conditional simulation of max-stable processesribatet/docs/dombry2011.pdf · in classical...

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Biometrika (2012), xx, x, pp. 1–22

C© 2007 Biometrika Trust

Printed in Great Britain

Conditional simulation of max-stable processes

BY C. DOMBRY, F. EYI-MINKO,

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 cedex

2 Montpellier, France

[email protected]

SUMMARY

Since many environmental processes are spatial in extent, a single extreme 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 max-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 extreme rainfall around Zurich and

extreme temperatures. The proposed framework provides accurate conditional simulations and

can handle real-sized problems.

Some key words: Conditional simulation; Markov chain Monte Carlo; Max-stable process; Precipitation; Regular

conditional distribution; Temperature.

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

1. INTRODUCTION

Max-stable processes arise naturally when studying extremes of stochastic processes and

therefore play a major role in the statistical modelling of spatial extremes (Buishand et al, 2008;

Padoan et al., 2010; Davison et al., 2012). Although a different spectral characterization of

max-stable processes exists (de Haan, 1984), for our purposes the most useful representation

is (Schlather, 2002)

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 and

the Yi are independent replicates of a non-negative stochastic process Y such that E{Y (x)} = 1

for all x ∈ Rd. It is well known that Z is a max-stable process on Rd with unit Frechet margins

(Schlather, 2002; de Haan & Fereira, 2006, page 307). Although (1) takes the pointwise maxi-

mum over an infinite number of points {ζi} and processes Yi, it is possible to get approximate

realizations from Z (Schlather, 2002; Oesting et al., 2012).

Based on (1) several parametric max-stable models have been proposed (Schlather, 2002;

Brown & Resnick, 1977; Kabluchko et al., 2009; Davison et al., 2012) that share the same finite-

dimensional distribution functions

pr{Z(x1) ≤ z1, . . . , Z(xk) ≤ zk} = exp

[−E

{maxj=1,...,k

Y (xj)

zj

}],

where k ∈ N, z1, . . . , zk > 0 and x1, . . . , xk ∈ Rd.

Apart from the Smith model (Genton et al., 2011), only the bivariate cumulative distribution

functions are explicitly known. To bypass this hurdle to inference based on max-stable processes,

de Haan & Pereira (2006) propose a semi-parametric estimator and Padoan et al. (2010) suggest

the use of the maximum pairwise likelihood estimator.

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Conditional simulation of max-stable processes 3

Paralleling the use of the variogram in classical geostatistics, the extremal coefficient function

(Schlather & Tawn, 2003; Cooley et al., 2006)

θ(x1 − x2) = −z log pr{Z(x1) ≤ z, Z(x2) ≤ z}

is widely used to summarize the spatial dependence of extremes for stationary processes. It takes

values in the interval [1, 2]; the lower bound indicates complete dependence, and the upper bound

independence.

The last decade has seen many advances to develop geostatistics for extremes, and software is

available to practitioners (Wang, 2010; Schlather, 2012; Ribatet, 2011). However an important

tool currently missing is conditional simulation of max-stable processes. In classical geostatistic

based on Gaussian models, conditional simulation is well established (Chiles & Delfiner, 1999)

and provides a framework to assess the distribution of a Gaussian random field given values

observed at fixed locations. For example, conditional simulations of Gaussian processes have

been used to model land topography (Mandelbrot, 1982).

Conditional simulation of max-stable processes is a long-standing problem (Davis & Resnick,

1989, 1993). Wang & Stoev (2011) 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 complex situations.

Based on the recent developments on the regular conditional distribution of max-infinitely di-

visible processes, the aim of this paper is to provide a methodology for conditional simulation

of max-stable processes with continuous spectral measures. More formally for a study region

X ⊂ Rd, our goal is to derive an algorithm to sample from the regular conditional distribu-

tion ofZ | {Z(x1) = z1, . . . , Z(xk) = zk} for some z1, . . . , zk > 0 and k conditioning locations

x1, . . . , xk ∈ X .

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2. CONDITIONAL SIMULATION OF MAX-STABLE PROCESSES

2·1. General framework

This section reviews some key results of an unpublished paper available from the first author

with a particular emphasis on max-stable processes. Our goal is to give a more practical inter-

pretation of their results from a simulation perspective. To this aim, we recall two key results and

propose a procedure to generate conditional realizations of max-stable processes.

Let RX be the space of real-valued functions on X ⊂ Rd and let Φ = {ϕi}i≥1 be a Poisson

point process on RX where ϕi(x) = ζiYi(x) (i = 1, 2, . . .) with ζi and Yi as in (1). We write

f(x) = {f(x1), . . . , f(xk)} for all random functions f : X → R and x = (x1, . . . , xk) ∈ X k. It

is not difficult to show that for all Borel sets A ⊂ Rk, the Poisson process {ϕi(x)}i≥1 defined

on Rk has intensity measure

Λx(A) =

∫ ∞0

pr {ζY (x) ∈ A} ζ−2dζ.

