View
233
Download
2
Category
Preview:
Citation preview
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Extreme Value Analysis and Spatial Extremes
Whitney Huang
Department of StatisticsPurdue University
11/07/2013
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Outline
1 Motivation
2 Extreme Value Theorem and GeostatisticsUnivariate ExtremesMultivariate ExtremesGeostatistics
3 Spatial ExtremesBayesian Hierarchical ModelsCopula ModelsMax-stable Models
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
What are (Spatial) Extremes?
Extreme events are defined to be rare and unexpected100–year flood–the level of flood water expected to beequaled or exceeded every 100 years on averageMany extreme events of interest are spatio-temporal innatural
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
What are (Spatial) Extremes?
Extreme events are defined to be rare and unexpected100–year flood–the level of flood water expected to beequaled or exceeded every 100 years on averageMany extreme events of interest are spatio-temporal innatural
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
What are (Spatial) Extremes?
Extreme events are defined to be rare and unexpected100–year flood–the level of flood water expected to beequaled or exceeded every 100 years on averageMany extreme events of interest are spatio-temporal innatural
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
What are (Spatial) Extremes?
Extreme events are defined to be rare and unexpected100–year flood–the level of flood water expected to beequaled or exceeded every 100 years on averageMany extreme events of interest are spatio-temporal innatural
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Why study extremes?
Although infrequent, extremes have large human impact.2003 European heat wave example. Around 70,000 werekilled!
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Why study extreme?
"There is always going to be an element of doubt, as one isextrapolating into into areas one does not know about. Butwhat EVT is doing is making the best use of whatever data youhave about extreme phenomena." – Richard Smith
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Usual vs Extremes
Relies on asymptotic theory to provide models for the tailUses only the "extreme" observations to fit the model
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
History
1920’s: Foundations of asymptotic argument developed byFisher and Tippett1940’s: Asymptotic theory unified and extended byGnedenko and von Mises1950’s: Use of asymptotic distributions for statisticalmodelling by Gumbel and Jenkinson1970’s: Classic limit laws generalized by Pickands
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
History
1980’s: Leadbetter (and others) extend theory to stationaryprocesses1990’s: Multivariate and other techniques explored as ameans to improve inference2000’s: Interest in spatial and spatio-temporal applications,and in finance
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Probability Framework
Let X1, · · · ,Xniid∼ F and define Mn = maxX1, · · · ,Xn
Then the distribution function of Mn is
P(Mn ≤ x) = P(X1 ≤ x , · · · ,Xn ≤ x)
= P(X1 ≤ x)× · · · × P(Xn ≤ x) = F n(x)
Remark
F n(x)n→∞
=
0 if F (x) < 11 if F (x) = 1
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Classical Limit Laws
Recall the Central Limit Theorem:
Xn − µσ√n
d→ N(0,1)
⇒ rescaling is the key to obtain a non-degenerate distribution
Question: Can we get the limiting distribution of
Mn − bn
an
for suitable sequence an > 0 and bn?
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Classical Limit Laws
Recall the Central Limit Theorem:
Xn − µσ√n
d→ N(0,1)
⇒ rescaling is the key to obtain a non-degenerate distribution
Question: Can we get the limiting distribution of
Mn − bn
an
for suitable sequence an > 0 and bn?
