Multivariate Max-Stable Spatial Processes
Marc G. Genton, Simone Padoan, Huiyan Sang
Spatio-Temporal Statistics and Data Analysis:
stsda.kaust.edu.sa
[email protected]
2014
1
Introduction
2
Maxima of Independent Replicates of Multivariate Processes
3
A Poisson Process Construction
4
Inference
5
Monte Carlo Simulations
6
Data Analysis
7
Discussion
1. Introduction
Spatial statistics of extreme events:
Extreme rainfall
Extreme heat
Extreme wind
Extreme weather events
Spatial extremes: looking at maxima
Central limit theorem: sums or means have Gaussian limiting
distribution
What is the limiting distribution for maxima?
Generalized extreme value distribution: Gumbel, Fréchet,
Weibull
Spatial statistics: analogy for random process Z (s)?
Need to study max-stable random fields
New research: multivariate max-stable random fields
Spatial extremes
Much work has been done on spatial extremes
From the theoretical point of view: de Haan and Ferreira
(2006), Falk, Hüsler and Reiss (2011), Kabluchko, Schlather
and de Haan (2009), Davis, Kluppelberg and Steinkohl
(2012), etc.
From the inferential point of view: Padoan, Ribatet and
Sisson (2010), Zhang and Smith (2010), Huser and Davison
(2013), Davis, Kluppelberg and Steinkohl (2013), etc.
Applications: extreme temperatures, rainfall, snowfall,
sea-levels, winds, pollution, etc.
Recent reviews: Davison et al. (2012), Cooley et al. (2012),
Davison et al. (2013), Ribatet (2013)
Motivation
All these works are concerned with the modeling of one
variable (e.g. rainfall, temperature, etc.) observed at some
locations s1 , . . . , sq over a continuous region S ⊂ R2 .
However, many statistical analyses focus on studying the
relation among several variables. For example, the analysis of
pollution data, environmental data, or climate data.
Can we model the extremes of multiple pollutants or multiple
climate variables observed at many locations?
The aim of this work is to provide a framework for modeling
spatial extremes of several variables.
Overall Dependence Structure
Three type of pairs:
(a) is purely spatial;
(b) is cross-component;
(c) is spatial cross-component.
Notation
Index sets:
I = {1, . . . , p}, variables;
K = {1, . . . , q}, locations;
J = I × K is the Cartesian product with N = p × q elements;
Jik = {(j, l) ∈ J\(i, k)}.
Let {Y(s)}s∈S , with Y(s) = {Yi (s)}i∈I , be a p-dimensional
vector-valued process.
Let M(s) = {Mi (s)}i∈I be a p-dimensional vector of
pointwise maxima,
(m)
Mi (s) = max Yi
m=1,...,n
(s),
∀ s ∈ S.
2. Maxima of Indep. Rep. of Multivariate Processes
A Y(s) is a stationary zero-mean, unit-variance Gaussian
process with matrix-valued correlation ρ(h) = {ρij (h)}i,j∈I ,
ρij (h) is the spatial cross-correlation between processes i
and j; see Genton and Kleiber (2013) for review
B Triangular array scheme (Hüsler–Reiss, 1989):
1
Assume that for each n ∈ N, Yn (s) is Gaussian with correlation
ρ(h; n) and in particular ρij (h; n) satisfies the condition
2bn2 {1 − ρij (h; n)} → λ2ij (h) ∈ (0, ∞),
n → ∞,
∀ h and i, j, ∈ I
2
Assume that for each n, indep copies Yn (s)(1) , . . . , Yn (s)(n) are
available.
3
Define Zn (s) = bn {Mn (s) − bn 1}, where bn is a sequence such
√
2
that 2πbn e bn /2 ∼ n as n → ∞.
Result (Multivariate Spatial Hüsler–Reiss Model)
Then, the finite dimensional distribution of Zn (s) for n → ∞:
[N]
Pr{Zi (sk ) ≤ zi (sk ), ∀ (i, k) ∈ J} = exp[−V1···p ({zi (sk )}(i,k)∈J )]
[N]
where the exponent function V1···p ({zi (sk )}(i,k)∈J ) is
X
(i,k)∈J


zj (sl ) 

log
1
λij (sk − sl )
zi (sk )
ΦN−1 
+

zi (sk )
2
λij (sk − sl ) 

