Conditional simulation of max-stable processes

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1 Biometrika (1),,, pp C 7 Biometrika Trust Printed in Great Britain Conditional simulation of ma-stable processes BY C. DOMBRY, F. ÉYI-MINKO, Laboratoire de Mathématiques et Application, Université de Poitiers, Téléport, BP 3179, F-8696 Futuroscope-Chasseneuil cede, France 8 clement.dombry@math.univ-poitiers.fr, frederic.eyi.minko@math.univ-poitiers.fr AND M. RIBATET Department of Mathematics, Université Montpellier, 4 place Eugène Bataillon, 3495 cede Montpellier, France mathieu.ribatet@math.univ-montp.fr SUMMARY Since many environmental processes are spatial in etent, a single etreme event may affect several locations, and their spatial dependence has to be appropriately taken into account. This paper proposes a framework for conditional simulation of ma-stable processes and gives closed forms for the regular conditional distributions of Brown Resnick and Schlather processes. We test the method on simulated data and give an application to etreme rainfall around Zurich and etreme temperatures. The proposed framework provides accurate conditional simulations and can handle real-sized problems. Some key words: Conditional simulation; Markov chain Monte Carlo; Ma-stable process; Precipitation; Regular conditional distribution; Temperature

2 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET 1. INTRODUCTION Ma-stable processes arise naturally when studying etremes of stochastic processes and therefore play a major role in the statistical modelling of spatial etremes (Buishand et al, 8; Padoan et al., 1; Davison et al., 1). Although a different spectral characterization of ma-stable processes eists (de Haan, 1984), for our purposes the most useful representation is (Schlather, ) Z() = ma i 1 ζ iy i (), R d, (1) where {ζ i } i 1 are the points of a Poisson process on (, ) with intensity dλ(ζ) = ζ dζ and the Y i are independent replicates of a non-negative stochastic process Y such that E{Y ()} = 1 for all R d. It is well known that Z is a ma-stable process on R d with unit Fréchet margins (Schlather, ; de Haan & Fereira, 6, page 37). Although (1) takes the pointwise maimum over an infinite number of points {ζ i } and processes Y i, it is possible to get approimate realizations from Z (Schlather, ; Oesting et al., 1). Based on (1) several parametric ma-stable models have been proposed (Schlather, ; Brown & Resnick, 1977; Kabluchko et al., 9; Davison et al., 1) that share the same finitedimensional distribution functions [ { pr{z( 1 ) z 1,..., Z( k ) z k } = ep E where k N, z 1,..., z k > and 1,..., k R d. ma j=1,...,k }] Y ( j ), z j Apart from the Smith model (Genton et al., 11), only the bivariate cumulative distribution functions are eplicitly known. To bypass this hurdle to inference based on ma-stable processes, de Haan & Pereira (6) propose a semi-parametric estimator and Padoan et al. (1) suggest the use of the maimum pairwise likelihood estimator

3 Conditional simulation of ma-stable processes 3 Paralleling the use of the variogram in classical geostatistics, the etremal coefficient function (Schlather & Tawn, 3; Cooley et al., 6) θ( 1 ) = z log pr{z( 1 ) z, Z( ) z} is widely used to summarize the spatial dependence of etremes for stationary processes. It takes values in the interval [1, ]; the lower bound indicates complete dependence, and the upper bound independence. The last decade has seen many advances to develop geostatistics for etremes, and software is available to practitioners (Wang, 1; Schlather, 1; Ribatet, 11). However an important tool currently missing is conditional simulation of ma-stable processes. In classical geostatistic based on Gaussian models, conditional simulation is well established (Chilès & Delfiner, 1999) and provides a framework to assess the distribution of a Gaussian random field given values observed at fied locations. For eample, conditional simulations of Gaussian processes have been used to model land topography (Mandelbrot, 198). Conditional simulation of ma-stable processes is a long-standing problem (Davis & Resnick, 1989, 1993). Wang & Stoev (11) provide a first solution, but their framework is limited to processes having a discrete spectral measure and thus may be too restrictive to appropriately model spatial dependence in comple situations. Based on the recent developments on the regular conditional distribution of ma-infinitely divisible processes, the aim of this paper is to provide a methodology for conditional simulation of ma-stable processes with continuous spectral measures. More formally for a study region X R d, our goal is to derive an algorithm to sample from the regular conditional distribution of Z {Z( 1 ) = z 1,..., Z( k ) = z k } for some z 1,..., z k > and k conditioning locations 1,..., k X