The point process Φ is called regular if the intensity measure Λx has an intensity function λx,

i.e., Λx(dz) = λx(z)dz, for all x ∈ X k.

The first key result is that provided the Poisson process Φ is regular, the intensity function λx

and the conditional intensity function

λs|x,z(u) =λ(s,x)(u, z)

λx(z), (s, x) ∈ Xm+k, u ∈ Rm, z ∈ (0,+∞)k, (2)

drive how the conditioning terms {Z(xj) = zj} (j = 1, . . . , k) are met, see Steps 1 and 2 in

Theorem 1.

The second key result is that, conditionally on Z(x) = z, the Poisson process Φ can be de-

composed into two independent point processes, say Φ = Φ− ∪ Φ+, where

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

{ϕ ∈ Φ: ϕ(xi) = zi} .

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Before introducing a procedure to get conditional realizations of max-stable processes, we

introduce notation and make connections with the pioneering work of Wang & Stoev (2011).

A function ϕ ∈ Φ+ such that ϕ(xi) = zi for some i ∈ {1, . . . , k} is called an extremal func-

tion associated to xi and denoted by ϕ+xi . It is easy to show that there exists almost surely a

unique extremal function associated to xi. Although Φ+ = {ϕ+x1 , . . . , ϕ

+xk} almost surely, it

might happen that a single extremal function contributes to the random vector Z(x) at sev-

eral locations xi, e.g., ϕ+x1 = ϕ+

x2 . To take such repetitions into account, we define a random

partition θ = (θ1, . . . , θ`) of the set {x1, . . . , xk} into ` = |θ| blocks, and extremal functions

(ϕ+1 , . . . , ϕ

+` ) such that Φ+ = {ϕ+

1 , . . . , ϕ+` } and ϕ+

j (xi) = zi if xi ∈ θj and ϕ+j (xi) < zi if

xi /∈ θj (i = 1, . . . , k; j = 1, . . . , `). Wang & Stoev (2011) call the partition θ the hitting sce-

nario. The set of all possible partitions of {x1, . . . , xk}, denoted Pk, identifies all possible hitting

scenarios.

From a simulation perspective, the fact that Φ− and Φ+ are independent given Z(x) = z

is especially convenient and suggests a three–step procedure to sample from the conditional

distribution of Z given Z(x) = z.

THEOREM 1. Suppose that the point process Φ is regular and let (x, s) ∈ X k+m. For τ =

(τ1, . . . , τ`) ∈Pk and j = 1, . . . , `, define Ij = {i : xi ∈ τj}, xτj = (xi)i∈Ij , zτj = (zi)i∈Ij ,

xτcj = (xi)i/∈Ij and zτcj = (zi)i/∈Ij . Consider the three–step procedure:

Step 1. Draw a random partition θ ∈Pk with distribution

πx(z, τ) = pr {θ = τ | Z(x) = z} =1

C(x, z)

|τ |∏j=1

λxτj (zτj )

∫{uj<zτc

j}λxτc

j|xτj ,zτj (uj)duj ,

where the normalization constant is

C(x, z) =∑τ∈Pk

|τ |∏j=1

λxτj (zθj )

∫{uj<zτc

j}λxτc

j|xτj ,zτj (uj)duj .

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Step 2. Given τ = (τ1, . . . , τ`), draw ` independent random vectors ϕ+1 (s), . . . , ϕ+

` (s) from

the distribution

pr{ϕ+j (s) ∈ dv | Z(x) = z, θ = τ

}=

1

Cj

{∫1{u<zτc

j}λ(s,xτc

j)|xτj ,zτj (v, u)du

}dv,

where 1{·} is the indicator function and

Cj =

∫1{u<zτc

j}λ(s,xτc

j)|xτj ,zτj (v, u)dudv,

and define the random vector Z+(s) = maxj=1,...,` ϕ+j (s).

Step 3. Independently draw a Poisson point process {ζi}i≥1 on (0,∞) with intensity ζ−2dζ

and {Yi}i≥1 independent copies of Y , and define the random vector

Z−(s) = maxi≥1

ζiYi(s)1{ζiYi(x)<z}.

Then the random vector Z(s) = max {Z+(s), Z−(s)} follows the conditional distribution of

Z(s) given Z(x) = z.