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Theorem (Fisher–Tippett–Gnedenko theorem)If there exist sequences of constants an > 0 and bn such that,as n→∞
P(Mn − bn
an≤ x)
d→ G(x)
for some non-degenerate distribution G, then G belongs toeither the Gumbel, the Frechet or the Weibull family
Gumbel : G(x) = exp(exp(−x)) −∞ < x <∞;
Frechet : G(x) =
0 x ≤ 0,exp(−x−α) x > 0, α > 0;
Weibull : G(x) =
exp(−(−x)α) x < 0, α > 0,1 x ≥ 0;
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Example: Exponential Maxima
X ∼ Exp(1)
F n(x + log n) = (1− exp(−x − log n))n =
(1− 1n
exp(−x))n n→∞−→ exp−exp(−x)
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Generalized Extreme Value Distribution (GEV)
This family encompasses all three extreme value limit families:
G(x) = exp−[1 + ξ(
x − µσ
)]−1ξ
+
where x+ = max(x ,0)
µ and σ are location and scale parametersξ is a shape parameter determining the rate of tail decay,with
ξ > 0 giving the heavy-tailed (Frechet) caseξ = 0 giving the light-tailed (Gumbel) caseξ < 0 giving the bounded-tailed (Weibull) case
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Quantiles and Return Levels
In terms of quantiles, take 0 < p < 1 and define xp such that:
G(xp) = exp−[1 + ξ(
xp − µσ
)]−1ξ
+
= 1− p
⇒ xp = µ− σ
ξ
[1− − log(1− p)−ξ]
In the extreme value terminology, xp is the return levelassociated with the return period 1
p
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Max-Stability
DefinitionA distribution G is said to be max-stable if
Gk (akx + bk ) = G(x), k ∈ N
for some constants ak > 0 and bk
Taking powers of G results only in a change of location andscaleA distribution is max-stable ⇐⇒ it is a GEV distribution
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Some Remarks
There has been some work on the convergence rate of Mnto the limiting regime, which depends on the underlingdistributionFor statisticians, we use the GEV as an approximatedistribution for sample maximal for "finite" n –assess the fitempiricallyDirect use the GEV rather than three types separately
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Block–Maximum Approach
Determine the block size and compute maximal forblocks–usually annual maximalFit the GEV to the maximal and assess fit– usually vialikelihood– based techniquesPerform inference for return levels, probabilities, etc
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Diagnostics
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Point Process Approach
MotivationThe block maximum method ignores much of the data whichmay also relevant to extreme–we would like to use the datamore efficient.Alternatives:
peaks over thresholdsr-largest order statistics
Both are special cases of a point process representation, underwhich we approximate the exceedances over a threshold by atwo-dimensional Poisson process
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Point Process Limit: Basic Idea
Suppose X1, · · · ,Xniid∼ F and Mn−bn
an converges to GEV
distributionConstruct a sequence of point processes on R2 by
Pn =
(
in + 1
,Xi − bn
an) : i = 1, · · · ,n
Pn
n→∞−→ P, where P is a Poisson process
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Formally Linking Extremes and Point Processes
Given: P(Mn−bnan≤ x)→ exp
− (1 + ξx)
−1ξ
logP(Mn − bn
an≤ x)→ −(1 + ξx)
−1ξ
logPn(X − bn
an≤ x)→ −(1 + ξx)
−1ξ
n log(1− P(X − bn
an> x))→ −(1 + ξx)
−1ξ
nP(X − bn
an> x)→ (1 + ξx)
−1ξ
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Extremes and Point Processes Cond’t
nP(X − bn
an> x)→ (1 + ξx)
−1ξ
Therefore, creating a series of point processesXi − bn
an: i = 1, · · · ,n
n→∞
these will converge to an inhomogeneous Poisson process withmeasure
ν((x ,∞)) = (1 + ξx)−1ξ
for sets bounded away from zero (exceedance example).
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Generalized Pareto Distribution (GPD) forExceedances
P(Xi > x + u|Xi > u) =nP(Xi > x + u)
nP(Xi > u)
→(1 + ξ x+u−bn
an
1 + ξ x−bnan
)−1ξ
=
(1 +
ξxan + ξ(u − bn)
)−1ξ
⇒ Survival function of generalized Pareto distribution
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Theorem (Pickands–Balkema–de Haan theorem)
Let X1, · · ·iid∼ F, and let Fu be their conditional excess
distribution function. Pickands (1975), Balkema and de Haan(1974) posed that for a large class of underlying distributionfunctions F , and large u, Fu is well approximated by thegeneralized Pareto distribution GPD. That is:
Fu(y)→ GPDξ,σ(u),u(y) u →∞
where
GPDξ,σ(u),u(y) =
1− (1 + ξy
σ(u))−1ξ ξ 6= 0,
1− exp( −yσ(u)) ξ = 0;
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Fort Collins Data: Block Maximum vs.Threshold-exceedance
GPD has lower standard error for ξ, lower estimate as well.GPD has narrower confidence interval.Have not yet discussed threshold selection procedure.
