Conditional simulations of Brown–Resnick processescdombry.perso.math.cnrs.fr/[D20].pdf ·...

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Biometrika(2011),xx, x, pp. 1–15C© 2007 Biometrika TrustPrinted in Great Britain

Conditional simulations of Brown–Resnick processes

BY C. DOMBRY, F. EYI -M INKO ,

Laboratoire de Mathematiques et Application, Universite de Poitiers, Teleport 2, BP 30179,F-86962 Futuroscope-Chasseneuil cedex, France

[email protected], [email protected]

AND M. RIBATET

Department of Mathematics, Universite Montpellier 2, 4 place Eugene Bataillon, 34095 cedex2 Montpellier, France

[email protected]

SUMMARY

Since many environmental processes such as heat waves or precipitation are spatial in extent, itis likely that a single extreme event affects several locations and the areal modelling of extremesis therefore essential if the spatial dependence of extremes has to be appropriately taken into ac-count. Although some progress has been made to develop a geostatistic of extremes, conditionalsimulation of max-stable processes is still in its early stage. This paper proposes a frameworkto get conditional simulations of Brown–Resnick processes. Although closed forms for the reg-ular conditional distribution of Brown–Resnick processeswere recently found, sampling fromthis conditional distribution is a considerable challengeas it leads quickly to a combinatorialexplosion. To bypass this computational burden, a Markov chain Monte–Carlo algorithm is pre-sented. We test the method on simulated data and give an application to extreme rainfall aroundZurich. Results show that the proposed framework provides accurate conditional simulations ofBrown–Resnick processes and can handle real-sized problems.

Some key words: Brown–Resnick process; Conditional simulation; Gibbs sampler; Markov chain Monte–Carlo; Pre-cipitation; Regular conditional distribution.

1. INTRODUCTION

Max-stable processes are a natural extension of the extremevalue theory to the infinite dimen-sional case, i.e., extremes of stochastic processes, and therefore play a major role in the statisticalmodelling of spatial extremes (Padoan et al., 2010; Davisonet al., 2011). Although a differentspectral characterization of max-stable processes exists(de Haan, 1984), for our purposes themost useful representation is the one of Schlather (2002) that considers the process

Z(x) = maxi≥1

ζiYi(x), x ∈ Rd, (1)

where{ζi}i≥1 are the points of a Poisson process on(0,∞) with intensity dΛ(ζ) = ζ−2dζ andYi(·) are independent replicates of a non-negative continuous sample path stochastic processY (·) such thatE[Y (x)] = 1 for all x ∈ R

d. It is well known thatZ(·) is a max-stable process onRd with unit Frechet margins (de Haan & Fereira, 2006; Schlather, 2002). Although (1) amounts

http://arxiv.org/abs/1112.3891v1

2 C. DOMBRY, F. EYI -M INKO AND M. RIBATET

to compute the pointwise maximum over an infinite number of points {ζi} and processesYi(·),it is possible to get (approximate) realizations fromZ(·) (Schlather, 2002; Oesting et al., 2011).

Based on (1) several parametric max-stable models have beenproposed (Schlather, 2002;Smith, 1990; Davison et al., 2011) but in this paper we focus on Brown–Resnick processes(Brown & Resnick, 1977; Kabluchko et al., 2009) for which

Y (x) = exp {W (x)− γ(x)} , x ∈ Rd, (2)

whereW (·) is a centered Gaussian process with continuous sample path,stationary increments,(semi) variogramγ(·) and such thatW (o) = 0 almost surely. Closed forms for the bivariatedistribution function of (2) are known for a long time (Brown& Resnick, 1977; Kabluchko et al.,2009; Davison et al., 2011), i.e.,

− log pr[Z(x1) ≤ z1, Z(x2) ≤ z2] =1

z1Φ

(

a

2+

1

alog

z2z1

)

+1

z2Φ

(

a

2+

1

alog

z1z2

)

, (3)

wherea = {2γ(x1 − x2)}1/2 andΦ denotes the standard normal cumulative distribution func-

tion. Inferential procedures for fitting max-stable processes have been missing for a long timebut recently Padoan et al. (2010) suggest the use of the maximum pairwise likelihood estimatorsince closed forms for thek-variate distribution function whenk > 2 are not available.

Paralleling the use of the variogram in classical geostatistics, the extremal coefficient function(Schlather & Tawn, 2003; Cooley et al., 2006) is a widely usedsummary statistic to analyze thespatial dependence of extremes. This function takes valuesin [1, 2] and the lower bound indicatescomplete dependence while the upper one independence. From(3) it is straightforward to see thatthe extremal coefficient function for Brown–Resnick processes isθ(h) = 2Φ[{γ(h)/2}1/2 ] forall h ∈ R

d.As suggested by the preceding brief review on max-stable processes, the last decade has

seen many advances to develop a geostatistic of extremes andsoftware are already availableto practitioners (Wang, 2010; Schlather, 2011; Ribatet, 2011). However one important tool iscurrently missing: conditional simulation of max-stable processes. In classical geostatistic basedon Gaussian models, conditional simulations are well established (Chiles & Delfiner, 1999) andplay a key role since it provides a framework to assess the distribution of a Gaussian randomfield given that some values have been observed at some fixed locations. For example, condi-tional simulations of Gaussian processes have been successfully used to model oil reservoirs(Delfiner & Chiles, 1997), land topography (Mandelbrot, 1982) or even the length of a subma-rine cable to be laid between two coastal cities (Alfaro, 1979).