The corresponding conditional cumulative distribution function is

pr {Z(s) ≤ a | Z(x) = z} =

∑τ∈Pk

πx(z, τ)

|τ |∏j=1

Fτ,j(a)

pr{Z(s) ≤ a, Z(x) ≤ z}pr{Z(x) ≤ z}

, (3)

where

Fτ,j(a) = pr{ϕ+j (s) ≤ a | Z(x) = z, θ = τ

}=

∫{u<zτc

j,v<a} λ(s,xτc

j)|xτj ,zτj (v, u)dudv∫

{u<zτcj} λxτc

j|xτj ,zτj (u)du

.

The first term in the right hand side of (3) corresponds to Steps 1 and 2 while the ratio is a

consequence of Step 3. It is clear from (3) that the conditional random field Z | {Z(x) = z} is

not max-stable.

2·2. Distribution of the extremal functions

In this section we derive closed forms for the intensity function λx(z) and the conditional in-

tensity function λs|x,z(u) for two widely used max-stable processes; the Brown–Resnick (Brown

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& Resnick, 1977; Kabluchko et al., 2009) and the Schlather (2002) processes. Details of the

derivations are given in the Appendix.

The Brown–Resnick process corresponds to the case where Y (x) = exp{W (x)− γ(x)} (x ∈

Rd) in (1) with W a centered Gaussian process with stationary increments, semivariogram γ and

such that W (o) = 0 almost surely where o denotes the origin of Rd. For x ∈ X k and provided

the covariance matrix Σx of the random vector W (x) is positive definite, the intensity function

is

λx(z) = Cx exp

(−1

2log zTQx log z + Lx log z

) k∏i=1

z−1i , z ∈ (0,∞)k,

with 1k = (1)i=1,...,k, γx = {γ(xi)}i=1,...,k,

Qx = Σ−1x −Σ−1x 1k1

Tk Σ−1x

1Tk Σ−1x 1k, Lx =

(1Tk Σ−1x γx − 1

1Tk Σ−1x 1k1k − γx

)TΣ−1x ,

Cx = (2π)(1−k)/2|Σx|−1/2(1Tk Σ−1x 1k)−1/2 exp

{1

2

(1Tk Σ−1x γx − 1)2

1Tk Σ−1x 1k− 1

2γTx Σ−1x γx

},

and for all (s, x) ∈ Xm+k, (u, z) ∈ (0,∞)m+k and provided the covariance matrix Σ(s,x) is pos-

itive definite, the conditional intensity function corresponds to a multivariate log-normal proba-

bility density function

λs|x,z(u) = (2π)−m/2|Σs|x|−1/2 exp

{−1

2(log u− µs|x,z)TΣ−1s|x(log u− µs|x,z)

} m∏i=1

u−1i ,

where µs|x,z ∈ Rm and Σs|x are the mean and covariance matrix of the underlying normal dis-

tribution and are given by

Σ−1s|x = JTm,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 =

Im

0k,m

, Jm,k =

0m,k

Ik

,

where Ik denotes the k × k identity matrix and 0m,k the m× n null matrix.

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The Schlather process considers the case Y (x) = (2π)1/2 max{0, ε(x)} (x ∈ Rd) 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 (x) =

(2π)1/2ε(x) (x ∈ Rd). For x ∈ X k and provided the covariance matrix Σx of the random vector

ε(x) is positive definite, the intensity function is

λx(z) = π−(k−1)/2|Σx|−1/2ax(z)−(k+1)/2Γ

(k + 1

2

), z ∈ Rk,

where ax(z) = zTΣ−1x z.

For (s, x) ∈ Xm+k, (u, z) ∈ Rm+k and provided that the covariance matrix Σ(s,x) is positive

definite, the conditional intensity function λs|x,z(u) corresponds to the density of a multivariate

Student distribution with k + 1 degrees of freedom, location parameter µ = Σs:xΣ−1x z, and scale

matrix

Σ =ax(z)

k + 1

(Σs − Σs:xΣ−1x Σx:s

), Σ(s,x) =

Σs Σs:x

Σx:s Σx

.

3. MARKOV CHAIN MONTE CARLO SAMPLER

Section 2 introduced a procedure to get realizations from the regular conditional distribution

of max-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 Pk becomes very large and the distribution πx(z, ·) cannot be computed exactly. A Gibbs

sampler is especially convenient for Monte Carlo sampling from πx(z, ·).

For τ ∈Pk, let τ−j be the restriction of τ to the set {x1, . . . , xk} \ {xj}. Our goal is to

simulate from the conditional distribution

pr(θ ∈ · | θ−j = τ−j), (4)

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where θ ∈Pk is a random partition which follows the target distribution πx(z, ·).

Since the number of possible updates is always less than k, a combinatorial explosion is

avoided. Indeed for τ ∈Pk of size `, the number of partitions τ∗ ∈Pk such that τ∗−j = τ−j

for some j ∈ {1, . . . , k} is

b+ =

` if {xj} is a partitioning set of τ ,

`+ 1 if {xj} is not a partitioning set of τ ,

since the point xj may be reallocated to any partitioning set of τ−j or to a new one.

