Conditional Simulation of Max-Stable Processes

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1 1 Conditional Simulation of Ma-Stable Processes Clément Dombry UniversitédeFranche-Comté Marco Oesting INRA, AgroParisTech Mathieu Ribatet UniversitéMontpellier2andISFA,UniversitéLyon1 CONTENTS Abstract Introduction: the prediction problem and conditional distribution Conditional distribution of ma-stable processes Etremal functions, subetremal functions and hitting scenarios The general structure of conditional distributions Eamples of regular models Distribution of the etremal functions Conditional sampling of ma-stable processes A 3-step procedure for conditional sampling Gibbs sampling for the conditional hitting scenario Simulation Study Gibbs Sampler Conditional simulations Acknowledgements References Abstract Ma-stable processes are well established models for spatial etremes. In this chapter, we address the prediction problem: suppose a ma-stable processisobservedat some locations only, how can we use these observations to predict the behavior of 1

2 2 Etreme Value Modeling and Risk Analysis: Methods and Applications the process at other unobserved locations? Mathematically,thepredictionproblemis related to the conditional distribution of the process giventheobservations.recently, Dombry and Eyi-Minko (2013) provided an eplicit theoretical formula for the conditional distributions of ma-stable processes. The resultreliesonthespectralrepresentation of the ma-stable process as the pointwise maima over an infinite number of spectral functions belonging to a Poisson point process. The effect of conditioning on the Poisson point process is analyzed, resulting in the notions of hitting scenario and etremal or subetremal functions. Due to the compleity ofthestructureofthe conditional distributions, conditional simulation appears at the same time challenging and important to assess characteristics that are analytically intractable such as the conditional median or quantiles. The issue of conditional simulation was considered by Dombry et al. (2012) who proposed a three step procedure for conditionalsampling. As the conditional simulation of the hitting scenario becomes computationally very demanding even for a moderate number of conditioning points, a Gibbs sampler approach was proposed for this step. The results are illustrated on some simulation studies and we propose several diagnostics to check the performance. 1.1 Introduction: the prediction problem and conditional distribution In classical geostatistics, Gaussian random fields play a central role in the statistical theory based on the Central Limit Theorem. In a similar manner, ma-stable random fields turn out to be fundamental models for spatial etremes since they etend the well known univariate and multivariate etreme value theory to the infinite dimensional setting. Ma-stable random fields arise naturally when considering the component-wise maima of a large number of independent and identically random fields and seeking for a limit under a suitable affine normalization. Inthismultivariate or functional setting,thenotion of dependence is crucial: how does an etreme event occurring in some region affect the behavior of the random field at other locations? This is related to the prediction problem which is an important and long-standing challenge in etreme value theory. Suppose that a ma-stable random field Z =(Z()) X is observed at some stations 1,..., k Xonly, yielding Z( 1 )=z 1,...,Z( k )=z k. (1.1) How can we take benefit from these observations and predict the random field Z at other places? We are naturally lead to consider the conditional distribution of (Z()) X given the observations (1.1). In the classical Gaussian framework, i.e., if Z is a centered 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: for eample, the conditional epectation of Z() can be obtained as the L 2 -proection of the random field η

3 Conditional Simulation of Ma-Stable Processes 3 onto the Gaussian subspace generated by the variables {Z( i ), 1 i k},resulting in the linear combination E [Z() Z( 1 ),...,Z( k )] = α 1 ()Z( 1 )+ + α k ()Z( k ), X, for some weight functions α i (), i = 1,...,n, that are obtained via the kriging theory (cf. Chilès and Delfiner (1999), for eample). A conditional simulation of Z given the observations (1.1) is then easily performed by setting Z() = Z()+α 1 ()(z 1 Z( 1 )) + + α k ()(z k Z( k )), X, where Z denotes the realization of an unconditional simulation of the random field Z. In etreme value theory, the prediction problem turns out to be more difficult. A similarkriging theory for ma-stable processes isvery appealing and a first approach in that direction was done by Davis and Resnick (1989, 1993). They introduced a L 1 - metric between ma-stable variables and proposed a kind of proection onto mastable spaces. To some etent, this work mimics the corresponding L 2 -theory for Gaussian spaces. However, unlike the eceptional Gaussian case, there is no clear relationship between the predictor obtained by proection onto the ma-stable space generated by the variables {Z( i ), 1 i k} and the conditional distributions of η with respect to these variables. Conditional distributions have been considered first by Weintraub (1991) in the case of a bivariate vector. A maor contribution is the work by Wang and Stoev (2011) where the authors consider ma-linear random fields, a special class of ma-stable random fields with discrete spectral measure, and give an eact epression of the conditional distributions as well as efficient algorithms. The ma-linear structure plays an essential role in their work and provides maor simplifications since in this case Z admits the simple representation q Z() = F f (), =1 X, where the symbol denotes the maimum, f 1,...,f q are deterministic functions and F 1,...,F q are i.i.d. random variables with unit Fréchet distribution. The authors determine the conditional distributions of (F ) 1 q given the observations (1.1) and deduce the conditional distribution of Z.AnotherapproachbyOestingand Schlather (2014) deals with ma-stable random fields with a mied moving maima representation. We present and discuss here the recent result from Dombry and Eyi-Minko (2013) anddombryetal. (2013) providingformulasfortheconditional distributions of ma-stable random fields as well as efficient algorithm for conditional sampling. Note that the results can be stated in the more general framework of ma-infinitely divisible processes, but for the sake of simplicity, we stick here to the ma-stable case. Section 2 reviews the theoretical results on conditional distribution: we introduce the notion of etremal functions and hitting scenarios and eplicit the eact distribution of the ma-stable process given the observations. Here, we focus on the

