Regular conditional distributions of continuous max-infinitely divisible random fields

This paper is devoted to the  prediction problem in extreme value theory. Our main result is an explicit expression of the  regular conditional distribution of a max-stable (or max-infinitely divisible) process $\{\eta(t)\}_{t\in T}$ given observations $\{\eta(t_i)=y_i,\ 1\leq i\leq k\}$. Our starting point is the point process representation of max-infinitely divisible processes by Gine, Hahn and Vatan (1990). We carefully analyze the structure of the underlying point process, introduce the notions of extremal function, sub-extremal function and hitting scenario associated to the constraints and derive the associated distributions. This allows us to explicit the conditional distribution as a mixture over all hitting scenarios compatible with the conditioning constraints. This formula extends a recent result by Wang and Stoev (2011) dealing with the case of spectrally discrete max-stable random fields. This paper offers new tools and perspective or prediction in extreme value theory together with numerous potential applications.


Motivations
Max-stable random fields turn out to be fundamental models for spatial extremes since they arise as the the limit of rescaled maxima.More precisely, consider the component-wise maxima η n (t) = max 1≤i≤n X i (t), t ∈ T, of independent and identically distributed (i.i.d.) random fields {X i (t)} t∈T , i ≥ 1.If the random field η n = {η n (t)} t∈T converges in distribution, as n → ∞, under suitable affine normalization, then its limit η = {η(t)} t∈T is necessarily max-stable (see e.g.[7,10]).
Therefore, max-stable random fields play a central role in extreme value theory, just like Gaussian random fields do in the classical statistical theory based on the Central Limit Theorem.
In this framework, the prediction problem arises as an important and long-standing challenge in extreme value theory.Suppose that we already have a suitable maxstable model for the dependence structure of a random field η = {η(t)} t∈T and that the field is observed at some locations t 1 , . . ., t k ∈ T .How can we take benefit from these observations and predict the random field η at other locations ?We are naturally lead to consider the conditional distribution of {η(t)} t∈T given the observations {η(t i ) = y i , 1 ≤ i ≤ k}.A formal definition of the notion of regular conditional distribution is deferred to the Appendix A.2.
In the classical Gaussian framework, i.e., if η is a Gaussian random field, it is well known that the corresponding conditional distribution remains Gaussian and simple formulas give the conditional mean and covariance structure.This theory is strongly linked with the theory of Hilbert spaces: the conditional expectation, for example, can be obtained as the L 2 -projection of the random field η onto a suitable Gaussian subspace.In extreme value theory, the prediction problem turns out to be difficult.A first approach by Davis and Resnick [4,5] is based on a L 1 -metric between max-stable variables and on a kind of projection onto max-stable spaces.To some extent, this work mimics the corresponding L 2 -theory for Gaussian spaces.However, unlike the Gaussian case wich is an exception, there is no clear relationship between the predictor obtained by projection onto the max-stable space generated by the variables {η(t i ), 1 ≤ i ≤ k} and the conditional distributions of η with respect to these variables.A first major contribution to the conditional distribution problem is the work by Wang and Stoev [12].The authors consider max-linear random fields, a special class of max-stable random fields with discrete spectral measure, and give an exact expression of the conditional distributions as well as efficient algorithms.The max-linear structure plays an essential role in their work and provides major simplifications since in this case η admits the simple representation η(t) = q j=1 Z j f j (t), t ∈ T, where the symbol denotes the maximum, f 1 , . . ., f q are deterministic functions and Z 1 , . . ., Z q are i.i.d.random variables with unit Fréchet distribution.The authors determine the conditional distributions of (Z j ) 1≤j≤q given observations {η(t i ) = y i , 1 ≤ i ≤ k}.Their result relies on the important notion of hitting scenario defined as the subset of indices j ∈ [ [1, q]] such that η(t i ) = Z j f (t i ) for some i ∈ [ [1, k]], where, for n ≥ 1, we note [[1, n]] = {1, . . ., n}.The conditional distribution of (Z j ) 1≤j≤q is expressed as a mixture over all admissible hitting scenarios with minimal rank.
The purpose of the present paper is to propose a general theoretical framework for conditional distributions in extreme value theory covering not only the whole class of sample continuous max-stable random fields but also the class of sample continuous max-infinitely divisible (max-i.d.) random fields (see Balkema and Resnick [1]).This paper is mostly theoretical and establishes exact formulas for the conditional distribution of a max-i.d.continuous random field given its value at finitely many points.More practical aspects of the theory such as sampling algorithm or application on real data sets are presented in the subsequent paper [8].There we obtain closed formulas for Brown-Resnick processes and extremal Gaussian processes that are used in the study of extremal rainfall or temperatures in Switzerland.
The motivations and applications come from spatial extreme value theory where the max-stable case plays a central role.However, from the theoretical point of view, there is no additional difficulty considering the more general framework of max-i.d.random fields.Our starting point is the general representation by Giné, Hahn and Vatan [9] of max-i.d.sample continuous random fields (see also de Haan [6] for the max-stable case).It is possible to construct a Poisson random measure Φ = N i=1 δ φi on the space of continuous functions on T such that η(t) Here the random variable N is equal to the total mass of Φ that may be finite or infinite and L = stands for equality of probability laws (see Theorem 1.1 below for a precise statement).We denote by [ otherwise it is called sub-extremal.We show that under some mild condition, one can define a random partition Θ = (θ 1 , . . ., θ ) of {t 1 , . . ., t k } and extremal functions ϕ + 1 , . . ., ϕ + ∈ [Φ] such that the point t i belongs to the component θ j if and only if ϕ + j (t i ) = η(t i ).Using the terminology of Wang and Stoev [12], we call hitting scenario a partition of {t 1 , • • • , t k } that reflects the way how the extremal functions ϕ + 1 , . . ., ϕ + hit the constraints ϕ + j (t i ) ≤ η(t i ), 1 ≤ i ≤ k.The main results of this paper are Theorems 3.2 and 3.3, where the conditional distribution of η given {η(t i ) = y i , 1 ≤ i ≤ k} is expressed as a mixture over all possible hitting scenarios.
The paper is structured as follows.In Section 2, the distribution of extremal and sub-extremal functions is analyzed and a characterization of the hitting scenario distribution is given.In Section 3, we focus on conditional distributions: we compute the conditional distribution of the hitting scenario and extremal functions and then derive the conditional distribution of η.Section 4 is devoted to examples: we specify our results in the simple case of a single conditioning point and consider max-stable models.The proofs are collected in Section 5 and some technical details are postponed to an appendix.

