����E
+� ¯lj−1
+�
+i=1
+exp
+�sU j+(i−1)k
+¯lj
+��
+E
+�
+exp
+�sU j+(¯lj−1)k
+¯lj
+����Fj+(¯lj−2)k
+�
+− E
+�
+exp
+�sU j+(¯lj−1)k
+¯lj
+���������
++
+�����E
+�
+exp
+�sU j+(¯lj−1)k
+¯lj
+�������
+�����E
+� ¯lj−1
+�
+i=1
+exp
+�sU j+(i−1)k
+¯lj
+��
+−
+¯lj−1
+�
+i=1
+E
+�
+exp
+�sU j+(i−1)k
+¯lj
+�������
+≤
+�����
+¯lj−1
+�
+i=1
+exp
+�sU j+(i−1)k
+¯lj
+������
+∞
+E
+����E
+�
+exp
+�sU j+(¯lj−1)k
+¯lj
+����Fj+(¯lj−2)k
+�
+− E
+�
+exp
+�sU j+(¯lj−1)k
+¯lj
+�����
+�
++
+����� exp
+�sU j+(¯lj−1)k
+¯lj
+������
+∞
+M¯lj−1
+The above is then less than or equal to
+exp
+�s(¯lj − 1)|b − a|
+¯lj
+�
+exp
+�−s|a|
+¯lj
+�
+E
+������E
+�
+exp
+�
+sf(Xj+(¯lj−1)k)
+¯lj
+������Fj+(¯lj−2)k
+�
+(96)
+− E
+�
+exp
+�
+sf(Xj+(¯lj−1)k)
+¯lj
+�������
+�
++ exp
+�s|b − a|
+¯lj
+�
+M¯lj−1.
+Note that the function g(x) =
+exp
+�
+sx
+¯lj
+�
+exp
+�
+s|b|
+¯lj
+�
+s
+¯lj
+1A(x) is in L1 for each s, where A = {x : |x| ≤ |b − a|}. We
+51
+
+then use the mixingales-type representation of the θ-coefficients of f(X), and obtain that
+M¯lj ≤ exp
+�s¯lj|b − a|
+¯lj
+�
+θ(k) s
+¯lj
++ exp
+�s|b − a|
+¯lj
+�
+M¯lj−1.
+Let now, u = exp
+�
+s|b−a|
+¯lj
+�
+, we have that
+M¯lj ≤ θ(k)u
+¯lj s
+¯lj
++ uM¯lj−1
+≤ (¯lj − 2)θ(k)u
+¯lj s
+¯lj
++ u
+¯lj−2���E
+�
+exp
+�sU j
+¯lj
+�
+exp
+�sU j+k
+¯lj
+��
+− E
+�
+exp
+�sU j
+¯lj
+��
+E
+�
+exp
+�sU j+k
+¯lj
+�����
+≤ (¯lj − 2)θ(k)u
+¯lj s
+¯lj
++ u
+¯lj−1���E
+�
+exp
+�sU j+k
+¯lj
+�
+|Fj
+�
+− E
+�
+exp
+�sUj+k
+¯lj
+�����
+≤ (¯lj − 1)θ(k)u
+¯lj s
+¯lj
+≤ exp
+�
+¯lj − 2 + s|b − a|
+�
+θ(k) s
+¯lj
+.
+By assumption ¯lj ≥ 2. Moreover, we apply the following estimation in the last inequality: for all x > 0,
+we have that log(x) ≤ x−1. In conclusion, for all j = 1, . . . , k (and remembering that ¯lj = l or ¯lj = l +1)
+M¯lj ≤ exp(l − 1 + s |b − a|)θ(k)s
+l .
+Similar calculations apply when U i = E[f(Xi)] − f(Xi).
+Remark B.3. Note that by showing Lemma B.2 for U i = E[f(Xi)] − f(Xi) there is a slight change in
+the proof at point (96). However, in the end, the result (95) equal holds.
+Lemma B.4. Let the Assumptions of Proposition 3.8 hold and define the process (U i)i∈Z such that
+U i = f(Xi) − E[f(Xi)]. For all j = 1, 2, . . . , k, l ≥ 2 and 0 < s <
+3l
+|b−a|
+E
+�
+exp
+�sV j,m
+¯lj
+��
+≤ exp
+�
+s2E[U 2
+1]
+2l
+�
+1 − s|b−a|
+3l
+�
+�
++ s
+l exp(l − 1 + s|b − a|)θ(k)
+The same result holds when defining the process (U i)i∈Z for U i = E[f(Xi)] − f(Xi).