Conditional simulation of max-stable processes is known tobe a long-standing problem(Davis & Resnick, 1989, 1993) but recently Wang & Stoev (2011) provide a first answer to thisproblem. However their framework is limited to processes having a discrete spectral measure andmight be too restrictive to appropriately model the spatialdependence in complex situations.

The aim of this paper is to provide a methodology to get conditional simulations of Brown–Resnick processes, which have a continuous spectral measure, based on the recent developmentson the regular conditional distribution of infinitely divisible processes (Dombry & Eyi-Minko,2011). More formally for a study regionX ⊂ R

d, our goal is to derive an algorithm to sam-ple from the regular conditional distribution ofZ(·) | {Z(x1) = z1, . . . , Z(xk) = zk} for somez1, . . . , zk > 0 andk conditioning locationsx1, . . . , xk ∈ X .

In Section 2 we provide an overview of the main results of Dombry & Eyi-Minko (2011)and propose a procedure to get conditional realizations of Brown–Resnick processes. Section 3puts the emphasis on the combinatorial explosion of the regular conditional distribution andintroduces a Markov chain Monte Carlo sampler to bypass thiscomputational burden, which is

Conditional simulations of Brown–Resnick processes 3

then analyzed on a simulation study in Section 4. The paper ends with an application on extremerainfall around Zurich followed by a brief discussion.

2. CONDITIONAL SIMULATION OF BROWN–RESNICK PROCESSES

This section reviews some key results of Dombry & Eyi-Minko (2011) with a particular em-phasis on Brown–Resnick processes. Our goal is to give a morepractical interpretation of theirresults from a simulation perspective. To this aim, we recall two key results and propose a pro-cedure to get conditional realizations of Brown–Resnick processes.

Let C0 be the space of positive continuous functions onX ⊂ Rd andΦ = {ϕi}i≥1 a Poisson

point process onC0 where

ϕi(x) = ζi exp {Wi(x)− γ(x)} , (i = 1, 2, . . .)

with ζi andWi as in (1) and (2). To shorten the notations we writef(x) = {f(x1), . . . , f(xk)}for all (random) functionf : X → R andx = (x1, . . . , xk) ∈ X k. It is not difficult to show thatthe Poisson point process{ϕi(x)}i≥1 defined on(0,+∞)k has intensity measure

Λx(A) =

∫ ∞

0pr [ζ exp {W (x)− γ(x)} ∈ A] ζ−2dζ, A ⊂ (0,+∞)k Borel set.

Dombry & Eyi-Minko (2011) show that provided the distribution of the random vectorW (x)is non degenerate, i.e., the covariance matrixΣx = E[W (x)W (x)T ] is positive definite, the in-tensity measureΛx is absolutely continuous with respect to the Lebesgue measure and the den-sity of Λx is

λx(z) = Cx exp

(

−1

2log zTQx log z+ Lx log z

) k∏

i=1

z−1i , z ∈ (0,∞)k,

where

Qx = Σ−1x

−Σ−1x

1k1TkΣ

−1x

1TkΣ−1x 1k

,

Lx =1

2

(

1TkΣ−1x

σ2x− 2

1TkΣ−1x 1k

1Tk − σ2x

T

)

Σ−1x

,

Cx = (2π)(1−k)/2|Σx|−1/2(1TkΣ

−1x

1k)−1/2 exp

{

1

2

(1TkΣ−1x

σ2x− 1)2

1TkΣ−1x 1k

−1

2σ2x

TΣ−1x

σ2x

}

,

and1k = (1)i=1,...,k, σ2x= {σ2(xi)}i=1,...,k.

Interestinglyλx is closely related to a multivariate log-normal probability density functionsince for all(s,x) ∈ X (m+k) andz ∈ (0,∞)k, the functionλ

s|x,z(u) = λ(s,x)(u, z)/λx(z),u ∈(0,∞)m, is a multivariate log-normal density, i.e.,

λs|x,z(u) = (2π)−m/2|Σ

s|x|−1/2 exp

{

−1

2(log u− µ

s|x,z)TΣ−1

s|x(log u− µs|x,z)

} m∏

i=1

u−1i ,

(4)whereµ

s|x,z ∈ Rm andΣ

s|x are the mean and covariance matrix of the underlying normal dis-tribution and are given by