4 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET. CONDITIONAL SIMULATION OF MAX-STABLE PROCESSES 1. General framework This section reviews some key results of an unpublished paper available from the first author with a particular emphasis on ma-stable processes. Our goal is to give a more practical interpretation of their results from a simulation perspective. To this aim, we recall two key results and propose a procedure to generate conditional realizations of ma-stable processes. Let R X be the space of real-valued functions on X R d and let Φ = {ϕ i } i 1 be a Poisson point process on R X where ϕ i () = ζ i Y i () (i = 1,,...) with ζ i and Y i as in (1). We write f() = {f( 1 ),..., f( k )} for all random functions f : X R and = ( 1,..., k ) X k. It is not difficult to show that for all Borel sets A R k, the Poisson process {ϕ i ()} i 1 defined on R k has intensity measure Λ (A) = pr {ζy () A} ζ dζ. The point process Φ is called regular if the intensity measure Λ has an intensity function λ, i.e., Λ (dz) = λ (z)dz, for all X k. The first key result is that provided the Poisson process Φ is regular, the intensity function λ and the conditional intensity function λ s,z (u) = λ (s,)(u, z), (s, ) X m+k, u R m, z (, + ) k, () λ (z) drive how the conditioning terms {Z( j ) = z j } (j = 1,..., k) are met, see Steps 1 and in Theorem 1. The second key result is that, conditionally on Z() = z, the Poisson process Φ can be decomposed into two independent point processes, say Φ = Φ Φ +, where Φ = {ϕ Φ: ϕ( i ) < z i, i = 1,..., k}, Φ + = k {ϕ Φ: ϕ( i ) = z i }. i=

5 Conditional simulation of ma-stable processes 5 Before introducing a procedure to get conditional realizations of ma-stable processes, we introduce notation and make connections with the pioneering work of Wang & Stoev (11). A function ϕ Φ + such that ϕ( i ) = z i for some i {1,..., k} is called an etremal function associated to i and denoted by ϕ + i. It is easy to show that there eists almost surely a unique etremal function associated to i. Although Φ + = {ϕ + 1,..., ϕ + k } almost surely, it might happen that a single etremal function contributes to the random vector Z() at several locations i, e.g., ϕ + 1 = ϕ +. To take such repetitions into account, we define a random partition θ = (θ 1,..., θ l ) of the set { 1,..., k } into l = θ blocks, and etremal functions (ϕ + 1,..., ϕ+ l ) such that Φ+ = {ϕ + 1,..., ϕ+ l } and ϕ+ j ( i) = z i if i θ j and ϕ + j ( i) < z i if i / θ j (i = 1,..., k; j = 1,..., l). Wang & Stoev (11) call the partition θ the hitting scenario. The set of all possible partitions of { 1,..., k }, denoted P k, identifies all possible hitting scenarios. From a simulation perspective, the fact that Φ and Φ + are independent given Z() = z is especially convenient and suggests a three step procedure to sample from the conditional distribution of Z given Z() = z. THEOREM 1. Suppose that the point process Φ is regular and let (, s) X k+m. For τ = (τ 1,..., τ l ) P k and j = 1,..., l, define I j = {i: i τ j }, τj = ( i ) i Ij, z τj = (z i ) i Ij, τ c j = ( i ) i/ Ij and z τ c j = (z i ) i/ Ij. Consider the three step procedure: Step 1. Draw a random partition θ P k with distribution π (z, τ) = pr {θ = τ Z() = z} = where the normalization constant is C(, z) = τ τ P k j=1 1 C(, z) τ j=1 λ τj (z τj ) λ τ c τj,z τj (u j )du j, {u j <z τ c } j j λ τj (z θj ) λ τ c (u τj,z τj j )du j. {u j <z τ c } j j