+Proof.
+E
+�
+exp
+�sV j,m
+¯lj
+��
+= E
+�
+exp
+�
+¯lj
+�
+i=1
+sU j+(i−1) k
+¯lj
+��
+≤
+¯lj
+�
+i=1
+E
+�
+exp
+�sU j+(i−1) k
+¯lj
+��
++
+���E
+� ¯lj
+�
+i=1
+exp
+�sU j+(i−1) k
+¯lj
+��
+−
+¯lj
+�
+i=1
+E
+�
+exp
+�sU j+(i−1) k
+¯lj
+�����
+= E
+�
+exp
+�sU j+(i−1) k
+¯lj
+��¯lj
++ M¯lj
+(97)
+We have that E[U j+(i−1) k] = 0 by definition of the process U, and
+U j+(i−1) k
+¯lj
+satisfies the Bernstein
+52
+
+moment condition (Remark A1 [44]) with K1 = |b−a|
+3¯lj . Hence, for ¯lj ≥ 2 and 0 < s <
+3¯lj
+|b−a|
+E
+�
+exp
+�sU j+(i−1) k
+¯lj
+��
+≤ exp
+�
+s2E[(U j+(i−1)k/¯lj)2]
+2
+�
+1 − s|b−a|
+3¯lj
+�
+�
+.
+(98)
+Because ¯lj ≥ l, we can conclude that the inequality above holds for l ≥ 2. Moreover, by stationarity of
+the process U and since for all j = 1, 2, . . . , k, we can bound (97) uniformly with respect to the index j
+by using Lemma B.2, and noticing that
+0 < s <
+3l
+|b − a| ≤
+3¯lj
+|b − a|,
+and then
+�
+1 − s|b − a|
+3¯lj
+�
+≥
+�
+1 − s|b − a|
+3l
+�
+.
+The same proof applies when defining U i = E[f(Xi)] − f(Xi).
+Proof of Proposition 3.8. By combining Lemmas B.1, B.2, B.4, we can show the bound for the Laplace
+transform of the process U := f(X) − E[f(X)] for 0 < s <
+3l
+|b−a| is given by:
+E
+�
+exp
+�
+s 1
+m
+m
+�
+i=1
+f(Xi) − E[f(Xi)]
+��
+= E
+�
+exp
+�
+s 1
+m
+m
+�
+i=1
+U i
+��
+= E
+�
+exp
+� s
+m
+k
+�
+j=1
+V j,m
+��
+= E
+�
+exp
+� s
+m
+k
+�
+j=1
+¯lj
+�
+i=1
+U j+(i−1) k
+��
+≤
+k
+�
+j=1
+pjE
+�
+exp
+�sV j,m
+¯lj
+��
+(99)
+≤ exp
+�
+s2V ar(f(X1))
+2l
+�
+1 − s|b−a|
+3l
+�
+�
++ s
+l exp(l − 1 + s|b − a|)θ(k),
+(100)
+where V j,m = �¯lj
+i=1 U j+(i−1) k. The inequalities (99) and (100) hold because of Lemma B.1 and Lemma
+B.4, respectively. We have then proved the inequality (42). The same proof applies for showing the bound
+(43) by defining U i = E[f(Xi)] − f(Xi) .
+Proof of Proposition 3.10. Let us choose f(s) =
+s
+ϵ for 0 < s < ϵ , which satisfies the assumptions of
+Proposition 3.8 and has support in [0, 1]. Note, that ∆(β) =
+ϵ
+m(�m
+i=1 E[f(Lϵ
+i)] − f(Lϵ
+i)). We have in this
+case that U i = E[f(Li)] − f(Lϵ
+i) such that E[U i] = 0 and |U i| ≤ 1. Equally, ∆′(β) = rϵ(β) − Rϵ(β) =
+ϵ
+m(�m
+i=1 f(Lϵ
+i) − E[f(Lϵ
+i)]), and again by defining U ′
+i = f(Lϵ
+i) − E[f(Li
+ϵ)] has E[U ′
+i] = 0 and |U ′
+i| ≤ 1.