4 4 Etreme Value Modeling and Risk Analysis: Methods and Applications framework of regular models that yields tractable formulas. InSection3,weputthe emphasis on efficient simulation and discuss a 3-step procedure for conditional sampling. A difficult step is the conditional sampling for the hitting scenario for which a Gibbs sampler approach is proposed. Section 4 is devoted to simulation studies and we present several diagnostics to check the performance of the methods suggested. 1.2 Conditional distribution of ma-stable processes In the sequel, we consider Z asamplecontinuousma-stablerandomfieldonx R d.wecanassumewithoutlossofgeneralitythatz has unit Fréchet margins. Thus, the process Z possesses a spectral representation (see for eample de Haan (1984), Penrose (1992), Schlather (2002) or de Haan and Ferreira (2006)) Z() =ma i 1 ζ iy i (), X, (1.2) where {ζ i } i 1 are the points of a Poisson process on (0, ) with intensity ζ 2 d ζ, (Y i ) i 1 are independent replicates of a non-negative continuous sample path stochastic process Y such that E[Y ()] = 1 for all X, (ζ i ) i 1 and (Y i ) i 1 are independent. We now introduce a fundamental obect in our analysis which isafunction- valued Poisson point process associated with the representation (1.2). Let C = C(X, [0, + )) be the space of continuous non-negative functions on X.Weconsider the C-valued point process Φ={φ i } i 1 where φ i () =ζ i Y i () with ζ i and Y i as in (1.2). It is well known (de Haan and Ferreira, 2006) to verify that Φ is a Poisson point process with intensity measure Λ given by Λ(A) = 0 P[ζY A]ζ 2 dζ, A CBorel. Our strategy is to work on the level of the Poisson point process Φ rather than on the level of the ma-stable process Z and to derive the conditional distribution of Φ given the observations (1.1). The conditional distribution of Z is thendeduced easily Etremal functions, subetremal functions and hitting scenarios The observations (1.1) together with the spectral representation (1.2) yield the constraint ma φ i( 1 )=z 1,...,ma φ i( k )=z k. i 1 i 1 A first step in our strategy is the analysis of the way how these constraints are met, i.e. the way how the maima are attained. An important preliminary remark is that for each = 1,,k,themaimumZ( ) = ma i 1 φ i ( ) is almost surely attained by a unique function φ i,whichcanbeeasilyseenbythefactthatthepoint

5 Conditional Simulation of Ma-Stable Processes 5 process {ϕ i ( ),i 1} on (0, ) has intensity ζ 2 dζ. Thisleadstothefollowing definition. Definition The (almost surely) unique function φ Φ such that φ( ) = Z( ) is called the etremal function at and denoted by ϕ +. Furthermore, we define the etremal point process Φ + = {ϕ + } 1 k as the set of etremal functions with respect to the conditioning points { } 1 k. Note that the notation Φ + = {ϕ + } 1 k may include the multiple occurrence of some functions for eample if ϕ + 1 = ϕ + 2.Thisisnottakenintoaccountinthe etremal point process Φ + where each point has multiplicity one. The possible redundancies are captured by the so-called hitting scenario. We first introduce this notion with simple eamples. Assume first that there are k =2conditioning points 1, 2 and hence two etremal functions ϕ + 1 and ϕ + 2.Twocasescanoccur: either ϕ + 1 = ϕ + 2,i.e.themaimaatpoints 1 and 2 are reached by the same etremal function, in this case the hitting scenario is θ = { 1, 2 }; or ϕ + 1 ϕ + 2,i.e.themaimaatpoints 1 and 2 are reached by different etremal functions, in this case the hitting scenario is θ =({ 1 }, { 2 }). In the case of k = 3 conditioning points, there are 5 different possibilities for the hitting scenario: ({ 1, 2, 3 }), ({ 1, 2 }, { 3 }), ({ 1, 3 }, { 2 }), ({ 1 }, { 2, 3 }) and ({ 1 }, { 2 }, { 3 }). The interpretation is straightforward: for instance, the hitting scenario θ = ({ 1, 3 }, { 2 }) corresponds to the case when the maima at 1 and 3 are attained by the same etremal function but the maimum at 2 corresponds to a different etremal event, i.e. ϕ + 1 = ϕ + 3 ϕ + 2.Thegeneraldefinitionisasfollows. Definition The hitting scenario θ is a random partition (θ 1,...,θ l ) of the conditioning points { 1,, k } such that for any 1 2, 1 and 2 are in the same component of θ if and only if ϕ + 1 = ϕ + 2. The hitting scenario θ takes into account the redundancies in Φ + = {ϕ + 1,,ϕ + k } and it is straightforward that the number of blocks l of θ is eactly the number of distinct etremal functions and hence the cardinality of Φ +.Hencewe can rewrite Φ + = {ϕ + 1,,ϕ+ l } where ϕ+ = ϕ+ for all θ. So far, we considered only those functions φ i that hit the maimum Z at some conditioning points { } 1 k.theremainingfunctionsarecalledsub-etremal,as in the following definition. Definition Afunctionφ Φ satisfying φ( ) <Z( ) for all 1 k is called subetremal. The set of subetremal functions is called the subetremal point process Φ.