To illustrate consider the set {x1, x2, x3} and let τ = ({x1, x2}, {x3}). Then the possible

partitions τ∗ such that τ∗−2 = τ−2 are ({x1, x2}, {x3}), ({x1}, {x2}, {x3}), ({x1}, {x2, x3}),

while there exist only two partitions such that τ∗−3 = τ−3, i.e., ({x1, x2}, {x3}), ({x1, x2, x3}).

The distribution (4) has nice properties. Since for all τ∗ ∈Pk such that τ∗−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

, (5)

where

wτ,j = λxτj (zτj )

∫{u<zτc

j}λxτc

j|xτj ,zτj (u)du.

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<zτc

j}λxτc

j|xτj ,zτj (u)du.

For the Brown–Resnick and Schlather processes, we follow the lines of Genz (1992) 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.

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Since it is not obvious how to implement a Gibbs sampler whose target distribution has support

Pk, the remainder of this section gives practical details. For any fixed locations x1, . . . , xk ∈ X ,

we first describe how each partition of {x1, . . . , xk} is stored. To illustrate this consider the set

{x1, x2, x3} and the partition ({x1, x2}; {x3}). This partition is defined as (1, 1, 2), indicating

that x1 and x2 belong to the partitioning set labeled 1 and x3 belongs to the partitioning set 2.

There exist several equivalent notations for this partition: for example one can use (2, 2, 1) or

(1, 1, 3). Since there is a one-one mapping between Pk and the set

P∗k =

{(a1, . . . , ak) : i ∈ {2, . . . , k}, 1 = a1 ≤ ai ≤ max

1≤j<iaj + 1, ai ∈ Z

},

we shall restrict our attention to the partitions that live in P∗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 size `, let r1 =

∑ki=1 1{τi=aj} and r2 =

∑ki=1 1{τi=b}, i.e., the number of

conditioning locations that belong to the partitioning sets aj and b where b ∈ {1, . . . , b+} with

b+ =

` (r1 = 1),

`+ 1 (r1 6= 1).

Then the conditional probability distribution (5) satisfies

pr(τj = b | τi = ai, i 6= j) ∝

1 (b = aj), (6a)

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

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

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

where τ∗ = (a1, . . . , aj−1, b, aj+1, . . . , ak). Although τ∗ may not belong to P∗k , it corresponds

to a unique partition of Pk and we can use the bijection Pk →P∗k to recode τ∗ into an element

of P∗k . In (6a)–(6d) the event {r1 = 1, r2 = 0, b 6= aj} is missing since {r1 = 1, r2 = 0} implies

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Table 1. Sample path properties of the max-stable models. For the Brown–Resnick

model, the variogram parameters are set to ensure that the extremal coefficient function

satisfies θ(115) = 1·7 while for the Schlather model the correlation function parame-

ters are set to ensure that θ(100) = 1·5.

Brown–Resnick: γ(h) = (h/λ)κ Schlather: ρ(h) = exp{−(h/λ)κ}

γ1: Very wiggly γ2: Wiggly γ3: Smooth ρ1: Very wiggly ρ2: Wiggly ρ3: Smooth

λ 25 54 69 208 144 128

κ 0·5 1·0 1·5 0·5 1·0 1·5

that τ∗ = τ , where the equality has to be understood in terms of elements of Pk, and this case

has been already taken into account with (6a).

Once these conditional weights have been computed, the Gibbs sampler proceeds by up-

dating 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 = (p1, . . . , pk), 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).

4. 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(x) = z} for each configuration, we generated 1000 conditional realizations with standard

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−100 −50 0 50 100

−1

0

1

2

3

4

x

Z(x

)

−100 −50 0 50 100

−1

0

1

2

3

4

xZ

(x)

−100 −50 0 50 100

−1

0

1

2

3

4

x

Z(x

)

−100 −50 0 50 100

−1

0

1

2

3

4

x

Z(x

)

−100 −50 0 50 100

−1

0

1

2

3

4

x

Z(x

)

−100 −50 0 50 100

−1

0

1

2

3

4

xZ

(x)

Fig. 1. Pointwise sample quantiles estimated from 1000

conditional simulations of max-stable processes with stan-

dard Gumbel margins with k = 5, 10, 15 conditioning lo-

cations. The top row shows results for the Brown–Resnick

models with semivariograms γ3, γ2, γ1, from left to right.