+Note that the process f(Lϵ
+i)i∈Z has the same θ-weak coefficients of the process (Lϵ
+i)i∈Z because f is a
+Lipschitz function. We follow below the notations of Table 1.
+53
+
+(i) By Proposition 3.8 applied for 0 < ϵ
+√
+l < 3
+√
+l, and the bound (41),
+E
+�
+exp
+�√
+l ∆(β)
+��
+= E
+�
+exp
+�
+ϵ
+√
+l 1
+m
+m
+�
+i=1
+U i
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2¯α(∥β1∥1 + 1) exp(l − 1 + 3
+√
+l − λr)
+where the last equality holds because of the particular shape of the chosen function f. Let us
+determine the conditions on the parameter at and k such that exp(l − 1 + 3
+√
+l − λr) = exp(l − 1 +
+3
+√
+l − λ(ka − p)) ≤ 1. Given that l ≥ 9, the inequality above holds if
+2 N
+atk − 1 − λathtk + λp ≤ 0
+The above is equivalent to
+λhta2
+tk2 − (λp − 1)atk − 2N ≥ 0.
+That holds, if
+atk ≥ (λp − 1) +
+�
+(λp − 1)2 + 8λhtN
+2λht
+.
+(ii) As in (i), we have that
+E
+�
+exp
+�√
+l ∆(β)
+��
+= E
+�
+exp
+�
+ϵ
+√
+l 1
+m
+m
+�
+i=1
+U i
+��
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2¯α(∥β1∥1 + 1) exp(l − 1 + 3
+√
+l)r−λ
+≤ exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2¯α(∥β1∥1 + 1) exp(l − 1 + 3
+√
+l − λ log(ak − p))
+We have that exp(l − 1 + 3
+√
+l − λ log(ak − p)) ≤ 1 holds if
+2 N
+atk − 1 − λ log(ak − p) ≤ 0,
+given that l ≥ 9. The latter inequality holds if and only if the parameters at and k are chosen such
+that
+2N − atk
+atk log(ak − p) ≤ λ,
+Proof of Corollary 3.13. We drop the bold notations indicating random fields and stochastic processes
+in this proof. When working with the family of predictors hnet,w for w ∈ B′, we have that the proof of
+Proposition 3.4 changes in the estimation of the bound (92). In particular, for a given w ∈ B′ we have
+that
+E[|h(Xj) − h(X(ψ)
+j
+)|] = E[|
+K
+�
+l=1
+αlσ(β⊤
+l Xj + γl) −
+K
+�
+l=1
+αlσ(β⊤
+l X(ψ)
+j
++ γl)|] ≤
+�
+l
+|αl|∥βl∥1E[|Zt1(x1) − Zψ
+t1(x1)|]
+by the stationarity of the field Z. The proof of the Corollary proceeds then identically as in the proof of
+Proposition 3.10 by modifying the bound (40) with the above calculations.
+54
+
+We remind the reader that the proof of Proposition 3.18 and 3.21 make use of the below Lemma
+that we recall just for completeness.
+Lemma B.5 (Legendre transform of the Kullback-Leibler divergence function). For any π ∈ M1
++(B),
+for any measurable function h : B → R such that π[exp(h)] ≤ ∞, we have that
+π[exp(h)] = exp
+�
+sup
+ρ∈M1
++(B)
+ρ[h] − KL(ρ||π)
+�
+,
+with the convention ∞ − ∞ = −∞. Moreover, as soon as h is upper bounded on the support of π, the
+supremum with respect to ρ in the right-hand side is reached for the Gibbs distribution with Radon-
+Nikodym derivative w.r.t. π equal to
+exp(h)
+π[exp(h)].
+The proof of Lemma (B.5) has been known since the work of Kullback [42] in the case of a finite
+space B, whereas the general case has been proved by Donsker and Varadhan [27]. Given this result, we
+are now ready to prove our PAC Bayesian bounds.
+Proof of Proposition 3.18. We follow a standard proof scheme developed in [12].
+√
+l (ˆρ[Rϵ(β)] − ˆρ[rϵ(β)]) = ˆρ[
+√
+l (Rϵ(β) − rϵ(β))]
+≤ KL(ˆρ||π) + log(π[exp(
+√
+l (Rϵ(β) − rϵ(β)))])
+(P-a.s. by Lemma B.5).