6 6 Etreme Value Modeling and Risk Analysis: Methods and Applications This yields a disoint decomposition of the point process Φ=Φ + Φ into its etremal and subetremal parts. This is illustrated in Figure where the etremal functions (black) and the subetremal functions (gray) are depicted as well as the corresponding hitting scenarios FIGURE 1.1 Two realizations of the Poisson point process Φ and of the corresponding hitting scenario θ with conditioning points i = i, i =1,, 4 represented by the circles. Left: the hitting scenario is θ =({ 1 }, { 2, 3 }, { 4 }). Right:thehittingscenario is θ =({ 1, 2 }, { 3, 4 }). Akeyresultinthestudyoftheconditionaldistributionisthe following theorem providing the oint distribution of (θ, Φ +, Φ ).WedenotebyP k the set of partitions of { 1,..., k } and by M p (C) the set of C-valued point measures. We introduce some vectorial notation. Let =( 1,..., k ) and for τ =(τ 1,...,τ l ) P k,we introduce τ =() τ.foranyvectors =(s 1,...,s m ) X m and any function f : X R, wenotef(s) =(f(s 1 ),...,f(s m )). Iff 1,f 2 : X R m are two functions, the notation f 1 (s) <f 2 (s) means that f 1 (s ) <f 2 (s ), =1,...,m. Theorem For any partition τ =(τ 1,...,τ l ) in P k and any Borel sets A C l and B M p (C), wehave = P[θ = τ,(ϕ + 1,...,ϕ+ l ) A, Φ B] 1 {ma C l f ( τ )<f ( τ ),=1,...,l,}1 {(f1,...,f l ) A} [ ] P Φ B and φ Φ, φ() < ma f () Λ(df 1 ) Λ(df l ). 1 l The main technical tool for this proof is the Mecke-Slivniack formula from stochastic geometry, see e.g. Stoyan et al. (1987). We sketch herethemainlines of the proof. Proof. First note that the event {θ = τ,(ϕ + 1,...,ϕ+ l ) A, Φ B} is realized if and only if there eists a l-tuple (φ 1,,φ l ) Φ l satisfying the following conditions:

7 Conditional Simulation of Ma-Stable Processes 7 i) Φ + = {φ 1,...,φ l } ii) Φ =Φ\{φ 1,...,φ l }; iii) θ = τ; iv) (φ 1,...,φ l ) A; v) Φ B. Clearly, if such a l-tuple does eist, it is necessarily unique and equal to (ϕ + 1,...,ϕ+ l ).Wededucethattheprobabilityofinterestcanbewritteninthe form P[θ = τ,(ϕ + 1,...,ϕ+ l ) A, Φ B] = E F (φ 1,...,φ l, Φ \{φ 1,...,φ l }) (1.3) (φ 1,,φ l ) Φ l with F : C l M p (C) {0, 1} defined by { 1 if conditions i)-v) are satisfied F (φ 1,...,φ l, Φ \{φ 1,...,φ l })= 0 otherwise. By the Slyvniak-Mecke formula (Stoyan et al., 1987), we deduce that the epectation (1.3) is equal to C l E[F (f 1,...,f l, Φ)Λ(df 1 ) Λ(df l ). The theorem follows after a more eplicit description of the functional F.Condition i), ii) and iii) together are equivalent to and ma φ ( τ ) <φ ( τ ),=1,...,l, φ Φ \{φ 1,...,φ l },φ() < ma 1 l φ (). Condition iv) is clear and condition v) is equivalent to Φ \{φ 1,...,φ l } B The general structure of conditional distributions In the previous section, we have seen that we are able to compute the oint distribution of the hitting scenario θ, theetremalfunctionsφ + and the subetremal functions Φ.Wenowconsidertheconditionalointdistributiongiventhe observations (1.1). It can be computed on the basis of Theorem only, because the observations (Z( 1 ),...,Z( k )) can be epressed as a function of θ and Φ +,i.e. Z( i )=ϕ + ( i) with such that i θ.