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Threshold Selection
Bias–variance trade–off: threshold too low–bias because of themodel asymptotics being invalid; threshold too high–variance islarge due to few data points
Figure: Mean residual life plot(MRL): MRL is linear when GPD holds
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Temporal Dependence
Question: Is the GEV still the limiting distribution for blockmaxima of a stationary (but not independent) sequence Xi?Answer: Yes, so long as mixing conditions hold. (Leadbetter etal., 1983)What does this mean for inference?Block maximum approach: GEV still correct for marginal. Sinceblock maximum data likely have negligible dependence,proceed as usualThreshold exceedance approach: GPD is correct for themarginal. If extremes occur in clusters, estimation affected aslikelihood assumes independence of threshold exceedances
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Temporal Dependence
Question: Is the GEV still the limiting distribution for blockmaxima of a stationary (but not independent) sequence Xi?Answer: Yes, so long as mixing conditions hold. (Leadbetter etal., 1983)What does this mean for inference?Block maximum approach: GEV still correct for marginal. Sinceblock maximum data likely have negligible dependence,proceed as usualThreshold exceedance approach: GPD is correct for themarginal. If extremes occur in clusters, estimation affected aslikelihood assumes independence of threshold exceedances
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Temporal Dependence
Question: Is the GEV still the limiting distribution for blockmaxima of a stationary (but not independent) sequence Xi?Answer: Yes, so long as mixing conditions hold. (Leadbetter etal., 1983)What does this mean for inference?Block maximum approach: GEV still correct for marginal. Sinceblock maximum data likely have negligible dependence,proceed as usualThreshold exceedance approach: GPD is correct for themarginal. If extremes occur in clusters, estimation affected aslikelihood assumes independence of threshold exceedances
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Temporal Dependence
Question: Is the GEV still the limiting distribution for blockmaxima of a stationary (but not independent) sequence Xi?Answer: Yes, so long as mixing conditions hold. (Leadbetter etal., 1983)What does this mean for inference?Block maximum approach: GEV still correct for marginal. Sinceblock maximum data likely have negligible dependence,proceed as usualThreshold exceedance approach: GPD is correct for themarginal. If extremes occur in clusters, estimation affected aslikelihood assumes independence of threshold exceedances
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Remarks on Univariate Extremes
To estimate the tail, EVT uses only extreme observationsTail parameter ξ is extremely important but hard to estimateGPD is the limiting distribution for threshold exceedancesThreshold exceedance approaches allow the user to retainmore data than block-maximum approaches, therebyreducing the uncertainty with parameter estimatesTemporal dependence in the data is more of an issue inthreshold exceedance models. One can either decluster,or alternatively, adjust inference
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Remarks on Univariate Extremes
To estimate the tail, EVT uses only extreme observationsTail parameter ξ is extremely important but hard to estimateGPD is the limiting distribution for threshold exceedancesThreshold exceedance approaches allow the user to retainmore data than block-maximum approaches, therebyreducing the uncertainty with parameter estimatesTemporal dependence in the data is more of an issue inthreshold exceedance models. One can either decluster,or alternatively, adjust inference
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Remarks on Univariate Extremes
To estimate the tail, EVT uses only extreme observationsTail parameter ξ is extremely important but hard to estimateGPD is the limiting distribution for threshold exceedancesThreshold exceedance approaches allow the user to retainmore data than block-maximum approaches, therebyreducing the uncertainty with parameter estimatesTemporal dependence in the data is more of an issue inthreshold exceedance models. One can either decluster,or alternatively, adjust inference
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Remarks on Univariate Extremes
To estimate the tail, EVT uses only extreme observationsTail parameter ξ is extremely important but hard to estimateGPD is the limiting distribution for threshold exceedancesThreshold exceedance approaches allow the user to retainmore data than block-maximum approaches, therebyreducing the uncertainty with parameter estimatesTemporal dependence in the data is more of an issue inthreshold exceedance models. One can either decluster,or alternatively, adjust inference
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Remarks on Univariate Extremes
To estimate the tail, EVT uses only extreme observationsTail parameter ξ is extremely important but hard to estimateGPD is the limiting distribution for threshold exceedancesThreshold exceedance approaches allow the user to retainmore data than block-maximum approaches, therebyreducing the uncertainty with parameter estimatesTemporal dependence in the data is more of an issue inthreshold exceedance models. One can either decluster,or alternatively, adjust inference
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Multivariate Extremes Examples
A central aim of multivariate extremes is trying to find anappropriate structure to describe tail dependence
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
What is a Multivariate Extreme?
Let Zm = (Zm,1, · · · ,Zm,d )T , m ∈ N be an iid sequence ofrandom vectors. Want to extract a subset of data considered"extreme"
Block-maximum: Mn = (∨n
m=1 Zm,1, · · · ,∨n
m=1 Zm,d )T
Leads to modeling with multivariate max-stabledistributionsMarginal-exceedance: For each marginal i = 1, · · · ,d , findan appropriate threshold ui and retain data whereZm,i > ui . Leads to multivariate generalized ParetodistributionNorm-exceedance: For a given norm retain data where||Zm|| > z. Leads to description by multivariate regularvariation
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
What is a Multivariate Extreme?