; Λ̄ik 
j,l∈Jik
and where ΦN−1 is an N-1-dimensional Gaussian cdf with zero
mean and partial correlation matrix Λ̄ik .
The related spatial cross-component extremal coefficient:
[N]
θ1...p ({sk − sl }k,l∈K ) =
X
(i,k)∈J
(
ΦN−1
λij (sk − sl )
2
)
; Λ̄ik
(j,l)∈Jik
.
Examples
1
For (Zi , Zj ) and (s, s + h) then we deal with the set of random
components {Zi (s), Zi (s + h), Zj (s), Zj (s + h)} whose
coefficient is
λi (h) λij λij (h)
λi (h) λij (−h) λij
,
,
; Λ̄ik + Φ3
,
,
; Λ̄il
2
2
2
2
2
2
λij λji (h) λj (h)
λij (h) λij λj (h)
+ Φ3
,
,
; Λ̄jk + Φ3
,
,
; Λ̄jl ∈ [1, 4].
2
2
2
2
2
2
[4]
θij (h) =Φ3
2
For (Zi , Zj , Zv ) and (s, s + h, s + h0 ) . . . and for the particular
sub-sequence {Zi (s), Zj (s + h), Zv (s + h0 )} its coefficient is
[3]
λij (h) λvj (h00 )
λiv (h0 ) λjv (h00 )
,
; Λ̄jl + Φ2
,
; Λ̄vw
2
2
2
2
λji (h) λvi (h0 )
+ Φ2
,
; Λ̄ik ∈ [1, 3].
2
2
θijv (h, h0 , h00 ) =Φ2
Correlation Models
Example (separable). Let ρij (h) = ρij ρ(h), where
−1 ≤ ρij ≤ 1,
ρ(h) ≡ ρ(h; φ, κ) = exp {− (khk/φ)κ } ,
φ > 0, 0 < κ ≤ 2,
where κ and φ are smoothness and scaling parameters.
If Yn has spatial cross-correlation function
ρij (h; n) = ρij (n)ρ(cnκ h),
where cn = (2bn2 )−1 , ρij (n) = 1 − cn λ2ij + o(cn ) for cn → 0 as
n → ∞ and λ2ij is a non-negative constant. Then
ρij (h; n) = 1 − cn λ2ij − cn (khk/φ)κ + o(cn2 ),
therefore
λij (h) =
where λ2 (h) = (khk/φ)κ .
q
λ2ij + λ2 (h)
n → ∞,
Simulation of Bivariate Spatial Hüsler–Reiss
80
100
Z2(s); λ12=1.5
60
80
100
Z2(s); λ12=0.8
60
60
80
100
Z2(s); λ12=0.3
20
40
60
80
100
20
40
60
80
100
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
40
0
0
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
60
60
0
20
40
60
80
100
0
20
40
60
80
100
0
20
40
60
80
100
40
60
80
100
0
20
40
60
80
100
0
20
40
20
0
100
80
60
0
−2.5
s1
−1.5
−0.5
0.5
s1
1.5
40
0
20
40
0
20
40
20
0
0
20
40
s2
60
80
100
0
20
40
s2
60
20
40
40
100
Z1(s);φ=30,κ=1.8
20
80
0
0
60
100
100
80
80
60
80
40
80
20
0
20
40
20
0
Z1(s);φ=15,κ=1.8
100
0
100
0
20
40
s2
60
80
100
Z1(s);φ=15,κ=0.3
2.5
s1
3.5
4.5
s1
5.5
[3]
250
200
150
2
100
150
150
h'
3
6
2.2
1.9
100
φ=15 , κ=0.3 , λ=1.5
100
250
φ=15 , κ=0.3 , λ=0.8
200
φ=15 , κ=0.3 , λ=0.3
200
250
Extremal coefficient θijv (h, h, h0 ) with h = khk, h0 ∈ (0, 2h]
2.
50
1.9
1.85
60
20
2.14
0
1.8
100
20
200
200
150
150
2.16
60
2
φ=15 , κ=1.8 , λ=1.5
2
100
50
100
100
20
200
100
50
2.3
0
2.2
0
50
50
60
h
100
3
2.
2
20
2.8
2.7
2.6
2
0
2.6
100
1.5
1
2.2
60
φ=30 , κ=1.8 , λ=1.5
2.
h
100
7
2.
2
60
2.6
2.4
9
1.
1.7
1.8
5
1
2.
6
