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
Université de Lyon, Université Lyon 1, Institut de Science Financière 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 use
the spectral representation due to Schlather [14] and assume that the stationary process used in
the representation is proportional to a power of a max-stable random field. We derive the finite
dimensional distributions and explain the relationship between distributions of both max-stable
random fields. For the second class, we consider a multiplicative factor model and a Poisson-Voronoı̈
tessellation of Rd to construct new max-stable random fields. We provide explicit expressions for
the pairwise distribution function.
Keywords: Max-stable random fields; Poisson-Voronoı̈ 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
d
Z(x) = lim
n→∞
maxi=1,...,n Yi (x) − βn (x)
,
αn (x)
x ∈ Rd .
We assume that Z has unit Fréchet marginal distributions, usually referred as a simple max-stable
random 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 representation due to Schlather [14] (see also [5]). Let (ζj )j≥1 be the points of a Poisson process on R+ with
intensity ds/s2 , and let (Vj )j≥1 be independent replicates of a stationary random field V on Rd ,
independent on the Poisson process and satisfying E [max (0, V (x))] = 1. Then
W (x) = max ζj Vi (x),
j≥1
x ∈ Rd ,
(1)
is a simple max-stable random field. The finite dimensional distributions of W are characterized
by
V (x1 )
V (xm )
P (W (x1 ) ≤ z1 , . . . , W (xm ) ≤ zm ) = exp −E max
,...,
,
z1
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: [email protected] (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)− σ 2 /2}, σ > 0, where ε
is a stationary standard Gaussian random field. A second choice proposed by Smith [15] is to take
Vj (x) = g(x − Xj ), where g is a probability density function on Rd (e.g. the standard multivariate
Gaussian or Student distribution) and (Xj )j≥1 are the points of a homogeneous Poisson process on
Rd . 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] and
Wadsworth and Tawn [16] is to take Vj (x) = f (x − Xj )Bj (x), where f is a probability density
function on Rd 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 by
Lantuejoul, Bacro and Bel [10] is to take V as the indicator function of a Poisson polytope. For
most 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 simple
max-stable random field and 0 < β < 1. We derive the finite dimensional distributions and explain
the relationship between distributions of both max-stable random fields. In Section 3, we consider
a multiplicative factor model and a Poisson-Voronoı̈ tessellation of Rd to construct a second class
of simple max-stable random fields. We provide explicit expressions for the pairwise distribution
function. New models are illustrated by simulations of sample paths of the random fields in R2 .
2. The max-max-stable random fields
In this section, we consider a stationary max-stable random field W with spectral representation
(1) and denote by H(x1 ,...,xn ) the function given by
V (x1 )
V (xm )
H(x1 ,...,xm ) (z1 , . . . , zm ) = E max
,...,
,
z1 , . . . , zm ∈ R+ ,
z1
zm
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-stable
random field Zβ the following way
Zβ (x) =
1
max ξj Wjβ (x),
Γ (1 − β) j≥1
x ∈ Rd ,
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+ ,
h
iβ 1/β
1/β
P (Zβ (x1 ) ≤ z1 , . . . , Zβ (xm ) ≤ zm ) = exp − H(x1 ,...,xm ) z1 , . . . , zm
.
Proof: By using the spectral representation of Zβ , we derive that
β
W (x1 )
1
W β (xm )
P (Zβ (x1 ) ≤ z1 , . . . , Zβ (xm ) ≤ zm ) = exp −
E max
,...,
Γ (1 − β)
z1
zm
2
with
β
X
m
1
W (x1 )
W β (xm )
W β (xi )
E max
I{∩j6=i (W β (xi )≥W β (xj )zi /zj )} .
