statistical modelling of extreme values

Statistical Modelling of Extreme Values

Stuart Coles and Anthony Davison

c©2008

Based on ‘An Introduction to Statistical Modeling of Extreme Values’,

by Stuart Coles, Springer, 2001

http://stat.epfl.ch

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Introduction 3Basic problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Brief history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Basic Notions 17Principles of Stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Probability framework for block maxima distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Classical limit laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Extremal types theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Using the limit law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Generalized extreme value distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Quantiles and return levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Max-stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Outline proof of GEV as limit law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Domains of attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Convergence and approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Interest and nuisance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Modelling with the GEV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Port Pirie Annual Maxima data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Bayesian Extremes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42MCMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Metropolis–Hastings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Port Pirie: MCMC analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1

Limitations of the GEV limit law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Point Process Approach 51Improved Inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Example: Eskdalemuir rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Poisson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Statistical application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Example: Eskdalemuir rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Threshold methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Alternative inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Mean residual life plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Threshold selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70r–largest order statistics method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Venice Sea Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Modelling Issues 73D(un) condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Short-term dependence: Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Moving maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Extremal index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80D’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Wooster example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Return levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Rain example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88Non-stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92Model reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Other extreme value models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Semiparametric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Example: Swiss winter temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Multivariate Extremes 111Multivariate extremes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Componentwise maxima. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Parametric models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Structure variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120Sea levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Point process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129Consistency of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Poisson likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Exchange rate data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133Oceanographic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Higher-dimensional models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Asymptotic dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Bivariate normal variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

2

Spatial Extremes 143Spatiotemporal extremes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Geostatistics 145Geostatistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Temperature data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Latent variable approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Heffernan–Tawn (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151Max-stable processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Extremal coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153Madogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Spectral representations I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155Spectral representations II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Pairwise likelihood. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157Example: Temperature data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158Swiss summer temperature data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Fit of marginal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160Estimated correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Fit of correlation curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162Pairwise fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Simulated fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3

Overview

1. Basic notions: maximum analysis, and threshold models

2. Alternative charactizations—improved inferences

3. Modelling issues

4. Multivariate extremes

5. Spatial models for extremes

Statistics of Extremes January 2008 – slide 2

Introduction slide 3

Basic problem

Simplest case: X1, . . . Xniid∼ F . Require accurate inferences on tail of F .

Key issues:

there are very few observations in the tail of the distribution;

estimates are often required beyond the largest observed data value;

standard density estimation techniques fit well where the data have greatest density, but can beseverely biased in estimating tail probabilities.

Usual lack of physical or empirical basis for extrapolation leads to the extreme value paradigm:

Base tail models on asymptotically-motivated distributions.


Applications

Historically there have been two main application areas of extreme value theory:

Environmental:

– sea levels

– wind speeds

– pollution concentrations

– river flow

– . . .

Reliability Modelling:

– weakest-link type models

Growing areas of application include: finance, insurance, telecommunications, athletic records,microarrays, . . .


4

Examples: Annual maximum sea levels

Year

Sea

-Lev

el (

met

res)

1930 1940 1950 1960 1970 1980

3.6

3.8

4.0

4.2

4.4

4.6

Annual maximum sea levels at Port Pirie, Western Australia.


Examples: Breaking strengths of glass fibres

0.0 0.5 1.0 1.5 2.0 2.5

05

1015

2025

Breaking Strength

Fre

quen

cy

Here minima are of interest.


5

Examples: Dependence on covariates

Year

Sea

-leve

l (m

eter

s)

1900 1920 1940 1960 1980

1.2

1.4

1.6

1.8

Fremantle annual maximum sea levels: apparent trend in time.

SOI

Sea

-leve

l (m

eter

s)

-1 0 1 2

1.2

1.4

1.6

1.8

Apparent dependence on the Southern Oscillation Index (an indicator of El Nino).


Examples: Fastest annual times for women’s 1500m

Year

Rac

e T

ime

(sec

s.)

1975 1980 1985 1990

232

234

236

238

240

242

244

246

There is an obvious time trend.


6

Examples: Largest order statistics

Year

Sea

-leve

l (cm

)

1930 1940 1950 1960 1970 1980

6080

100

120

140

160

180

200

Largest ten (available) annual sea levels at Venice.


Examples: Threshold exceedances

Year

Dai

ly R

ainf

all (

mm

)

1920 1930 1940 1950 1960

020

4060

80

Daily rainfall accumulation (location in south-west England)


7

Examples: Nonstationarity

Year

Dai

ly M

inim

um T

empe

ratu

re (

Deg

rees

Bel

ow 0

F.)

1983 1984 1985 1986 1987 1988

-60

-40

-20

020

Daily minimum temperatures (Wooster, Canada).


Examples: Financial time series

Year

Inde

x

1996 1997 1998 1999 2000

6000

8000

1000

012

000

Year

Inde

x

1996 1997 1998 1999 2000

-0.0

6-0

.04

-0.0

20.

00.

020.

04

Dow Jones Index. (Left Panel: Raw. Right Panel: After transformation to induce approximatestationarity)


8

Examples: Multivariate extremes

Annual Maximum Wind Speed (knots) at Albany (NY)

Ann

ual M

axim

um W

ind

Spe

ed (

knot

s) a

t Har

tford

(C

T)

50 60 70 80

4045

5055

6065

Annual maximum wind speeds at 2 different locations (North East U.S.)


Brief history

1920’s: Foundations of asymptotic argument developed by Fisher and Tippett

1940’s: Asymptotic theory unified and extended by Gnedenko and von Mises

1950’s: Use of asymptotic distributions for statistical modelling by Gumbel and Jenkinson

1970’s: Classic limit laws generalized by Pickands

1980’s: Leadbetter (and others) extend theory to stationary processes

1990’s: Multivariate and other techniques explored as a means to improve inference

2000’s: Interest in spatial and spatio-temporal applications, and in finance


Resources Books

– Galambos (1987) The Asymptotic Theory of Extreme Order Statistics, Krieger

– Resnick (1987) Extreme Values, Regular Variation, and Point Processes, SV

– Leadbetter, Lindgren and Rootzen (1983) Extremes and Related Properties of Random Sequences and Processes,SV

– de Haan and Ferreira (2006) Extreme Value Theory: An Introduction, SV

– Gumbel (1958) Statistics of Extremes, Columbia University Press

– Embrechts, Kluppelberg and Mikosch (1997) Modelling Extreme Events for Insurance and Finance, SV

– Kotz and Nadarajah (2000) Extreme Value Distributions, Imperial College Press

– Coles (2001) An Introduction to Statistical Modelling of Extreme Values, SV

– Beirlant, Goegebeur, Segers, and Teugels (2004) Statistics of Extremes: Theory and Applications, Wiley

– Finkenstadt and Rootzen (2004) Extreme Values in Finance, Telecommunications and the Environment, CRC

– Reiss and Thomas (2007) Statistical Analysis of Extreme Values, Birkhauser

R packages evd, evdbayes, evir, extRemes, fExtremes, POT

Journal Extremes (published by Springer)


9

Basic Notions slide 17

Principles of Stability

Fundamental to all characterizations of extreme value processes is the concept of stability.

For example, we might propose one model for the annual maximum of a process, and another forthe 5-year maximum. Since the 5-year maximum will be the maximum of 5 annual maxima, themodels should be mutually consistent.

Similarly, a model for exceedances over a high threshold should remain valid (in a precise sense)for exceedances of higher threshold.

The expression of such stability requirements as mathematical statements leads to asymptoticmodels.


Probability framework for block maxima distribution

Let X1, . . . ,Xniid∼ F and define

Mn = maxX1, . . . ,Xn.

Then the distribution function of Mn is

PrMn ≤ x = PrX1 ≤ x, . . . ,Xn ≤ x= PrX1 ≤ x × · · · × PrXn ≤ x= F (x)n.

But F is unknown, so approximate Fn by limit distributions as n→∞.

What distributions can arise?

Obviously, if F (x) < 1, then F (x)n → 0, as n→ ∞.


Classical limit laws

Question, as stated, is trivial: Mn will converge to the upper endpoint of the distribution of X asn→∞. But

recall the Central Limit Theorem: with µn = µ and σn = σ/√n,

Xn − µn

σn→N(0, 1),

so rescaling is needed to obtain a non-degenerate limit.

Same applies here: we seek limits of

Mn − bnan

for suitable sequences an > 0 and bn.


10

Extremal types theorem

Definition 1 The distributions F and F ∗ are of the same type if there are constants a > 0 and b such that

F ∗(ax + b) = F (x) for all x.

Theorem 2 (Extremal types theorem) If there exist sequences of constants an > 0 and bn such that, as n → ∞,

Pr(Mn − bn)/an ≤ x → G(x)

for some non–degenerate distribution G, then G has the same type as one of the following distributions:

I : G(x) = exp− exp(−x), −∞ < x < ∞;

II : G(x) =

0, x ≤ 0,exp(−x−α), x > 0, α > 0;

III : G(x) =

exp−(−x)α, x < 0, α > 0,1, x ≥ 0.

Conversely, each of these G’s may appear as a limit for the distribution of (Mn − bn)/an, and does so when G itself is

the distribution of X.


Three limiting distributions

−10 −5 0 5 10

0.0

0.1

0.2

0.3

Gumbel

x

PD

F

−10 −5 0 5 10

0.0

0.1

0.2

0.3

0.4

Frechet, alpha=0.5

x

PD

F

−10 −5 0 5 10

0.0

0.1

0.2

0.3

0.4

Negative Weibull, alpha=0.5

x

PD

F

The three types are known as the Gumbel, Frechet and Weibull (strictly, negative Weibull),respectively.

