Some new classes of stationary max-stable random fields - Christian ROBERT (Université Lyon 1, Laboratoire SAF)

2012.17

Laboratoire SAF – 50 Avenue Tony Garnier - 69366 Lyon cedex 07 http://www.isfa.fr/la_recherche

Some new classes of stationary max-stable random fields

Christian Y. Robert

Universite de Lyon, Universite Lyon 1, Institut de Science Financiere et d’Assurances,

50 Avenue Tony Garnier, F-69007 Lyon, France

Abstract

We present two new classes of stationary max-stable random fields. For the first class, we usethe spectral representation due to Schlather [14] and assume that the stationary process used inthe representation is proportional to a power of a max-stable random field. We derive the finitedimensional distributions and explain the relationship between distributions of both max-stablerandom fields. For the second class, we consider a multiplicative factor model and a Poisson-Voronoıtessellation of Rd to construct new max-stable random fields. We provide explicit expressions forthe pairwise distribution function.

Keywords: Max-stable random fields; Poisson-Voronoı tessellation; Spectral representation.

1. Introduction

A random field Z in Rd is a max-stable random field if there exist a sequence of independent

and identically distributed random fields (Yi)i≥1 in Rd, and two sequences of non-random functions

αn (·) > 0 and βn (·) ∈ R such that

Z(x)d= lim

n→∞

maxi=1,...,n Yi(x)− βn (x)

αn (x), x ∈ R

d.

We assume that Z has unit Frechet marginal distributions, usually referred as a simple max-stablerandom field, and we focus here on stationary max-stable random fields.

A useful representation of simple max-stable random fields is the so-called spectral representa-tion due to Schlather [14] (see also [5]). Let (ζj)j≥1 be the points of a Poisson process on R

+ withintensity ds/s2, and let (Vj)j≥1 be independent replicates of a stationary random field V on R

d,independent on the Poisson process and satisfying E [max (0, V (x))] = 1. Then

W (x) = maxj≥1

ζjVi(x), x ∈ Rd, (1)

is a simple max-stable random field. The finite dimensional distributions of W are characterizedby

P (W (x1) ≤ z1, . . . ,W (xm) ≤ zm) = exp

(

−E

[

max

(

V (x1)

z1, . . . ,

V (xm)

zm

)])

,

for x1, . . . , xm ∈ Rd and z1, . . . , zm ∈ R+.

Different choices for the random field V lead to some well-known and useful max-stable models(see the well-written review [4] for some applications to statistical modeling of spatial extremes). A

Email address: christian.robert@univ-lyon1.fr (Christian Y. Robert)

Preprint submitted to Elsevier December 18, 2012

first choice proposed by Brown and Resnick [2] is to take V (x) = exp{σε(x)−σ2/2}, σ > 0, where εis a stationary standard Gaussian random field. A second choice proposed by Smith [15] is to takeVj(x) = g(x−Xj), where g is a probability density function on R

d (e.g. the standard multivariateGaussian or Student distribution) and (Xj)j≥1 are the points of a homogeneous Poisson process onRd. A third choice proposed by Schlather [14] is to take V to be a stationary standard Gaussian

random field, scaled so that E [max (0, V (x))] = 1. A fourth choice proposed by Schlather [14] andWadsworth and Tawn [16] is to take Vj(x) = f(x − Xj)Bj (x), where f is a probability densityfunction on R

d or the indicator function of a compact random set in Rd, (Xj)j≥1 are the points

of a homogeneous Poisson process on Rd and (Bj)j≥1 are independent replicates of a stationary

standard Gaussian random field, scaled so that E [max (0, Vj(x))] = 1. A fifth choice proposed byLantuejoul, Bacro and Bel [10] is to take V as the indicator function of a Poisson polytope. Formost of these models, only the pairwise distributions are known (an exception is Smith’s model,see [7], for which it is possible to give the p−multivariate distributions with p ≤ d+ 1).

In this paper, we introduce two new classes of simple max-stable random fields. In Section 2,we construct a first class by assuming that V is proportional to W β where W is itself a simplemax-stable random field and 0 < β < 1. We derive the finite dimensional distributions and explainthe relationship between distributions of both max-stable random fields. In Section 3, we considera multiplicative factor model and a Poisson-Voronoı tessellation of Rd to construct a second classof simple max-stable random fields. We provide explicit expressions for the pairwise distributionfunction. New models are illustrated by simulations of sample paths of the random fields in R

2.