6 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET Step. Given τ = (τ 1,..., τ l ), draw l independent random vectors ϕ + 1 (s),..., ϕ+ l (s) from the distribution } pr {ϕ + j (s) dv Z() = z, θ = τ = 1 { C j where 1 { } is the indicator function and C j = 1 {u<zτ c j }λ (s,τ c j ) τj,z τj (v, u)dudv, and define the random vector Z + (s) = ma j=1,...,l ϕ + j (s). } 1 {u<zτ c j }λ (s,τ c ) τj,z τj (v, u)du dv, j Step 3. Independently draw a Poisson point process {ζ i } i 1 on (, ) with intensity ζ dζ and {Y i } i 1 independent copies of Y, and define the random vector Z (s) = ma i 1 ζ iy i (s)1 {ζi Y i ()<z}. Then the random vector Z(s) = ma {Z + (s), Z (s)} follows the conditional distribution of Z(s) given Z() = z. The corresponding conditional cumulative distribution function is τ pr{z(s) a, Z() z} pr {Z(s) a Z() = z} = π (z, τ) F τ,j (a), (3) pr{z() z} τ P k j=1 where } F τ,j (a) = pr {ϕ + j (s) a Z() = z, θ = τ = {u<z τ c,v<a} λ (s, τ c ) τj,z τj (v, u)dudv j j {u<z τ c } λ. τ c τj,z τj (u)du j j The first term in the right hand side of (3) corresponds to Steps 1 and while the ratio is a consequence of Step 3. It is clear from (3) that the conditional random field Z {Z() = z} is not ma-stable.. Distribution of the etremal functions In this section we derive closed forms for the intensity function λ (z) and the conditional intensity function λ s,z (u) for two widely used ma-stable processes; the Brown Resnick (Brown

7 Conditional simulation of ma-stable processes 7 & Resnick, 1977; Kabluchko et al., 9) and the Schlather () processes. Details of the derivations are given in the Appendi. The Brown Resnick process corresponds to the case where Y () = ep{w () γ()} ( R d ) in (1) with W a centered Gaussian process with stationary increments, semivariogram γ and such that W (o) = almost surely where o denotes the origin of R d. For X k and provided the covariance matri Σ of the random vector W () is positive definite, the intensity function is ( λ (z) = C ep 1 ) k log zt Q log z + L log z zi 1, z (, ) k, with 1 k = (1) i=1,...,k, γ = {γ( i )} i=1,...,k, Q = Σ 1 Σ 1 1 k 1 T k Σ 1 1 T, L = k Σ 1 1 k C = (π) (1 k)/ Σ 1/ (1 T k Σ 1 1 k ) 1/ ep ( 1 T k Σ 1 γ 1 1 T k Σ 1 i=1 1 k γ 1 k { 1 ) T Σ 1, } (1 T k Σ 1 γ 1) 1 T 1 k Σ 1 1 k γt Σ 1 γ, and for all (s, ) X m+k, (u, z) (, ) 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) = (π) m/ Σ s 1/ ep 1 } (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 with } Σ 1 s = J m,k T Q (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, J m,k = I m k,m, Jm,k = m,k I k, where I k denotes the k k identity matri and m,k the m n null matri. i=