,...,
=
zi−1 E
Γ (1 − β)
z1
zm
Γ (1 − β)
i=1
Let us use the notation H (·) = H(x1 ,...,xm ) (·) and note that
P (∩j6=i (W (xj ) ≤ zj ) |W (xi ) = zi ) = −Hi (z1 , . . . , zm ) zi2 exp −H (z1 , . . . , zm ) + zi−1
where Hi (z1 , . . . , zm ) = ∂H (z1 , . . . , zm ) /∂zi . We have
W β (xi )
I
E
β
β
Γ (1 − β) {∩j6=i (W (xi )≥W (xj )zi /zj )}
W β (xi ) 1/β P ∩j6=i W (xj ) ≤ W (xi ) (zj /zi )
= E
W (xi )
Γ (1 − β)
1
1/β
β
= −
E W (xi )Hi
W (xi ) (zj /zi )
, W (xi )
Γ (1 − β)
j6=i
1/β
2
−1
W (xi ) (zj /zi )
, W (xi ) + W (xi )
×W (xi ) exp −H
j6=i
1
Hi
(zj /zi )1/β
, 1 E W β (xi ) exp − H
(zj /zi )1/β
, 1 − 1 W (xi )−1
= −
Γ (1 − β)
j6=i
j6=i
Z ∞
1
1/β
1/β
β−2
−1
= −
Hi
(zj /zi )
,1
w
exp −H
(zj /zi )
,1 w
dw
Γ (1 − β)
j6=i
j6=i
0
β−1
1/β
1/β
= −Hi
(zj /zi )
,1 H
(zj /zi )
,1
j6=i
j6=i
h iβ−1
2/β
1/β
1−1/β
1/β
1/β
1/β
= −zi Hi z1 , . . . , zm
) zi
H z1 , . . . , zm
)
h iβ−1
1+1/β
1/β
1/β
1/β
1/β
= −zi
Hi z1 , . . . , zm
) H z1 , . . . , zm
)
.
It follows by Euler’s formula that
−
and
m
X
i=1
1/β
zi
h iβ−1 β
1/β
1/β
1/β
1/β
1/β
1/β
Hi z1 , . . . , zm
) H z1 , . . . , zm
)
= H z1 , . . . , zm
)
β
β
1
W (x1 )
W β (xm )
1/β
1/β
E max
,...,
= H z1 , . . . , zm
)
,
Γ (1 − β)
z1
zm
which concludes the proof. If all the components are less than z, then
P (W (x1 ) ≤ z, . . . , W (xm ) ≤ z) = exp −z −1 H(x1 ,...,xm ) (1, . . . , 1)
= exp −z −1 θW (x1 , . . . , xm )
where θW (x1 , . . . , xm ) is called the extremal coefficient of the vector (W (x1 ), . . . , W (xm )) and
measures the extremal dependence: it varies from 1 when the observations are fully dependent to
m when they are independent. We derive from the previous proposition that
β
θZβ (x1 , . . . , xm ) = θW
(x1 , . . . , xm ) .
3
This equation shows that the extremal dependence is stronger for the max-max-stable random field
Zβ than for the max-stable random field W . Figure 1 gives an illustration of this. We plotted in R2
samples paths of Smith’s model and Schlather’s model and sample paths of their associated maxmax-stable random fields with β = 0.1, 0.5, 0.9. The sample paths of the max-max-stable random
fields have been computed with the same points for the Poisson process and the same replicates of
W . We see that the coefficient β smooths and squeezes the extreme values of the replicates of W
and 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’s
model, MMSRF with β = 0.9, MMSRF with β = 0.5, MMSRF with β = 0.1. Bottom, from left to right: Schlather’s
model with exponential correlation, MMSRF with β = 0.9, MMSRF with β = 0.5, MMSRF with β = 0.1.
3. The Voronoı̈ max-stable random fields
Let U be a stationary positive random field in Rd and Γ be a random variable with a Paretotype distribution, i.e. its survival probability function is given by P (Γ > γ) = γ −α l (γ) for γ > 0,
where α > 0 and l is a slowly varying function. U and Γ are assumed to be independent and there
exists 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(U α (x))P (Γ > y) ,
Let Yi be independent replicates of Y and define
1
Yi (x) α
Mn (x) = max
,
i=1,...,n E(U α (x))
UΓ (n)
where UΓ (y) = inf γ : P (Γ ≤ γ) ≥ 1 − y −1 .