2. The max-max-stable random fields

In this section, we consider a stationary max-stable random fieldW with spectral representation(1) and denote by H(x1,...,xn) the function given by

H(x1,...,xm) (z1, . . . , zm) = E

[

max

(

V (x1)

z1, . . . ,

V (xm)

zm

)]

, z1, . . . , zm ∈ R+,

for any x1, . . . , xm ∈ Rd.

Let (ξj)j≥1 be the points of a Poisson process on R+ with intensity ds/s2, and let (Wj)j≥1 be

independent replicates of W , independent on the Poisson process. We define the max-max-stablerandom field Zβ the following way

Zβ(x) =1

Γ (1− β)maxj≥1

ξjWβj (x), x ∈ R

d,

with 0 < β < 1.In the following theorem, we give the relationship between the finite dimensional distributions

of W and Zβ.

Proposition 1. For x1, . . . , xm ∈ Rd and z1, . . . , zm ∈ R+,

P (Zβ(x1) ≤ z1, . . . , Zβ(xm) ≤ zm) = exp

(

−[

H(x1,...,xm)

(

z1/β1 , . . . , z1/βm

)]β)

.

Proof: By using the spectral representation of Zβ, we derive that

P (Zβ(x1) ≤ z1, . . . , Zβ(xm) ≤ zm) = exp

(

− 1

Γ (1− β)E

[

max

(

W β(x1)

z1, . . . ,

W β(xm)

zm

)])

2

with

1

Γ (1− β)E

[

max

(

W β(x1)

z1, . . . ,

W β(xm)

zm

)]

=

m∑

i=1

z−1i E

[

W β(xi)

Γ (1− β)I{∩j 6=i(W β(xi)≥W β(xj)zi/zj)}

]

.

Let us use the notation H (·) = H(x1,...,xm) (·) and note that

P (∩j 6=i (W (xj) ≤ zj) |W (xi) = zi) = −Hi (z1, . . . , zm) z2i exp(

−H (z1, . . . , zm) + z−1i

)

where Hi (z1, . . . , zm) = ∂H (z1, . . . , zm) /∂zi. We have

E

[

W β(xi)

Γ (1− β)I{∩j 6=i(W β(xi)≥W β(xj)zi/zj)}

]

= E

[

W β(xi)

Γ (1− β)P(

∩j 6=i

(

W (xj) ≤ W (xi) (zj/zi)1/β)∣

∣

∣W (xi))

]

= − 1

Γ (1− β)E

[

W β(xi)Hi

(

(

W (xi) (zj/zi)1/β)

j 6=i,W (xi)

)

×W (xi)2 exp

(

−H

(

(

W (xi) (zj/zi)1/β)

j 6=i,W (xi)

)

+W (xi)−1

)]

= − 1

Γ (1− β)Hi

(

(

(zj/zi)1/β)

j 6=i, 1

)

E

[

W β(xi) exp

(

−[

H

(

(

(zj/zi)1/β)

j 6=i, 1

)

− 1

]

W (xi)−1

)]

= − 1

Γ (1− β)Hi

(

(

(zj/zi)1/β)

j 6=i, 1

)∫ ∞

0wβ−2 exp

(

−H

(

(

(zj/zi)1/β)

j 6=i, 1

)

w−1

)

dw

= −Hi

(

(

(zj/zi)1/β)

j 6=i, 1

)[

H

(

(

(zj/zi)1/β)

j 6=i, 1

)]β−1

= −z2/βi Hi

(

z1/β1 , . . . , z1/βm )

)

z1−1/βi

[

H(

z1/β1 , . . . , z1/βm )

)]β−1

= −z1+1/βi Hi

(

z1/β1 , . . . , z1/βm )

) [

H(

z1/β1 , . . . , z1/βm )

)]β−1.

It follows by Euler’s formula that

−m∑

i=1

z1/βi Hi

(

z1/β1 , . . . , z1/βm )

) [

H(

z1/β1 , . . . , z1/βm )

)]β−1=(

H(

z1/β1 , . . . , z1/βm )

))β

and1

Γ (1− β)E

[

max

(

W β(x1)

z1, . . . ,

W β(xm)

zm

)]

=(

H(

z1/β1 , . . . , z1/βm )

))β,

which concludes the proof. �

If all the components are less than z, then

P (W (x1) ≤ z, . . . ,W (xm) ≤ z) = exp(

−z−1H(x1,...,xm) (1, . . . , 1))

= exp(

−z−1θW (x1, . . . , xm))

where θW (x1, . . . , xm) is called the extremal coefficient of the vector (W (x1), . . . ,W (xm)) andmeasures the extremal dependence: it varies from 1 when the observations are fully dependent tom when they are independent. We derive from the previous proposition that

θZβ(x1, . . . , xm) = θβW (x1, . . . , xm) .