The Frechet (Type II) is bounded below, and the negative Weibull (Type III) is bounded above.

The standard Weibull is a distribution for minima.


11

Example: Exponential maxima

If F is the standard exponential distribution, then

F (x + log n)n = (1 − e−x−log n)n =

„

1 −1

ne−x

«n

→ exp− exp(−x).

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

x

CD

F

−4 −2 0 2 4 6 8

0.0

0.2

0.4

0.6

0.8

1.0

x

CD

F

Distributions of maxima and renormalised maxima of n = 1, 7, 30, 365, 3650 standard exponential variables (from left to right), with Gumbel distribution (heavy).


Example: Normal maxima

Maxima of standard normal variables also converge to a Gumbel limit, with

an = (2 log n)−0.5, bn = (2 log n)0.5 − 0.5(2 log n)−0.5(log log n + log 4π),

but convergence is extremely slow.

y

CD

F

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

y

CD

F

-4 -2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

Distributions of maxima and renormalised maxima of n = 1, 7, 30, 365, 3650 standard normal variables (from left to right), with Gumbel distribution (heavy).


12

Using the limit law

We assume that for some a > 0 and b,

Pr(Mn − b)/a ≤ x ≈ G(x),

or equivalently,PrMn ≤ x ≈ G(x − b)/a = G∗(x),

where G∗ is of the same type as G.

That is, the family of extreme value distributions may be fitted directly to a series of observationsof Mn.

However, it is inconvenient to have to work with three possible limiting families.


Generalized extreme value distribution

This family encompasses all three of the previous extreme value limit families:

G(x) = exp

−[1 + ξ

(x− µ

σ

)]−1/ξ

+

,

defined on x : 1 + ξ(x− µ)/σ > 0. From now on let x+ = max(x, 0).

µ and σ are location and scale parameters

ξ is a shape parameter determining the rate of tail decay, with

– ξ > 0 giving the heavy-tailed (Frechet) case

– ξ = 0 giving the light-tailed (Gumbel) case

– ξ < 0 giving the short-tailed (negative Weibull) case


13

Quantiles and return levels

In terms of quantiles, take 0 < p < 1 and define

xp = µ− σ

ξ

[1 − − log(1 − p)−ξ

],

where G(xp) = 1 − p.

In extreme value terminology, xp is the return level associated with the return period 1/p.

Log y

Qua

ntile

-2 0 2 4 6

05

1015

Shape=0.2

Shape=0

Shape=-0.2


Max-stability

Some insight into these results is obtained using the concept of max-stability.

Definition 3 A distribution G is said to be max-stable if

Gk(akx+ bk) = G(x), k = 1, 2, . . . ,

for some constants ak and bk.

In other words, taking powers of G results only in a change of location and scale.

The connection with extremes is that a distribution is max-stable if and only if it is a GEVdistribution.


14

Outline proof of GEV as limit law

Suppose that (Mn − bn)/an has limit distribution G. Then, for large n,

Pr(Mn − bn)/an ≤ x ≈ G(x)

and so for any k,

Pr(Mnk − bnk)/ank ≤ x ≈ G(x).

But

Pr(Mnk − bnk)/ank ≤ x = [Pr(Mn − bnk)/ank ≤ x]k,

giving two expressions for the distribution of Mn:

Pr(Mn ≤ x) ≈ G(x−bn

an

), Pr(Mn ≤ x) ≈ G1/k

(x−bnk

ank

),

so that G and G1/k are identical apart from scaling coefficients. Thus, G is max-stable and therefore GEV.


Domains of attraction

For a given distribution function F , is it easy to determine suitable sequences an and bn and to know what limitG will occur?For sufficiently smooth distributions, defining the reciprocal hazard function

r(x) =1 − F (x)

f(x).

and letting

bn = F−1(1 − 1n ), an = r(bn), ξ = lim

x→∞r′(x)

the limit distribution of (Mn − bn)/an is

exp−(1 + ξx)−1/ξ+ if ξ 6= 0

and

exp(−e−x) if ξ = 0.


15

Convergence and approximation

There has been a good deal of work on the speed of convergence of Mn to the limiting regime, whichdepends on the underlying distribution F—for example, convergence is slow for maxima of n Gaussianvariables.

From a statistical viewpoint, this is not so useful: we use the GEV as an approximate distribution forsample maxima for finite (small?) n, so the key question is whether the GEV fits the available data—assessthis empirically.

Direct use of the GEV rather than the three types separately allows for flexible modelling, and ducks thequestion of which type is most appropriate—the data decide.

Testing for fit of one type or another is usually unhelpful, because setting (say) ξ = 0 can giveunrealistically precise inferences. Often uncertainty in extremes is (appropriately) large, and it can bemisleading to constrain inferences artificially.


Minima

All the ideas apply equally to minima, because

min(X1, . . . ,Xn) = −max(−X1, . . . ,Xn).

Our general discussion is for maxima, and we make this transformation without comment when wemodel minima.


Inference

Given observed ‘annual maxima’ X1, . . . ,Xk, aim now is to make inferences on the GEV parameters(µ, σ, ξ) . Possibilities include:

graphical techniques—historically important, remain useful for model-checking;

moment-based estimators—usually inefficient for extremes, as moments may not exist;

probability-weighted moments—widely used in hydrology, but difficult to extend to complexdata;

likelihood-based techniques—widely used in statistics because

– intuitive paradigm often readily adapted to complex data;

– have unifying general approximate estimation and testing theory;

– incorporation of prior information via Bayes’ theorem also uses likelihood as key ingredient.


16

Likelihood

Definition 4 Let y be a data set, assumed to be the realisation of a random variable Y ∼ f(y; θ), where θ isunknown. Then the likelihood and log likelihood are

L(θ) = L(θ; y) = fY (y; θ), ℓ(θ) = logL(θ), θ ∈ Ωθ.

The maximum likelihood estimate (MLE) θ satisfies ℓ(θ) ≥ ℓ(θ), for all θ ∈ Ωθ.

Often θ is unique and in many cases it satisfies the score (or likelihood) equation

∂ℓ(θ)

∂θ= 0,

which is interpreted as a p× 1 vector equation if θ is a p× 1 vector.The observed information defined as

J(θ) = −∂2ℓ(θ)

∂θ∂θT

is a p× p matrix if θ has dimension p.The likelihood ratio statistic (Wilks’ statistic) is W (θ) = 2ℓ(θ) − ℓ(θ) ≥ 0.


Likelihood approximations

For large n, and if the data were generated from f(y; θ0, then

θ·∼ Np

(θ0, J(θ)−1

),

so a standard error for θr is the square root of the rth diagonal element of J(θ)−1.

To obtain θ and J(θ), we just have to maximise ℓ(θ) and to obtain the Hessian matrix at themaximum—purely numerically, if a routine to compute ℓ(θ) is available;

W (θ0)·∼ χ2

p.

We base tests and/or confidence intervals on these results, even when the sample is ‘small’ (e.g. n = 20 orless). Often the approximations are adequate for practical work.

For details, see for example Chapter 4 of Davison (2003) Statistical Models, CUP.


17

Interest and nuisance parameters

In practice θ usually divides into

interest parameters ψq×1 central to the problem (often q = 1 in practice);

nuisance parameters λp−q×1 whose values are not of real importance.

Let λψ denote the maximum likelihood estimator of λ when ψ is fixed:

ℓ(ψ, λψ) ≥ ℓ(ψ, λ),

and define the (generalised) likelihood ratio statistic

Wp(ψ) = 2ℓ(ψ, λ) − ℓ(ψ, λψ)

= 2

ℓ(θ) − ℓ(θψ)

.

If ψ0 is the value of ψ that generated the data from a regular model,

Wp(ψ)·∼ χ2

q, for large n.

We often base confidence intervals and tests for ψ0 on this, particularly if the profile log likelihood for ψ,ℓp(ψ) = ℓ(θψ) = ℓ(ψ, λψ) is asymmetric—generally the case for quantiles, Value-At-Risk, and similar quantities.


Non-regular models

With extremal models there is a potential difficulty the endpoint of the distribution (if finite) is a functionof the parameters, so usual asymptotic results may not hold.

Smith (1985, Biometrika) established that the limiting behaviour of the MLE depends on the value of shapeparameter ξ:

– when ξ > −0.5, maximum likelihood estimators obey standard theory above;

– when −1 < ξ < −1/2, the MLE satisfies the score equation;

– when ξ < −1, the MLE does not satisfy the score equation.

When ξ < −1/2, the MLE for the endpoint converges to the endpoint faster than when ξ > −1/2 (good)at a convergence rate that depends on ξ (bad), and other parameters converge at the usual rate (OK), sothere is no simple theory.

In most environmental problems −1 < ξ < 1 (often ξ ≈ 0) so maximum likelihood works fine.

If not, Bayesian methods (which do not depend on these regularity conditions) may be preferable.


18

Modelling with the GEV

Specification (programming) of log-likelihood function:

ℓ(µ, σ, ξ) =

k∑

i=1

− log σ − (1 + 1/ξ) log

[1 + ξ

(xi−µσ

)]+

−[1 + ξ

(xi−µσ

)]−1/ξ

+

;

note this equals −∞ if any 1 + ξ(xi − µ)/σ < 0.

Numerical maximization of log-likelihood.

Calculation of standard errors from inverse of observed information matrix (also obtained numerically).

Diagnostic checks: probability plots, quantile plots, return level plots.