Σ−1s|x = JT

m,kQ(s,x)Jm,k, µs|x,z =

(

L(s,x)Jm,k − log zT J Tm,kQ(s,x)Jm,k

)

Σs|x,


with

Jm,k =

[

Idm

0k,m

]

, z =

[

z

1m

]

, and Jm,k =

[

0m,k

Idk

]

,

where Idk denotes thek × k identity matrix and0m,k them× n null matrix.For our work, equation (4) is the first key result of Dombry & Eyi-Minko (2011) since it drives

how the conditioning terms{Z(xi) = zi}i=1,...,k are met. The second key point is that, condi-tionally on Z(x) = z, the Poisson point processΦ can be decomposed into two independentPoisson point processes, sayΦ = Φ− ∪ Φ+, where

Φ− = {ϕ ∈ Φ: ϕ(xi) < zi for all i = 1, . . . , k},

Φ+ = {ϕ ∈ Φ: ϕ(xi) = zi for somei = 1, . . . , k}.

Before introducing a procedure to get conditional realizations of Brown–Resnick pro-cesses, we introduce some notations and make some connections with the pioneer work ofWang & Stoev (2011) on the regular conditional distributionof spectrally discrete max-stableprocesses.

A functionϕ ∈ Φ+ such thatϕ(xi) = zi for somei ∈ {1, . . . , k} is called an extremal func-tion associated toxi and denoted byϕ+

xi. SinceΦ is a simple point process, there exists al-

most surely a unique extremal function associated toxi. Although Φ+ = {ϕ+x1, . . . , ϕ+

xk} al-

most surely, it might happen that a single extremal functioncontributes to the random vectorZ(x) at several locationsxi, e.g.,ϕ+

x1= ϕ+

x2. To take such repetitions into account, we define

a (random) partitionθ = (θ1, . . . , θℓ) of the set{x1, . . . , xk} into ℓ = |θ| blocks and extremalfunctions(ϕ+

1 , . . . , ϕ+ℓ ) such thatΦ+ = {ϕ+

1 , . . . , ϕ+ℓ } and

ϕ+j (xi) = zi if xi ∈ θj and ϕ+

j (xi) < zi if xi /∈ θj, (i = 1, . . . , k; j = 1, . . . , ℓ).

Using the terminology of Wang & Stoev (2011), the partitionθ is called the hitting scenario. Theset of all possible partitions of{x1, . . . , xk}, denotedPk, identifies all possible hitting scenarios.

From a simulation perspective, the fact that, givenZ(x) = z, Φ− andΦ+ are independentis especially convenient and suggests a three step procedure to sample from the conditionaldistribution ofZ(·) givenZ(x) = z.

As we will see,Φ− andΦ+ are conditionally independent givenZ(x) = z. This is especiallyconvenient from a simulation perspective and suggests a three step procedure to sample from theconditional distribution ofZ(·) givenZ(x) = z.

THEOREM 1. Let (x, s) ∈ X k+m and suppose that the covariance matrixΣ(s,x) is positivedefinite. Forτ = (τ1, . . . , τℓ) ∈ Pk and j = 1, . . . , ℓ, defineIj = {i : xi ∈ τj} and notexτj =(xi)i∈Ij , zτj = (zi)i∈Ij , xτc

j= (xi)i/∈Ij and zτc

j= (zi)i/∈Ij . Consider the following three step

procedure:

Step 1.Draw a random partitionθ ∈ Pk with distribution

πx(z, τ) = pr [θ = τ | Z(x) = z] =1

C(x, z)

|τ |∏

j=1

λxτj(zτj )

∫

{uj<zτcj}λxτc

j|xτj

,zτj(uj)duj ,

where the normalization constantC(x, z) is given by

C(x, z) =∑

τ∈Pk

|τ |∏

j=1

λxτj(zθj )

∫

{uj<zτcj}λxτc

j|xτj

,zτj(uj)duj .


Step 2.Given τ = (τ1, . . . , τℓ), draw ℓ independent random vectorsϕ+1 (s), . . . , ϕ

+ℓ (s) from

the distribution

pr[

ϕ+j (s) ∈ dv | Z(x) = z, θ = τ

]

=1

Cj

{∫

1{u<zτcj}λ(s,xτc

j)|xτj

,zτj(v,u)du

}

dv

where1{·} is the indicator function and

Cj =

∫

1{u<zτcj}λ(s,xτc

j)|xτj

,zτj(v,u)dudv,

and define the random vectorZ+(s) = maxj=1,...,ℓ ϕ+j (s).

Step 3.Independently draw{ζi}i≥1 a Poisson point process on(0,∞) with intensityζ−2dζand{Wi(·)}i≥1 independent copies ofW (·) and define the random vector

Z−(s) = maxi≥1

ζi exp {Wi(s)− γ(s)} 1{ζieWi(x)−γ(x)<z}.

Then the random vectorZ(s) = max {Z+(s), Z−(s)} follows the conditional distribution ofZ(s) givenZ(x) = z.