8 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET The Schlather process considers the case Y () = (π) 1/ ma{, ε()} ( R d ) in (1) with ε a standard Gaussian process with correlation function ρ. The associated point process Φ is not regular and it is more convenient to consider the equivalent representation where Y () = (π) 1/ ε() ( R d ). For X k and provided the covariance matri Σ of the random vector ε() is positive definite, the intensity function is ( ) k + 1 λ (z) = π (k 1)/ Σ 1/ a (z) (k+1)/ Γ, z R k, where a (z) = z T Σ 1 z. 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 the density of a multivariate Student distribution with k + 1 degrees of freedom, location parameter µ = Σ s: Σ 1 z, and scale matri Σ = a (z) ( Σs Σ s: Σ 1 ) Σ :s, Σ(s,) = k + 1 Σ s Σ :s 3. MARKOV CHAIN MONTE CARLO SAMPLER Section introduced a procedure to get realizations from the regular conditional distribution of ma-stable processes. This sampling scheme amounts to sampling from a discrete distribution whose state space corresponds to all possible partitions of the set of conditioning points, see Step 1 of Theorem 1. Hence, even for a moderate number k of conditioning locations, the state space P k becomes very large and the distribution π (z, ) cannot be computed eactly. A Gibbs sampler is especially convenient for Monte Carlo sampling from π (z, ). For τ P k, let τ j be the restriction of τ to the set { 1,..., k } \ { j }. Our goal is to simulate from the conditional distribution Σ s: Σ. pr(θ θ j = τ j ), (4)

9 Conditional simulation of ma-stable processes 9 where θ P k is a random partition which follows the target distribution π (z, ). Since the number of possible updates is always less than k, a combinatorial eplosion is avoided. Indeed for τ P k of size l, the number of partitions τ P k such that τ j = τ j for some j {1,..., k} is l if { j } is a partitioning set of τ, b + = l + 1 if { j } is not a partitioning set of τ, since the point j may be reallocated to any partitioning set of τ j or to a new one. To illustrate consider the set { 1,, 3 } and let τ = ({ 1, }, { 3 }). Then the possible partitions τ such that τ = τ are ({ 1, }, { 3 }), ({ 1 }, { }, { 3 }), ({ 1 }, {, 3 }), while there eist only two partitions such that τ 3 = τ 3, i.e., ({ 1, }, { 3 }), ({ 1,, 3 }). The distribution (4) has nice properties. Since for all τ P k such that τ j = τ j we have where pr (θ = τ θ j = τ j ) = π (z, τ ) τ P k π (z, τ)1 { τ j =τ j } w τ,j = λ τj (z τj ) λ τ c τj,z τj (u)du. {u<z τ c } j j τ j=1 w τ,j τ j=1 w, (5) τ,j Since many factors cancel out on the right hand side of (5), the Gibbs sampler is especially convenient. The most computationally demanding part of (5) is the evaluation of the integral {u<z τ c j } λ τ c j τj,z τj (u)du. For the Brown Resnick and Schlather processes, we follow the lines of Genz (199) and compute these probabilities using a separation of variables method which provides a transformation of the original integration problem to the unit hyper-cube. A quasi Monte Carlo scheme and antithetic variables are used to improve efficiency

10 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET Since it is not obvious how to implement a Gibbs sampler whose target distribution has support P k, the remainder of this section gives practical details. For any fied locations 1,..., k X, we first describe how each partition of { 1,..., k } is stored. To illustrate this consider the set { 1,, 3 } and the partition ({ 1, }; { 3 }). This partition is defined as (1, 1, ), indicating that 1 and belong to the partitioning set labeled 1 and 3 belongs to the partitioning set. There eist several equivalent notations for this partition: for eample one can use (,, 1) or (1, 1, 3). Since there is a one-one mapping between P k and the set } Pk {(a = 1,..., a k ): i {,..., k}, 1 = a 1 a i ma a j + 1, a i Z, 1 j<i we shall restrict our attention to the partitions that live in Pk, and going back to our eample we see that (1, 1, ) is valid but (,, 1) and (1, 1, 3) are not. For τ P k of size l, let r 1 = k i=1 1 {τ i =a j } and r = k i=1 1 {τ i =b}, i.e., the number of conditioning locations that belong to the partitioning sets a j and b where b {1,..., b + } with l (r 1 = 1), b + = l + 1 (r 1 1). Then the conditional probability distribution (5) satisfies pr(τ j = b τ i = a i, i j) 1 (b = a j ), (6a) w τ,b/(w τ,b w τ,aj ) w τ,bw τ,a j /(w τ,b w τ,aj ) w τ,bw τ,a j /w τ,aj (r 1 = 1, r, b a j ), (6b) (r 1 1, r, b a j ), (6c) (r 1 1, r =, b a j ), (6d) where τ = (a 1,..., a j 1, b, a j+1,..., a k ). Although τ may not belong to Pk, it corresponds to a unique partition of P k and we can use the bijection P k P k to recode τ into an element of P k. In (6a) (6d) the event {r 1 = 1, r =, b a j } is missing since {r 1 = 1, r = } implies