Comparison of competing nested models through deviance (likelihood ratio statistic), and of non-nested

models by minimising the Akaike information criterion AIC = 2dim(θ) − ℓ, or other similar criteria.

Calculation of confidence intervals for return levels.


Port Pirie Annual Maxima data

Year

Sea

-Lev

el (

met

res)

1930 1940 1950 1960 1970 1980

3.6

3.8

4.0

4.2

4.4

4.6

portpirie.fit<- gev.fit(portpirie$SeaLevel)

$conv

[1] 0

$nllh

[1] -4.339058

$mle

[1] 3.87474692 0.19804120 -0.05008773

$se

[1] 0.02793211 0.02024610 0.09825633


19

Port Pirie data: Diagnostics

gev.diag(portpirie.fit)

Probability Plot

Empirical

Mod

el

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Quantile Plot

Model

Em

piric

al

3.6 3.8 4.0 4.2 4.4 4.6

3.6

3.8

4.0

4.2

4.4

4.6

Return Period

Ret

urn

Leve

l

0.1 1.0 10.0 100.0 1000.0

4.0

4.5

5.0

Return Level Plot

3.6 3.8 4.0 4.2 4.4 4.6

0.0

0.5

1.0

1.5

Density Plot

zf(

z)


Port Pirie data: Use of profile likelihood

Profile log-likelihood for ξ:

Shape Parameter

Pro

file

Log-

likel

ihoo

d

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

01

23

4

Profile log-likelihood for 100-year return level:

Return Level

Pro

file

Log-

likel

ihoo

d

4.5 5.0 5.5 6.0

-20

24


20

Bayesian Extremes

Markov chain Monte Carlo (MCMC), and other stochastic computation algorithms, have enabledBayesian techniques to be applied to extreme value problems.Advantages include

use of additional information via prior specification

no reliance on n→ ∞ asymptotics (except for model choice!)

ease of inference (for example, parameter transformations)

development of predictive inference through unified treatment of variation and uncertainty


Bayesian inference

Key idea: everything is treated as random variable, so probability calculus is applied to unknown parametersand potential data

Use Bayes’ theorem to compute

Pr( unknowns | data ) =Pr( data | unknowns )Pr( unknowns )

Pr( data )

If the unknown is a continuous parameter θ, then compute

Pr(θ | data ) =Pr( data | θ)π(θ)∫Pr( data | θ)π(θ) dθ

,

where the prior density π(θ) summarises our prior knowledge/belief about θ

Main problem is to compute integrals like the one in the denominator, for realistic models

Often feasible using Markov chain Monte Carlo (MCMC) simulation


MCMC

Idea is that when direct simulation from a density f is difficult, the problem can be tackled bysetting up a Markov chain whose limit density is f .

Consequently, simulation of x1, x2, . . . from the chain yields a series with the property that themarginal density of xj for large enough j is approximately f .

In other words, for large enough N , xN , xN+1, . . . can be regarded as a dependent series withmarginal density f . Hence, for example, empirical moments of this series yield approximations ofthe moments of f .


21

Metropolis–Hastings

To generate a Markov chain xt with limit density f :

1. Set t = 0 and choose x(0).

2. Simulate x∗ with ’arbitrary’ transition density h(· | x(t)).

3. Calculate

α = α(x(t), x∗) = min

1,f(x∗)q(x(t) | x∗)f(x(t))q(x∗ | x(t))

4. Define

x(t+1) =

x∗ with probability α

x(t) with probability 1 − α.

5. Set t = t+ 1 and go to step 2.


Comments on Metropolis–Hastings

The algorithm remains valid when f is only proportional to a target density function. This is whythe algorithm is so important in Bayesian applications.

Subject to regularity conditions—in particular, the possibility of moving everywhere in thedomain—the choice of q is arbitrary. Simple families such as random walks are generally chosen,though poor choices will lead to poor algorithms.

A successful algorithm will:

– converge quickly: density of xt is near f for small t;

– mix well: have low autocorrelation.

These conditions facilitate the use of fairly short series of simulated data for inference, but use ofconvergence diagnostics is essential, as there is usually no theoretical guarantee of convergence


22

Port Pirie: MCMC analysis

MCMC chains for GEV parameters and (by transformation) 100-year return level in analysis of Port Pirie annual

maxima. Analysis based on vague, though proper, priors.

0 1000 3000 5000

3.8

4.2

4.6

5.0

Index

loca

tion

0 1000 3000 5000

0.2

0.4

0.6

0.8

Index

scal

e

0 1000 3000 5000

−0.

6−

0.2

0.2

Index

shap

e

0 1000 3000 5000

4.5

5.0

5.5

6.0

6.5

Index

100−

year

ret

urn

leve

l



Posterior means:

µ = 3.87, σ = 0.208, ξ = −0.045,

Posterior standard deviations:

SD(µ) = 0.029, SD(σ) = 0.022, SD(ξ) = 0.099


23


3.75 3.85 3.95 4.05

05

1015

20

location

Pro

babi

lity

dens

ity fu

nctio

n

Location

0.15 0.20 0.25 0.30

05

1020

scale

Pro

babi

lity

dens

ity fu

nctio

n

Scale

−0.4 0.0 0.2 0.4

01

23

45

6

shape

Pro

babi

lity

dens

ity fu

nctio

nShape

4.0 4.5 5.0 5.5 6.0 6.5 7.0

0.0

1.0

2.0

return level

Pro

babi

lity

dens

ity fu

nctio

n

100−year return level

Posterior densities in analysis of Port Pirie annual maxima.


Limitations of the GEV limit law

The GEV limit law for block maxima requires careful reading.

It states that when (normalized) maxima have a limit, that limit will be a member of the GEVfamily. It does not (by itself) guarantee the existence of a limit, and there exist wide classes ofmodels for which no limit exists.

One interesting case is the Poisson distribution. If Xi is a sequence of Poisson variables withmean λ, a sequence of constants In can be found such that

limn→∞

Pr

max1≤i≤n

= In or In + 1

= 1

and this degeneracy implies that no limit distribution for normalized maxima exists.

The discreteness in the Poisson distribution causes the oscillation between In and In + 1. It is notthe cause of the limit degeneracy, which is actually a consequence of the Poisson tail decay.


24

Point Process Approach slide 51

Improved Inferences

The annual maxima method can be inefficient if other data are available. Alternative methods include:

peaks over thresholds,

r-largest order statistics.

Both are special cases of a point process representation.


Example: Eskdalemuir rainfall

Time

Hou

rly r

ainf

all (

mm

)

1970 1975 1980 1985

05

1015



Time

Hou

rly r

ainf

all (

mm

)

1970 1975 1980 1985

05

1015


25


Time

Hou

rly r

ainf

all (

mm

)

1970 1975 1980 1985

05

1015


Point process

A point process P is simply a collection of points—times of avalanches, positions of stars in thesky, time/position of earthquake times/epicentres; . . .

For any suitable set A, we simply count the number of points in A, and for some (random) n wecan write the process as

P =n∑

j=1

δXj

where the Xj are the positions of the points, and δx puts unit mass at x.

The basic point process is the Poisson process.


Poisson process

Definition 5 Let X ⊂ Rd, and let a function Λ(A) ≥ 0 be defined for any measurable A ⊂ X . A

Poisson process P on X with intensity measure Λ satisfies

the number of points of P in A, denoted N(A), has the Poisson distribution with mean Λ(A);

If A,B ⊂ X are disjoint, then N(A) and N(B) are independent.

If A = [a1, x1] × · · · × [ad, xd], and if

λ(x1, . . . , xd) =∂dΛ(A)

∂x1 · · · ∂xd

exists, then λ is called the intensity (density) function of P.


26

Point process limit: Basic idea

Suppose F unknown, and take X1, . . . ,Xniid∼ F .

Form a 2-dimensional point process (i,Xi); i = 1, . . . , n and characterize the behaviour of thisprocess in regions of the form A = [t1, t2] × [u,∞).

More formally, suppose (Mn − bn)/an converges to the distribution

G(x) = exp− (1 + ξx)

−1/ξ+

.

Construct a sequence of point processes on R2 by

Pn =

(i

n+ 1,Xi − bnan

): i = 1, . . . , n

.

Then Pn → P as n→ ∞, where P is a Poisson process.


Poisson limit

Poisson limit when rescaling samples of sizes 10, 100, 1000, 10,000.

•• •

•• •

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(X-b

)/a

0.0 0.2 0.4 0.6 0.8 1.0

-8-6

-4-2

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•

t

(X-b

)/a

0.0 0.2 0.4 0.6 0.8 1.0

-8-6

-4-2

02


Limiting Poisson process

As n→∞, on regions bounded away from −∞, Pn→P, a Poisson process defined by its intensitymeasure Λ.

Consider the set Ax = [0, 1] × [x,∞), for some x > u.

The Poisson property gives

Prno points in Ax = exp−Λ(Ax)= PrMn ≤ x≈ exp

− (1 + ξx)

−1/ξ+

,

and so, by time-homogeneity of the process,

Λ(t1, t2) × [x,∞) = (t2 − t1)(1 + ξx)−1/ξ+ .


27

Statistical application

For statistical application, it is convenient to:

Assume the limiting process is a reasonable approximation for finite n above a high threshold, u.

To absorb the unknown scaling coefficients into the intensity function and so work directly withthe original series.

To rescale the intensity so that the annual maximum has the GEV distribution with parameters(µ, σ, ξ) .

This leads to modelling the series (i,Xi); i = 1, . . . , n as a Poisson process above u with intensityfunction

Λ(t1, t2) × (x,∞) = ny(t2 − t1)1 + ξ(x− µ)/σ−1/ξ+

where ny is the number of years of observation, and the time limits are rescaled to 0 ≤ t1 < t2 ≤ 1.