8 8 Etreme Value Modeling and Risk Analysis: Methods and Applications Nevertheless, this computation is rather tedious and we give here the result without proof and we consider only the so-called regular case. We say that the intensity measure Λ of the point process Φ is regular at some point s =(s 1,...,s p ) X p if the marginal spectral measure Λ is absolutely regular with respect to the Lebesgue measure on (0, + ) k.moreprecisely,wedefinethemarginalspectralmeasure Λ s (dz 1,...,dz p )=Λ(f(s 1 ) dz 1,...,f(s p ) dz p ) and assume that on (0, + ) k Λ s (dz 1,...,dz p )=λ s (z 1,...,z p )dz 1 dz p. The proof of the following result and more details are to be found in Dombry and Eyi- Minko (2013). Note that, in the non-regular case, some further analysis of the different hitting scenarios is needed, as some hitting scenarios have to be ecluded due to the conditioning data. For eamples of non-regular cases and conditionalsampling procedures in these cases, see Wang and Stoev (2011) and Oesting and Schlather (2014) who consider ma-linear and mied moving maima processes, respectively. Theorem Assume that Λ is regular at = ( 1,..., k ). For τ = (τ 1,...,τ l ) P k and =1,...,l,defineI = {i: i τ }, τ =( i ) i I, z τ =(z i ) i I, τ c =( i ) i/ I and z τ c =(z i ) i/ I.Theconditionaldistributionof (θ, Φ +, Φ ) with respect to the observations (1.1) is obtained as follows: 1. For τ =(τ 1,...,τ l ) P k, = P [θ = τ Z() =z] 1 l λ (τ, τ c )(z τ, u )du, (1.4) C,z =1 {u <z τ c } where the normalization constant C,z is such that τ P k P [θ = τ Z() =z] =1. 2. Conditionally on the observations (1.1) and on the hitting scenario τ = (τ 1,...,τ l ) P k,theetremalfunctionsϕ + 1,...,ϕ+ l are independent with distribution P[ϕ + df Z() =z,θ = τ] = Λ[df f( τ )=z τ,f( τ c ) < z τ c ], =1,...,l. (1.5) 3. Conditionally on the observations (1.1), Φ is independent of Φ + = {ϕ + 1,...,ϕ+ l } and has the same distribution as a Poisson Point process on C with intensity 1 {f()<z} Λ(df).

9 Conditional Simulation of Ma-Stable Processes 9 This theorem allows to reconstruct the conditional distribution of the point process Φ given the observations Z() =z via a three step procedure: construct first the conditional hitting scenario θ and the etremal functions ϕ + 1,...,ϕ+ l (step 1 and 2), this yields Φ + ;independentlyconstructtheconditionalsubetremalpoint process Φ (step 3); finally set Φ=Φ + Φ to obtain the conditional point process Φ. The conditional probability appearing in steps 2 and 3 are quite natural. Given the observations Z() =z and the hitting scenario θ = τ, theetremalfunction ϕ + must satisfy the constraints ϕ+ ( τ )=z τ and ϕ + ( τ c) < z τ c so that it seems natural to obtain the conditional intensity Λ given these constraints. Similarly, given Z() =z, thesubetremalfunctionsϕ Φ must satisfy the constraint ϕ() < z which naturally leads to the point process intensity Λ restricted to the set of functions {f C,f() < z} Eamples of regular models Theorem requires the assumption of regularity of the intensity measure Λ at point =( 1,..., k ).Wenowrecallsomepopularmodelsforma-stableprocess that are regular and give the corresponding intensity function λ.thesemodelsare mainly based on Gaussian or related distributions and provide a nice framework for eplicit computations. Eample 1: Brown-Resnick process The Brown Resnick process introduced by Kabluchko et al. (2009) is a stationary ma-stable random field on X = R d corresponding to the case where Y () = ep{w () γ()} in (1.2) with W acenteredgaussianprocesswithstationary increments, semi-variogram γ and such that W (o) =0almost surely. One can show that the intensity measure Λ is regular at X k as long as the covariance matri Σ of the random vector W () is positive definite. We denote by g the Gaussian density { g (u) =(2π) k/2 det(σ ) 1/2 ep 1 2 ut Σ 1 u The marginal intensity measure is computed as follows: for all Borel set A (0, + ) k }. Λ (A) = = = = A P[ζY () A]ζ 2 dζ P[ζ ep{w () γ()} A]ζ 2 dζ A λ (z)dz g (log z log ζ + γ()) k i=1 z 1 i ζ 2 dζdz

10 10 Etreme Value Modeling and Risk Analysis: Methods and Applications with λ (z) = 0 g (log z log ζ + γ()) k i=1 z 1 i ζ 2 dζ. Some standard but tedious computations for Gaussian integrals reveal that ( λ (z) =C ep 1 ) 2 log k zt Q log z + L log z z 1 i, z (0, ) k, with 1 k =(1) i=1,...,k, γ = {γ( i )} i=1,...,k, Q =Σ 1 i=1 Σ 1 1 k 1 T ( k Σ 1 1 T, L 1 T = k Σ 1 ) T γ 1 1 k Σ 1 1 k 1 T k γ Σ 1 k Σ 1 1 k C =(2π) (1 k)/2 det(σ ) 1/2 (1 T k Σ 1 1 k ) 1/2 { 1 (1 T k ep Σ 1 γ 1) 2 1 } 2 1 T k Σ 1 1 k 2 γt Σ 1 γ. See Dombry et al. (2013) for more details. Eample 2: Schlather process The Schlather process (Schlather, 2002), also called etremal Gaussian process, is ama-stablerandomfieldonx = R d corresponding to the case where Y () = (2π) 1/2 ma{0,w()} in (1.2) with W astandardgaussianprocesswithcorrelation function ρ.for X k and provided the covariance matri Σ of the random vector W () is positive definite, the intensity function is ( ) k +1 λ (z) =π (k 1)/2 det(σ ) 1/2 a (z) (k+1)/2 Γ, z (0, + ) k, 2 where a (z) =z T Σ 1 z.seedombryetal.(2013)formoredetailsonthecomputation. Eample 3: etremal t-process The etremal t-process is a generalization of the Schlather process above obtained with an etra parameter ν > 0 that yields more fleibility in the model. It is a ma-stable random field on X = R d obtained with Y () = c ν ma{0,w() ν } in (1.2) with W astandardgaussianprocesswithcorrelationfunctionρ and c ν = π2 (ν 2)/2 Γ((ν +1)/2) 1.ProvidedthecovariancematriΣ of the random vector W () is positive definite, the intensity function is λ (z) = c ν ν k+1 2 (ν 2)/2 π k/2 det(σ ) 1/2 a (z,ν) (k+ν)/2 ( ) k + ν k Γ z (1 ν)/ν 2 =1 for z (0, + ) k and with a (z,ν)=(z 1/ν ) T Σ 1 z 1/ν.SeeRibatet(2013).,