11 Conditional simulation of ma-stable processes 11 Table 1. Sample path properties of the ma-stable models. For the Brown Resnick model, the variogram parameters are set to ensure that the etremal coefficient function satisfies θ(115) = 1 7 while for the Schlather model the correlation function parameters are set to ensure that θ(1) = 1 5. Brown Resnick: γ(h) = (h/λ) κ Schlather: ρ(h) = ep{ (h/λ) κ } γ 1: Very wiggly γ : Wiggly γ 3: Smooth ρ 1: Very wiggly ρ : Wiggly ρ 3: Smooth λ κ that τ = τ, where the equality has to be understood in terms of elements of P k, and this case has been already taken into account with (6a). Once these conditional weights have been computed, the Gibbs sampler proceeds by updating each element of τ successively. We use a random scan implementation (Liu et al., 1995). One iteration selects an element of τ at random according to a given distribution, say p = (p 1,..., p k ), and then updates this element. Throughout this paper we will use the uniform random scan Gibbs sampler for which the selection distribution is a discrete uniform distribution, i.e., p = (k 1,..., k 1 ) SIMULATION STUDY In this section we check if our algorithm is able to produce realistic conditional simulations of Brown Resnick and Schlather processes. For each model, we consider three different sample path properties, as summarized in Table 1. These configurations were chosen such that the spatial dependence structures are similar to our applications in Section 5. In order to check if our sampling procedure is accurate and given a single conditional event {Z() = z} for each configuration, we generated 1 conditional realizations with standard

12 59 1 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET Z() 3 1 Z() 3 1 Z() Z() 1 Z() 1 Z() Fig. 1. Pointwise sample quantiles estimated from 1 conditional simulations of ma-stable processes with standard Gumbel margins with k = 5, 1, 15 conditioning locations. The top row shows results for the Brown Resnick models with semivariograms γ 3, γ, γ 1, from left to right. The bottom row shows results for the Schlather models with correlation functions ρ 3, ρ, ρ 1, from left to right. The solid black lines shows the pointwise 5, 5, 975 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. The solid grey lines show the simulated paths used to get the conditioning events. Gumbel margins. Figure 1 shows the pointwise sample quantiles obtained from these 1 simulated paths and compares them to unit Gumbel quantiles. As epected the conditional sample paths inherit the regularity driven by the shape parameter κ and there is less variability in regions close to conditioning locations. Since the Brown Resnick processes considered are ergodic

13 Conditional simulation of ma-stable processes 13 Table. Timings for conditional simulations of ma-stable processes on a 5 5 grid defined on the square [, 1 1/ ] for a varying number k of conditioning locations uniformly distributed over the region. The times, in seconds, are mean values over 1 conditional simulations; standard deviations are reported in brackets. Brown Resnick: γ(h) = (h/5) 5 Schlather: ρ(h) = ep { (h/8) 5} Step 1 Step Step 3 Overall Step 1 Step Step 3 Overall k = 5 1 ( 1) 49 (11) 1 4 ( 1) 5 (11) 1 4 ( ) 1 9 ( 7) 9 ( 3) 4 ( 8) k = 1 8 () 76 (18) 1 4 ( 1) 85 (19) 1 (4) 4 ( 8) 1 ( 3) 15 (4) k = 5 95 (38) 117 (3) 1 4 ( 1) 14 (61) 86 (4) 4 (1) 1 ( 3) 9 (43) k = (313) 348 (391) 1 5 ( 1) 931 (535) 367 () 6 (113) 1 ( 3) 43 (6) Conditional simulations with k = 5 do not use a Gibbs sampler (Kabluchko and Schlather, 1), for regions far away from any conditioning location the sample quantiles converges to that of a standard Gumbel distribution, indicating that the conditional event has no influence. This is not the case for the non-ergodic Schlather process. Most of the time the sample paths used to get the conditional events belong to the 95% pointwise confidence intervals, corroborating that our sampling procedure seems to be accurate. Table gives timings for conditional simulations of ma-stable processes on a 5 5 grid with a varying number of conditioning locations. Due to the combinatorial compleity of the partition set P k, the timings increase rapidly with respect to the number of conditioning points k. It is however reassuring that the algorithm is tractable when k 5; this covers many practical situations