Likelihood

Maximum likelihood inference is most natural.

For a region of the form Av = [0, 1] × (v,∞) for v > u, the likelihood is

L(Av;µ, σ, ξ) = exp−Λ(Av)NAv∏

i=1

dΛ(ti, xi)

= exp−ny

(1 + ξ v−µσ

)−1/ξ

+

×NAv∏

i=1

1

σ

(1 + ξ xi−µ

σ

)−1/ξ−1

+,

where x1, . . . , xNAvis an enumeration of the NAv

points that exceed the threshold v.

The parameters are the same as for the maximum model.



Time

Hou

rly r

ainf

all (

mm

)

1970 1975 1980 1985

05

1015

Model is now applied to a time series of hourly rainfall aggregates measured in mm with a thresholdof u = 5mm.


28


(rain.fit <- fpot(esk.rain,threshold=5,model="pp",

start=list(loc=10,scale=1.2,shape=0.1),npp=365.25*24))

Threshold: 5

Number Above: 356

Proportion Above: 0.0024

Estimates

loc scale shape

10.13628 1.86637 0.06696

Standard Errors

loc scale shape

0.35380 0.23673 0.05379



0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Probability Plot

Empirical

Mod

el

6 8 10 12 14 16 18

510

1520

Quantile Plot

Model

Em

piric

al

6 8 10 12 14 16

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Density Plot

Quantile

Den

sity

0.05 0.20 1.00 5.00 20.00

510

1520

25

Return Level Plot

Return Period

Ret

urn

Leve

l


29

Threshold methods

Let X∗n,i = (Xi − bn)/an, for i = 1, . . . , n. Then, by the Poisson limit,

PrX∗n,i > u+ x | X∗

n,i > u ≈ Λ(0, 1)× (u+ x,∞)Λ(0, 1)× (u,∞)

=

1 + ξ

x

σ + ξ(u− µ)

−1/ξ

+

.

Absorbing the unknown scaling coefficients leads to the survivor function of the Generalized ParetoDistribution (GPD):

PrXn,i > u+ x | Xn,i > u =(1 + ξ

x

τ

)−1/ξ

+, x > 0,

where τ = σ + ξ(u − µ).

Taking the limit ξ→0 gives the exponential distribution as a special case.


Alternative inference

An equivalent inference to the point process method is to apply the GPD model to the exceedances ofa threshold u. For the rainfall data we obtain:

(rain.gpdfit <- fpot(esk.rain,threshold=5,npp=365*24))

Deviance: 1058.954

Threshold: 5

Number Above: 356


Estimates

scale shape

1.52239 0.06702

Standard Errors

scale shape

0.11488 0.05383


30

Interpretation

There are now just two parameters, since the model conditions on the exceedance (one parameteris lost corresponding to the crossing rate of u).

The estimate of ξ is (almost!) identical to the point process analysis, while the change ofparameterization leads to a different value of σ.

Provided ξ < 1, the mean residual life exists and satisfies

E(X − u | X > u) =σ + ξu

1 − ξ,

we obtain a simple diagnostic for threshold selection. The mean exceedance above u should belinear in u at levels for which the model is valid.Suggests looking for linearity in a plot of the empirical mean residual life.


Mean residual life plot

mrlplot(esk.rain) gives

0 2 4 6 8 10 12

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Mean Residual Life Plot

Threshold

Mea

n E

xces

s

Interpretation hard, but u > 2 looks reasonable; u = 5 looks high.


Threshold selection

Bias-variance trade-off: threshold too low—bias because of the model asymptotics being invalid; threshold toohigh—variance is large due to few data points.

An alternative approach is to fit the Poisson process model at many thresholds and look for parameter stability:

0 1 2 3 4 5 6 7

1416

1820

22

Threshold

Loca

tion

0 1 2 3 4 5 6 7

01

23

45

6

Threshold

Sca

le

0 1 2 3 4 5 6 7

−0.

20.

00.

10.

20.

3

Threshold

Sha

pe

Again, u = 5 looks too high.


31

r–largest order statistics method

Consider the vector of r largest order statistics in a block (say, year) of data:

(M

(1)n − bnan

,M

(2)n − bnan

, . . . ,M

(r)n − bnan

)

rescaled in the same way as just the maximum, M(1)n .

Setting u = M(r)n in the point process likelihood gives

L = exp

−(

1 + ξM

(r)n − µ

σ

)−1/ξ

r∏

i=1

1

σ

(

1 + ξM

(i)n − µ

σ

)−1/ξ−1

.

This is the likelihood contribution for a single block (year). The full likelihood is obtained bymultiplying such terms across the blocks.


Venice Sea Levels

rlarg.fit(venice.data)

nllh:

[1] 1149.268

mle:

[1] 120.3961096 12.6929941 -0.1148273

se:

[1] 1.34704941 0.52516203 0.01901015


32

Modelling Issues slide 73

Modelling issues

Extreme value data usually show:

dependence on covariate effects;

short term dependence (storms for example);

seasonality (due to annual cycles in meteorology);

long–term trends (due to gradual climatic change);

other forms of non–stationarity (for example, the deterministic effect of tides on sea–levels).

For temporal dependence there is a sufficiently wide-ranging theory which can be invoked. Otheraspects have to be handled at the modelling stage.We now discuss how to deal with

short-term dependence;

trends and seasonality.


D(un) condition

The usual (weak) condition which is adopted to eliminate the effect of long-range dependence isLeadbetter’s D(un) condition:For all i1 < . . . < ip < j1 < . . . < jq with j1 − ip > l

|PrXi1 ≤ un, . . . , Xip ≤ un, Xj1 ≤ un, . . . , Xjq ≤ un

−PrXi1 ≤ un, . . . , Xip ≤ unPrXj1 ≤ un, . . . , Xjq ≤ un| ≤ α(n, l),

where α(n, ln)→0 for some sequence ln = o(n). This implies that rare events that are sufficientlyseparated are independent.

Theorem 6 If D(un) is satisfied with un = anx+ bn, and if

PrMn ≤ anx+ bn→G(x),

then G is a GEV distribution.

Conclusion: Same models apply for processes with (restricted) long-range dependence.


33

Short-term dependence: Example

Suppose Y1, Y2, . . .iid∼ exp(1), and Xi = max(Yi, Yi+1):

0 20 40 60 80 100

01

23

45

6x

Extremes tend to cluster in pairs.


Exponential example

Marginal distribution:

PrXi < x = PrYi < x, Yi+1 < x = (1 − e−x)2.

Let X∗1 , X

∗2 , . . . be an independent series with the same marginal distribution, and M∗

n = maxX∗1 , . . . , X

∗n.

Then

PrM∗n − log(2n) < x = [1 − exp−x− log(2n)]2n =

(1 − 1

2ne−x)2n

→ exp(−e−x

)= G2(x),

while, for Mn = maxX1, . . . , Xn,

PrMn − log(2n) < x = PrY1 < x+ log(2n), . . . , Yn+1 < x+ log(2n)

=(1 − e−x

2n

)n+1

→ exp(− 1

2e−x)

= G1(x),

and G1(x) = G2(x)1/2.


34

Example: Moving maxima

Let Y0, Y1, Y2, . . .iid∼ F , with

F (y) = expn

− 1(a+1)y

o

, y > 0,

where 0 ≤ a ≤ 1 is a parameter, and define the process Xi by

X0 = Y0, Xi = maxaYi−1, Yi, i = 1, . . . , n.

i

x(i)

0 10 20 30 40 500.

10.

55.

050

.0

i

x(i)

0 10 20 30 40 50

0.1

0.5

5.0

50.0

i

x(i)

0 10 20 30 40 50

0.1

0.5

5.0

50.0

i

x(i)

0 10 20 30 40 50

0.1

0.5

5.0

50.0


Moving maxima: Asymptotic properties

PrXi ≤ x = PraYi−1 ≤ x, Yi ≤ x = exp(−1/x),

Now, let X∗1 , X

∗2 , . . . be a series of independent variables having a marginal standard Frechet distribution, and

define M∗n = maxX∗

1 , . . . , X∗n. Then,

PrM∗n ≤ nz = [exp−1/(nz)]n = exp(−1/z).

On the other hand, for Mn = maxX1, . . . , Xn,

PrMn ≤ nz = PrX1 ≤ nz, . . . , Xn ≤ nz= PrY1 ≤ nz, . . . , Yn ≤ nz=

[exp

− 1

(a+1)nz

]n

= exp (−1/z)1a+1 ,

It follows that

PrM∗n ≤ nz = [PrMn ≤ nz]

1a+1


35

Extremal index

The previous examples illustrate the following general result.

Theorem 7 Let Xi be a stationary process and X∗i be independent variables with the same marginal

distribution. Set Mn = maxX1, . . . , Xn and M∗n = maxX∗

1 , . . . , X∗n. Under suitable regularity conditions,

Pr (M∗n − bn)/an ≤ z → G1(z)

as n→ ∞ for normalizing sequences an > 0 and bn, where G1 is a non-degenerate distribution function, ifand only if

Pr (Mn − bn)/an ≤ z → G2(z),

where

G2(z) = Gθ1(z)

for a constant θ called the extremal index that satisfies 0 < θ ≤ 1.

Thus if G1 is GEV, then so is G2, with the same ξ—a strong robustness result.


Extremal index

The extremal index can also be defined as

θ = limn→∞

Prmax(X2, . . . ,Xpn) ≤ un | X1 ≥ un,

where pn = o(n), and the sequence un is such that PrMn ≤ un converges.