Preliminaries on max-i.d. processes
Let T be a compact metric space and C = C(T, R) be the space of continuous functions on T endowed with the sup norm Let (Ω, F, P) be a probability space.A random process η = {η(t)} t∈T is said to be maxi.d. on C if η has a version with continuous sample path and if, for each n ≥ 1, there exist {η ni , 1 ≤ i ≤ n} i.i.d.sample continuous random fields on T such that where denotes pointwise maximum.Giné, Hahn and Vatan (see [9] Theorem 2.4) give a representation of such processes in terms of Poisson random measure.For any function f on T and set A ⊂ T , we note f (A) = sup t∈A f (t).
and define C h = {f ∈ C; f = h, f ≥ h}.Under the condition that the vertex function h is continuous, there exists a locally-finite Borel measure µ on C h , such that if Φ is a Poisson random measure Φ on C h with intensity measure µ, then where [Φ] denotes the set of atoms of Φ.
Furthermore, the following relations hold: where n ∈ N, K i ⊂ T closed and Theorem 1.1 provides an almost complete description of max-i.d.continuous random processes, the only restriction being the continuity of the vertex function.Clearly, the distribution of η is completely characterized by the vertex function h and the so called exponent measure µ.If h > −∞, the random process η − h is continuous and max-i.d. and its vertex function is identically equal to 0. If h(t) = −∞ at some point t ∈ T , we may consider e η − e h which is max-i.d. with zero vertex function.Since the conditional distribution of η is easily deduced from that of η − h (or e η − e h ), we can assume without loss of generality that h ≡ 0. The corresponding set C 0 is the space of non-negative and non-null continuous functions on T .
We need some more notations from point process theory (see Daley and Vere-Jones [2,3]).It will be convenient to introduce a measurable enumeration of the atoms of Φ (see [3] We endow M p (C 0 ) with the σ-algebra M p generated by the applications } is non empty and has finitely many points in (ε, +∞) for all ε > 0 so that the maximum max{f (t); with the convention that max(M ) ≡ 0 if M = 0.By considering restrictions of the measure M to sets {f ∈ C 0 ; f > ε} and using uniform convergence, it is easy to show that the function max(M ) is continuous on T .