14 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET (m) (km) Maimum rainfall in Zurich (mm) θ(h) Years Fig.. Left: Map of Switzerland showing the stations of the 4 rainfall gauges used for the analysis, with an insert showing the altitude. The station marked with a triangle corresponds to Zurich. Middle: Summer maimum rainfall values for at Zurich. Right: Comparison between the pairwise etremal coefficient estimates for the 51 original weather stations and the etremal coefficient function derived from a fitted Brown Resnick processes having semi variogram γ(h) = (h/λ) κ. The grey points are pairwise estimates; the black ones are binned estimates and the curve is the fitted etremal coefficient function h (km) APPLICATION 5 1. Etreme precipitation around Zurich The data considered here were previously analyzed by Davison et al. (1), who showed that Brown Resnick processes were among the best models for these data. The data are summer maimum rainfall for the years at 51 weather stations in the Plateau region of Switzerland, provided by the national meteorological service, MeteoSuisse. To ensure strong dependence between the conditioning locations, we consider as conditional locations the 4 weather stations that are at most 3km distant from Zurich and set as the conditional values the rainfall amounts recorded in the year, the year of the largest precipitation event recorded there between

15 Conditional simulation of ma-stable processes 15 Table 3. Distribution of the partition size for the rainfall and temperature data estimated from simulated Markov chains of length 15 and 1 respectively 675 Rainfall Temperature Partition size Frequencies (%) < , see Figure. The largest and smallest distances between the conditioning locations are around 55km and just over 4km respectively. A Brown Resnick process having semivariogram γ(h) = (h/λ) κ was fitted using the maimum pairwise likelihood estimator introduced by Padoan et al. (1) to simultaneously estimate the marginal parameters and the spatial dependence parameters λ and κ. As in Davison et al. (1), the marginal parameters were described by η() = β,η + β 1,η lon() + β,η lat(), σ() = β,σ + β 1,σ lon() + β,σ lat(), ξ() = β,ξ, where η(), σ(), ξ() are the location, scale and shape parameters of the generalized etreme value distribution and lon(), lat() denote the longitude and latitude of the stations at which the data are observed. The maimum pairwise likelihood estimates and standard errors for λ and κ are 38 (14) and 69 ( 7) and give a practical etremal range, i.e., the distance h + such that θ(h + ) = 1 7, of around 115km; see the right panel of Figure. Table 3 shows the distribution of the partition size estimated from a Markov chain of length 15. Around 65% of the time the summer maima observed at the 4 conditioning locations belonged to a single etremal function, i.e., only one storm event, and around 3% of the time to two different storms. Since the simulated Markov chain keeps a trace of all the simulated partitions, we looked at the partitions of size two and saw that around 65% of the time, at least one of the four up north conditioning locations was impacted by one etremal function while the remaining locations were always influenced by another one

16 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET Fig. 3. From left to right, maps on a 5 5 grid of the pointwise 5, 5 and 975 sample quantiles for rainfall (mm) obtained from 1 conditional simulations of Brown Resnick processes having semi variogram γ(h) = (h/38) 69. The rightmost panel plots the ratio of the pointwise confidence intervals with and without taking estimation uncertainties into account. The squares show the conditional locations Figure 3 plots the pointwise 5, 5 and 975 sample quantiles obtained from 1 conditional simulations of our fitted Brown Resnick process. The conditional median provides a point estimate for the rainfall at an ungauged location and the 5 and 975 conditional quantiles a 95% pointwise confidence interval. As indicated by Figure 1, the shape parameter κ has a major impact on the regularity of paths and on the width of the confidence interval. The value ˆκ 69 corresponds to very wiggly sample paths and wider confidence intervals. To assess the impact of parameter uncertainties on conditional simulations, the ratio of the width of the confidence intervals with and without parameter uncertainty is shown in the right panel of Figure 3. The uncertainties were taken into account by sampling from the asymptotic distribution of the maimum composite likelihood estimator, and drawing one conditional simulation for each realization. These ratios show no clear spatial pattern and the width of the confidence interval is increased by at most 1%