1.
20
2.7
150
200
150
100
8
100
2.
8
2.
2.6
2.4
2.9
2.8
2.5
2.9
150
200
60
φ=30 , κ=1.8 , λ=0.8
250
20
0
0
100
φ=30 , κ=1.8 , λ=0.3
250
0
2.4
2.
h'
50
100
50
60
2.8
2.6
2.2
2.2
2.4
2
20
250
2.8
2.6
8
2
1.
2.5
2.18
100
h'
200
φ=15 , κ=1.8 , λ=0.8
2.2
2.2
5
150
φ=15 , κ=1.8 , λ=0.3
0
1.75
100
250
1.7
60
1.9
250
50
0
1.8
1.65
20
250
50
24
1.8
5
20
60
h
2.4
2.5
100
1
3. A Poisson Process Construction I
Let W(s) = {Wi (s)}i∈I be a non-negative random process on
S, such that
IE{Wi (s)} = τ ∈ (0, ∞), ∀ s ∈ S;
IE{sup Wi (s)} < ∞.
s∈S
Let {R (m) , W(m) }m≥1 be points of an inhomogeneous Poisson
point process on R+ × Rp+ with intensity measure
ϕ(dr , dw) = r −2 dr × τ −1 δ(dw)
where (R (m) , W(m) ) are copies of (R, W(s)) ∀ s ∈ S. Then,
Z(s) = max {R (m) W(m) (s)},
m=1,2,...
s∈S
defines a max-stable process on S with unit Fréchet margins.
If p = 1: well-known univariate definition of max-stable
processes (de Haan and Ferreira, 2006).
A Poisson Process Construction II
The finite dimensional distribution is equal to
[N]
Pr{Zi (sk ) ≤ zi (sk ), ∀ (i, k) ∈ J} = exp[−V1···p ({zi (sk )}(i,k)∈J )]
with exponent function
[N]
V1···p ({zi (sk )}(i,k)∈J )
1
=
τ
Z
max
(i,k)∈J
Wi (sk )
zi (sk )
δ(dw).
The overall spatial cross-component extremal coefficient
is
[N]
[N]
θ1···p ({sk − sl }k,l∈K ) = V1···p (1, . . . , 1) ∈ [1, N].
Remarks. Consider pairs (Zi , Zj ) and (s, s + h) then
[4]
[4]
θij (h)= θji (h);
[2]
[2]
[2]
Lower order are: θij (h), θi (h) and θij , where
[2]
[2]
θij (h) 6= θji (h)
Models
1
Multivariate extremal-Gaussian model (Schlather, 2002)
W(s) := max{0, X(s)}, X(s) is zero-mean, unit-variance
Gaussian with correlation ρ(h) = {ρij (h)}i,j∈I ;
2
Multivariate Brown–Resnick model (Kabluchko et al.,
2009)
W(s) := exp{X(s) − σ 2 (s)/2}, X(s) is zero-mean Gaussian
with matrix-valued variogram 2γ(h) = {2γij (h)}i,j∈I ,
σ 2 (s) = {σi2 (s)}i∈I ;
3
Multivariate extremal-t model (Davison et al., 2012; Opitz,
2013)
W(s) := max{0, X(s)}ν , X(s) is zero-mean, unit-variance
Gaussian with correlation ρ(h) = {ρij (h)}i,j∈I and ν > 0;
4
Multivariate Gaussian extreme-value model (Smith, 1990;
de Haan and Pereira, 2006)
W(s) := {fi (Xm − s)}i∈I , fi is a Gaussian density on Rd and
{Xm }m≥1 are points of a homogeneous Poisson process on Rd
with intensity measure δ(dx) (Lebesgue).
Example
[2]
When p = 1 and d = 1, Vj (h) of the Gaussian
extreme-value model is equal to that of the Hüsler-Reiss
model with λ(h) = h/σ.
When p = 2, it is no longer the case
[2]
Vij (h) =