3

This equation shows that the extremal dependence is stronger for the max-max-stable random fieldZβ than for the max-stable random field W . Figure 1 gives an illustration of this. We plotted in R

2

samples paths of Smith’s model and Schlather’s model and sample paths of their associated max-max-stable random fields with β = 0.1, 0.5, 0.9. The sample paths of the max-max-stable randomfields have been computed with the same points for the Poisson process and the same replicates ofW . We see that the coefficient β smooths and squeezes the extreme values of the replicates of Wand therefore increases the dependence in space of the random field as it decreases.

Figure 1: Simulations of Smith’s model and Schlather’s model with their associated max-max-stable random fields(MMSRF). The random fields have been transformed to unit Gumbel margins. Top, from left to right: Smith’smodel, MMSRF with β = 0.9, MMSRF with β = 0.5, MMSRF with β = 0.1. Bottom, from left to right: Schlather’smodel with exponential correlation, MMSRF with β = 0.9, MMSRF with β = 0.5, MMSRF with β = 0.1.

3. The Voronoı max-stable random fields

Let U be a stationary positive random field in Rd and Γ be a random variable with a Pareto-

type distribution, i.e. its survival probability function is given by P (Γ > γ) = γ−αl (γ) for γ > 0,where α > 0 and l is a slowly varying function. U and Γ are assumed to be independent and thereexists a constant δ > α such that E(U δ(x)) < ∞.

We define the stationary random field Y by

Y (x) = ΓU(x), x ∈ Rd.

By Breiman’s theorem (see e.g. [3]), it follows that

P (Y (x) > y) = (1 + o(1))E(Uα(x))P (Γ > y) , as y → ∞.

Let Yi be independent replicates of Y and define

Mn (x) = maxi=1,...,n

1

E(Uα(x))

(

Yi (x)

UΓ (n)

)α

, x ∈ Rd,

where UΓ (y) = inf{

γ : P (Γ ≤ γ) ≥ 1− y−1}

.

4

Proposition 2. The finite dimensional distributions of Mn converge to the finite dimensional dis-

tributions of a simple max-stable random field Z in Rd. They are characterized by

P (Z(x1) ≤ z1, . . . , Z(xm) ≤ zm) = exp

(

−E

[

max

(

1

z1

Uα(x1)

E(Uα(x1)), . . . ,

1

zm

Uα(xm)

E(Uα(xm))

)])

,

for x1, . . . , xm ∈ Rd and z1, . . . , zm ∈ R+. Z has the following spectral representation

Z(x)d= max

j≥1ζj

Uαj (x)

E(Uα(x)), x ∈ R

d,

where (ζj)j≥1 are the points of a Poisson process on R+ with intensity ds/s2, and (Uj)j≥1 are

independent replicates of U .

Proof: We have

limn→∞

logP (Mn (xj) ≤ zj, j = 1, ...,m)

= limn→∞

logPn(

∩mj=1

{

ΓU(xj) ≤ (zjE(Uα(xj)))

1/α UΓ (n)})

= limn→∞

logPn

(

Γ maxj=1,...,m

U(xj)

(zjE(Uα(xj)))1/α

≤ UΓ (n)

)

= − limn→∞

nP

(

Γ maxj=1,...,m

U(xj)

(zjE(Uα(xj)))1/α

> UΓ (n)

)

= − limn→∞

nE

[

maxj=1,...,m

Uα(xj)

zjE(Uα(xj))

]

P (Γ > UΓ (n))

= −E

[

maxj=1,...,m

Uα(xj)

zjE(Uα(xj))

]

.

The spectral representation follows from (1). �

We now present the construction of a particular random field U which we call a Voronoı randomfield.

Let (ξj)j≥1 denote the points of a homogeneous Poisson process with intensity λ in Rd, and let

V denote the Voronoı tessellation generated by (ξj)j≥1. The Voronoı cell, or Voronoı region, Rj,associated with the site ξj is the set of all points in R

d whose (Euclidean) distance to ξj is smallerthan their distance to the other sites ξk, where k is any index different from j. We denote by ∆the set of points in R

d for which the smallest distances to the sites (ξj)j≥1 are equal for at leasttwo sites. The tessellation V = {Rj, j ≥ 1} is known as the Poisson-Voronoı tessellation and wasintroduced by [12] (see e.g. [13] for further details).