It is not difficult to show that the corresponding conditional cumulative distribution functionis given by

pr [Z(s) ≤ a | Z(x) = z] =pr[Z(s) ≤ a, Z(x) ≤ z]

pr[Z(x) ≤ z]

∑

τ∈Pk

πx(z, τ)

|τ |∏

j=1

Fτ,j(a), (5)

whereFτ,j is expressed in terms of log-normal distributions as follows

Fτ,j(a) = pr[

ϕ+j (s) ≤ a | Z(x) = z, θ = τ

]

=

∫

{u<zτcj,v<a} λ(s,xτc

j)|xτj

,zτj(v,u)dudv

∫

{u<zτcj} λxτc

j|xτj

,zτj(u)du

.

Interestingly it is clear from (5) that the conditional random fieldZ(·) | {Z(x) = z} is not max-stable.

3. MARKOV CHAIN MONTE–CARLO SAMPLER

The previous section introduced a procedure to get realizations from the regular conditionaldistribution of Brown–Resnick processes. We saw that this sampling scheme requires to be ableto sample from a discrete distribution whose state space corresponds to all possible partitions ofthe set of conditioning points—see Theorem 1 step 1.

It is well known that the number of all possible partitions ofa set withk elements corre-sponds to Bell numbers and results in a combinatorial explosion even for a moderate number ofconditioning locationsk. For example whenk = 10 there are around 116000 possibilities. Evenworse, the current computer capacities prevent the storageof all possible partitions in memorywhenk > 12.

Due to this combinatorial explosion, the distributionπx(z, ·) cannot be computed and it seemssensible to use MCMC algorithms. It is questionable whethera Metropolis–Hastings algorithmwill be efficient since we aim at sampling from a discrete distribution whose state space is hugeand where typically only a few states have a significant probability to occur—see Figure 1; thus


resulting in a chain with poor mixing properties. A safest approach consists in using a Gibbssampler as we shall describe now.

Forτ ∈ Pk, let τ−j be the restriction ofτ to the set{x1, . . . , xk} \ {xj}. As usual with Gibbssamplers, our goal is to simulate from the conditional distribution

pr[θ ∈ · | θ−j = τ−j], (6)

whereθ ∈ Pk is a random partition which follows the target distributionπx(z, ·) andτ is typi-cally the current state of the Markov chain.

Since the number of possible updates is always less thank, the combinatorial explosion isavoided. Indeed forτ ∈ Pk of sizeℓ, the number of partitionsτ∗ ∈ Pk such thatτ∗−j = τ−j forsomej ∈ {1, . . . , k} is

b+ =

{

ℓ if {xj} is a partitioning set ofτ ,

ℓ+ 1 if {xj} is not a partitioning set ofτ ,

since the pointxj may be reallocated to any partitioning set ofτ−j or to a new one.To illustrate consider the set{x1, x2, x3} and letτ = ({x1, x2}, {x3}). Then the possible

partitionsτ∗ such thatτ∗−2 = τ−2 are

({x1, x2}, {x3}), ({x1}, {x2}, {x3}), ({x1}, {x2, x3}),

while there exists only two partitions such thatτ∗−3 = τ−3, i.e.,

({x1, x2}, {x3}), ({x1, x2, x3}).

The distribution (6) has nice properties. Since for allτ∗ ∈ Pk such thatτ∗−j = τ−j we have

pr[θ = τ∗ | θ−j = τ−j] =πx(z, τ

∗)∑

τ∈Pk

πx(z, τ )1{τ−j=τ−j}

∝

∏|τ∗|j=1wτ∗,j

∏|τ |j=1wτ,j

, (7)

where

wτ,j = λxτj(zτj )

∫

{u<zτcj}λxτc

j|xτj

,zτj(u)du,

it can be seen that in the right hand side of (7) many factors will cancel out except at most fourof them. From a simulation perspective, it makes the Gibbs sampler especially convenient.

The most CPU demanding part of (7) corresponds to the computation of the multivariate log-normal probabilities

∫

{u<zτcj}λxτc

j|xτj

,zτj(u)du.

Following Genz (1992), these probabilities are computed using a separation of variable methodwhich provide a transformation of the original integrationproblem to the unit hyper-cube. Furthervariable ordering combined with a quasi Monte Carlo scheme and an antithetic variable samplingis used to improve integration efficiency.

Since it is not obvious how to implement a Gibbs sampler whosetarget distribution has supportPk, the remainder of these section gives practical details on its implementation. For any fixedlocationsx1, . . . , xk ∈ X , we first describe how each partition of{x1, . . . , xk} is stored. Toillustrate consider the set{x1, x2, x3} and the partition{(x1, x2); (x3)}. With our convention,


this partition is defined as(1, 1, 2) indicating thatx1 andx2 belong to the same partitioning setlabeled ’1’ andx3 belongs to the partitioning set ’2’. There exist several equivalent notations forthis partition: for example one can use(2, 2, 1) or (1, 1, 3). However Orlov (2002) shows thatthere is a one-one mapping betweenPk and the set

P∗k =

{

(a1, . . . , ak), ∀i ∈ {2, . . . , k} : 1 = a1 ≤ ai ≤ max1≤j<i

aj + 1, ai ∈ Z

}

.