11 Conditional Simulation of Ma-Stable Processes Distribution of the etremal functions The distribution of the etremal function appearing in Equation (1.5) is still quite theoretical at this stage and it is unclear how one can sample from this distribution. There are also theoretical questions related to its definition: the intensity measure Λ has an infinite total mass and the conditioning event {f( τ )=z τ,f( τ c ) < z τ c } has zero measure. Awaytobypassthesedifficultiesistoconsideronlythefinitedimensionalmargins of the etremal functions ϕ + and not the full random process. In practice, this is enough for simulation purpose since one simulate the random field on a finite grid only and not on the whole state space X.Lets =(s 1,...,s m ) X m be the set of new locations for the conditional sampling, we focus on the conditional distribution of Z(s) Z() =z. Equation(1.5)canbesimplifiedifweassumetheregularity of the intensity measure Λ at (, s) X k+m.wecanthenintroducetheconditional intensity function Equation (1.5) can be rewritten as λ s,z (u) = λ (s,)(u, z), u (0, ) m. λ (z) P [ ϕ + (s) dv Z() =z,θ = τ] = 1 ( ) 1 {u <z C τ c }λ (s,τ c ) τ,z τ (v, u )du dv (1.6) with C the normalization constant C = 1 {u <z τ c }λ (s,τ c ) τ,z τ (v, u )du dv. In words, the conditional law of ϕ + (s) is equal to the distribution of the random variable V obtained as the first component of (V, U ) with density λ (s,τ c ) τ,z τ (v, u) conditioned to the event U < z τ c. Eample 1 continued: Brown-Resnick process In the case of Brown-Resnick process, for all (s, ) X m+k, (u, z) (0, ) m+k and provided the covariance matri Σ (s,) is positive definite, the conditional intensity function corresponds to a multivariate log-normal probability density function λ s,z (u) = (2π) m/2 det(σ s ) 1/2 { ep 1 } 2 (log u µ s,z) T Σ 1 s (log u µ m s,z) u 1 i, where µ s,z R m and Σ s are the mean and covariance matri of the underlying normal distribution and are given by Σ 1 s = Jm,kQ T (s,) J m,k } µ s,z = {L (s,) J m,k log z T T J m,k Q (s,) J m,k Σ s, i=1

12 12 Etreme Value Modeling and Risk Analysis: Methods and Applications with [ Idm J m,k = 0 k,m ], Jm,k = [ 0m,k where Id k denotes the k k identity matri and 0 m,k the m k null matri. Eample 2 continued: Schlather process For (s, ) X m+k, (u, z) R m+k and provided that the covariance matri Σ (s,) is positive definite, the conditional intensity function λ s,z (u) corresponds to a multivariate Student distribution λ s,z (u) = π m/2 (k +1) m/2 det( Σ) 1/2 { } (m+k+1)/2 1+ (u ( µ)t Σ 1 (u µ) Γ m+k+1 ) 2 k +1 Γ ( ), k+1 2 with k +1degrees of freedom, location parameter µ =Σ s: Σ 1 z and scale matri where Id k Σ = a (z) ( Σs Σ s: Σ 1 ) Σ :s k +1 [ ] Σs Σ Σ (s,) = s:. Σ :s Σ ], Eample 3 continued: etremal t-process The results are similar and generalize the case of Schlather processes. For (s, ) X m+k, (u, z) R m+k we set µ =Σ s: Σ 1 z 1/ν and Then we have Σ =(k + ν) 1 a (z,ν)(σ s Σ s: Σ 1 Σ :s ) λ s,z (u) = π m/2 (k + ν) m/2 det( Σ) 1/2 { } (m+k+ν)/2 1+ (u1/ν µ) T Σ 1 (u 1/ν µ) k + ν Γ ( ) m+k+ν m 2 Γ ( ) k+ν ν m u (ν 1)/ν. 2 The last term in bracket in the previous equation corresponds to the Jacobian of the mapping u u 1/ν.Hencewerecognizethattheconditionalintensityfunctionis the density of the random vector T ν where T is a Student random vector with k + ν degrees of freedom, mean µ and scale matri Σ. =1