Extremal points and related distributions
In the sequel, η denotes a sample continuous max-i.d.random process with vertex function h ≡ 0 and exponent measure µ on C 0 .On the same probability space, we suppose that an M p (C 0 )-valued Poisson random measure Φ = N i=1 δ φi with intensity measure µ is given and such that η = max(Φ).

Extremal and sub-extremal point measures
Let K ⊂ T be a closed subset of T .We introduce here the notion of K-extremal points that will play a key role in this work.We use the following notations: if f 1 , f 2 are two functions defined (at least) on K, we write and only if f < K max(M ) and K-extremal otherwise.In words, a sub-extremal atom has no contribution to the maximum max(M ) on K.
where g is any continuous function defined (at least) on K. Clearly, it always holds The following theorem fully characterizes the joint distribution of (Φ + K , Φ − K ) provided that Φ + K (C 0 ) is almost surely finite.We note δ 0 the Dirac mass at 0.
and, for k ≥ 1, We now focus on conditions ensuring Φ + K (C 0 ) to be a.s.finite.

Proposition 2.3.
The K-extremal point measure Φ + K is a.s.finite if and only if one of the following condition holds: It should be noted that any simple max-stable random field (with unit Fréchet margins) satisfies condition (ii) above.See for example Corollary 3.4 in [9].Remark 2.4.Using Theorem 2.2, it is easy to show that the distribution of (Φ + K , Φ − K ) has the following structure.Define the tail functional μK by μK (g) = µ({f ∈ C 0 ; f < K g}) for any continuous function g defined (at least) on K. Suppose that Φ + K is finite almost surely.Its distribution is then given by the so-called Janossy measures (see e.g.Daley and Vere-Jones [2] section 5.3).The Janossy measure of order k of the K-extremal point measure Φ + K is given by the distribution of a Poisson random measure with measure intensity 1 {f < K ∨ k i=1 fi} µ(df ).
These results are not used in the sequel and we omit their proof for the sake of brevity.

Extremal functions
Let t ∈ T .We denote by µ t the measure on (0, +∞) defined by Note that (2. 3) The following proposition states that, under a natural condition, there is almost surely a unique {t}-extremal point in Φ.This extremal point will be referred to as the t-extremal function and noted φ + t .
Proposition 2.5.For t ∈ T , the following statements are equivalent: An important class of processes satisfying the conditions of Proposition 2.5 is the class of max-stable processes (see section 4.2 below).