+We have that π[exp(
+√
+l (Rϵ(β) − rϵ(β)))] := AS is a random variable on S. By Markov’s inequality, for
+δ ∈ (0, 1)
+P
+�
+AS ≤ E[AS]
+δ
+�
+≥ 1 − δ.
+This in turn implies that with probability at least 1 − δ over S
+ˆρ[Rϵ(β)] − ˆρ[rϵ(β)] ≤ KL(ˆρ||π) + log 1
+δ
+√
+l
++ 1
+√
+l
+log
+�
+π
+�
+E
+�
+exp
+�√
+l��(β)
+����
+(101)
+≤ KL(ˆρ||π) + log 1
+δ
+√
+l
++ 1
+√
+l
+log
+�
+π
+�
+exp
+�
+3ϵ2
+2(3 − ϵ)
+�
++ 2(∥β1∥1 + 1)¯α
+��
+,
+(102)
+where (101) holds by swapping the expectation over S and over π using Fubini’s Theorem, and (102) is
+obtained by using Proposition 3.10. Similarly, the second inequality can easily be obtained.
+Proof of Proposition 3.21. Let h = −
+√
+lrϵ(β) and d¯ρ
+dπ =
+exp(h)
+π[exp(h)]. By Lemma B.5, we have that
+¯ρ = arg inf
+ˆρ
+�
+K(ˆρ||π) − ˆρ[h]
+�
+= arg inf
+ˆρ
+�K(ˆρ||π)
+√
+l
++ ˆρ[rϵ(β)]
+�
+,
+and by using (58), for all δ ∈ (0, 1)
+P
+�
+¯ρ[Rϵ(β)] ≤ inf
+ˆρ
+�
+ˆρ[rϵ(β)]+
+�
+KL(ˆρ||π)+log
+�1
+δ
+�
++
+3ϵ2
+2(3 − ϵ)
+� 1
+√
+l
++ 1
+√
+l
+log
+�
+π[1+2(∥β1∥1+1)¯α]
+���
+≥ 1−δ
+(103)
+55
+
+A union bound gives that the inequality (103) and the one in (59) hold with probability at least 1 − 2δ.
+By now substituting to ˆρ[rϵ(β)] in (103) its bound obtained in inequality (59), we obtain that
+P
+�
+¯ρ[Rϵ(β)] ≤ inf
+ˆρ
+�
+ˆρ[Rϵ(β)]+
+�
+KL(ˆρ||π)+log
+�1
+δ
+�
++
+3ϵ2
+2(3 − ϵ)
+� 1
+√
+l
++ 1
+√
+l
+log
+�
+π[1+2(∥β1∥1+1)¯α]
+���
+≥ 1−2δ.
+We conclude by replacing δ with δ
+2.
+The modeling selection procedure of the paper discussed in Section 3.3 can be proven by using the
+following two Lemmas
+Lemma B.6. Let 0 < ϵ < 3, and the assumptions of Proposition 3.10 hold. Moreover, let fp be a regular
+conditional probability measure such that fp << ¯ρp and ¯ρp << fp for each possible training data set S.
+Then,
+E
+�
+sup
+fp
+exp
+�
+fp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ) − log dfp
+d¯ρp
+���
+(104)
+≤ E
+�
+sup
+fp
+fp
+�
+exp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ) − log dfp
+d¯ρp
+���
+≤ 1
+Proof. By Proposition 3.10, it holds that
+E[exp(
+√
+l∆(β))] ≤ exp
+�
+3ϵ2
+2(3 − ϵ) + log(1 + 2(∥β1∥1 + 1)¯α)
+�
+.
+which is equivalent to
+E
+�
+exp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ)
+��
+≤ 1.
+(105)
+By using the tower property, for all p = 1, . . . , ⌊ N
+2 ⌋
+E
+�
+exp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ)
+��
+= E
+�
+¯ρp
+�
+exp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ)
+���
+.
+(106)
+Moreover, for any non-negative and measurable function b on Bp, it holds that
+¯ρp[b(β)] =
+�
+b(β) ¯ρp(dβ) ≥
+�
+dfp
+d¯
+ρp (β)>0
+b(β) ¯ρp(dβ) =
+�
+dfp
+d¯
+ρp (β)>0
+b(β) d¯ρp
+dfp
+fp(dβ)
+= fp
+�
+b(β) exp
+�
+− log dfp
+d¯ρp
+��
+.