The bottom row shows results for the Schlather models

with correlation functions ρ3, ρ2, ρ1, from left to right. The

solid black lines shows the pointwise 0·025, 0·5, 0·975

sample quantiles and the dashed grey lines that of a stan-

dard Gumbel distribution. The squares show the condi-

tional points {(xi, zi)}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 1000 sim-

ulated paths and compares them to unit Gumbel quantiles. As expected the conditional sample

paths inherit the regularity driven by the shape parameter κ and there is less variability in re-

gions close to conditioning locations. Since the Brown–Resnick processes considered are ergodic

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Table 2. Timings for conditional simulations of max-stable processes on a 50× 50 grid de-

fined on the square [0, 100× 21/2]2 for a varying number k of conditioning locations uniformly

distributed over the region. The times, in seconds, are mean values over 100 conditional simu-

lations; standard deviations are reported in brackets.

Brown–Resnick: γ(h) = (h/25)0·5 Schlather: ρ(h) = exp{−(h/208)0·50

}Step 1 Step 2 Step 3 Overall Step 1 Step 2 Step 3 Overall

k = 5 0·21 (0·01) 49 (11) 1·4 (0·1) 50 (11) 1·4 (0·02) 1·9 (0·7) 0·9 (0·3) 4·2 (0·8)

k = 10 8 (2) 76 (18) 1·4 (0·1) 85 (19) 12 (4) 2·4 (0·8) 1·0 (0·3) 15 (4)

k = 25 95 (38) 117 (30) 1·4 (0·1) 214 (61) 86 (42) 4 (1) 1·0 (0·3) 90 (43)

k = 50 583 (313) 348 (391) 1·5 (0·1) 931 (535) 367 (222) 62 (113) 1·0 (0·3) 430 (262)

Conditional simulations with k = 5 do not use a Gibbs sampler.

(Kabluchko and Schlather, 2010), for regions far away from any conditioning location the sam-

ple 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 2 gives timings for conditional simulations of max-stable processes on a 50× 50 grid

with a varying number of conditioning locations. Due to the combinatorial complexity of the

partition set Pk, the timings increase rapidly with respect to the number of conditioning points

k. It is however reassuring that the algorithm is tractable when k ≤ 50; this covers many practical

situations.

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1000 2000 3000 4000

0 25 50 75 100(km)

(m)

1970 1980 1990 2000

0

20

40

60

80

Years

Maxim

um

rain

fall

in Z

uri

ch (

mm

)

0 50 100 150 200

1.0

1.2

1.4

1.6

1.8

2.0

h (km)

θ(h

)

Fig. 2. Left: Map of Switzerland showing the stations of

the 24 rainfall gauges used for the analysis, with an insert

showing the altitude. The station marked with a triangle

corresponds to Zurich. Middle: Summer maximum rain-

fall values for 1962–2008 at Zurich. Right: Comparison

between the pairwise extremal coefficient estimates for the

51 original weather stations and the extremal 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 extremal coefficient function.

5. APPLICATION

5·1. Extreme precipitation around Zurich

The data considered here were previously analyzed by Davison et al. (2012), who showed that

Brown–Resnick processes were among the best models for these data. The data are summer max-

imum rainfall for the years 1962–2008 at 51 weather stations in the Plateau region of Switzerland,

provided by the national meteorological service, MeteoSuisse. To ensure strong dependence be-

tween the conditioning locations, we consider as conditional locations the 24 weather stations

that are at most 30km distant from Zurich and set as the conditional values the rainfall amounts

recorded in the year 2000, the year of the largest precipitation event recorded there between

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720


Table 3. Distribution of the partition size for the rainfall and temperature data estimated from

simulated Markov chains of length 15000 and 10000 respectively

Rainfall Temperature

Partition size 1 2 3 4 5 6 7–24 1 2 3 4 5–16

Frequencies (%) 66·2 28·0 4·8 0·5 0·2 0·2 <0·05 2·47 21·55 64·63 10·74 0·61

1962–2008, see Figure 2. 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 max-

imum pairwise likelihood estimator introduced by Padoan et al. (2010) to simultaneously es-

timate the marginal parameters and the spatial dependence parameters λ and κ. As in Davison

et al. (2012), the marginal parameters were described by η(x) = β0,η + β1,ηlon(x) + β2,ηlat(x),

σ(x) = β0,σ + β1,σlon(x) + β2,σlat(x), ξ(x) = β0,ξ, where η(x), σ(x), ξ(x) are the location,

scale and shape parameters of the generalized extreme value distribution and lon(x), lat(x) de-

note the longitude and latitude of the stations at which the data are observed. The maximum

pairwise likelihood estimates and standard errors for λ and κ are 38 (14) and 0·69 (0·07) and

give a practical extremal range, i.e., the distance h+ such that θ(h+) = 1·7, of around 115km;

see the right panel of Figure 2.