13 Conditional Simulation of Ma-Stable Processes Conditional sampling of ma-stable processes We now consider the conditional sampling of ma-stable processes following Dombry et al. (2013). We present a 3-step sampling procedure that is quite straightforwardly derived from Theorem We particularly focus on the first step and discuss a Gibbs sampling approach for the conditional hitting scenario A 3-step procedure for conditional sampling Theorem provides the conditional distribution of the point process Φ given the observation (1.1). In practice, we are rather interested in the conditional simulation of the ma-stable process Z(s) at some location s = (s 1,...,s m ) X m.the connection is rather straightforward and is made eplicit inthenetproposition. Proposition Assume that Λ is regular at (, s) X k+m.considerthethree step procedure: 1. Draw a random partition θ P k with distribution (1.4); 2. Given θ =(τ 1,...,τ l ),drawl independent random vectors ϕ + (s),...,ϕ+ l (s) with distribution (1.6) and define the random vector Z + (s) = ma =1,...,l ϕ+ (s). 3. Independently draw a Poisson point process {ζ i } i 1 on (0, ) with intensity ζ 2 dζ and {Y i } i 1 independent copies of Y,anddefinetherandomvector Z (s) =ma i 1 ζ iy i (s)1 {ζiy i()<z}. Then the random vector Z(s) =ma{z + (s),z (s)} follows the conditional distribution of Z(s) given Z() =z. This 3-step procedure which, in general, can also be applied inthe non-regular case is illustrated on Figure on the simple case of 1-D Smith storm process with Gaussian shape. In step 3, it is worth noting that Z (s) follows the conditional distribution of Z(s) given Z() < z. It is not difficult to show that the conditional cumulative distribution function of Z(s) given Z() =z is given by P[Z(s) a Z() =z] =P[Z(s) a Z() < z] τ P k π,z (τ) l F τ, (a), =1

14 14 Etreme Value Modeling and Risk Analysis: Methods and Applications FIGURE 1.2 The 3-step procedure for simulation sampling. First line: construction of the conditional hitting scenario (step 1, left), of the etremal functions (step 2, middle) and of the associated process Z + (right). Second line: construction of the subetremal functions (step 3, left), of the associated process Z (middle) and of the conditioned ma-stable process Z =ma(z +,Z ) (right). where π,z (τ) =P[θ = τ Z() =z] is given by Equation (1.4) and F τ, (a) = P[ϕ + (s) a Z() =z, θ= τ] = {u <z τ c,v<a} λ (s, τ c ) τ,z τ (v, u )du dv {u <z τc } λ. τ τ,z τ (u )du c Gibbs sampling for the conditional hitting scenario In the above 3-step procedure for conditional sampling, step2requirestosamplethe etremal functions. As we have seen in Eamples 1, 2 and 3 where log-normaland Student distributions appear, these can be of a relatively simple structure although the imposed equality and inequality constraints may cause additional difficulties. Step 3 can be performed by an unconditional simulation of the ma-stable process Z according to an appropriate spectral representation reecting all those functions that do not respect the constraints given by the conditioning data.fordetailson unconditional simulation of ma-stable processes we refer to corresponding chapter. However, the conditional sampling of the hitting scenario (step 1) remains the most challenging step. According to Equation (1.4), it is given by π,z (τ) =P[θ = τ Z() =z] = 1 C,z l =1 ω τ

15 Conditional Simulation of Ma-Stable Processes 15 where ω τ = λ τ (z τ ) λ τ c τ,z τ (u )du. {u <z τ c } The last integral is the multivariate cumulative distribution of the conditional intensity function λ c τ τ,z τ.hencethisisamultivariatelog-normalcdfinthebrown- Resnick model and a multivariate Student cdf in the Schlather oretremal-tmodels. In the following, we assume that the weights ω τ can be accurately computed numerically and we focus on how to sample from π,z,adiscreteprobabilitymeasureonthe set P k of partition of { 1,, k }.Themaindifficultyisthatthenormingconstant C,z is unknown and a naive computation of this constant requires the computation of the weight l =1 ω τ for all τ P k.thenumberofterms,orequivalentlythecardinality of P k is given by the so-called Bell number B k.thefirst10 Bell numbers are k B k Hence determining the discrete probability π,z requires the computation of 52 weights for k = 5 and for k = 10.Evenworse,B and B so that the storage of all partitions is beyond the memory capacities of a standard computer. This is the so called phenomenon of combinatorial eplosion and we need an alternative method to the naive discrete enumeration of all possibilities. It is customary to use Monte Carlo Markov Chain sampling when the target distribution is known up to a multiplicative constant only. The aim is to construct a Markov chain with stationary distribution equal to the target distribution and with good miing properties. Two main methods eist: the Metropolis-Hasting algorithm and the Gibbs sampler. We focus here on this second option. For τ P k and {1,...,k}, letτ be the restriction of τ to the set { 1,..., k }.AsusualwithGibbssamplers,ourgoalistosimulatefrom P[θ θ = τ ], (1.7) where θ P k is a random partition which follows the target distribution π,z ( ) and τ is typically the current state of the Markov chain. It is easy to see that the number of possible updates according to (1.7) is always less than k,sothatthecombinatorial eplosion is avoided. Indeed, the point can be reallocated to any of the components of τ or to a new component with a single point. We deduce that the number of possible updates τ P k such that τ = τ is { b + l if { } is a partitioning set of τ, = l +1 if { } is not a partitioning set of τ, For illustration, consider the set { 1, 2, 3 } and let τ =({ 1, 2 }, { 3 }).Thenthe possible partitions τ such that τ 2 = τ 2 are ({ 1, 2 }, { 3 }), ({ 1 }, { 2 }, { 3 }), ({ 1 }, { 2, 3 }), (1.8)