+(107)
+From (105), (106) and (107) we obtain that
+E
+�
+fp
+�
+exp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ) − log dfp
+d¯ρp
+���
+≤ 1.
+56
+
+By using now the Jensen inequality, we have that
+E
+�
+exp
+�
+fp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ) − log dfp
+d¯ρp
+���
+≤ E
+�
+fp
+�
+exp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ) − log dfp
+d¯ρp
+���
+≤ 1.
+(108)
+For a given p and by taking the supremum on every possible fp distribution, we conclude.
+Lemma B.7. Let the assumptions of Lemma B.6 hold. Moreover, let g be a probability distribution on
+the grid 1, . . . , ⌊ N
+2 ⌋, such that g = �
+p wpδp, where wp =
+1
+⌊ N
+2 ⌋. Then, we have that
+E
+� �
+p
+wp sup
+fp
+exp
+�
+fp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ) − log dfp
+d¯ρp
+���
+≤ 1,
+(109)
+and
+E
+� �
+p
+wp sup
+fp
+fp
+�
+exp
+�√
+l∆(β) − log(1 + 2(∥β1∥1 + 1)¯α) −
+3ϵ2
+2(3 − ϵ) − log dfp
+d¯ρp
+���
+≤ 1.
+(110)
+Proof. Let hp = supfp fp
+�
+exp
+�√
+l∆(β))−log(1+2(∥β1∥1 +1)¯α)−
+3ϵ2
+2(3−ϵ) −log dfp
+d¯ρp
+��
+, and h = �
+p hp1Bp.
+By definition of the probability measure g, and applying Lemma B.6,
+E[g[h]] = E
+� �
+p
+wp h
+�
+≤ 1.
+By substituting the explicit expression of h, we obtain that the inequality (110) holds and consequently
+also (109) again by Lemma B.6.
+Proof of Proposition 3.28. By inequality (109) and the Chernoff bound, we obtain that for all p =
+1, . . . ,
+�
+N
+2
+�
+and fp, and δ ∈ (0, 1), with probability greater than 1 − δ that
+fp[Rϵ(β)] ≤ fp
+�
+rϵ(β)+ log(2∥β1∥1 + 3)
+√
+l
+�
++ 1
+√
+l
+KL(fp||¯ρp)+ 1
+√
+l
+log
+� 1
+wp
+�
++ 1
+√
+l
+�
+log ¯α+
+3ϵ2
+2(3 − ϵ) +log 1
+δ
+�
+.
+For all Gibbs randomized estimator ¯ρp, with probability greater than 1 − δ, it holds that
+¯ρp[Rϵ(β)] ≤ ¯ρp
+�
+rϵ(β) + log(2∥β1∥1 + 3)
+√
+l
+�
++ 1
+√
+l
+log 1
+wp
++ 1
+√
+l
+�
+log ¯α +
+3ϵ2
+2(3 − ϵ) + log 1
+δ
+�
+,
+(111)
+being the KL-divergence equal to zero in this case. By the definition of the parameter p∗
+inf
+p
+�
+¯ρp
+�
+rϵ(β) + log(2∥β1∥1 + 3)
+√
+l
+�
++ 1
+√
+l
+log
+��N
+2
+���
+= ¯ρp∗
+�
+rϵ(β) + log(2∥β1∥1 + 3)
+√
+l∗
+�
++
+1
+√
+l∗ log
+��N
+2
+��
+.
+Then, by using the above equality in (111) computed for ¯ρp∗ we conclude.
+57
+
+Proof of Proposition 3.29. If the inequality (65) holds, then because KL(¯ρp||πp) ≥ 0 for all possible
+observed training sets
+P
+�
+¯ρp∗[Rϵ(β)] ≤ inf
+p
+�
+¯ρp
+�
+rϵ(β)
+�
++ 1
+√
+l
+2 + 3C
+C
++ 1
+√
+l
+KL(¯ρp||πp) + 1
+√
+l
+log
+��N
+2
+���
++
+3ϵ2
+2(3 − ϵ)
+√
+l∗ +
+1
+√
+l∗ log ¯α
+δ
+�
+≥ 1 − 2δ,
+(112)
+by choosing a δ ∈ (0, 1) and using a union bound. Because of Lemma B.5, the above is equivalent to
+P
+�
+¯ρp∗[Rϵ(β)] ≤ inf
+p
+�
+inf
+ˆρ∈M1
++(Bp) ˆρ
+�
+rϵ(β)
+�
++ 1
+√
+l
+2 + 3C
+C
++ 1
+√
+l
+KL(ˆρ||πp) + 1
+√
+l
+log
+��N
+2
+���
++
+3ϵ2
+2(3 − ϵ)
+√
+l∗ +
+1
+√
+l∗ log ¯α
+δ
+�
+≥ 1 − 2δ.