Loosely, θ is the probability that a high threshold exceedance is the final element in a cluster ofexceedances.

Thus extremes occur in clusters whose (limiting) mean cluster size is 1/θ.

The mean distance between clusters is increased by a factor 1/θ.

In fact the distribution of a cluster maximum is the same as the marginal distribution of anexceedance, so there is no bias in considering only cluster maxima, if we can identify clusters . . .


36

Consequences

When clustering occurs, the notion of return level is more complex:

– if θ = 1, then the ‘100-year-event’ has probability 0.368 of not appearing in the next 100 years;

– if θ = 1/10, then on average the event also occurs ten times in a millenium, but all together: it hasprobability 0.904 of not appearing in the next 100 years.

If we can estimate the tail of marginal distribution F (e.g. by fitting to block maxima), then

Pr(Mn ≤ x) ≈ F (x)nθ ≈ G(x),

where G is GEV with parameters µ, σ, ξ. The marginal quantiles are approximately

F−1(p) ≈ G−1(pnθ) > G−1(pn),

so may be much larger than would be the case with θ = 1.

A similar argument shows that ignoring θ can lead to over-estimating a return level estimated using G.


D′(un) condition

The D′(un) condition is said to hold for the process Xi if

lim supn→∞

n

[n/k]∑

j=2

PX1 > un,Xj > un → 0 as k → ∞

If a stationary process satisfies both the D(un) and D′(un) conditions, then its extremal index isθ = 1.

D′(un) holds for many well-known processes, for example Gaussian autoregressive processes. Thisraises the issue of how best to model the extremes of such processes, since at any realistic level,dependence is likely to be strong, while the asymptotic limit is independence.

Other conditions D′′, D(2), . . . facilitate extremal index calculation in special cases.


Modelling techniques

The above discussion tells us that under the weak condition D(un) and some mild additionalconditions, the maxima of stationary series can be modelling using the GEV, and that exceedancescan be modelling using the GPD. This is a strong robustness result, but it implies that extremes willoccur in clusters of correlated observations.Possible modelling strategies using exceedances and the GPD are:

to identify clusters and model cluster maxima only;

as above, but also estimate the extremal index empirically;

to ignore the dependence on the basis that the marginal model is valid, but to inflate standarderrors to account for reduction in independent information;

to specify explicit model for dependence, such as a first-order Markov chain.


37

Example: Wooster temperatures

Day Index

Deg

rees

Bel

ow 0

F.

0 10 20 30 40

-30

-20

-10

010

20

1 2 3 4

Day Index

Deg

rees

Bel

ow 0

F.

0 10 20 30 40

-30

-20

-10

010

20

1 2

Simple estimates of the extremal index are based on empirical means of clusters. Here the runs method is used:

a cluster is deemed to have terminated when there are r consecutive observations below the threshold. Left:

r = 1 gives 4 clusters. Right: r = 3 gives 2 clusters.


Calculation of return levels

The m-observation return level is

xm = u+ σξ

[(mζuθ)

ξ − 1],

where σ and ξ are the parameters of the threshold excess generalized Pareto distribution, ζu is theprobability of an exceedance of u, and θ is the extremal index.

ζu =nu

nand θ =

nc

nu.

where nc is number of clusters and nu is number of exceedances.

So simply estimate the component ζuθ by nc/n.


Example: Wooster temperatures

u = −10 u = −20r = 2 r = 4 r = 2 r = 4

nc 31 20 43 29σ 11.8 (3.0) 14.2 (5.2) 17.4 (3.6) 19.0 (4.9)

ξ −0.29 (0.19) −0.38 (0.30) −0.36 (0.15) −0.41 (0.19)x100 27.7 (12.0) 26.6 (14.4) 26.2 (9.3) 25.7 (9.9)

θ 0.42 0.27 0.24 0.16

Results may be sensitive to choice of threshold, u, and run length, r.


38


(rain.fit <- fpot(esk.rain,threshold=5,model="pp",


Threshold: 5

Number Above: 356


Estimates

loc scale shape

10.13628 1.86637 0.06696

Standard Errors

loc scale shape

0.35380 0.23673 0.05379



(rain.fit <- fpot(esk.rain,threshold=5,model="pp",cmax=T, r=0,


Threshold: 5

Number Above: 356


Clustering Interval: 0

Number of Clusters: 272

Extremal Index: 0.764

Estimates

loc scale shape

9.81557 1.83937 0.04178

Standard Errors

loc scale shape

0.3519 0.2406 0.0632


39


For finite threshold u, θ increases with u, suggesting that very extreme hourly rainfall totals occursingly.

0 1 2 3 4 5 6 7

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Threshold

Ext

rem

al In

dex


Non-stationarity

General results not broad enough for application — hence, model trends, seasonality and covariateeffects by parametric or nonparametric models for the usual extreme value model parameters.Some possibilities for parametric modelling:

µ(t) = α+ βt;

σ(t) = exp(α+ βt);

ξ(t) =

ξ1, t ≤ t0,ξ2, t > t0;

µ(t) = α+ βy(t).


Parameter estimation

Model specification (example)Zt ∼ GEV(µ(t), σ(t), ξ(t)),

Likelihood (for complete parameter set β,

L(β) =m∏

t=1

g(zt;µ(t), σ(t), ξ(t)),

where g is GEV model density.

Maximization of L yields maximum likelihood estimates.

Standard likelihood techniques also yield standard errors, confidence intervals etc.


40

Model reduction

For nested models M0 ⊂ M1, the deviance statistic is

D = 2ℓ1(M1) − ℓ0(M0),

Based on asymptotic likelihood theory, M0 is rejected by a test at the α-level of significance ifD > cα, where cα is the (1 − α) quantile of the χ2

k distribution, and k is the difference in thedimensionality of M1 and M0.


Model diagnostics

Assuming a fitted modelZt ∼ GEV(µ(t), σ(t), ξ(t)),

the standardized variables

Zt =1

ξ(t)log

1 + ξ(t)

Zt − µ(t)

σ(t)

,

each have the standard Gumbel distribution, with probability distribution function

PrZt ≤ z = exp(−e−z), z ∈ R.

Possible diagnostics:

probability plot:i/(m+ 1), exp(− exp(−z(i))); i = 1, . . . ,m

quantile plot:(

− log [− logi/(m+ 1)] , z(i)); i = 1, . . . ,m


Other extreme value models

Similar techniques are applicable for the threshold exceedance and point process models, butthreshold selection is likely to be a more sensitive issue.

Time-varying thresholds may also be appropriate, though there is little guidance on how to makesuch a choice.


Example: Fremantle sea levels

Model Log-likelihood

Constant 43.6Linear in µ 49.9Quadratic in µ 50.6Linear in µ and σ 50.7

Based on likelihood considerations, best model is linear in µ.

For this model, trend is around 2mm per year and ξ = −0.125 (0.070).

(Note: Model improves further by inclusion of SOI as a covariate for µ).


41


Year

Sea

-leve

l (m

etre

s)

1900 1920 1940 1960 1980

1.2

1.4

1.6

1.8

Fitted trend in location parameter for Fremantle annual maximum sea levels.



Empirical

Mod

el

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Residual Probability Plot

Model

Em

piric

al

-1 0 1 2 3 4

02

46

Residual Quantile Plot (Gumbel Scale)

Probability and quantile plots for nonstationary GEV analysis of Fremantle annual maximum sea levels.


Example: Race times

Model Log-likelihood β σ ξ

Constant −54.5 239.3 3.63 −0.469(0.9) (0.64) (0.141)

Linear −51.8 (242.9,−0.311) 2.72 −0.201(1.4, 0.101) (0.49) (0.172)

Quadratic −48.4 (247.0,−1.395, 0.049) 2.28 −0.182(2.3, 0.420, 0.018) (0.45) (0.232)

Quadratic model apparently preferable.


42

Example: Race times

Year

Rac

e T

ime

(sec

s.)

1975 1980 1985 1990

232

234

236

238

240

242

244

246

Fitted models for location parameter in womens 1500 metre race times. Note quadratic model wouldlead to slower races in recent and future events.


Example: Race times

Alternative exponential modelµ(t) = β0 + β1e

−β2t.

has log-likelihood −49.5. Not so good as quadratic model, though comparison via likelihood ratio testis invalid as models are not nested. Better behaviour for large t suggests a preferable model though.

Year

Rac

e T

ime

(sec

s.)

1975 1980 1985 1990

232

234

236

238

240

242

244

246


43

Example: Wooster Temperature Series

Day Index

Deg

rees

Bel

ow Z

ero

F.

0 100 200 300 400

-40

-20

020

Winter

Day Index

Deg

rees

Bel

ow Z

ero

F.

0 100 200 300 400

-60

-40

-20

0

Spring

Day Index

Deg

rees

Bel

ow Z

ero

F.

0 100 200 300 400

-70

-60

-50

-40

Summer

Day Index

Deg

rees

Bel

ow Z

ero

F.

0 100 200 300 400

-70

-50

-30

-10

Autumn

One approach to handle seasonality is to model seasons separately. This is not likely to be sufficienthere, at least with only 4 seasons.



Year

Dai

ly M

inim

um T

empe

ratu

re (

Deg

rees

Bel

ow 0

F.)

1983 1984 1985 1986 1987 1988

-60

-40

-20

020

Selection of a time-varying threshold for negated Wooster temperature series.