16 16 Etreme Value Modeling and Risk Analysis: Methods and Applications while there eists only two partitions such that τ 3 = τ 3,i.e., ({ 1, 2 }, { 3 }), ({ 1, 2, 3 }). For our work we use a random scan implementation of the Gibbs sampler (Liu et al., 1995), meaning that one iteration of the Gibbs sampler selects randomly an element of {1,...,k} and then updates the current state τ according to the proposal distribution (1.7). For the sake of simplicity, we use the uniform random scan, i.e. is selected according to the uniform distribution on {1,...,k}. Figure shows two successive iterations of this random scan Gibbs sampler. FIGURE 1.3 Two successive iterations of the Gibbs sampler for the conditional hitting scenario with k =8conditioning points: after choosing a random point, the arrows show the different possible reallocations, the bold arrow representing the chosen one. The distribution (1.7) has nice properties. For all τ P k with τ = τ we have P[θ = τ θ = τ ]= π,z (τ ) τ P k π,z ( τ)1 { τ =τ } τ k=1 w τ k τ k=1 w τ k. (1.9) Since, τ and τ share many components, it can be seen that many factors in the righthand side of (1.9) cancel out ecept at most four of them. In the previouseample (1.8), the corresponding weights are 1, w {1, 2} w {1}w {2}, w {1, 2}w {3} w {1}w {2, 3} respectively. This makes the Gibbs sampler especially convenient. We finally provide some practical details on the computation of these conditional weights as well as a detailed algorithm (see Algorithm 1 in thesupplementarymaterial).

17 Conditional Simulation of Ma-Stable Processes 17 We first describe how each partition of { 1,..., k } is stored. To illustrate consider the set { 1, 2, 3 } and the partition ({ 1, 2 }, { 3 }). Withourconvention, this partition is defined as (1, 1, 2) indicating that 1 and 2 belong to the same partitioning set labeled 1 and 3 belongs to the partitioning set 2. There eist several equivalent notations for this partition: for eample one can use(2, 2, 1) or (1, 1, 3). However there is a one-one mapping between P k and the set { } Pk = (a 1,...,a k ), i {2,...,k}: 1=a 1 a i ma a +1,a i Z. 1 <i Consequently we shall restrict our attention to the partitions that live in Pk and going back to our eample we see that (1, 1, 2) is valid but (2, 2, 1) and (1, 1, 3) are not. For τ Pk of size l, letr 1 = k i=1 δ τ i=a and r 2 = k i=1 δ τ i=b, i.e.,the number of conditioning locations that belong to the partitioning sets a and b where b {1,...,b + } and { b + l (r 1 =1), = l +1 (r 1 1). Then the conditional probability distribution P[τ = b τ i = a i, i ] given by Equation (1.9) is proportional to 1 (b = a ), (1.10a) w τ,b/(w τ,b w τ,a ) (r 1 =1,r 2 0,b a ), (1.10b) w τ,bw τ,a /(w τ,b w τ,a ) (r 1 1,r 2 0,b a ), (1.10c) w τ,bw τ,a /w τ,a (r 1 1,r 2 =0,b a ), (1.10d) where τ =(a 1,...,a 1,b,a +1,...,a k ) and w τ,ai = w {k : τ k =a i}. Itisworth stressing that although τ does not necessarily belongs to Pk,itcorrespondstoa unique partition of P k and we can use the biection P k Pk to recode τ into an element of Pk. 1.4 Simulation Study Gibbs Sampler In this section we check whether the Gibbs sampler is able to sample from π,z ( ). To illustrate a typical sample path, Figure 1 in the supplementary material shows the trace plot of a simulated chain of length 2000 with k =5conditioning locations and athinninglagof5 and compares the theoretical probabilities {π,z (τ),τ P k } to the empirical ones estimated from the Markov chain. As mentioned in the previous sectionit can be seen that only a fewstates have asignificant probability to occur and these states most often differs only slightly. As epected for this particular simulation the empirical probabilities match the theoretical ones.

18 18 Etreme Value Modeling and Risk Analysis: Methods and Applications To better assess if our uniform random scan Gibbs sampler is able to sample from π,z ( ) we simulate 250 Markov chains of length 1000 after removing a burnin period and thinning the chain, and similarly to Figure??, comparethetheoretical probabilities {π,z (τ),τ P k } to the empirical ones using a χ 2 test for each of these chains. Since the computation of these theoretical probabilities is CPU intensive, the number of conditioning locations is at most 5 in our simulation study. Sample χ 2 p values K.S. p value = Uniform quantiles Sample χ 2 p values K.S. p value = Uniform quantiles Sample χ 2 p values K.S. p value = Uniform quantiles Sample χ 2 p values K.S. p value = Uniform quantiles FIGURE 1.4 QQ-plots for the sample χ 2 test p-values against U(0, 1) quantiles with a varying number of conditioning locations k with an insert showing the p-values of a Kolmogorov Smirnov test for uniformity from left to right, k = 2, 3, 4 and 5. The dashed lines show the 95% confidence envelopes. Figure 1.4 plots the sample p-values of these χ 2 tests against the quantiles of a U(0, 1) distribution for a varying number of conditional locations. This figure corroborates what we saw in Figure?? since the sample p-values seem to follow a U(0, 1) distribution indicating that the sampler is able to sample from π,z ( ) Conditional simulations In this section we check if our algorithm is able to produce realistic conditional simulations of Brown Resnick processes with semi-variogram γ(h) =(h/λ) ν.inthis case, the spectral process Y ( ) in (1.2) is a fractional Brownian motion with Hurst inde ν.tohaveabroadoverview,weconsiderthreedifferentsamplepathproperties as summarized below. Sample path properties γ 1 :Wiggly γ 2 :Smooth γ 3 :Verysmooth λ ν The variogram parameters are set to ensure that the etremal coefficient function satisfies θ(115) = 1.7.Figure1.5showsonerealizationforeachsamplepathconfiguration as well as the corresponding etremal coefficient function. These realizations will serve as the basis for our conditioning events. In order to check if our sampling procedure is accurate, givenasingleconditional event {Z() = z}, we generated 1000 conditional realizations of a Brown Resnick processes with standard Gumbel margins and semi-variograms γ ( =1, 2, 3).