Table 3 shows the distribution of the partition size estimated from a Markov chain of length

15000. Around 65% of the time the summer maxima observed at the 24 conditioning locations

belonged to a single extremal function, i.e., only one storm event, and around 30% 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 extremal function while

the remaining 20 locations were always influenced by another one.

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20

40

60

80

30

40

50

60

70

80

90

40

60

80

100

1.00

1.02

1.04

1.06

1.08

1.10

Fig. 3. From left to right, maps on a 50× 50 grid of the

pointwise 0·025, 0·5 and 0·975 sample quantiles for rain-

fall (mm) obtained from 10000 conditional simulations of

Brown–Resnick processes having semi variogram γ(h) =

(h/38)0·69. The rightmost panel plots the ratio of the point-

wise confidence intervals with and without taking estima-

tion uncertainties into account. The squares show the con-

ditional locations.

Figure 3 plots the pointwise 0·025, 0·5 and 0·975 sample quantiles obtained from 10000 con-

ditional simulations of our fitted Brown–Resnick process. The conditional median provides a

point estimate for the rainfall at an ungauged location and the 0·025 and 0·975 conditional quan-

tiles 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

κ ≈ 0·69 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 confi-

dence 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

maximum composite likelihood estimator, and drawing one conditional simulation for each re-

alization. These ratios show no clear spatial pattern and the width of the confidence interval is

increased by at most 10%.

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1000 2000 3000 4000

Arosa (1840)

Bad Ragaz (496)

Basel (316)

Bern (565)

Chateau d’Oex (985)

Davos (1590)Engelberg (1035)

Gd−St−Bernard (2472)

Locarno−Monti (366)

Lugano (273)

Montana (1508)

Montreux (405)

Neuchatel (485)

Oeschberg (483)

Santis (2490)Zurich (556)

0 25 50 75 100(km)

(m)

2.5 3 3.5 4 4.5

(°C)

Fig. 4. Left: Topographical map of Switzerland showing

the sites and altitudes in metres above sea level of 16

weather stations for which annual maxima temperature

data are available. Right: Map of temperature anomalies

(◦C), i.e., the difference between the pointwise medians

obtained from 10000 conditional simulations and uncondi-

tional medians estimated from the fitted Schlather process.

5·2. Extreme temperatures in Switzerland

The data considered here are annual maximum temperatures recorded at 16 sites in Switzer-

land during the period 1961–2005, see Figure 4. Following Davison and Gholamrezaee (2012),

we fit a Schlather process with an isotropic powered exponential correlation function and trend

surfaces η(x) = β0,η + β1,ηalt(x), σ(x) = β0,σ, ξ(x) = β0,ξ + β1,ξalt(x), where alt(x) de-

notes the altitude above mean sea level in kilometres and {η(x), σ(x), ξ(x)} are the location,

scale and shape parameters of the generalized extreme value distribution at location x. The spatial

dependence parameter estimates and their standard errors are λ = 260 (149) and κ = 0.52 (0.12)

and yield a fitted extremal coefficient function similar to our test case ρ3 in Section 4.

In year 2003, western Europe endured a heatwave believed to be most severe recorded since

1540 (Beniston, 2004). Switzerland was severely impacted by this event: the nationwide record

temperature of 41·5◦C was recorded that year in Grono, Graubunden, near Lugano. For our anal-

ysis we condition on the maximum temperatures observed in 2003. Based on the fitted Schlather

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model, we simulate a Markov chain of effective length 10000 with a burn-in period of length

500 and a thinning lag of 100 iterations. The distribution of the partition size estimated from

these Markov chains is shown in Table 3. Around 90% of the time the conditional realizations

were a consequence of at most three extremal functions. Since our original observations were not

summer maxima but maximum daily values, a close inspection of the times series in year 2003

reveals that the hottest temperatures occurred between the 11th and 13th of August and, to some

extent, 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 10000 conditional simu-

lations and the pointwise unconditional medians estimated from the fitted Schlather model. As

expected, the largest deviations occur in the plateau region of Switzerland while appreciably

smaller values are found in the Alps. The differences range between 2·5◦C and 4·75◦C and are

consistent with the values reported by climatologists for mean values (Beniston, 2004).

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. Oest-

ing for pointing out an error in the computations for the Brown–Resnick model.

APPENDIX

Brown–Resnick model

For all x ∈ X k and Borel set A ⊂ Rk

Λx(A) =

∫ ∞0

pr [ζ exp{W (x)− γ(x)} ∈ A] ζ−2dζ =

∫ ∞0

∫Rk

1{ζ exp{y−γ(x)}∈A}fx(y)dyζ−2dζ,

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where fx denotes the density of the random vector W (x), i.e., a centered Gaussian random vector with

covariance matrix Σx and variance 2γ(x). The change of variables z = ζ exp{y − γ(x)} and r = log ζ

yields

Λx(A) =

∫ ∞−∞

∫A

fx{log z − r + γ(x)}k∏i=1

z−1i dze−rdr =

∫A

λx(z)dz

with

λx(z) =

k∏i=1

z−1i

∫ ∞−∞

fx{log z − r + γ(x)}e−rdr.