4
as y → ∞.
x ∈ Rd ,
Proposition 2. The finite dimensional distributions of Mn converge to the finite dimensional distributions of a simple max-stable random field Z in Rd . They are characterized by
1 U α (x1 )
1 U α (xm )
P (Z(x1 ) ≤ z1 , . . . , Z(xm ) ≤ zm ) = exp −E max
,...,
,
z1 E(U α (x1 ))
zm E(U α (xm ))
for x1 , . . . , xm ∈ Rd and z1 , . . . , zm ∈ R+ . Z has the following spectral representation
d
Z(x) = max ζj
j≥1
Ujα (x)
,
E(U α (x))
x ∈ Rd ,
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
lim log P (Mn (xj ) ≤ zj , j = 1, ..., m)
n
o
1/α
α
lim log P n ∩m
ΓU
(x
)
≤
(z
E(U
(x
)))
U
(n)
j
j
j
Γ
j=1
n→∞
!
U
(x
)
j
≤ UΓ (n)
lim log P n Γ max
n→∞
j=1,...,m (z E(U α (x )))1/α
j
j
!
U (xj )
− lim nP Γ max
> UΓ (n)
n→∞
j=1,...,m (z E(U α (x )))1/α
j
j
U α (xj )
− lim nE max
P (Γ > UΓ (n))
n→∞
j=1,...,m zj E(U α (xj ))
U α (xj )
.
−E max
j=1,...,m zj E(U α (xj ))
n→∞
=
=
=
=
=
The spectral representation follows from (1). We now present the construction of a particular random field U which we call a Voronoı̈ random
field.
Let (ξj )j≥1 denote the points of a homogeneous Poisson process with intensity λ in Rd , and let
V denote the Voronoı̈ tessellation generated by (ξj )j≥1 . The Voronoı̈ cell, or Voronoı̈ region, Rj ,
associated with the site ξj is the set of all points in Rd whose (Euclidean) distance to ξj is smaller
than their distance to the other sites ξk , where k is any index different from j. We denote by ∆
the set of points in Rd for which the smallest distances to the sites (ξj )j≥1 are equal for at least
two sites. The tessellation V = {Rj , j ≥ 1} is known as the Poisson-Voronoı̈ tessellation and was
introduced by [12] (see e.g. [13] for further details).
The Voronoı̈ random field is then defined by
P
x ∈ Rd /∆,
j≥1 ηj I{x∈Rj } ,
U (x) =
(2)
0,
∆,
where the (ηj )j≥1 are independent and identically distributed positive random variables, independent on the (ξj )j≥1 . We assume that there exists a constant δ > α such that E(ηjδ ) < ∞.
The construction of the random field Y may have a natural explanation in hydrology: the
random 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 all
points closer to that generator than to any other and for which the level of rainfall is given by the
level of its generator.
We now want to characterize the finite dimensional distributions of Z. For this purpose, we
introduce some notation. We first define, for m ≥ 1 and z1 , . . . , zm ∈ R+ ,


X
1/α !
m
α
α
α Y
η
z
η
η
s
m
−1
=
zl−1 E 
H
η ,
Λm (z1 , ..., zm ) = E max z1−1 1α , . . . , zm
α)
E(η1 )
E(ηm
E(η α )
zl
l=1
s6=l
where H is the common probability distribution function of the ηj . Let us give some examples 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 η α
is also a Bernoulli distribution with the same parameter and
−1
Λm (z1 , ..., zm ) = pm−1 max z1−1 , ..., zm
.
For m = 2, G is the Marshall-Olkin distribution [11].