Consequently we shall restrict our attention to the partitions that live inP∗k and going back to

our example we see that(1, 1, 2) is valid but(2, 2, 1) and(1, 1, 3) are not.For τ ∈ P∗

k of sizeℓ, let r1 =∑k

i=1 δτi=aj andr2 =∑k

i=1 δτi=b, i.e., the number of condi-tioning locations that belong to the partitioning sets ’aj ’ and ’b’ whereb ∈ {1, . . . , b+} and

b+ =

{

ℓ (r1 = 1),

ℓ+ 1 (r1 6= 1).

Then the conditional probability distribution (7) satisfies

pr[τj = b | τi = ai, ∀i 6= j] ∝

1 (b = aj), (8a)

wτ∗,b/(wτ,bwτ,aj ) (r1 = 1, r2 6= 0, b 6= aj), (8b)

wτ∗,bwτ∗,aj/(wτ,bwτ,aj ) (r1 6= 1, r2 6= 0, b 6= aj), (8c)

wτ∗,bwτ∗,aj/wτ,aj (r1 6= 1, r2 = 0, b 6= aj), (8d)

whereτ∗ = (a1, . . . , aj−1, b, aj+1, . . . , ak). It is worth stressing that althoughτ∗ does not nec-essarily belongs toP∗

k , it corresponds to a unique partition ofPk and we can use the bijectionPk → P∗

k to recodeτ∗ into an element ofP∗k .

In (8a)–(8d) the event{r1 = 1, r2 = 0, b 6= aj} is missing since{r1 = 1, r2 = 0} implies thatτ∗ = τ , where the equality has to be understood in terms of elementsof Pk, and this casehas been already taken into account with (8a). Practical details on the computation of theseconditional weights are given in Algorithm 1 and deferred toAppendix 1.

Once these conditional weights have been computed, the Gibbs sampler proceeds as usual byupdating each element ofτ successively. However for our work we use a random scan imple-mentation of the Gibbs sampler (Liu et al., 1995). More precisely, one iteration of the randomscan Gibbs sampler selects an element ofτ at random according to a given distribution, sayp = (p1, . . . , pk), and then updates this element. A widely used sampler is the uniform randomscan Gibbs sampler for which the selection distribution is assumed to be a discrete uniform dis-tribution, e.g.,p = (k−1, . . . , k−1).

Recently Latuszynski & Rosenthal (2010) show under mild regularity conditions the ergodic-ity of the adaptive random scan Gibbs sampler, i.e., a randomscan Gibbs sampler such that theselection distributionp changes as the sampler proceeds. For our purposes the use of an adaptiveGibbs sampler might be appealing since typically only a few states ofπx(z, ·) have a significantprobability to occur—see Figure 3. Although the quality of the simulated Markov chain couldbe largely improved, the implementation of such a sampler requires many manual tuning andwe were not able to implement it appropriately to improve thequality of the simulated Markovchains.


0 500 1000 1500 2000

t

τ t

(1,1,1,1,1)

(1,1,2,1,1)

(1,2,1,2,2)

(1,2,2,2,2)

(1,2,3,2,2)

Partition size1 2 3 4 5

Probability mass function

0.0 0.2 0.4 0.6 0.8 1.0

Weights (χ62 = 5.94, p − value = 0.43)

Theoretical Empirical

Fig. 1. Left: Trace plot of one Markov chain generated us-ing the uniform random scan Gibbs sampler and whosetarget distribution isπx(z, ·) with k = 5 conditioning lo-cations. The labels on they-axis indicates the partitionshaving a probability greater than 0·01 to occur. Right:Comparison of the theoretical probabilities{πx(z, τ ), τ ∈Pk} and the empirical ones estimated from the simulatedMarkov chain with an insert showing the sampleχ2 statis-

tic and the associatedp-value.

4. SIMULATION STUDY

4·1. Gibbs SamplerIn this section we check whether the Gibbs sampler is able to sample fromπx(z, ·). To illus-

trate a typical sample path, Figure 1 shows the trace plot of asimulated chain of length 2000 withk = 5 conditioning locations and a thinning lag of5 and compares the theoretical probabilities{πx(z, τ), τ ∈ Pk} to the empirical ones estimated from the Markov chain. As mentioned inthe previous section it can be seen that only a few states havea significant probability to occurand these states most often differs by a small bit. As expected for this particular simulation theempirical probabilities match the theoretical ones.