Since

fx{log z − r + γ(x)}e−r = (2π)−k/2|Σx|−1/2 exp

{−1

2P (r)

},

with

P (r) = r21Tk Σ−1x 1k − 2r[1Tk Σ−1x {log z + γ(x)} − 1

]+ {log z + γ(x)}TΣ−1x {log z + γ(x)},

standard computations for Gaussian integrals give

λx(z) = Cx exp

(−1

2log zTQx log z + Lx log z

) k∏i=1

z−1i .

The conditional intensity function is

λs|x,z(u) =C(s,x)

Cxexp

{−1

2log (u, z)

TQ(s,x) log(u, z) + L(s,x) log(u, z) +

1

2log zTQx log z − Lx log z

} m∏i=1

u−1i ,

and since log(u, z) = Jm,k log u+ Jm,k log z, it is not difficult to show that

λs|x,z(u) =C(s,x)

Cxexp

{−1

2(log u− µs|x,z)TΣ−1s|x(log u− µs|x,z)

} m∏i=1

u−1i .

Finally, the relation C(s,x)/Cx = (2π)−m/2|Σs|x|−1/2 is a simple consequence of the normalization∫λs|x,z(u)du = 1.

Schlather model

For all x ∈ X k and Borel sets A ⊂ Rk

Λx(A) =

∫ ∞0

pr{

(2π)1/2ζε(x) ∈ A}ζ−2dζ =

∫ ∞0

∫Rk

1{(2π)1/2ζy∈A}fx(y)dyζ−2dζ,

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where fx denotes the density of the random vector ε(x), i.e., a centered Gaussian random vector with

covariance matrix Σx. The change of variable z = (2π)1/2ζy gives

Λx(A) = (2π)−k/2∫ ∞0

∫A

fx

{z

(2π)1/2ζ

}ζ−(k+2)dzdζ

= (2π)−k|Σx|−1/2∫ ∞0

∫A

exp

(− 1

4πζ2zTΣ−1x z

)ζ−(k+2)dzdζ

= (2π)−k|Σx|−1/2∫A

2π

zTΣ−1x zE[Xk−1]dz, X ∼Weibull

{(4π

zTΣ−1x z

)1/2

, 2

}

= (2π)−k|Σx|−1/2∫A

2π

zTΣ−1x z

(4π

zTΣ−1x z

)(k−1)/2

Γ

(k + 1

2

)dz

=

∫A

λx(z)dz,

where λx(z) = π−(k−1)/2|Σx|−1/2ax(z)−(k+1)/2Γ {(k + 1)/2} and ax(z) = zTΣ−1x z.

For all u ∈ Rm the conditional intensity function is

λs|x,z(u) = π−m/2|Σ(s,x)|−1/2

|Σx|−1/2

{a(s,x)(u, z)

ax(z)

}−(m+k+1)/2

ax(z)−m/2Γ(m+k+1

2

)Γ(k+12

) .

We start by focusing on the ratio a(s,x)(u, z)/ax(z). Since Σs Σs:x

Σx:s Σx

−1

=

(Σs − Σs:xΣ−1x Σx:s)−1 −(Σs − Σs:xΣ−1x Σx:s)

−1Σs:xΣ−1x

−Σ−1x Σx:s(Σs − Σs:xΣ−1x Σx:s)−1 Σ−1x + Σ−1x Σx:s(Σs − Σs:xΣ−1x Σx:s)

−1Σs:xΣ−1x

,

straightforward algebra shows that

a(s,x)(u, z)

ax(z)= 1 +

(u− µ)T Σ−1(u− µ)

k + 1, µ = Σs:xΣ−1x z, Σ =

ax(z)

k + 1

(Σs − Σs:xΣ−1x Σx:s

).

We now simplify the ratio |Σ(s,x)|/|Σx|. Using the fact that

Σ(s,x) =

Σs Σs:x

Σx:s Σx

=

Im Σs:x

0k,m Σx

Σs − Σs:xΣ−1x Σx:s 0m,k

Σ−1x Σx:s Ik

,

combined with some more algebra yields

|Σ(s,x)||Σx|

= |Σs − Σs:xΣ−1x Σx:s| ={k + 1

ax(z)

}m|Σ|.