Model p ℓ

1. Time-homogeneous 3 −143.62. As 1. but periodic in µ 5 126.73. As 2. but periodic in log σ 7 143.64. As 3. but periodic in ξ 9 145.95. As 3. plus linear trend in µ and log σ 9 145.16. As 3. with separate ξ for each season 10 143.9

Likelihood considerations lead to Model 3 as preferable.


44

Semiparametric regression

We may relax parametric assumptions, or mix them with more flexible forms. For example, we may want tomodel temperatures at sites in the Alps as dependent on altitude a(x), where x is location, plus a smoothfunction of time t, giving

µ(x, t) = β0 + β1a(x) + g(t), or µ(x, t) = β0 + β1a(x) + gt, a(x).


Formulation

A standard approach is then to include basis functions in the model, so

β0 + β1a(x) + g(t) ≈ β0 + β1a(x) + γ1b1(t) + · · · + γpbp(t),

where the functions b1(t), . . . , bp(t) may be splines, polynomials, etc., depending on the properties sought.

Often take splines, which have optimality properties, and which correspond to prediction of stochasticprocesses.

Usually penalise the smoothing parameters γ = (γ1, . . . , γp)T, for example, maximising a log likelihood of

formℓ(β, γ) + λγγ

TKγ,

where smoothing parameter λγ > 0 controls smoothness of result (equivalent to effective degrees offreedom).

Interpretation in terms of mixed model, by representing γ ∼ Np(0,K−1λ−1

γ ), allows data to choose degreeof smoothing.


45

Example: Swiss winter temperatures

−20

−15

−10

−5

0Rheinfelden

1970 1980 1990

Grono Altstatten

1970 1980 1990

Thun Ebnat Kappel

1970 1980 1990

Meiringen

Fribourg Haidenhaus Heiden Einsredeln Vattis

−20

−15

−10

−5

0Elm

−20

−15

−10

−5

0Chateau d’Oex La Brevine Guttannen Oberiberg Chaumont Andermatt

Grachen Sils Maria1970 1980 1990

−20

−15

−10

−5

0Arosa

Year

Tem

pera

ture

bel

ow th

resh

old

(deg

rees

Cel

sius

)

December–February temperatures (C) below thresholds at 21 Swiss weather stations, for winters1970–71 to 1997–98.


Swiss winter temperatures

Day

lam

bda

0 20 40 60 80

0.0

0.1

0.2

0.3

0.4

North Atlantic oscillation index

lam

bda

−4 −2 0 2

0.0

0.1

0.2

0.3

0.4

Day

sigm

a

0 20 40 60 80

24

68


sigm

a

−4 −2 0 2

24

68

Altitude

xi

500 1000 1500

−0.

25−

0.20

−0.

15−

0.10

−0.

05


xi

−4 −2 0 2

−0.

25−

0.20

−0.

15−

0.10

−0.

05


46

Swiss winter temperatures

Estimated 0.05 quantiles for winter temperatures at Rheinfelden (298m), Vattis (957m), and Arosa(1821m), as a function of the North Atlantic oscillation index

•

•••

•

••

•

•

••

•••

••

•

•••

•

•

••

•

•

•

•••

•

•

•

•

••

•

•

••

•

•

••

•• ••••

•

••

•

•

••

••

••

••••

••

•


Tem

pera

ture

(de

gree

s C

elsi

us)

−4 −2 0 2

−30

−25

−20

−15

−10

−5

Rheinfelden

•

•

•••

•••

•

•

•

•• •

•

•

• •

•

•

•••

•

••

•••

•

•

• •••

•

•

•

••••

••••

•• • •

••

•

••

••

•

••

••

•

•

North Atlantic oscillation indexT

empe

ratu

re (

degr

ees

Cel

sius

)

−4 −2 0 2

−30

−25

−20

−15

−10

−5

Vattis

•••

•

•

••••

••

•

••

•

•••

•

• ••

• •

• •

• •

••••

•

•

••

•

•

•

•

•

••

•••

••

•

•

•

•

••

•• ••

•

•

•

••

•

• ••


Tem

pera

ture

(de

gree

s C

elsi

us)

−4 −2 0 2

−30

−25

−20

−15

−10

−5

Arosa


Issues with smoothing extremes

Care needed with choice of model: with GPD, change of threshold u 7→ u′ changes scaleparameter:

τu 7→ τu′ = τu + ξ(u′ − u),

so, for example, the formulation

(τu, ξ) = (expg(t), h(x))

at threshold u will become

(τu′ , ξ) =(expg(t) + h(x)(u− u′), h(x)

),

at threshold u′, so interpretation depends on threshold—should avoid.

Better to fit on GEV scale, or using point process formulation, for which parametrization isinvariant.


47

Multivariate Extremes slide 111

Multivariate extremes

Lack of data means the precision of extreme value estimates is often poor.The only way to overcome this is to incorporate additional information, suggesting the use ofmultivariate models.Questions include:

What issues are important when contemplating multivariate extremes?

What are appropriate ways to summarize dependence in extremes?

What models are suggested by asymptotic theory?

How should inference be carried out?


Componentwise maxima

If (X1, Y1), (X2, Y2)iid∼ F (x, y), define

Mx,n = maxi=1,...,n

Xi and My,n = maxi=1,...,n

Yi,

thenMn = (Mx,n,My,n)

is the vector of componentwise maxima.

The asymptotic theory of multivariate extremes begins with an analysis of Mn as n→ ∞.

The issue is partly resolved by recognizing that Xi and Yi considered separately are sequencesof independent, univariate random variables, to which the earlier theory may be applied.


Marginal standardization

Representations are especially simple if we assume that both Xi and Yi have the standard Frechetdistribution, with distribution function

F (z) = exp(−1/z), z > 0.

Then defining

M∗n =

(max

i=1,...,nXi/n, max

i=1,...,nYi/n

)

the marginal distributions of M ∗n are standard Frechet for all n. Remaining questions concern

dependence of limit distribution only.


48

Limit distribution of componentwise maxima

Let M ∗

n = (M∗

x,n, M∗

y,n) be componentwise maxima of independent vectors with standard Frechet marginaldistributions. Then if

PrM∗

x,n ≤ x,M∗

y,n ≤ yd→ G(x, y),

where G is a non-degenerate distribution function, G has the form

G(x, y) = exp−V (x, y), x > 0, y > 0

where

V (x, y) = 2

Z 1

0

max“

wx

, 1−wy

”

dH(w),

and H is a distribution function on [0, 1] satisfying the mean constraint

Z 1

0

wdH(w) = 1/2.

If H is differentiable with density h, then

V (x, y) = 2

Z 1

0

max“

wx

, 1−wy

”

h(w)dw.


Special cases

Independence: When H is a measure with masses 0.5 on w = 0 and w = 1

G(x, y) = exp−(x−1 + y−1), x > 0, y > 0.

Perfect dependence: When H is a measure that places unit mass on w = 0.5

G(x, y) = exp−max(x−1, y−1), x > 0, y > 0,

which is the distribution function of variables that are marginally standard Frechet, but which areperfectly dependent: X = Y with probability 1.


Parametric models

For modelling purposes, it is usual to specify a parametric family for H or h that encompasses awide range of dependence types over the parametric domain.

Standard example is the logistic model, with

h(w) = 12(α−1 − 1)w(1 − w)−1−1/αw−1/α + (1 − w)−1/αα−2

for 0 < α < 1.

In this caseG(x, y) = exp

−(x−1/α + y−1/α

)α, x > 0, y > 0.

Independence and perfect dependence arise as limits as α ↑ 1 and α ↓ 0 respectively.


49

Alternative models

A limitation of the logistic model is its symmetry. Asymmetric alternatives include the

bilogistic model

h(w) = 12(1 − α)(1 − w)−1w−2(1 − u)u1−αα(1 − u) + βu−1

on 0 < w < 1, where 0 < α < 1 and 0 < β < 1, and u = u(w,α, β) is the solution of

(1 − α)(1 − w)(1 − u)β − (1 − β)wuα = 0;

and the Dirichlet model

h(w) =αβΓ(α+ β + 1)(αw)α−1β(1 − w)β−1

2Γ(α)Γ(β)αw + β(1 − w)α+β+1, 0 < w < 1,

for parameters α > 0 and β > 0.


Inference for multivariate extreme value models

Inference consists of the following steps:

1. estimation of marginal distributions and transformation to standard Frechet;

2. choice of the dependence model H;

3. estimation of the parameters of H by maximum likelihood;

4. model assessment.

There is a potential gain in efficiency by estimating marginal and dependence parameters in a singlelikelihood maximization.


Structure variables

Though processes may be multivariate, it may be that a univariate function—a so-calledstructure variable—is the quantity of interest.

This suggests two possible methods of analysis:

1. univariate extreme value analysis of the sturcture variable process; or

2. multivariate analysis of the full process, followed by marginalization to the structure variableof interest.


50

Inference for structure variables

Univariate inference is trivial.

For multivariate inference, if Z = φ(Mx,My) is the structure variable of interest, andM = (Mx,My) has density g, then

PrZ ≤ z =

∫

Az

g(x, y)dxdy,

where Az = (x, y) : φ(x, y) ≤ z. Sometimes integration can be avoided. For example, if Z = maxMx,My,

PrZ ≤ z = PrMx ≤ z,My ≤ z = G(z, z),

where G is the joint distribution function of M .


Example: Sea levels

Freemantle Annual Maximum Sea-level (m)

Por

t Piri

e A

nnua

l Max

imum

Sea

-leve

l (m

)

1.3 1.4 1.5 1.6 1.7 1.8 1.9

3.6

3.8

4.0

4.2

4.4

4.6

Annual maximum sea levels at Port Pirie and Fremantle. Logistic α = 0.922 (0.087), implyingnear-independence.