17 Basel (316) Zurich (556) Santis (49) Oeschberg (483) Neuchatel (485) Bad Ragaz (496) Engelberg (135) Bern (565) Chateau d Oe (985) Montreu (45) Montana (158) Gd St Bernard (47) (m) Conditional simulation of ma-stable processes 17 Locarno Monti (366) Lugano (73) Davos (159) Arosa (184) (km) Fig. 4. Left: Topographical map of Switzerland showing the sites and altitudes in metres above sea level of 16 weather stations for which annual maima temperature data are available. Right: Map of temperature anomalies ( C), i.e., the difference between the pointwise medians obtained from 1 conditional simulations and unconditional medians estimated from the fitted Schlather process. 5. Etreme temperatures in Switzerland The data considered here are annual maimum temperatures recorded at 16 sites in Switzerland during the period , see Figure 4. Following Davison and Gholamrezaee (1), we fit a Schlather process with an isotropic powered eponential correlation function and trend surfaces η() = β,η + β 1,η alt(), σ() = β,σ, ξ() = β,ξ + β 1,ξ alt(), where alt() denotes the altitude above mean sea level in kilometres and {η(), σ(), ξ()} are the location, scale and shape parameters of the generalized etreme value distribution at location. The spatial dependence parameter estimates and their standard errors are ˆλ = 6 (149) and ˆκ =.5 (.1) and yield a fitted etremal coefficient function similar to our test case ρ 3 in Section 4. In year 3, western Europe endured a heatwave believed to be most severe recorded since 154 (Beniston, 4). Switzerland was severely impacted by this event: the nationwide record temperature of 41 5 C was recorded that year in Grono, Graubünden, near Lugano. For our analysis we condition on the maimum temperatures observed in 3. Based on the fitted Schlather ( C)

18 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET model, we simulate a Markov chain of effective length 1 with a burn-in period of length 5 and a thinning lag of 1 iterations. The distribution of the partition size estimated from these Markov chains is shown in Table 3. Around 9% of the time the conditional realizations were a consequence of at most three etremal functions. Since our original observations were not summer maima but maimum daily values, a close inspection of the times series in year 3 reveals that the hottest temperatures occurred between the 11th and 13th of August and, to some etent, corroborates the distribution of Table 3. The right panel of Figure 4 shows the spatial distribution of temperature anomalies, i.e., the difference between the pointwise conditional medians obtained from 1 conditional simulations and the pointwise unconditional medians estimated from the fitted Schlather model. As epected, the largest deviations occur in the plateau region of Switzerland while appreciably smaller values are found in the Alps. The differences range between 5 C and 4 75 C and are consistent with the values reported by climatologists for mean values (Beniston, 4) ACKNOWLEDGEMENTS M. Ribatet was partly funded by the MIRACCLE-GICC and McSim ANR projects. The authors thank MeteoSuisse and Dr. S. A. Padoan for providing the precipitation data, Prof. A. C. Davison and Dr. M. M. Gholamrezaee for providing the temperature data and Dr. M. Oesting for pointing out an error in the computations for the Brown Resnick model APPENDIX Brown Resnick model For all X k and Borel set A R k Λ (A) = pr [ζ ep{w () γ()} A] ζ dζ = R k 1 {ζ ep{y γ()} A} f (y)dyζ dζ,