• Assume that η = eX where X has a Gaussian distribution N µ, σ 2 . Then η α has a Lognormal distribution with parameter αµ and α2 σ 2 and
Λm (z1 , ..., zm ) =
m
X
zl−1 ΦΣ(m−1)
θ −1 + θ log zj zl−1
l=1
/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 Hüsler-Reiss distribution [9].
• Assume that H is a Weibull distribution (W ei (c, τ )), c > 0 and τ > 0, i.e. H̄ (x) =
exp (−cxτ ). Then η α has a Weibull distribution W ei (c, τ /α) and
Λm (z1 , ..., zm ) =
m
X
(−1)m−1
X
1≤j1 <...<jl ≤m
l=1

−1/θ
l
X

zjθ 
,
l
j=1
where θ = τ /α. For m = 2, G is the Galambos distribution [6].
• Assume that H is a Fréchet distribution (F ré (c, τ )), c > 0 and τ > 0, i.e. H (x) =
exp (−cx−τ ). If τ /α > 1 then η α has a Fréchet distribution F ré (c, τ /α) with a finite mean
and
!1/θ
m
X
Λm (z1 , ..., zm ) =
zl−θ
,
l=1
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
X
m
X
U α (xi )
E max
P
=
i=1,...,m zi E(U α (xi ))
h=1 I1 ,...,Ih
!
h
\
′
jxi = jxi′ ; i, i ∈ Il
Λh
l=1
1
mini∈I1 zi
, ...,
1
mini∈Ih zi
.
Proof: First note that
"
#
ηjxi
U α (xi )
E max
= E max
.
i=1,...,m zi E(U α (xi ))
i=1,...,m zi E(ηjx )
i
If jxi = jxi′ for i, i′ ∈ {1, . . . , m}, then ηjxi = ηjx ′ and if jxi 6= jxi′ , then ηjxi and ηjx ′ are
i
i
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
"
#
ηjxi 1
1
E max
, ...,
.
AIh = Λh
i=1,...,m zi E(ηjx ) mini∈I1 zi
mini∈Ih zi
i
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 distribution
functions. Let vd (l) denote the volume of the d-dimensional sphere with radius l and σd (l) the
volume of the d-dimensional sphere with radius l, i.e.
vd (l) =
π d/2
ld ,
Γ (d/2 + 1)
σd (l) = 2
π d/2 d−1
l .
Γ (d/2)
Let sd (r, α, l) denote the volume of two d-dimensional spheres, with their centres a distance l apart,
1/2
where one has radius r and the other has radius r 2 + l2 − 2lr cos α
. We have
!
Z π
Z π
d/2
sd (r, α, l) = vd−1 (1) r d
sind (θ) dθ + r 2 + l2 − 2lr cos α
sind (θ) dθ
α
t(r,α,l)
where
l − r cos α
t(r, α, l) = arccos
(r 2 + l2 − 2lr cos α)1/2
!
.
Corollary 1. For x1 , x2 ∈ Rd , the pairwise distribution function of (Z(x1 ), Z(x2 )) is given by
1
P (Z(x1 ) ≤ z1 , Z(x2 ) ≤ z2 ) = exp − pd (l)
+ (1 − pd (l)) Λ2 (z1 , z2 ) , z1 , z2 ∈ R+ .
min(z1 , z2 )
(3)
where
Z π
Z ∞
d−1
d−1
pd (l) = λσd−1 (1)
r
sin
(α) exp (−λsd (r, α, l)) dα dr
0
0
with l = kx1 − x2 k2 .
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 Voronoı̈ 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
1/2
r 2 + l2 − 2lr cos α
(α ∈ [0, π)) the distance between x2 and ξj . Let S1 (r) be the d-dimensional
sphere with center x1 , and let S2 (r, α) be the d-dimensional sphere with center x2 and radius
1/2
. There must be no other generator ξi , i 6= j, in the union S1 (r) ∪ S2 (r, α).
r 2 + l2 − 2lr cos α
This is done with probability exp (−λsd (r, α, l)).