1
zj
1
zi
1
zi
1
zj
,
a+
1
zj
(1 − b),
(1 − a) +
1
zj
b,
,
where
(
c(u, σi ; h) = exp
−
for
0 < zj < c(u, σi ; h) zi ,
u > 1,
for
zi c(u, σi ; h) ≤ zj ≤ zi ,
u > 1,
for
zi ≤ zj < c(u, σi ; h) zi ,
u < 1,
for
zj ≥ c(u, σi ; h) zi ,
u < 1,
h2
2
2σi (1 − u)
)
,
u = σj2 /σi2 ,


v
u




u h2 u
2σj2
1
σ
z
h
i
i
t
≤
+
log
,
a = Pr Y −
2

σi (1 − u)
σi
(1 − u )
1−u
σj zj 




v
u




u h2 u
2σj2
h
u
1
σ
z
i
i
t
≤
+
log
.
b = Pr Y −
2

σj (1 − u)
σj
(1 − u )
1−u
σj zj 


4. Inference
Standard-likelihood inference in the univariate case is
impractical. The composite-likelihood approach, see e.g.
Padoan et al. (2010) and Davison and Gholamrezaee (2012)
is a successful alternative.
Using marginal likelihoods of higher-order than 2 provides
more efficient estimators than those of the pairwise
(Genton et al., 2011; Huser and Davison, 2013).
Is this still the case in the multivariate context?
Can we obtain estimates in a reasonable time?
The numericalmaximization of the CL is demanding, since it is
a sum of p×q
marginal likelihoods (m = 2, 3, . . .).
m
We aim to define a composite that addresses these questions.
Composite likelihoods
CL2-CI and CL2-C:
log f {zi (sk ), zj (sl ); ϑ}
CL3-CI and CL3-C:
log f {zi (sk ), zj (sl ), zv (sw ); ϑ}
CL-CI:
log f {zi (sk ), zi (sl ), zj (sk ); ϑ} + log f {zi (sk ), zi (sl ), zj (sl ); ϑ} +
log f {zi (sk ), zj (sk ), zj (sl ); ϑ} + log f {zi (sl ), zj (sk ), zj (sl ); ϑ}
5. Monte Carlo Simulations I
30 iid replicates of bivariate and trivariate Hüsler-Reiss
random field are simulated at 15 random points in [0, 100]2 .
Different parameter settings for φ, κ, λ12 , λ13 and λ23 .
Simulations are repeated 500 times.
Findings:
1
b CL2−CI is more efficient than ϑ
b CL2−C ;
ϑ
2
b CL3−CI is more efficient than ϑ
b CL2−CI ;
ϑ
3
b CL−CI is the most efficient;
ϑ
4
b CL2−CI rel. to
42% is the smallest relative efficiency of ϑ
b
ϑCL−CI ;
5
More gain in efficiency is obtained for:
shorter spatial scales φ(1) < φ(2) ;
smoother or rougher spatial fields κ(1) > κ(2) or κ(1) < κ(2) ;
(1)
(2)
weaker cross-component dependence λ12 < λ12 .
Monte Carlo Simulations II
Bivariate:
True
CL-CI
θ[2] -CI
νF -CI
CL2-C
CL2-CI
True
CL-CI
θ[2] -CI
νF -CI
CL2-C
CL2-CI
True
CL-CI
θ[2] -CI
νF -CI
CL2-C
CL2-CI
log(φ)
log(30)
log(31.37)(0.60)
RElog(φ)
0.30
0.95
0.88
0.90
log(30)
log(30.56)(0.59)
RElog(φ)
0.35
1.08
0.38
0.98
log(30)
log(29.77)(0.60)
RElog(φ)
0.58
1.48
0.33
0.95
κ
0.5
0.51(0.09)
REκ
0.15
0.64
0.06
0.93
0.5
0.51(0.09)
REκ
0.13
0.63
0.05
0.88
0.5
0.50(0.10)
REκ
0.12
0.63
0.04
0.58
λ12
0.3
0.30(0.05)
REλ
0.68
1.03
0.03
0.94
0.8
0.80(0.14)
REλ
0.60
0.83
0.16
0.74
1.5
1.51(0.29)
REλ
0.55
0.68
0.49
0.55
log(φ)
log(30)
log(28.40)(0.92)