Hitting scenarios
Proposition 2.5 gives the distribution of Φ + K when K = {t} is reduced to a single point.Going a step further, we consider the case when K is finite.In the sequel, we suppose that the following assumption is satisfied: (A) K = {t 1 , . . ., t k } is finite and, for all t ∈ K, μt is continuous and μt (0 + ) = +∞.
According to Proposition 2.5, Assumption (A) holds true if and only if for all i ∈ {1, . . ., k}, the law of η(t i ) has no atom .It is always satisfied in the max-stable case.Roughly speaking, Assumption (A) ensures that the maximum η(t) = max(Φ)(t) is uniquely reached for all t ∈ K.This will provide combinatorial simplifications.More precisely, under Assumption (A), the event is of probability 1 and the extremal functions φ + t1 , . . ., φ + t k are well defined.In the next definition, we introduce the notion of hitting scenario that reflects the way how these extremal functions hit the maximum η on K.
Let P K be the set of partitions of K.It is convenient to think about K as an ordered set, say t 1 < • • • < t k .Then each partition τ can be written uniquely in the standardized form τ = (τ 1 , . . ., τ ) where = (τ ) is the length of the partition, τ 1 ⊂ K is the component of t 1 , τ 2 ⊂ K is the component containing min(K \ τ 1 ) and so on.With this convention, the components τ 1 , . . ., τ of the partition are labeled so that min τ 1 < • • • < min τ .We illustrate the definition with two examples in Figure 3. Clearly a point φ ∈ [Φ] is K-extremal if and only if it is t-extremal for some t ∈ K, so that [Φ + K ] = {φ + t , t ∈ K}.Furthermore, the random measure Φ + K is almost surely simple, i.e. any atoms have a simple multiplicity, otherwise the condition Φ + {t} (C 0 ) = 1 a.s.would not be satisfied for some t ∈ K.These considerations entail that (2.5) In particular, the length (Θ) of the hitting scenario is equal to Φ + K (C 0 ).Furthermore the extremal functions satisfy (2.6) The distribution of the hitting scenario and extremal functions is given by the following proposition.The proof relies on Theorem 2.2.Proposition 2.7.Suppose Assumption (A) is met.

Regular conditional distribution of max-id processes
We now focus on conditional distributions.We will need some notations.
If s = (s 1 , . . ., s l ) ∈ T l and f ∈ C 0 , we note f (s) = (f (s 1 ), . . ., f (s l )).Let µ s be the exponent measure of the max-i.d.random vector η(s), i.e. the measure on [0, +∞) l \ {0} defined by Define the corresponding tail function Let {P s (x, df ); x ∈ [0, +∞) l \ {0}} be a regular version of the conditional measure µ(df ) given f (s) = x (see Lemma A.2 in Appendix A.2). Then for any measurable function Let t = (t 1 , . . ., t k ) and y = (y 1 , . . ., y k ) ∈ [0, +∞) k .Before considering the conditional distribution of η with respect to η(t) = y, we give in the next theorem an explicit expression of the distribution of η(t).We note K = {t 1 , . . ., t k }.For any non empty Then, and the distribution ν t of η(t) is equal to Under some extra regularity assumptions, one can even get an explicit density function for ν t (see the section 4.3 on regular models below).
We are now ready to state our main result.In Theorem 3.2 below, we consider the regular conditional distribution of the point process Φ with respect to η(t) = y.Then, thanks to the relation η = max(Φ), we deduce easily in Corollary 3.3 below the regular conditional distribution of η with respect to η(t) = y.
We briefly comment on these formulas.The fact that the distribution Q t (y, τ, •) in Equation (3.4) factorizes into a tensorial product means that the extremal functions ϕ + 1 , . . ., ϕ + are independent conditionally on η(t) = y and Θ = τ .The fact that the distribution R t (y, τ, (f j ), •) in Equation (3.5) does not depend on τ and (f j ) means that conditionally on η(t) = y, Φ − K is independent of Θ and (ϕ + 1 , . . ., ϕ + (Θ) ).The distribution R t (y, •) can be seen as the distribution of the Poisson point measure Φ conditioned to lie in C − t (y), i.e., to have no atom in {f ∈ C 0 ; f (t) < y}.It is equal to the distribution of a Poisson point measure with intensity 1 {f (t)<y} µ(df ).
As a consequence of Theorem 3.2, we deduce the regular conditional distribution of η with respect to η(t) = y.Theorem 3.3.It holds ν t (dy)-a.e.
Remark 3.4.The formulas in Theorems 3.2 and 3.3 are quite theoretical.It should be noted that for Brown-Resnick processes or extremal Gaussian processes, explicit closed expressions are available, see [8].
Remark 3.5.Let us mention that Theorem 3.2 suggests a three-step procedure for sampling from the conditional distribution of η given η(t) = y: 1. Draw a random partition τ with distribution π t (y, •).
3. Independently of the above two steps, draw i∈I δ φi a Poisson point measure on C 0 with intensity 1 {f (t)<y} µ(df ).It can be obtained from a Poisson point measure with intensity µ(df ) by removing those points not satisfying the constraint f (t) < y.
Then, the random field η(t) = max{ψ 1 (t), . . ., ψ (t)} ∨ max{φ i (t), i ∈ I}, t ∈ T, has the required conditional distribution.The issues and computational aspects of conditional sampling are addressed in the paper [8].In particular, a Monte-Carlo Markov chain can be used in order to sample the random partition from the conditional distribution π t (y, •).