Second, for x ∈ Rd , let ϕx be the angle between the straight line (x1 x2 ) and the straight line
(x1 x). 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
Z ∞Z πh
i
pd (l) =
λσd−1 (1)r d−1 sind−1 (α) exp (−λsd (r, α, l)) dαdr
0
0
Z π
Z ∞
d−1
d−1
= λσd−1 (1)
r
sin
(α) exp (−λsd (r, α, l)) dα dr.
0
0
and the result follows. Figure 2: Top-left: simulation of the Voronoı̈ random field U with lognormal distributed η. Top-right: simulation
of the Voronoı̈ max-stable random field Z with lognormal distributed η. Bottom-left: simulation of the Voronoı̈
max-stable random field Z with exponentially distributed η. Bottom-right: simulation of the Voronoı̈ max-stable
random field Z with Fréchet distributed η. The max-stable random fields have been transformed to unit Gumbel
margins.
The extremal coefficient of Z is given by θZ (x1 , x2 ) = pd (l) + (1 − pd (l)) Λ2 (1, 1). The dependence 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 to
Λ2 (1, 1). This is due to the common factor Γ.
Figure 2 shows a sample path of the Voronoı̈ random field U in R2 and three sample paths of
the Voronoı̈ max-stable random field U when η is respectively Lognormal, Exponential and Fréchet
(the max-stable random fields have been transformed to unit Gumbel margins). We see that, as
for the Voronoı̈ random field, the values of the max-stable random fields are divided into several
regions, but these regions are no more Voronoı̈ regions. They provide new interesting algorithms
for creating tessellations of R2 .
References
[1] Bacro, J.N., Gaetan, C. (2012). A review on spatial extreme modelling. Advances and Challenges in Space-time Modelling of Natural Events, Lecture Notes in Statistics, 207, 103-124.
[2] Brown, B., Resnick, S. (1977). Extremes values of independent stochastic processes. Journal
of Applied Probability, 14, 732-739.
[3] Cline, D., Samorodnitsky, G. (1994). Subexponentiality of the product of independent random
variables. Stochastic Processes and their Applications, 49, 75-98.
[4] Davison, A., Padoan, S., Ribatet, M. (2012). Statistical modelling of spatial extremes. Statistical Science, 27, 161-186.
[5] de Haan. L. (1984): A spectral representation for max-stable processes. Annals of Probability,
12, 1194–1204.
[6] Galambos, J. (1975). Order statistics of samples from multivariate distributions. Journal of
the American Statistical Association, 70, 674-680.
[7] Genton, M.G., Ma, Y., Sang, H. (2011). On the likelihood function of Gaussian max-stable
processes. Biometrika, 98, 481-488.
[8] Gumbel, E.J. (1960). Distributions des valeurs extrêmes en plusieurs dimensions. Publication
de l’ISUP, 9, 171-173.
[9] Hüsler, J., Reiss, R. D. (1989). Maxima of normal random vectors: between independence and
complete dependence. Statistics and Probability Letters, 9, 283-286.
[10] Lantuejoul, C., Bacro, J.N., Bel, L. (2011). Storm processes and stochastic geometry. Extremes,
14, 413-428.
[11] Marshall, A.W., Olkin, I. (1967). A multivariate exponential distribution. Journal of the American Statistical Association, 62, 30-44.
[12] Meijering, J. L. (1953). Interface area, edge length and number of vertices in crystal aggregates
with random nucleation. Philips Res. Rep., 8, 270-290.
[13] Möller, J. (1994). Lectures on Random Voronoı̈ Tessellations. Springer, New-York.
[14] Schlather, M. (2002). Models for stationary max-stable random fields. Extremes, 5, 33-44.
[15] Smith, R.L. (1990). Max-stable processes and spatial extremes. Preprint, University of Surrey,
Surrey.
[16] Wadsworth, J.L., Tawn J.A. (2012). Dependence modelling for spatial extremes. Biometrika,
99, 253-272.
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