RElog(φ)
0.36
0.93
0.53
1.01
log(30)
log(30.80)(0.14)
RElog(φ)
0.40
0.91
0.49
0.93
log(30)
log(30.21)(0.13)
RElog(φ)
0.47
0.89
0.12
0.88
κ
0.3
0.31(0.08)
REκ
0.21
0.90
0.04
1.00
1.8
1.81(0.12)
REκ
0.17
0.40
0.23
0.77
1.8
1.78(0.11)
REκ
0.17
0.40
0.10
0.84
λ12
0.8
0.81(0.15)
REλ
0.65
0.88
0.14
0.80
0.8
0.79(0.12)
REλ
0.56
0.83
0.70
0.85
1.5
1.50(0.26)
REλ
0.49
0.72
0.73
0.64
log(φ)
log(5)
log(6.57)(0.77)
RElog(φ)
0.42
0.71
0.80
0.91
log(5)
log(5.26)(0.28)
RElog(φ)
0.20
0.56
0.19
0.42
log(15)
log(15.10)(0.25)
RElog(φ)
0.24
0.98
0.28
0.55
κ
0.3
0.32(0.08)
REκ
0.19
0.73
0.15
0.97
1
1.04(0.18)
REκ
0.17
0.65
0.16
0.44
1
1.02(0.13)
REκ
0.13
0.48
0.13
0.65
λ12
0.3
0.29(0.05)
REλ
0.71
1.26
0.02
0.95
0.8
0.78(0.08)
REλ
0.54
0.92
0.03
0.97
0.8
0.79(0.11)
REλ
0.60
0.90
0.14
0.91
Monte Carlo Simulations III
Trivariate:
Estimation of κ
6
2.0
Estimation of log(φ)
5
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CL3−Cross
CL3−Cross−Inter
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Estimation of λ23
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2.0
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Estimation of λ13
Estimation of λ12
1.0
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0.0
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Figure: T = 30 indep realizations from trivariate Hüsler–Reiss max-stable
process at q = 15 locations uniform in [0, 100]2 :
φ = 30, κ = 0.5, λ12 = 0.3, λ13 = λ23 = 1.5, with 500 replicates.
6. Data Analysis: Extreme Temperatures
From 1998-2012 for 20 monitoring stations surrounding
Oklahoma City
Extreme temperature indices: the maximum of daily maxima
(Tmax) and daily minima (Tmin) of summer temperatures
Data from National Climatic Data Center (NCDC)
Data at each location transformed to a unit Fréchet
distribution for each variable separately
Generalized extreme value parameters (µ, σ, ξ) modeled as
linear regression functions of longitude, latitude and altitude
The goal is to investigate the spatial extremal dependence
structure within and between Tmax and Tmin
Modeling Details
We considered three multivariate max-stable process models:
The multivariate Hüsler-Reiss model with a bivariate power
exponential correlation
The multivariate Schlather model with a bivariate
parsimonious Matérn correlation (Gneiting et al., 2010)
The multivariate extremal-t model with a bivariate
parsimonious Matérn correlation
We also compared with three univariate models, the
Hüsler-Reiss, the Schlather and the extremal-t
Results: Estimated extremal dependence param., 95% CI
CL2
φ (Tmax/Tmin)
42(21,68)/20(7,42)
CL2-CI
CL-CI
φ
30(17,50)
31(16,49)
CL2
φ (Tmax/Tmin)
175(130,229)/227(180,292)
CL2-CI
φ