Specific cases
We apply in this section our general results to specific cases.More specific and elaborate examples are considered in [8] where we obtain closed formulas for Brown-Resnick processes and extremal Gaussian processes and consider applications to temperatures and rainfall in Switzerland based on real data sets.

The case of a single conditioning point
It is worth noting that the case of a single conditioning point, i.e. k = 1, gives rise to major simplifications.There exists indeed a unique partition of the set K = {t} so that the notion of hitting scenario is irrelevant.Furthermore, there is a.s. a single K-extremal function ϕ + 1 which is equal to the t-extremal function φ + t .In this case, Theorems 3.2 and 3.3 simplify into the following proposition.

Max-stable models
We put the emphasis here on max-stable random fields.For convenience and without loss of generality, we focus on simple max-stable random fields η, i.e., with standard unit Fréchet margins A random field η is said to be simple max-stable if for any n ≥ 1, where {η i , i ≥ 1} are i.i.d.copies of η.Any general max-stable random field can be related to such a simple max-stable random field η by simple transformation of the margins, see e.g.Corollary 3.6 in [9].Furthermore, Corollary 4.5.6 in [7] states that η can be represented as where (Γ i ) i≥1 is the nonincreasing enumeration of the points of a Poisson point process on (0, ∞) with intensity x −2 dx, (Y i ) i≥1 is an i.i.d.sequence of continuous random processes on T , independent of (Γ i ) i≥1 and such that Since a continuous simple max-stable random field is max-i.d., it has a Poisson point measure representation (1.1).The normalization to unit Fréchet margins entails that the vertex function h is equal to 0 and that the exponent measure µ satifies, for all t ∈ T , µ t (dy) = y −2 1 {y>0} dy and μt (y) = y −1 , y > 0.
The correspondence between the two representations (1.1) and (4.2) is the following: the point measure Φ = i≥1 δ ΓiYi is a Poisson point measure on C 0 with intensity The distribution of the Y i 's, denoted by σ, is called the spectral measure and is related to the exponent measure µ by the relation Taking into account this particular form of the exponent measure, we can relate the kernel P t (y, •) to the spectral measure σ.For x ∈ R, we note (x) + = max(x, 0).
Furthermore, for l ≥ 1, s ∈ T l and z ∈ [0, +∞) l , Equation (4.3) extends Lemma 3.4 in Weintraub [13] where only the bivariate case l = 1 is considered.Note the author considers min-stability rather than max-stability; the correspondence is straightforward since, if η is simple max-stable, then η −1 is minstable with exponential margins.

Regular models
We have considered so far the case of a single conditioning point which allows for major simplifications.In the general case, there are several conditioning points and the hitting scenario is non trivial.This introduces more complexity since the conditional distribution is expressed as a mixture over any possible hitting scenarios and involves an abstract Radon-Nikodym derivative.The framework of regular models can be helpful to get more tractable formulas.
The exponent measure µ is said to be regular (with respect to the Lebesgue measure) if for any l ≥ 1 and s ∈ T l with pairwise distinct components, the measure µ s (dz) is absolutely continuous with respect to the Lebesgue measure dz on [0, +∞) l .We denote by h s the corresponding Radon-Nikodym derivative, i.e., µ s (dz) = h s (z)dz.
This approach is exploited in [8] for Brown-Resnick max-stable processes.Indeed, the model turns out to be regular.