The Voronoı random field is then defined by

U(x) =

{ ∑

j≥1 ηjI{x∈Rj}, x ∈ Rd/∆,

0, ∆,(2)

where the (ηj)j≥1 are independent and identically distributed positive random variables, indepen-dent on the (ξj)j≥1. We assume that there exists a constant δ > α such that E(ηδj ) < ∞.

The construction of the random field Y may have a natural explanation in hydrology: therandom points (ξj)j≥1 may be interpreted as the generators of rainfalls with ferocities (Γηj)j≥1

5

where Γ is a common factor; for each generator there is a corresponding region consisting of allpoints closer to that generator than to any other and for which the level of rainfall is given by thelevel of its generator.

We now want to characterize the finite dimensional distributions of Z. For this purpose, weintroduce some notation. We first define, for m ≥ 1 and z1, . . . , zm ∈ R+,

Λm (z1, ..., zm) = E

[

max

(

z−11

ηα1E(ηα1 )

, . . . , z−1m

ηαmE(ηαm)

)]

=m∑

l=1

z−1l E

ηα

E(ηα)

∏

s 6=l

H

(

(

zszl

)1/α

η

)

,

where H is the common probability distribution function of the ηj. Let us give some exam-ples by considering different distributions H and their associated distribution G (z1, ..., zm) =exp (−Λm (z1, ..., zm)).

• Assume that H is a Bernoulli distribution with parameter 0 < p < 1. The distribution of ηα

is also a Bernoulli distribution with the same parameter and

Λm (z1, ..., zm) = pm−1max(

z−11 , ..., z−1

m

)

.

For m = 2, G is the Marshall-Olkin distribution [11].

• Assume that η = eX where X has a Gaussian distribution N(

µ, σ2)

. Then ηα has a Lognor-mal distribution with parameter αµ and α2σ2 and

Λm (z1, ..., zm) =

m∑

l=1

z−1l ΦΣ(m−1)

((

θ−1 +(

θ log(

zjz−1l

))

/2)

; j 6= l)

,

where θ =√2/ (ασ) and ΦΣ(m−1) is the (m− 1)-variate Gaussian probability distribution

function with mean vector equal to zero and correlation matrix Σ (m− 1) = (σl,j (m− 1))given by σl,l (m− 1) = 1 for 1 ≤ l ≤ m− 1 and σl,j (m− 1) = 1/2 for 1 ≤ l < j ≤ m− 1.

For m = 2, G is the Husler-Reiss distribution [9].

• Assume that H is a Weibull distribution (Wei (c, τ)), c > 0 and τ > 0, i.e. H (x) =exp (−cxτ ). Then ηα has a Weibull distribution Wei (c, τ/α) and

Λm (z1, ..., zm) =m∑

l=1

(−1)m−1∑

1≤j1<...<jl≤m

l∑

j=1

zθjl

−1/θ

,

where θ = τ/α. For m = 2, G is the Galambos distribution [6].

• Assume that H is a Frechet distribution (Fre (c, τ)), c > 0 and τ > 0, i.e. H (x) =exp (−cx−τ ). If τ/α > 1 then ηα has a Frechet distribution Fre (c, τ/α) with a finite meanand

Λm (z1, ..., zm) =

(

m∑

l=1

z−θl

)1/θ

,

where θ = τ/α. For m = 2, G is the Logistic or the Gumbel distribution [8].

Let jx be defined by I{x∈Rjx}= 1. Let I1, . . . , Ih denote a partition of {1, . . . ,m} in h subsets.

6

Proposition 3. For x1, . . . , xm ∈ Rd, we have

E

[

maxi=1,...,m

Uα(xi)

ziE(Uα(xi))

]

=m∑

h=1

∑

I1,...,Ih

P

(

h⋂

l=1

{

jxi= jxi′

; i, i′ ∈ Il}

)

Λh

(

1

mini∈I1 zi, ...,

1

mini∈Ih zi

)

.

Proof: First note that

E

[

maxi=1,...,m

Uα(xi)

ziE(Uα(xi))

]

= E

[

maxi=1,...,m

ηjxiziE(ηjxi )

]

.

If jxi= jxi′

for i, i′ ∈ {1, . . . ,m}, then ηjxi = ηjxi′

and if jxi6= jxi′

, then ηjxi and ηjxi′

are

independent. Therefore, if we split {1, . . . ,m} into a partition Ih of h subsets I1, . . . , Ih, and if AIh

is the event such that the jx are the same for each subset, but different between the subsets, then

E

[

maxi=1,...,m

ηjxiziE(ηjxi )

∣

∣

∣

∣

∣

AIh

]

= Λh

(

1

mini∈I1 zi, ...,

1

mini∈Ih zi

)

.