19 Conditional Simulation of Ma-Stable Processes 19 Z() Z() Z() θ(h) h FIGURE 1.5 Three realizations of a Brown Resnick process with standard Gumbel margins and semi-variograms γ 1, γ 2 and γ 3 from left to right. The squares correspond to the 15 conditioning values that will be used in the simulation study. The right panel shows the associated etremal coefficient functions where the solid, dashed and dotted lines correspond respectively to γ 1,γ 2 and γ 3. Figure 1.6 shows the pointwise sample quantiles obtained from these 1000 simulated paths in comparison to unit Gumbel quantiles. As epected the conditional sample paths inherit the regularity driven by the Hurst inde ν/2 of the process Y ( ) in (1.2) and there is less variability in regions close to some conditioninglocations. On the opposite, although for this specific type of variogram θ(h) 2 as h, for regions far away from any conditioning location the sample quantiles converges to that of a standard Gumbel distribution indicating that the conditionaleventdoes not have any influence anymore. In addition the sample paths used to get the conditional events, see Figure 1.5, lay most of the time in the 95% pointwise confidence intervals corroborating that our sampling procedure seems to be accurate the coverage ranges between 0 93 and 1 00 with a mean value of So far we have checked that the proposed sampling procedure yields the epected coverage as well as the right marginal properties as we move far away from any conditioning location. The last point to be fulfilled is to assess whether the simulation procedure honors the spatial dependence driven by the semi-variogram γ( ). To this aim we use the F -madogram (Cooley et al., 2006) to compare the pairwise etremal coefficient estimates to the theoretical etremal coefficient function. Since Z( ) {Z() =z} is not ma-stable the F -madogram cannot be used. However, by integrating outtheconditional event werecover theoriginal Brown-Resnick distribution and the ma-stability property. So we generate independently 1000 conditional events {Z() =z}, being fied, and for each conditional event one conditional realization of a Brown Resnick process. Figure 1.7 compares thepairwiseetremal coefficient estimates based on these simulations to the theoretical etremal coefficient function. As epected, whatever the number of conditioning location and the semi-variograms are, the (binned) pairwise estimates match thetheoreticalcurveindicating that the spatial dependence is honored.

20 20 Etreme Value Modeling and Risk Analysis: Methods and Applications Z() Z() Z() Coverage (%): Coverage (%): Coverage (%): Z() Z() Z() Coverage (%): Coverage (%): Coverage (%): Z() Z() Z() Coverage (%): 100 Coverage (%): Coverage (%): FIGURE 1.6 Pointwise sample quantiles estimated from 1000 conditional simulations of Brown Resnick processes with standard Gumbel margins and semi-variograms γ 1, γ 2 and γ 3 (leftto right)and with k = 5, 10, 15 conditioning locations top to bottom. The solid black lines show the pointwise 0 025, 0 5, sample quantiles and the dashed grey lines that of a standard Gumbel distribution. The squares show the conditional points {( i,z i )} i=1,...,k.thesolidgreylinescorrespondthesimulatedpathsusedto get the conditioning events. The inserts give the proportion of points lying in the 95% pointwise confidence intervals.

21 Conditional Simulation of Ma-Stable Processes 21 θ(h) h θ(h) h FIGURE 1.7 Comparison of the etremal coefficient estimates (using a binned F -madogram with 250 bins) and the theoretical etremal coefficient function for a varying number of conditioning locations and different semi-variograms with k =5(left) or k =15 (right) conditioning points. The o, + and symbolscorrespondrespectivelyto γ 1, γ 2 and γ 3.Thesolid,dashedanddottedgreylinescorrespondtothetheoretical etremal coefficient functions for γ 1,γ 2 and γ 3. θ(h) h Acknowledgements M. Oesting and M. Ribatet acknowledge the ANR proect McSim for financial support. References Chilès, J.-P. and Delfiner, P. (1999), Geostatistics, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley&Sons,Inc.,NewYork, modeling spatial uncertainty, A Wiley-Interscience Publication. Cooley, D., Naveau, P., and Poncet, P. (2006), Variograms for spatial ma-stable random fields, in Dependence in probability and statistics, Springer,NewYork, vol. 187 of Lecture Notes in Statist.,pp Davis, R. A. and Resnick, S. I. (1989), Basic properties and prediction of ma- ARMA processes, Adv. in Appl. Probab., 21, (1993), Predictionofstationaryma-stableprocesses, Ann. Appl. Probab., 3, de Haan, L. (1984), A spectral representation for ma-stable processes, Ann. Probab., 12,

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