51

Analysis of structure variable

Return Period (Years)

Ret

urn

Leve

l (m

)

1 5 10 50 100 500 1000

1.0

1.5

2.0

2.5

3.0

Return levels of structure variable Z = maxMx, (My − 2.5) using both univariate and multivariatemethods.


Impact of dependence on structure variables I


Ret

urn

Leve

l (m

)

1 5 10 50 100 500 1000

1.6

1.8

2.0

2.2

2.4

Z = maxMx, (My − 2.5) in logistic model analysis of Fremantle and Port Pirie annual maximumsea-level series with α = 0, 0.25, 0.5, 0.75, 1 (α = 0 is lowest curve).


52

Impact of dependence on structure variables II


Ret

urn

Leve

l (m

)1 5 10 50 100 500 1000

1.2

1.4

1.6

1.8

2.0

Z = minMx, (My − 2.5) in logistic model analysis of Fremantle and Port Pirie annual maximumsea-level series with α = 0, 0.25, 0.5, 0.75, 1 (α = 0 is highest curve).


Point process representation

As in the univariate case, there are threshold exceedance and point process characterizations ofextremes that enable a greater use of available information.

The point process representation, in particular, also provides some insight into the measurefunction H that appears in the componentwise limit law.


Point process construction

(X1, Y1), (X2, Y2) . . . a sequence of independent variables with standard Frechet margins.

Assume componentwise maxima convergence

PrM∗x,n ≤ x,M∗

y,n ≤ y → G(x, y).

Define sequence of point processes Nn by

Nn = (n−1X1, n−1Y1), . . . , (n

−1Xn, n−1Yn).

Normalization by n of standard Frechet variables is required to obtain marginal convergence.


53

Point process limit

On regions bounded from the origin (0, 0), we have

Nnd→ N,

where N is a non-homogeneous Poisson process on (0,∞) × (0,∞).

Moreover, in terms of pseudo-polar coordinates

r = x+ y and w = xx+y ,

the intensity function of the limiting process N is

λ(r, w) = 2 dH(w)r2 .

The function H is related to G by

G(x, y) = exp−2

∫ 1

0

max(wx ,

1−wy

)dH(w)


Comments

The intensity function of the limit process factorizes across r and w. Loosely, the relativemagnitude of extreme events in the two variables is independent of the magnitude itself.

The limit provides an interpretation of H as the distribution of relative magnitudes of extremeevents between the two variables.

In practice—as usual—the limit process is taken to be exact for extreme enough events: in thiscase for events which are sufficiently far from (0, 0).


Consistency of results


y,n ≤ y = PrNn(A) = 0,where

A = (0,∞) × (0,∞)\(0, x) × (0, y).So,


y,n ≤ y → PrN(A) = 0 = exp−Λ(A),where

Λ(A) =

∫

A2dr

r2dH(w)

=

∫ 1

w=0

∫ ∞

r=minx/w,y/(1−w)2dr

r2dH(w)

= 2

∫ 1

w=0max

(wx ,

1−wy

)dH(w).


54

Likelihood for Poisson process model

Choosing A = (x, y) : x/n+ y/n > r0, for large r0,

Λ(A) = 2

∫

A

dr

r2dH(w) = 2

∫ ∞

r=r0

dr

r2

∫ 1

w=0dH(w) = 2/r0,

which is constant with respect to the parameters of H.Hence, assuming H has density h,

L(θ; (x1, y1), . . . , (xn, yn)) = exp−Λ(A)NA∏

i=1

λ(x(i)/n, y(i)/n)

∝NA∏

i=1

h(wi),

where wi = x(i)/(x(i) + y(i)) for the NA points (x(i), y(i)) falling in A.


Comments

Choice of threshold r0 is based, as usual, on empirical measures of model stability.

Alternative forms of threshold curve lead to more complicated likelihood expressions.

Simultaneous estimation of marginal and dependence features is possibly advantageous, but againresults in a considerably more complicated likelihood.


Example: Exchange rate data

Year

Log-

daily

ret

urn

1997 1998 1999 2000 2001

-0.0

20.

00.

02

Year

Log-

daily

ret

urn

1997 1998 1999 2000 2001

-0.0

20.

00.

02

Log-daily returns of exchange rates. Top panel: UK sterling/US dollar exchange rate. Bottom panel:UK sterling/Canadian dollar exchange rate.


55


Sterling v. US Dollars

Ste

rling

v. C

anad

ian

Dol

lars

-0.02 -0.01 0.0 0.01 0.02

-0.0

2-0

.01

0.0

0.01

0.02

Concurrent values of exchange rates.



Standardized US/UK Daily Returns

Sta

ndar

dize

d C

anad

a/U

K D

aily

Ret

urns

1 10 100 1000

110

100

1000

Concurrent values of exchange rates after transformation to Frechet scale.


56


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

w

Rel

ativ

e F

requ

ency

Fitted logistic model density h in point process analysis of exchange rate data. (α = 0.434 (0.025))


Example: Oceanographic data

Wave Height (m)

Sur

ge (

m)

0 2 4 6 8 10

-0.2

0.0

0.2

0.4

0.6

0.8

Simultaneous values of wave and surge height.


57


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

w

Rel

ativ

e F

requ

ency

Various fitted models h in point process analysis of wave-surge data.



Model ℓ α β

Logistic 227.2 0.659Bilogistic 230.2 0.704 0.603Dirichlet 238.2 0.852 0.502

Details of various fits to oceanographic data give support for asymmetric dependence structure.


Higher-dimensional models

Limit results and characterizations have analogous forms in higher dimensions.

Extremal dependence is chracterized by a dependence function H on the simplex

sd = (w1, . . . , wd) : wj ≥ 0, j = 1, . . . , d,

dX

j=1

wj = 1

subject toZ

Sd

wjdH(w) = 1, j = 1, . . . , d

In principle, inference techniques remain the same, but

1. model specification is more difficult;

2. likelihood calculation is more difficult;

3. reliability of asymptotic results is more questionable.


58

Asymptotic dependence

A limitation of multivariate extreme value models (and point process equivalents) is that apart fromthe degenerate case of independence, all such models are asymptotically dependent; the intuition isthat because these are asymptotic (limiting) models, the extent of dependence cannot vary with therareness of the event, whereas this happens in practice.

Two variables X and Y with the same distribution are said to be asymptotically independent if

limz→∞

PrY > z | X > z = 0,

otherwise they are asymptotically dependent.

Loosely, two variables are asymptotically independent if an extreme value in one has zeroprobability of happening on the occurrence of an extreme in the other variable.


Bivariate normal variables

The class of asymptotically independent variables is non-trivial, and includes, for example, allbivariate normal variables with positive correlation.

Using extreme value models for such distributions is misleading.

– The limit is independence, but this provides a poor approximation to dependence at finitelevels.

– A fitted model at any specific threshold will overestimate strength of dependence onexrapolation.

Much recent work has looked at inference for asymptotic independent models: Ledford and Tawn,Heffernan, Coles and Tawn, Heffernan and Tawn.


59

Spatial Extremes slide 143

Spatiotemporal extremes

Many environmental extremal problems are spatial or temporal in nature, or both:

heatwaves

rainfall

avalanches

forest fires

storms

sea levels

Likely to be more important in the future, ‘prediction’ (on different scales) needed for long-rangeplanning, short-range evacuation, . . .


Geostatistics slide 145

Geostatistics

Statistics of spatially-defined variables

Mostly a multivariate normal theory:

– remove trends in mean and dispersion in space and time

– transform residuals to standard normal margins

– fit ‘suitable’ spatial/space-time correlation functions

– inferences using WLS, (likelihood), or Bayes (McMC)

More generally, set up model for response variable Y (x) with x ∈ X :

Y (x) | S(x)ind∼ f(y; θ), S(x) ∼ Nµ(x),Ω(x),

and use Metropolis–Hastings algorithm for inference on S(x), predictions of future Y , etc.(Diggle, Tawn, Moyeed, 1998), lots of Bayesians


60

Example: Temperature data



Lugano (273 m)

Basel!Binningen (316 m)

Locarno!Monti (366 m)

Montreux!Clarens (405 m)

Oeschberg!Koppigen (483 m)

Neuchatel (485 m)

Bad Ragaz (496 m)

Zurich!MeteoSchweiz (556 m)

Bern!Liebefeld (565 m)

Chateau d'Oex (985 m)

Engelberg (1035 m)

Montana (1508 m)

Davos!Dorf (1590 m)

Arosa (1840 m)

Gd!St!Bernard (2472 m)

Santis (2490 m)

Jungfraujoch (3580 m)

2001 2002 2003 2004 2005

Temperature anomaly (degrees Celsius)

Maximum temperature: June, July, August, 2001!2005

! ! !!

!

! !!

!!

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!

!!

!!

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!!

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!!

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!!

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!!