19 Conditional simulation of ma-stable processes 19 where f denotes the density of the random vector W (), i.e., a centered Gaussian random vector with covariance matri Σ and variance γ(). The change of variables z = ζ ep{y γ()} and r = log ζ yields with Since with Λ (A) = A λ (z) = f {log z r + γ()} k i=1 k i=1 z 1 i dze r dr = λ (z)dz A z 1 i f {log z r + γ()}e r dr. f {log z r + γ()}e r = (π) k/ Σ 1/ ep { 1 } P (r), P (r) = r 1 T k Σ 1 1 k r [ 1 T k Σ 1 {log z + γ()} 1 ] + {log z + γ()} T Σ 1 {log z + γ()}, standard computations for Gaussian integrals give The conditional intensity function is λ s,z (u) = C (s,) C ( λ (z) = C ep 1 ) log k zt Q log z + L log z z 1 i. { ep 1 log (u, z)t Q (s,) log(u, z) + L (s,) log(u, z) + 1 } log m zt Q log z L log z u 1 i, and since log(u, z) = J m,k log u + J m,k log z, it is not difficult to show that λ s,z (u) = C (s,) C i=1 { ep 1 } (log u µ s,z) T Σ 1 s (log u µ m s,z) u 1 i. Finally, the relation C (s,) /C = (π) m/ Σ s 1/ is a simple consequence of the normalization λs,z (u)du = 1. For all X k and Borel sets A R k Λ (A) = Schlather model { } pr (π) 1/ ζε() A ζ dζ = i=1 R k 1 {(π) 1/ ζy A} f (y)dyζ dζ, i=

20 C. DOMBRY, F. ÉYI-MINKO AND M. RIBATET where f denotes the density of the random vector ε(), i.e., a centered Gaussian random vector with covariance matri Σ. The change of variable z = (π) 1/ ζy gives { } Λ (A) = (π) k/ z f ζ (k+) dzdζ A (π) 1/ ζ ( = (π) k Σ 1/ ep 1 ) A 4πζ zt Σ 1 z ζ (k+) dzdζ { ( ) 1/ = (π) k Σ 1/ π 4π A z T Σ 1 z E[Xk 1 ]dz, X Weibull z T Σ 1, } z ( ) (k 1)/ ( ) = (π) k Σ 1/ π 4π k + 1 A z T Σ 1 z z T Σ 1 Γ dz z = λ (z)dz, A where λ (z) = π (k 1)/ Σ 1/ a (z) (k+1)/ Γ {(k + 1)/} and a (z) = z T Σ 1 z. For all u R m the conditional intensity function is λ s,z (u) = π m/ Σ (s,) 1/ { } (m+k+1)/ a(s,) (u, z) a Σ 1/ (z) m/ Γ ( ) m+k+1 a (z) Γ ( ). k+1 We start by focusing on the ratio a (s,) (u, z)/a (z). Since Σ s Σ :s Σ s: Σ 1 = Σ 1 straightforward algebra shows that a (s,) (u, z) a (z) (Σ s Σ s: Σ 1 Σ :s ) 1 (Σ s Σ s: Σ 1 Σ :s ) 1 Σ s: Σ 1, Σ :s (Σ s Σ s: Σ 1 Σ :s ) 1 Σ 1 Σ :s (Σ s Σ s: Σ 1 Σ :s ) 1 Σ s: Σ 1 + Σ 1 = 1 + (u µ)t Σ 1 (u µ), µ = Σ s: Σ 1 a (z) ( z, Σ = Σs Σ s: Σ 1 ) Σ :s. k + 1 k + 1 We now simplify the ratio Σ (s,) / Σ. Using the fact that Σ (s,) = Σ s Σ :s Σ s: Σ combined with some more algebra yields Σ (s,) Σ = I m Σ s: Σ s Σ s: Σ 1 Σ :s m,k, k,m Σ Σ 1 Σ :s I k = Σ s Σ s: Σ 1 Σ :s = { } m k + 1 Σ. a (z) Using the two previous results it is easily found that { } (m+k+1)/ λ s,z (u) = π m/ (k + 1) m/ Σ 1/ 1 + (u ( µ)t Σ 1 (u µ) Γ m+k+1 ) k + 1 Γ ( ), k

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