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Multivariate Models
Let X 1,X 2, · · · iid d-dimensional random vectors withdistribution FOur interest is in the (non degenerate) limiting distribution
P
maxi=1,··· ,n
X i − bi
an≤ x
n→∞−→ G(x)
for some sequences an > 0 and bn ∈ Rd . G is called amultivariate extreme value distributionTransform to common marginals Z with unit Frechet i.e.,Z (x ,∞, · · · ,∞) = · · · = Z (∞, · · · ,∞, x) = exp(−1
x ) tomodel the dependence structure
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Multivariate Models cond’t
P(Z1 ≤ z1, · · · ,Zd ≤ zd ) =
exp− V (z1, · · · , zd )
z1, · · · , zd > 0
The exponent measure V (z1, · · · , zd ) satisfiesV (tz1, · · · , tzd ) = t−1V (z1, · · · , zd ) =⇒ V is homogeneousof order -1
V (z1, · · · , zd ) =
1z1
+ · · ·+ 1zD
if Z1, · · · , Zd are independent
1min(z1,··· ,zD)
if Z1, · · · , Zd are entirely dependent
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Spectral Representations
Pickands, 1981
V (z1, · · · , zd ) =∫
SDmax(w1
z1, · · · , wD
zD) dM(w1, · · · ,wD)
where M is a measure on the D-dimensional simplex SD∫wd dM(w1, · · · ,wD) = 1 for each d
No simple parametric forms for V
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Summary measure of extremal dependence
P(Z1 ≤ z, · · · ,ZD ≤ z) = exp−V (1,··· ,1)
z
≡ exp(−θD
z ) z >
0θD is the extrmal coefficient
θD = 1⇒ fully dependentθD = D ⇒ independent
In bivariate case limz→∞ P(Z2 > z|Z1 > z) = 2− θD
When D = 2 madogram [Cooley, Naveau and Poncet,2006]
µF =12E|F (Z1 − F (Z2))|
θ2 =1 + 2νF
1− 2νF
provide a good estimator of θ2
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Geostatistics
Developed originally to predict probability distributions ofore grades for mining operationsGeostatistics is based largely on the theory of Gaussianrandom processes
Y (x) = µ(x) + e(x) + ε(x), x ∈ D ⊂ R2
E(Y (x)) = µ(x), Cov(Y (x),Y (y)) = C(||x − y ||)
The main propose of geostatistics is interpolation
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Univariate ExtremesMultivariate ExtremesGeostatistics
Figure: A realization of Gaussian random processes
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Three-Level Spatial Hierarchical Model
Data level: Likelihood which characterizes the distributionof the observed data given the parameters at the processlevel
Yi(xd )|µ(xd ), σ(xd ), ξ(xd ) ∼ GEVµ(xd ), σ(xd )
, ξ(xd ) i = 1, · · · ,n,d = 1, · · · ,D
Process level: Latent process captured by spatial model forthe data level parametersPrior level: Prior distributions put on the parameters
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Three-Level Spatial Hierarchical Model
Suppose the response variables Y (x) are independentconditionally on an unobserved latent process S(x)Assume the S(x) follows a Gaussian process andinduce spatial dependence in Y (x) by integration overthe latent processCommon approach in geostatistics with non-normalresponse [Diggle et al. 1998,2007]. Usually performed in aBayesian setting using Markov chain Monte Carlo (MCMC)
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Pros and Cons
The quantile surfaces are realisticAfter averaging over S(x) the marginal distribution ofY (x) is NOT GEVThe spatial dependence is ignored because of conditionalindependence
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Pros and Cons
Figure: One realisation of the latent variable model, showing the lackof local spatial structure
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Copula
The D− dimensional joint distribution F of any randomvariable (Y1, · · · ,YD) may be written as
F (y1, · · · , yn) = CF1(y1), · · · ,FD(yD)
Where F1, · · · ,FD are univariate marginal distributions ofY1, · · · ,YD and C is a copulaThe copula is uniquely determined for distributions F withabsolutely continuous marginalWith continuous and strictly increasing marginals, thecopula corresponds toF (y1, · · · , yn) = CF1(y1), · · · ,FD(yD) is as follows:
C(u1, · · · ,uD) = F
F1−1(u1), · · · ,FD
−1(uD)