61

Geostatistics of extremes

Basic setup:

want to model extremes of process Y (x, t), (x, t) ∈ X × T ⊂ R3

time series (maybe intermittent)—could be annual maxima, or daily values, or . . .

data available at

– sites xd ∈ XD = x1, . . . , xD ⊂ X– times Td = td,1, . . . , td,nd

, for d ∈ 1, . . . , D exposition simplified if Td ≡ T ′ = t1, . . . , tnAim to compute distributions of quantities such as

R(t) =

∫

X

r(x, t)I Y (x, t) ≥ ydanger dx

where r(x, t) is population at risk if Y (x, t) exceeds some level ydanger at time t


Approaches

Four (?) main approaches:

latent variable (often Bayesian) approach

copulas

Heffernan–Tawn

multivariate extremes (max-stable processes, Smith 1990)


Latent variable approach

Conditional on underlying process S(x), observations Y (x), for x ∈ X follow an extremal distribution

Examples:

Y (x) | S(x) = (η(x), τ (x), ξ(x))ind∼ GEVy; S(x), S(x) ∼ N3µ(x), Ω(x)

Y (x) | S(x) = (σ(x), ξ(x))ind∼ GPDy; S(x), S(x) ∼ N2µ(x), Ω(x)

Examples: Casson and Coles (1999, Extremes); Cooley et al. (2007, JASA)

Could use copulas to transform margins to Gaussian, then fit geostatistical models—Gaussiananamorphesis

Advantages: computationally feasible for large-scale problems using standard simulation techniques(Metropolis–Hastings algorithm, . . .), possibility of estimating probabilities for complex events

Disadvantages: all extremal dependencies are incorporated through S(x); marginal distributions are notextremal; difficult to incorporate full range of possible extremal dependencies


62

Heffernan–Tawn (2004)

Use decomposition

Pr(Y ∈ C) =

D∑

d=1

EYdPr(Y−d ∈ Cd | Yd)f(Yd) ,

where C =⋃d Cd and Cd ∩ Cc = ∅, for c 6= d, with

Cd = C ∩ y ∈ RD : FYd

(yd) > FYc(yc), c = 1, . . . , D, c 6= d

Use GPD for Yd and nonparametric linear model for conditional probability, and simulate to estimatePr(Y ∈ C).

Advantages: can deal with large D, extends to near-independence models

Disadvantages: models on different Cd need not be coherent, uniqueness of representation for conditionalprobability unclear, inference messy


Max-stable processes

General setup:

Z(x) ∼ Frechet(1), for x ∈ X , so if the Zm(·) are independent and identically distributed, then

m−1 maxZ1(x), . . . , Zm(x) ∼ Frechet(1)

Joint distribution at any subset D = x1, . . . , xd of sites has multivariate extreme-valuedistribution, and extremal coefficient

θD =

∫

SD

maxd∈D

wd dH(w1, . . . , wD),

where D = |D|, SD is the unit simplex in RD, and H is a measure on SD with all its marginal

expectations equal to 1; we have 1 ≤ θD ≤ D.


Extremal coefficient

Summary of dependence in subset of variables

If Z1, . . . , Zd all marginally unit Frechet, and joint distribution is max-stable, then

PrZ1 ≤ z, . . . , Zd ≤ z = exp(−θD/z), z > 0,

where extremal coefficient θD depends on subset.

Interpretation: 1 ≤ θD ≤ d, where

– θ = 1 corresponds to complete dependence of maxima in subset,

– θ = d corresponds to independence of maxima in subset

Not a complete summary of joint distribution, but basis of useful diagnostics, as can estimate θD quiteeasily.

Rainfall example (sketch!)


63

Madogram

Exploratory tool (analogous to variogram), designed for dealing with many extremes on a map

F -madogram (Poncet, Cooley, Naveau, 2006/7?)

12E[|FY (x+ h) − FY (x)|] =

1

2

θ(h) − 1

θ(h) + 1,

where θ(h) is pairwise extremal coefficient at range h

Extended by Naveau et al. (2006) to estimation of dependence function Vh in expression

Pr Y (x) ≤ y1, Y (x+ h) ≤ y2 = exp −Vh(y1, y2) ,

where

Vh(y1, y2) = 2

∫ 1

0

max

(w

y1,1 − w

y2

)dHh(w).


Spectral representations I

de Haan (1984):

A continuous (in probability) max-stable process Z(x) may be expressed as

Z(x)D= max

kUkf(x− Tk),

where (Uk, Tk) are the points of a Poisson process on R+ × [0, 1] with measure du/u2 × ρ(dt),where ρ is a finite measure on [0, 1] and fk are non-negative L1 functions.

Interpretation in terms of storms of sizes Uk and shape f centred at Tk

Exploited by Smith (1990), Coles (1993), Coles and Walshaw (1994), de Haan and Pereira (2006)with various choices of f (normal, t, Laplace, circular)

Can compute joint density of Z(x1), Z(x2), but no further


64

Spectral representations II

Schlather (2002):

Let V (x) be a stationary process on Rd with µ = E max0, V (x) <∞, and let Π be a Poisson process on

(0,∞) with intensity µ−1ds/s2. If the Vs(x) are independent copies of V (x), then

Z(x) = maxs∈Π

sVs(x),

is a stationary max-stable random process with unit Frechet margins.

Interpretation in terms of maxima of random sheets

Example: Vs(·) stationary istropic Gaussian processes with correlation ρ(h), then

− log PrZ(x1) ≤ z1, Z(x2) ≤ z2 = 12

„

1

z1+

1

z2

«

1 +

»

1 − 2ρ(h) + 1z1z2

(z1 + z2)2

–1/2!

Corresponding extremal coefficient θ(h) = 1 + 2−1/21 − ρ(h)1/2 has natural bounds (Schlather andTawn, 2003)

Can only represent positive dependence—but most likely in practice


Pairwise likelihood

Data Z1, . . . , Zn are from (unavailable) full joint density f(z1, . . . , zn; θ), yielding log likelihood

ℓ(θ) = log f(z1, . . . , zn; θ).

If lower-order marginal densities are available, construct composite likelihood by taking product of(non-independent!) densities.

Simplest example is pairwise log likelihood

ℓ2(θ) =X

i>j

log f(zi, zj ; θ),

constructed from all distinct disjoint pairs of observations.

If θ is identifiable from the pairwise marginal densities, and if Z1, . . . , Zn iid∼ f(z; θ), with Zj = (Zj

1 , . . . , ZjD), then

under mild regularity conditions the maximum pairwise likelihood estimator θ is consistent, and satisfies

θ·

∼ Nn

θ, J(θ)−1K(θ)J(θ)−1o

as n → ∞.


65



Swiss summer temperature data

For illustration:

annual maximum temperature data at D = 17 Swiss sites, 1961–2006

fit GEV to standardised values Y (xd, tj) with

ηdtj = βd0 + βd1tj , τd, ξd, d = 1, . . . , 17, j = 1, . . . , 46,

and obtain Zdtjiid∼ Frechet(1) using estimated probability integral transform

estimate surfaces for β0, β1, τ , ξ using splines

use likelihood estimation to obtain ρij for each pair of sites (model checking)

use pairwise likelihood to fit stationary isotropic covariance function

ρ(u) = (1 − γ1) + γ1 exp−(u/γ2)γ3, 0 ≤ γ1 ≤ 1, γ2, γ3 > 0,

with estimated values γ1 = 0.62 (0.2), γ2 = 360 (100)km, γ3 = 1.59 (1.29)

simulate max-stable random fields Z∗(x) from fitted model, then transform back to ‘real’ scale Y ∗(x) andassess . . .


66

Fit of marginal model

0.5 2.0 10.0 50.0

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2.0

10.0

50.0

Station Arosa

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52.

010

.050

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Station Bad Ragaz

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Station Bern−Liebefeld

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Station Satis

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Station Zrich−MeteoSchweiz


Estimated correlations

Individual points: MLEs of correlations ±2 SE (grey bars) as a function of distance (103km) between pairs ofpointsSolid line: fitted correlation estimated using pairwise likelihoodLeft: temperature data; right: simulated data

0.0 0.2 0.4 0.6 0.8 1.0

−1.

0−

0.5

0.0

0.5

1.0

Schlather model coefficients (red points) and their confidence intervals (gray error bars) and Weibull model curve vs. distance.Distance between two stations

Coe

ffici

ent e

stim

ated

from

Sch

lath

er m

odel

0.0 0.2 0.4 0.6 0.8 1.0

−1.

0−

0.5

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0.5

1.0

Distance between two stations

Coe

ffici

ent e

stim

ated

from

Sch

lath

er m

odel


67

Fit of correlation curve

Normal QQ plot of g(ρij) − g(ρij)/SE, for Fisher z transform g

−2 −1 0 1 2

−2

−1

01

23

Normal Q−Q Plot

Theoretical Quantiles

Sam

ple

Qua

ntile

s


68

Pairwisefits

0.1 0.5 5.0 50.0

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50.0

pair(1,2) altitudes=(1840,496)

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StatisticsofExtremesJanuary2008–slide163

Simulatedfields

Forillustration,followingpagesshowsimulatedfieldsy∗(x,2007)orderedaccordingtovaluesof

∫

Xy∗(x,2007)dx

Scaleisstandardizedrelativetorobustlocationandscaleateachsite


69

0

1

2

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4

5

6 7 8 9 10

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500th hottest year out of 1000 realisations for 2007.500th hottest year out of 1000 realisations for 2007.

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Discussion

Schlatherrepresentationisverygeneral,canenvisageforothertypesofdata:

–rainfalltimeseriesatasinglesite,usingadditionofarandomset(Mehdi)

–rainfallinspaceandtime?

–snowfallinspace(Juliette?)

–etc.

Modellingissues:

–Whatprocessesaresensible?Gaussianrandomfields?Levyprocesses?

–Howtoextendtonear-independencecases?

–Howtobuildincovariates?Smoothing?Pre-whiteningornot?

–Howbesttoperformestimationandtesting?

–Bayesianinference?

–Howtobuildinphysicalknowledgeofunderlyingprocesses?


70

statistical modelling of extreme values

Documents

multivariate extremes

extreme value models

gev limit law

spatial extremes

spatiotemporal extremes

moving maxima

componentwise maxima

wooster example