404(270,692)
CL2
φ (Tmax/Tmin)
1469(676,2394)/531(175,2001)
CL2-CI
φ
1157(864,2345)
Univariate Hüsler–Reiss
κ (Tmax/Tmin)
0.58(0.41,0.75)/0.50(0.29,0.76)
Multivariate Hüsler–Reiss
κ
λ12
0.51(0.39,0.65)
1.54(1.30,1.85)
0.53(0.39,0.67)
1.86(1.56,2.12)
Univariate extremal-Gaussian
κ (Tmax/Tmin)
0.57(0.46,0.73)/0.32(0.22,0.42)
Multivariate extremal-Gaussian
κ (Tmax/Tmin)
ρ12
0.31(0.23,0.38)/0.22(0.14,0.30) 0.18(-0.27,0.45)
Univariate Extremal-t
κ (Tmax/Tmin)
0.36(0.29,0.42)/0.19(0.13,0.25)
Multivariate Extremal-t
κ (Tmax/Tmin)
ρ12
0.32(0.24,0.40)/0.26(0.19,0.34) 0.61(0.37,0.72)
CLIC∗
12686
CLIC∗
12591
261029
CLIC∗
12611
CLIC∗
12553
ν (Tmax/Tmin)
3.23 (2.18,4.14)/1.06(0.76,1.71)
CLIC∗
12606
ν
2.38(1.38,2.96)
CLIC∗
12551
1
Univariate models indicate range and smoothness parameters
are not significantly different between Tmax and Tmin fields
2
Cross-correlation parameters from all three multivariate
models indicate moderate correlation between Tmax and Tmin
3
Small smoothness parameters suggesting rough fields
Empirical/Fitted Extremal Coefficients
[2]
2.0
Mul. Husler−Reiss
Mul. Extremal−Gaussian
Mul. Extremal−t
2.0
Mul. Husler−Reiss
Mul. Extremal−Gaussian
Mul. Extremal−t
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2.2
2.2
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Mul. Extremal−Gaussian
Mul. Extremal−t
2.0
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distance(km)
distance(km)
distance(km)
37.0
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Oklahoma City OK
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25−Year R.L.(Tmax)
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37.0
Return Levels vs Conditional Return Levels
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−100
−98
−96
37.0
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34.0
−98
−96
25−Year C.R.L.(Tmin|Tmax)
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Oklahoma City OK
425
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240
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Similar patterns but right-hand-side map about 1-3C higher.
−96
7. Discussion
A We extended some univariate spatial max-stable models to the
multivariate setting;
B Several multivariate spatial models are available:
1
2
3
4
Hüsler–Reiss or Brown–Resnick
Gaussian extreme-value
extremal-Gaussian
extremal-t
C Different types of dependence structures can be taken into
account, based on:
1
2
separable correlation models
non-separable correlation models (e.g. multivariate Matérn)
D For the inference of those models we define a composite
likelihood that provides more efficient estimates than those of
the pairwise with an acceptable computational cost.