To better assess if our uniform random scan Gibbs sampler is able to sample fromπx(z, ·)we simulate 250 Markov chains of length 1000—after removinga burn-in period and thinningthe chain, and similarly to Figure 1, compare the theoretical probabilities{πx(z, τ), τ ∈ Pk}to the empirical ones using aχ2 test for each of these chains. Since the computation of thesetheoretical probabilities is CPU intensive, the number of conditioning locations is at most 5 inour simulation study.

Figure 2 plots the samplep-values of theseχ2 tests against the quantiles of aU(0, 1) dis-tribution for a varying number of conditional locations. This figure corroborates what we saw


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Uniform quantiles

Sam

ple

χ2 p−

valu

esK.S. p − value = 0.47

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Uniform quantiles

Sam

ple

χ2 p−

valu

es

K.S. p − value = 0.52

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Uniform quantiles

Sam

ple

χ2 p−

valu

es

K.S. p − value = 0.63

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Uniform quantiles

Sam

ple

χ2 p−

valu

es

K.S. p − value = 0.46

Fig. 2. QQ-plots for the sampleχ2 test p-values againstU(0, 1) quantiles with a varying number of conditioninglocationsk with an insert showing thep-values of a Kol-mogorov Smirnov test for uniformity—from left to right,k = 2, 3, 4 and5. The dashed lines show the 95% confi-

dence envelopes.

Table 1. Spatial dependence structures ofBrown–Resnick processes with (semi) vari-ogram γ(h) = (h/λ)ν . The variogram pa-rameters are set to ensure that the extremalcoefficient function satisfiesθ(115) = 1·7.

Sample path propertiesγ1: Wiggly γ2: Smooth γ3: Very smooth

λ 25 54 69ν 0·5 1·0 1·5

−100 −50 0 50 100

−2

−1

01

23

x

Z(x

)

−100 −50 0 50 100

−2

−1

01

23

x

Z(x

)

−100 −50 0 50 100

−2

−1

01

23

x

Z(x

)

0 50 100 150 200

1.0

1.2

1.4

1.6

1.8

2.0

h

θ(h)

Fig. 3. Three realizations of a Brown–Resnick process withstandard Gumbel margins and (semi) variogramsγ1, γ2andγ3—from left to right. The squares correspond to the15 conditioning values that will be used in the simulationstudy. The right panel shows the associated extremal coef-ficient functions where the solid, dashed and dotted lines

correspond respectively toγ1, γ2 andγ3.

in Figure 1 since the samplep-values seem to follow aU(0, 1) distribution indicating that thesampler is able to sample fromπx(z, ·).

4·2. Conditional simulationsIn this section we check if our algorithm is able to produce realistic conditional simulations

of Brown–Resnick processes with (semi) variogramγ(h) = (h/λ)ν . To have a broad overview,we consider three different sample path properties as summarized in Table 1. Figure 3 shows onerealization for each sample path configuration as well as thecorresponding extremal coefficientfunction. These realizations will serve as the basis for ourconditioning events.


−100 −50 0 50 100

−2

−1

01

23

4

x

Z(x

)

Coverage (%): 95.67

−100 −50 0 50 100

−2

−1

01

23

4

x

Z(x

)

Coverage (%): 99.13

−100 −50 0 50 100

−2

−1

01

23

4

x

Z(x

)

Coverage (%): 100

−100 −50 0 50 100

−2

−1

01

23

4

x

Z(x

)

Coverage (%): 93.64

−100 −50 0 50 100

−2

−1

01

23

4

x

Z(x

)Coverage (%): 98.73

−100 −50 0 50 100

−2

−1

01

23

4

x

Z(x

)

Coverage (%): 97.46

−100 −50 0 50 100

−2

−1

01

23

4

x

Z(x

)

Coverage (%): 92.53

−100 −50 0 50 100

−2

−1

01

23

4

x

Z(x

)

Coverage (%): 96.27

−100 −50 0 50 100

−2

−1

01

23

4

x

Z(x

)

Coverage (%): 92.95

Fig. 4. Pointwise sample quantiles estimated from 1000conditional simulations of Brown–Resnick processes withstandard Gumbel margins and (semi) variogramsγ1, γ2and γ3 (left to right) and with k = 5, 10, 15 condi-tioning locations—top to bottom. The solid black linesshow the pointwise 0·025, 0·5, 0·975 sample quantilesand the dashed grey lines that of a standard Gumbeldistribution. The squares show the conditional points{(xi, zi)}i=1,...,k. The solid grey lines correspond the sim-ulated paths used to get the conditioning events. The insertsgive the proportion of points lying in the 95% pointwise

confidence intervals.

In order to check if our sampling procedure is accurate and given a single conditional event{Z(x) = z}, we generated1000 conditional realizations of a Brown–Resnick processes withstandard Gumbel margins and (semi) variogramsγj (j = 1, 2, 3).

