Extreme Value Analysis and Spatial Extremeshuang251/stat695o_pres_slide.pdfExtreme events are defined to be rare and unexpected 100–year flood–the level of flood water expected

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



Outline

1 Motivation

2 Extreme Value Theorem and GeostatisticsUnivariate ExtremesMultivariate ExtremesGeostatistics

3 Spatial ExtremesBayesian Hierarchical ModelsCopula ModelsMax-stable Models




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




Why study extremes?

Although infrequent, extremes have large human impact.2003 European heat wave example. Around 70,000 werekilled!




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




Usual vs Extremes

Relies on asymptotic theory to provide models for the tailUses only the "extreme" observations to fit the model




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




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




Univariate ExtremesMultivariate ExtremesGeostatistics

Probability Framework

Let X1, · · · ,Xniid∼ F and define Mn = max{X1, · · · ,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





Classical Limit Laws

Recall the Central Limit Theorem:

X̄n − 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 − bnan

for suitable sequence {an} > 0 and {bn}?





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 Fréchet or the Weibull family

Gumbel : G(x) = exp(exp(−x)) −∞ < x 0, α > 0;

Weibull : G(x) ={

exp(−(−x)α) x < 0, α > 0,1 x ≥ 0;





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)}





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 (Fréchet) caseξ = 0 giving the light-tailed (Gumbel) caseξ < 0 giving the bounded-tailed (Weibull) case





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 1p





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





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





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





Diagnostics





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





Point Process Limit: Basic Idea

Suppose X1, · · · ,Xniid∼ F and {Mn−bnan } converges to GEV

distributionConstruct a sequence of point processes on R2 by

Pn ={

(i

n + 1,Xi − bn

an) : i = 1, · · · ,n

}Pn

n→∞−→ P, where P is a Poisson process





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 − bnan

> x))→ −(1 + ξx)−1ξ

nP(X − bn

an> x)→ (1 + ξx)

−1ξ





Extremes and Point Processes Cond’t

nP(X − bn

an> x)→ (1 + ξx)

−1ξ

Therefore, creating a series of point processes{Xi − 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).





Generalized Pareto Distribution (GPD) forExceedances

P(Xi > x + u|Xi > u) =nP(Xi > x + u)

nP(Xi > u)

→(1 + ξ x+u−bnan

1 + ξ x−bnan

)−1ξ

=

(1 +

ξxan + ξ(u − bn)

)−1ξ

⇒ Survival function of generalized Pareto distribution





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 + ξyσ(u))

−1ξ ξ 6= 0,

1− exp( −yσ(u)) ξ = 0;





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.





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





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





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





Multivariate Extremes Examples

A central aim of multivariate extremes is trying to find anappropriate structure to describe tail dependence





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





What is a Multivariate Extreme?





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 − bian

≤ 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 Fréchet i.e.,Z (x ,∞, · · · ,∞) = · · · = Z (∞, · · · ,∞, x) = exp(−1x ) tomodel the dependence structure





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 independent1

min(z1,··· ,zD) if Z1, · · · , Zd are entirely dependent





Spectral Representations

Pickands, 1981

V (z1, · · · , zd ) =∫

SDmax(w1z1 , · · · ,

wDzD

) dM(w1, · · · ,wD)where M is a measure on the D-dimensional simplex SD∫

wd dM(w1, · · · ,wD) = 1 for each dNo simple parametric forms for V





Summary measure of extremal dependence

P(Z1 ≤ z, · · · ,ZD ≤ z) = exp{−V (1,··· ,1)

z

}≡ exp(−θDz ) z >

0θD is the extrmal coefficient

θD = 1⇒ fully dependentθD = D ⇒ independent

In bivariate case limz→∞ P(Z2 > z|Z1 > z) = 2− θDWhen D = 2 madogram [Cooley, Naveau and Poncet,2006]

µF =12E{|F (Z1 − F (Z2))|}

θ2 =1 + 2νF1− 2νF

provide a good estimator of θ2Whitney Huang EVA and Spatial Extreme




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





Figure: A realization of Gaussian random processes




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 )} ∼ GEV{µ(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





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)





Pros and Cons

The quantile surfaces are realisticAfter averaging over S(x) the marginal distribution of{Y (x)} is NOT GEVThe spatial dependence is ignored because of conditionalindependence





Pros and Cons

Figure: One realisation of the latent variable model, showing the lackof local spatial structure





Copula

The D− dimensional joint distribution F of any randomvariable (Y1, · · · ,YD) may be written as

F (y1, · · · , yn) = C{F1(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) = C{F1(y1), · · · ,FD(yD)} is as follows:

C(u1, · · · ,uD) = F{

F1−1(u1), · · · ,FD−1(uD)}





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−1ν (uD) : Ω}

Extremal Copula:

Y1, · · · ,YD ∼ multivariate GEV

By max-stability⇒ C(um1 , · · · ,umD ) = Cm(u1, · · · ,uD), 0 <u1, · · · ,ud < 1, m ∈ N





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




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))





Spectral Representations

Theorem (Schlather, 2002)

Z (x), x ∈ D ⊂ R2 is max-stable with unit Fréchet marginals⇐⇒ There exist iid positive stochastic processesV1(x),V2(x), · · · with E[Vi(x)] = 1 ∀x ∈ D andE[supx∈D V (x)]




Some Models

Smith: Yi(x) = φ(xi − Ui)(), {Ui}i≤1 points of ahomogeneous Poisson process on R2Schlather: Yi(x) = 2πεi(x), εi(·) standard GaussianprocessGeometric: Yi(x) = exp{σεi(x)− σ

2

2 }Brown–Res: Yi(x) = exp{ε

′

i (x)− γ(x)}, ε′

i (·) intrinsicallystationary Gaussian process with (semi) variogram γ

Figure: One realization of max-stable processes





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, fjm; θ)

Not a true likelihoodOver–uses the data – each observation appears K-1 times





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




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





MotivationExtreme Value Theorem and GeostatisticsUnivariate ExtremesMultivariate ExtremesGeostatistics

Spatial ExtremesBayesian Hierarchical ModelsCopula ModelsMax-stable Models

Documents

Extreme Value Analysis and Spatial Extremeshuang251/stat695o_pres_slide.pdfExtreme events are defined to be rare and unexpected 100–year flood–the level of flood water expected