Using the two previous results it is easily found that

λs|x,z(u) = π−m/2(k + 1)−m/2|Σ|−1/2{

1 +(u− µ)T Σ−1(u− µ)

k + 1

}−(m+k+1)/2Γ(m+k+1

2

)Γ(k+12

) ,

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1008


which corresponds to the density of a multivariate Student distribution with the stated parameters.

REFERENCES

BENISTON, M. (2004). The 2003 heat wave in Europe: A shape of things to come? An analysis based on Swiss

climatological data and model simulations. Geophys. Res. Lett. 31, L02202.

BROWN, B. M. & RESNICK, S. (1977). Extreme values of independent stochastic processes. J. Appl. Prob. 14,

732–739.

BUISHAND, T. A., DE HAAN, L. & ZHOU, C. (2008). On spatial extremes: With application to a rainfall problem.

Annals of Applied Statistics 2, 624–642.

CHILES, J.-P. & DELFINER, P. (1999). Geostatistics: Modelling Spatial Uncertainty. New York: Wiley.

COOLEY, D., NAVEAU, P. & PONCET, P. (2006). Variograms for spatial max-stable random fields. In Dependence in

Probability and Statistics, P. Bertail, P. Soulier, P. Doukhan, P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin

& S. Zeger, eds., vol. 187 of Lecture Notes in Statistics. Springer New York, pp. 373–390.

DAVIS, R. & RESNICK, S. (1989). Basic properties and prediction of max-ARMA processes. Advances in Applied

Probability 21, 781–803.

DAVIS, R. & RESNICK, S. (1993). Prediction of stationary max-stable processes. Annals Of Applied Probability 3,

497–525.

DAVISON, A. C. & GHOLAMREZAEE, M. M. (2012). Geostatistics of extremes. Proceedings of the Royal Society

A: Mathematical, Physical and Engineering Science 468, 581–608.

DAVISON, A. C., PADOAN, S. A. & RIBATET, M. (2012). Statistical modelling of spatial extremes. Statistical

Science 7, 161–186.

DE HAAN, L. (1984). A spectral representation for max-stable processes. The Annals of Probability 12, 1194–1204.

DE HAAN, L. & PEREIRA, T. T. (2006). Spatial extremes: Models for the stationary case. The Annals of Statistics

34, 146–168.

DE HAAN, L. & FEREIRA, A. (2006). Extreme Value Theory: An Introduction. Springer, New-York.

GENTON, M. G., MA, Y. & SANG, H. (2011). On the likelihood function of Gaussian max-stable processes.

Biometrika 98, 481–488.

GENZ, A. (1992). Numerical computation of multivariate normal probabilities. J. Comp. Graph Stat 1, 141–149.

KABLUCHKO, Z., SCHLATHER, M. & DE HAAN, L. (2009). Stationary max-stable fields associated to negative

definite functions. Ann. Prob. 37, 2042–2065.

1009

1010

1011

1012

1013

1014

1015

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1017

1018

1019

1020

1021

1022

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1038

1039

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1041

1042

1043

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1045

1046

1047

1048

1049

1050

1051

1052

1053

1054

1055

1056


KABLUCHKO, Z. & SCHLATHER, M. (2010). Ergodic properties of max-infinitely divisible processes. Stochastic

Processes and their Applications, 120, 281–295.

LIU, J., WONG, W. & KONG, A. (1995). Correlation structure and convergence rate of the Gibbs sampler with

various scans. Journal of the Royal Statistical Society Series B 57, 157–169.

MANDELBROT, B. (1982). The Fractal Geometry of Nature. New York: W. H. Freeman.

OESTING, M., KABLUCHKO, Z. & SCHLATHER, M. (2012). Simulation of Brown–Resnick processes. Extremes

15, 89–107.

PADOAN, S. A., RIBATET, M. & SISSON, S. (2010). Likelihood-based inference for max-stable processes. Journal

of the American Statistical Association 105, 263–277.

R DEVELOPMENT CORE TEAM (2012). R: A Language and Environment for Statistical Computing. R Foundation

for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.

RIBATET, M. (2011). SpatialExtremes: Modelling Spatial Extremes. R package version 1.8-5.

SCHLATHER, M. (2002). Models for stationary max-stable random fields. Extremes 5, 33–44.

SCHLATHER, M. (2012). RandomFields: Simulation and Analysis of Random Fields. R package version 2.0.54.

SCHLATHER, M. & TAWN, J. (2003). A dependence measure for multivariate and spatial extremes: Properties and

inference. Biometrika 90, 139–156.

WANG, Y. (2010). maxLinear: Conditional sampling for max-linear models. R package version 1.0.

WANG, Y. & STOEV, S. A. (2011). Conditional sampling for spectrally discrete max-stable random fields. Advances

in Applied Probability 43, 461–483.

conditional simulation of max-stable processesribatet/docs/dombry2011.pdf · in classical...

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