Figure 4 shows the pointwise sample quantiles obtained fromthese 1000 simulated paths andcompare them to unit Gumbel quantiles. As expected the conditional sample paths inherit theregularity driven by the Hurst indexν/2 of the processY (·) in (2) and there is less variability inregions close to some conditioning locations. On the opposite, although for this specific type ofvariogramθ(h) → 2 ash → ∞, for regions far away from any conditioning location the samplequantiles converges to that of a standard Gumbel distribution indicating that the conditional event


0 50 100 150 200

1.0

1.2

1.4

1.6

1.8

2.0

h

θ(h)

0 50 100 150 200

1.0

1.2

1.4

1.6

1.8

2.0

h

θ(h)

0 50 100 150 200

1.0

1.2

1.4

1.6

1.8

2.0

h

θ(h)

Fig. 5. Comparison of the extremal coefficient estimates(using a binnedF -madogram with 250 bins) and the theo-retical extremal coefficient function for a varying numberof conditioning locations and different (semi) variograms.From left to right,k = 5, 10, 15. The ’o’, ’ +’ and ’x’ sym-bols correspond respectively toγ1, γ2 andγ3. The solid,dashed and dotted grey lines correspond to the theoretical

extremal coefficient functions forγ1, γ2 andγ3

does not have any influence anymore. In addition the sample paths used to get the conditionalevents, see Figure 3, lay most of the time in the 95% pointwiseconfidence intervals corroboratingthat our sampling procedure seems to be accurate—the coverage range between 0·93 and 1·00with a mean value of 0·96.

So far we have checked that the proposed sampling procedure yields the expected coverage aswell as the right marginal properties as we move far away fromany conditioning location. Thelast point to be fulfilled is to assess whether the simulationprocedure honours the spatial depen-dence driven by the (semi) variogramγ(·). To this aim we use theF -madogram (Cooley et al.,2006) to compare the pairwise extremal coefficient estimates to the theoretical extremal coef-ficient function. As mentioned in Section 2, sinceZ(·) | {Z(x) = z} is not max-stable theF -madogram cannot be used. However, by integrating out the conditional event we recover theoriginal Brown-Resnick distribution and the max-stability property. So we generate indepen-dently 1000 conditional events{Z(x) = z}, x, and for each conditional event one conditionalrealization of a Brown–Resnick process. Figure 5 compares the pairwise extremal coefficient es-timates based on these simulations to the theoretical extremal coefficient function. As expected,whatever the number of conditioning location and the (semi)variograms are, the (binned) pair-wise estimates match the theoretical curve indicating thatthe spatial dependence is honoured.

5. APPLICATION

In this section we apply our framework to get conditional simulations of extreme precipitationfields. The data considered here were previously analyzed byDavison et al. (2011) where it wasshown that Brown–Resnick processes were one of the most competitive models among variousstatistical models for spatial extremes.

The data are summer maximum daily rainfall for the years 1962–2008 at 51 weather stations inthe Plateau region of Switzerland, provided by the nationalmeteorological service, MeteoSuisse.For our study we consider as conditional points the 24 weather stations that are at most 30kmapart from Zurich and set as the conditional values the rainfall amounts recorded in the year 2000,i.e., the year of the largest precipitation event ever recorded in Zurich between 1962–2008—see


1000 2000 3000 40000 25 50 75 100

(km)(m)

1970 1980 1990 2000

0

20

40

60

80

Years

Max

imum

rai

nfal

l in

Zur

ich

(mm

)

0 50 100 150 200

1.0

1.2

1.4

1.6

1.8

2.0

h

θ(h)

Fig. 6. Left: Map of Switzerland showing the stations ofthe 24 rainfall gauges used for the analysis, with an insertshowing the altitude. The station marked with a blue squarecorresponds to Zurich. Middle: Summer maximum dailyrainfall values for 1962–2008 at Zurich. Right: Compar-ison between the pairwise extremal coefficient estimatesfor the 51 original weather stations and the extremal coef-ficient function derived from a fitted Brown–Resnick pro-cesses having (semi) variogramγ(h) = (h/λ)ν . The greypoints are pairwise estimates; the black ones are binned es-timates and the red curve is the fitted extremal coefficient

function.

Table 2.Empirical distribution of the partition size for the rainfalldata estimated from a simulated Markov chain of length 15000.Partition size 1 2 3 4 5 6 7–24Empirical probabilities (%) 66·2 28·0 4·8 0·5 0·2 0·2 <0·05

Figure 6. The largest and smallest distances between the conditioning locations are around 55kmand just over 4km respectively.

Prior to generate conditional realizations, a Brown–Resnick process having (semi) variogramγ(h) = (h/λ)ν has to be fitted and in this work the maximum pairwise likelihood estimatorintroduced by Padoan et al. (2010) was used to simultaneously fit the marginal parameters andthe spatial dependence parametersλ andν. In accordance to the previous work of Davison et al.(2011), the marginal parameters were described by the following trend surfaces

η(x) = β0,η + β1,ηlon(x) + β2,ηlat(x),

σ(x) = β0,σ + β1,σlon(x) + β2,σlat(x),

ξ(x) = β0,ξ ,

whereη(x), σ(x), ξ(x) are the location, scale and shape parameters of the generalized extremevalue distribution andlon(x), lat(x) the longitude and latitude of the stations at which the dataare observed. The maximum pairwise likelihood estimates for λ and ν are 38 (14) and 0·69(0·07) and give a practical extremal range, i.e., the distanceh+ such thatθ(h+) = 1·7, around115km—see the right panel of Figure 6.

