A Survey of Statistical Methods for Climate Extremes

Chris Ferro

Climate Analysis Group

Department of Meteorology

University of Reading, UK

9th International Meeting on Statistical Climatology, Cape Town, 26 May 2004

Overview

Climate extremes– Aims and issues– PRUDENCE project

Extreme-value theory– Fundamental idea– Spatial modelling– Clustering

Concluding remarks

Aims and Issues

Description

– Statistical properties

Comparison

– Space, time, model, obs

Prediction

– Space, time, magnitude

Non-stationarity

– Space, time

Dependence

– Space, time

Data

– Size, inhomogeneity

PRUDENCE

European climate

Control 1961–1990

Scenarios 2071–2100

10 high-resolution, limited domain regional GCMs

6 driving global GCMs

Fundamental Idea

Data sparsity requires efficient methods

Extrapolation must be justified by theory

Probability theory identifies appropriate models

Example: X1 + … + Xn Normal

max{X1, …, Xn} GEV

Spatial Statistical Models

Single-site models

Conditioned independence: Y(s', t) Y(s, t) | (s)

– Deterministically linked parameters

– Stochastically linked parameters

Residual dependence: Y(s', t) Y(s, t) | (s)

– Multivariate extremes

– Max-stable processes

Generalised Extreme Value (GEV)

Block maximum Mn = max{X1, …, Xn} for iid Xi

Pr(Mn x) G(x) = exp[–{1 + (x – ) / }–1/ ] for large n

5

1n

20 100

Single-site Model

Annual maximum Y(s, t) at site s in year t

Assume Y(s, t) | (s) = ((s), (s), (s)) iid GEV((s)) for all t

m-year return level satisfies G(ym(s) ; (s)) = 1 – 1 / m

Daily max 2m air temperature (ºC) at 35 grid points over Switzerland from control run of HIRHAM in HadAM3H

Temperature – Single-site Model

y100

Generalised Pareto (GP)

Points (i / n, Xi), 1 i n, for which Xi exceeds a high threshold approximately follow a Poisson process

Pr(Xi – u > x | Xi > u) (1 + x / u)–1/ for large u

Deterministic Links

Assume Y(s, t) | (s) = ((s), (s), (s)) iid GEV((s)) for all t

Global model (s) = h(x(s) ; 0) for all s

e.g. (s) = 0 + 1 ALT(s)

Local model (s) = h(x(s) ; 0) for all s N(s0)

Spline model (s) = h(x(s) ; 0) + (s) for all s

Temperature – Global Model

(s) = 0 + 1ALT(s)

0 = 31.8ºC (0.2)

1 = –6.1ºC/km (0.1)

p = 0.03

sin

gle

site

(y 1

00)

altitude (km)

glob

al (

y 100

)

Stochastic Links

Model l((s)) = h(x(s) ; 0) + Z(s ; 1), random process Z

Continuous Gaussian process, i.e.

{Z(sj) : j = 1, …, J } ~ N(0, (1)), jk(1) = cov{Z(sj), Z(sk)}

Discrete Markov random field, e.g.

Z(s) | {Z(s') : s' s} ~ N((s) + (s, s'){Z(s') – (s)}, 2)s'N(s)

Stochastic Links – Example

Model (s) = 0 + 1 ALT(s) + Z(s | a , b , c)

log (s) = log 0 + Z(s | a , b , c)

(s) = 0 + Z(s | a , b , c)

cov{Z*(sj), Z*

(sk)} = a*

2 exp[–{b* d(sj , sk)}c*]

Independent, diffuse priors on a*, b

*, c

*, 0, 1, 0 and 0

Metropolis-Hastings with random-walk updates

Temperature – Stochastic Links0 1

late

nt

(y 1

00)

glob

al (

y 100

)

Multivariate Extremes

Maxima Mnj = max{X1j, …, Xnj} for iid Xi = (Xi1, …, XiJ)

Pr(Mnj xj for j = 1, …, J ) MEV for large n

e.g. logistic Pr(Mn1 x1, Mn2 x2) = exp{–(z1–1/ + z2

–1/)}

Model {Y(s, t) : s N(s0)} | {, (s) : s N(s0)} ~ MEV

Temperature – Multivariate Extremes

Assume Y(s, t) Y(s', t) |

Y(s0, t) for all s, s' N(s0)

and locally constant

sin

gle

site

(y 1

00)

mu

ltiv

ar (

y 100

)

Max-stable Processes

Maxima Mn(s) = max{X1(s), …, Xn(s)} for iid {X(s) : s S}

Pr{Mn(s) x(s) for s S} max-stable for large n

Model Y*(s, t) = max{ri k(s, si) : i 1} where {(ri , si) : i 1} is a Poisson process on (0, ) S

e.g. k(s, si) exp{ – (s – si)' (1)–1 (s – si) / 2}

Precipitation – Max-stable Process

Estimate Pr{Y(sj , t) y(sj) for j = 1, …, J }

Max-stable model 0.16

Spatial independence 0.54

Rea

lisa

tion

of

Y*

Clustering

Extremes can cluster in stationary sequences X1, …, Xn

Points i / n, 1 i n, for which Xi exceeds a high threshold approximately follow a compound Poisson process

Zurich Temperature (June – July)

Extremal Index

Threshold Percentile

Pr(cluster size > 1)

Threshold Percentile

Review

Linkage efficiency, continuous space,description, interpretation,

bias, expense comparison

Multivariate discrete space, model choice,description dimension

limitation

Max-stable continuous space, estimation,prediction model choice

Future Directions

Wider application of EV theory in climate science

– combine with physical understanding

– shortcomings of models, new applications

Improved methods for non-identically distributed data

– especially threshold methods with dependent data

Further Information

Climate Analysis Group www.met.rdg.ac.uk/cag/extremes

NCAR www.esig.ucar.edu/extremevalues/extreme.html

Alec Stephenson’s R software http://cran.r-project.org

PRUDENCE http://prudence.dmi.dk

ECA&D project www.knmi.nl/samenw/eca

My personal web-site www.met.rdg.ac.uk/~sws02caf