Proof of Theorem 2.2 and Proposition 2.3
For the proof of Theorem 2.2, we need the following lemma giving a useful charac- Conversely for f ∈ [ Φ], the condition Φ ∈ C + K implies the existence of t 0 ∈ K such that f (t 0 ) = max( Φ)(t 0 ).Hence f (t 0 ) = max(Φ)(t 0 ) and f is K-extremal in Φ. space with its associated σ-algebra of Borel sets, (X , F) an arbitrary measurable space, and π a probability measure on the product space (X × Y, F ⊗ G).Let π X denote the X -marginal of π, i.e. π X (A) = π(A × Y) for any A ∈ F. Then there exists a family of kernels K(x, B) such that -K(x, •) is a probability measure on (Y, G) for any fixed x ∈ X ; -K(•, B) is an F-measurable function on X for each fixed B ∈ G; -π(A × B) = A K(x, B)π X (dx) for any A ∈ F and B ∈ G.
These three properties define the notion of regular conditional probability.When π is the joint distribution of the random variable (X, Y ), we may write K(x, •) = P(Y ∈ •|X = x).
The existence of the regular conditional probability relies on the assumption that Y is a complete and separable metric space.Furthermore, for any F ⊗ G-measurable non-negative function f on X × Y , it follows that X ×Y f (x, y)π(dx, dy) = X Y f (x, y)K(x, dy)π X (dx).
The following Lemma states the existence of the kernel {P s (x, df ); x ∈ [0, +∞) l \ {0}} satisfying Equation (3.1).This is not straightforward since the measure µ is not a probability measure and may be infinite.A is equal to the disjoint union of the A i 's.We note µ i (•) = µ(• ∩ A i ) the measure on the complete and separable space C 0 ∪ {0}.Equation (1.4) ensures that µ i is a finite measure (and hence a probability measure up to a normalization constant) and there exists a regular conditional probability kernel P i s (x, df ) with respect to f (s) = x.We obtain, for all F : [0, +∞) l × C 0 , Ai F (f (s), f )µ(df ) = Ãi C0 F (x, f )P i s (x, df )µ s (dx), where Ãi = {x ∈ [0, +∞) k ; (i + 1) −1 ≤ |x| < i −1 }.Let us define P s (x, df ) a probability measure on C 0 by P s (x, df ) = i≥1 1 {x∈ Ãi} P i s (x, df ).

Theorem 1 . 1 .
(Giné, Hahn and Vatan [9]) Let h be the vertex function of a sample continuous max-i.d.process η defined by

(1. 4 )
Consequently, we have Φ ∈ M p (C 0 ) almost surely and η L = max(Φ).An illustration of Theorem 1.1 is given in Figure 1 with a representation of the Poisson point measure Φ and of the corresponding maximum process η = max(Φ) in the moving maximum max-stable model based on the Gaussian density function.EJP 18 (2013), paper 7.

Figure 1 :
Figure 1: A representation of the point process Φ (left) and of the associated maximum process η = max(Φ) (right) in the moving maximum max-stable model based on the Gaussian density function.Here T = [0, 5].

Definition 2 . 1 .
Define the K-extremal random point measure Φ + K and the K-subextremal random point measure Φ − K by

Figure 2
Figure 2 provides an illustration of the definition.It should be noted that Φ + K and Φ − K are well defined measurable random point measures (see Lemma A.3 in Appendix A.3). Furthermore, it is straightforward from the definition that
Proposition 4.2.Let η be a continuous simple max-stable random field with spectral measure σ and t ∈ T .The {t}-extremal function φ + t has conditional distribution Under this assumption, we can reformulate Theorems 3.1 and 3.2.For example, Equation (3.2) implies that the distribution ν t of η(t) is absolutely continuous with respect to the Lebesgue measure with density τj , z j )dz j .(tτ j ,t τ j c ) (y τ j , z j )dz j .The conditional distribution of the extremal functions Q t (y, τ, •) in Equation (3.4) is based on the kernel P t (y, df ).Using the existence of a Radon-Nikodym derivative for the finite dimensional margins of µ, we obtain