Table 2 shows the empirical distribution of the partition size. We can see that around 65% ofthe time the summer maxima observed at the 24 conditioning locations were a consequence of asingle extremal function, i.e., only one storm event, and around 30% of the time a consequenceof two different storms. Since the simulated Markov chain keeps a trace of all the simulatedpartitions, we looked at the partitions of size two and saw that around 65% of the time, at least


20

40

60

80

30

40

50

60

70

80

90

40

60

80

100

120

Fig. 7. From left to right, maps on a50× 50 grid ofthe pointwise 0·025, 0·5 and 0·975 sample quantiles forrainfall (mm) obtained from1000 conditional simula-tions of Brown–Resnick processes having (semi) vari-ogramγ(h) = (h/38)0·69. The squares show the condi-

tional locations.

one of the four up-north conditioning locations was impacted by one extremal function while theremaining 20 locations were always influenced by another one.

Figure 7 plots the pointwise 0·025, 0·5 and 0·975 sample quantiles obtained from1000 con-ditional simulations of our fitted Brown–Resnick process. The conditional median provides apoint estimate for the rainfall at an unknown location and the 0·025 and 0·975 conditional quan-tiles a 95% pointwise confidence interval. As indicated by our simulation study—see Figures 3and 4, the Hurst indexν/2 has a major impact on the regularity of paths and on the width of theconfidence interval. Here, the valueν ≈ 0.69 corresponds to wiggly sample paths and wider con-fidence intervals. It is however reassuring that even on a complex case study like this one, i.e.,simulation on a50× 50 grid and withk = 24 conditioning locations, the maps are consistentwith one might expect.

6. CONCLUSION

The last decade has seen many major advances for building up amathematically well foundedgeostatistic of extremes. This work makes a contribution tothis field by allowing conditionalsimulations of Brown–Resnick processes. Although closed forms for the regular conditional dis-tributions of max infinitely divisible processes were foundrecently, simulating conditional real-izations of Brown–Resnick processes appears to be a considerable challenge since these closedforms are intractable even for a moderate number of conditioning locations.

To bypass this computational burden a Markov chain Monte–Carlo sampler was proposed.Using a simulation study and an application to extreme rainfall around Zurich, we show thatour procedure is able to draw accurate conditional simulations for real sized problems and henceopens many avenues for a better description of spatial extremes. The simulations were performedusing the statistical softwareR (R Development Core Team, 2011) and the implemented routineswill be collected in theSpatialExtremes package (Ribatet, 2011).

Future works could focus on different conditioning events.For instance, an interesting openproblem would be to get a procedure to sample from a max-stable processZ(·) conditionally onmaxx∈K Z(x) = z, K ⊂ X . Indeed since regional/global circulation models are widely used to


assess the potential consequences of a global warming,K could typically represent a grid cell ofa given atmospheric model.

ACKNOWLEDGEMENTS

M. Ribatet was partly funded by the MIRACCLE-GICC and McSim ANR projects. The au-thors thank the national meteorological service of Switzerland MeteoSuisse for providing thedata.

APPENDIX. ALGORITHM TO COMPUTEpr[θ = τ∗ | θ−j = τ−j]

Algorithm 1: Computing the probabilities of (7).Input : A partition τ ∈P

∗k of sizeℓ andj ∈ {1, . . . , k}

Output : The conditional weightsw∗ = (w∗1 , . . . , w

∗b+

)

//Identify which partitioning set xj belongs to1 s← τj ;//Compute the size of this partition set

2 r1 ←∑k

m=1 δτm=s;//Create a new partition that will be updated

3 τ∗ ← τ ;//Get the number of possible states for τ∗

4 b+ ← ℓ+ δr1 6=1;5 for b← 1 to b+ do

//b refers to the partitioning set where xj is moved to//Update the partition τ∗

6 τ∗j = b;//Compute the conditional weights for this new partition

7 r2 ←∑k

m=1 δτm=b;8 if b = s then

//This is case (8a)9 w∗

b ← 1;10 continue;11 end12 caser1 = 1 andr2 6= 0

//This is case (8b)13 w∗

b ← wτ∗,b/{wτ,bwτ,s};14

15 caser1 6= 1 andr2 6= 0//This is case (8c)

16 w∗b ← wτ∗,bwτ∗,s/{wτ,bwτ,s};

17

18 caser1 6= 1 andr2 = 0//This is case (8d)

19 w∗b ← wτ∗,bwτ∗,s/wτ,s;

20

21 end22 Normalize the conditional weights to sum to 1;23 return w∗;


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