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Copula
The D− dimensional joint distribution F of any randomvariable (Y1, · · · ,YD) may be written as
F (y1, · · · , yn) = CF1(y1), · · · ,FD(yD)
Where F1, · · · ,FD are univariate marginal distributions ofY1, · · · ,YD and C is a copulaThe copula is uniquely determined for distributions F withabsolutely continuous marginalWith continuous and strictly increasing marginals, thecopula corresponds toF (y1, · · · , yn) = CF1(y1), · · · ,FD(yD) is as follows:
C(u1, · · · ,uD) = F
F1−1(u1), · · · ,FD
−1(uD)
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Copula
The D− dimensional joint distribution F of any randomvariable (Y1, · · · ,YD) may be written as
F (y1, · · · , yn) = CF1(y1), · · · ,FD(yD)
Where F1, · · · ,FD are univariate marginal distributions ofY1, · · · ,YD and C is a copulaThe copula is uniquely determined for distributions F withabsolutely continuous marginalWith continuous and strictly increasing marginals, thecopula corresponds toF (y1, · · · , yn) = CF1(y1), · · · ,FD(yD) is as follows:
C(u1, · · · ,uD) = F
F1−1(u1), · · · ,FD
−1(uD)
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Example
Gaussian copula:
Y1, · · · ,YD ∼ Nd (0, Ω)
C(u1, · · · ,ud ) = Φ
Φ−1(u1), · · · ,Φ−1(ud ) : Ω
Student t copula:
C(u1, · · · ,ud ) = Tν
T−1ν (u1), · · · ,T−1
ν (uD) : Ω
Extremal Copula:
Y1, · · · ,YD ∼ multivariate GEV
By max-stability⇒ C(um1 , · · · ,um
D ) = Cm(u1, · · · ,uD), 0 <u1, · · · ,ud < 1, m ∈ N
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Pros and Cons
On the "copula scale’: the dependence seems more orless OKBut this is no longer true at the original, i.e., extremal scale.
Figure: One simulation from the fitted Gaussian copula modelWhitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Max-stable Processes
Definition
Let Ym(x), x ∈ D ⊂ R2, m = 1, · · · ,n be independent copies ofY (x), and let Mn(x) = max Ym(x). Y (x) is termed max-stable ifthere exist an(x) and bn(x) such that
P(Mn(x)− bn(x)
an(x)≤ y(x))
n→∞−→ P(G(x) ≤ y(x))
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Spectral Representations
Theorem (Schlather, 2002)
Z (x), x ∈ D ⊂ R2 is max-stable with unit Frechet marginals⇐⇒ There exist iid positive stochastic processesV1(x),V2(x), · · · with E[Vi(x)] = 1 ∀x ∈ D andE[supx∈D V (x)] <∞ and an independent point process si∞i=1on R+ with intensity measure r−2 dr such thatZ (x)
d= maxi∈N siVi
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Some Models
Smith: Yi(x) = φ(xi − Ui)(), Uii≤1 points of ahomogeneous Poisson process on R2
Schlather: Yi(x) = 2πεi(x), εi(·) standard GaussianprocessGeometric: Yi(x) = expσεi(x)− σ2
2 Brown–Res: Yi(x) = expε′i (x)− γ(x), ε′i (·) intrinsicallystationary Gaussian process with (semi) variogram γ
Figure: One realization of max-stable processes
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Inference
For the max-stable process models, only the bivariatedistributions are knownComposite likelihoods [Lindsay, 1988] are used to obtainestimationSince we have the bivariate distributions we will use thepair–wise likelihood
lp(θ,y) =n∑
m=1
K−1∑i=1
k∑j=i+1
log f (y im, f
jm; θ)
Not a true likelihoodOver–uses the data – each observation appears K-1 times
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Bayesian Hierarchical ModelsCopula ModelsMax-stable Models
Pros and Cons
Justified by extreme value theoryAble to describe asymptotic dependenceDescribe everything that is asymptotically independent asexactly independentOnly suitable for observations that are annual maximal atthis point
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Summary
Although infrequent, extremes have large human impactUses only the "extreme" observations to fit the model–Norole for normal distribution!Spatial extremes modeling is challenge–both in theoreticaland computational aspects
Whitney Huang EVA and Spatial Extreme
MotivationEVT and Geostatistics
Spatial ExtremesSummary
Whitney Huang EVA and Spatial Extreme
Recommended