The result follows by the formula of total probability. �

In the case m = 2, it is possible to give an explicit expression for the pairwise distributionfunctions. Let vd(l) denote the volume of the d-dimensional sphere with radius l and σd(l) thevolume of the d-dimensional sphere with radius l, i.e.

vd(l) =πd/2

Γ (d/2 + 1)ld, σd(l) = 2

πd/2

Γ (d/2)ld−1.

Let sd(r, α, l) denote the volume of two d-dimensional spheres, with their centres a distance l apart,

where one has radius r and the other has radius(

r2 + l2 − 2lr cosα)1/2

. We have

sd(r, α, l) = vd−1(1)

(

rd∫ π

αsind (θ) dθ +

(

r2 + l2 − 2lr cosα)d/2

∫ π

t(r,α,l)sind (θ) dθ

)

where

t(r, α, l) = arccos

(

l − r cosα

(r2 + l2 − 2lr cosα)1/2

)

.

Corollary 1. For x1, x2 ∈ Rd, the pairwise distribution function of (Z(x1), Z(x2)) is given by

P (Z(x1) ≤ z1, Z(x2) ≤ z2) = exp

(

−[

pd(l)1

min(z1, z2)+ (1− pd(l)) Λ2 (z1, z2)

])

, z1, z2 ∈ R+.

(3)where

pd(l) = λσd−1(1)

∫ ∞

0rd−1

[∫ π

0sind−1 (α) exp (−λsd(r, α, l)) dα

]

dr

with l = ‖x1 − x2‖2.

Proof: By the previous proposition, we only have to calculate P (jx1= jx2

) = pd(l).First note that x1 and x2 belongs to the same Voronoı cell if they are closer to the same

generator ξj = ξjx1 = ξjx2 than to any other. Let r denote the distance between x1 and ξj and

7

(

r2 + l2 − 2lr cosα)1/2

(α ∈ [0, π)) the distance between x2 and ξj. Let S1(r) be the d-dimensionalsphere with center x1, and let S2(r, α) be the d-dimensional sphere with center x2 and radius(

r2 + l2 − 2lr cosα)1/2

. There must be no other generator ξi, i 6= j, in the union S1(r) ∪ S2(r, α).This is done with probability exp (−λsd(r, α, l)).

Second, for x ∈ Rd, let ϕx be the angle between the straight line (x1x2) and the straight line

(x1x). If dr and dα are small variations in r and α, then the probability that ξj belongs to the set{x ∈ S1(r + dr)\S1(r), ϕx ∈ [α,α + dα)} is at the first order equal to λσd−1(1)r

d−1 sind−1 (α) dαdr.Therefore, we have by the formula of total probability

pd(l) =

∫ ∞

0

∫ π

0

[

λσd−1(1)rd−1 sind−1 (α)

]

exp (−λsd(r, α, l)) dαdr

= λσd−1(1)

∫ ∞

0rd−1

[∫ π

0sind−1 (α) exp (−λsd(r, α, l)) dα

]

dr.

and the result follows. �

Figure 2: Top-left: simulation of the Voronoı random field U with lognormal distributed η. Top-right: simulationof the Voronoı max-stable random field Z with lognormal distributed η. Bottom-left: simulation of the Voronoımax-stable random field Z with exponentially distributed η. Bottom-right: simulation of the Voronoı max-stablerandom field Z with Frechet distributed η. The max-stable random fields have been transformed to unit Gumbelmargins.

The extremal coefficient of Z is given by θZ (x1, x2) = pd(l) + (1− pd(l)) Λ2 (1, 1). The depen-dence decreases monotically and continuously as l increases or the intensity of the Poisson process

8

λ increases. But there is no asymptotic dependence since, as l or λ tend to infinity, θZ tends toΛ2 (1, 1). This is due to the common factor Γ.

Figure 2 shows a sample path of the Voronoı random field U in R2 and three sample paths of

the Voronoı max-stable random field U when η is respectively Lognormal, Exponential and Frechet(the max-stable random fields have been transformed to unit Gumbel margins). We see that, asfor the Voronoı random field, the values of the max-stable random fields are divided into severalregions, but these regions are no more Voronoı regions. They provide new interesting algorithmsfor creating tessellations of R2.

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