+(113)
+Following the line of proof in Proposition 3.18 applied to a reference distribution πp defined on M1
++(Bp)
+and such that πp[∥β1∥1] ≤ 1, it is straightforward to prove that for all ˆρ ∈ M1
++(Bp) and δ ∈ (0, 1).
+P
+�
+ˆρ[rϵ(β)] ≤ ˆρ[Rϵ(β)]+
+�
+KL(ˆρ||πp)+log
+�1
+δ
+�
++
+3ϵ2
+2(3 − ϵ) l1−2α
+� 1
+√
+l
++ 1
+√
+l
+log
+�
+πp
+�
+1+2(∥β1∥1+1)¯α
+���
+≥ 1−δ
+(114)
+By using a union bound, and replacing ˆρ[rϵ(β)] in (113) with the inequality appearing in (114),
+P
+�
+¯ρp∗[Rϵ(β)] ≤ inf
+p
+�
+inf
+ˆρ∈M1
++(Bp) ˆρ
+�
+Rϵ(β)
+�
++ 2
+√
+l
+2 + 3C
+C
++ 2
+√
+l
+KL(ˆρ||πp) + 1
+√
+l
+log
+��N
+2
+���
++
+3ϵ2
+(3 − ϵ)
+√
+l∗ +
+2
+√
+l∗ log ¯α
+δ
+�
+≥ 1 − 3δ.
+(115)
+By choosing a δ equal to δ
+3 we conclude.
+Proof of Corollary 3.31. First of all, let us remark that P¯ρp∗ is a well-defined probability measure as the
+Gibbs estimator is defined conditionally on the observations in the training set S. By using inequality
+(110) and the Chernoff bound, we can show that
+P¯ρp∗
+�
+Rϵ(ˆβ) ≤ rϵ(β) + 1
+√
+l
+�
+3ϵ2
+2(3 − ϵ) + log(1 + 2(∥β1∥1 + 1)¯α) + log 1
+wp
++ log 1
+δ
+��
+≥ 1 − δ.
+(116)
+It is straightforward to see that Lemma B.6 and B.7 hold also if we substitute at ∆(β) the expression
+∆′(β). Therefore, with probability 1 − δ it holds that
+P¯ρp∗
+�
+rϵ(ˆβ) ≤ Rϵ(β) + 1
+√
+l
+�
+3ϵ2
+2(3 − ϵ) + log(1 + 2(∥β1∥1 + 1)¯α) + log 1
+wp
++ log 1
+δ
+��
+≥ 1 − δ.
+(117)
+58
+
+Moreover, for a β ∈ B
+Rϵ(β) − Rϵ(¯β) ≤ E
+����Y − β0 −
+a(c∗,p∗)
+�
+i=1
+β1,iXi
+��� −
+���Y − ¯β0 −
+a(c∗,p∗)
+�
+i=1
+¯β1,iXi
+���
+�
+≤ E
+����(¯β0 − β0) −
+a(c∗,p∗)
+�
+i=1
+(¯β1,i − β1,i)Xi
+���
+�
+≤ ∥¯β − β∥ E[|Z|]
+(118)
+≤ 1
+C E[|Z|].
+(119)
+Note that inequality (118) holds because the model underlying the data is stationary, whereas (119) holds
+because of the assumptions made on B.
+Let us now plug in (116), the estimations (117) and (118), by using a union bound we obtain that
+P¯ρp∗
+�
+Rϵ(ˆβ) ≤ 1
+C E[|Z|] + Rϵ(¯β) + 2
+√
+l
+�
+3ϵ2
+2(3 − ϵ) + log(1 + 2(∥β1∥1 + 1)¯α) + log 1
+wp
++ log 1
+δ
+��
+≥ 1 − 3δ.
+By choosing δ equal to δ
+3 we obtain the thesis.
+59
+