24
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

A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

Embed Size (px)

Citation preview

Page 1: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 2: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

Overview

Climate extremes– Aims and issues– PRUDENCE project

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

Concluding remarks

Page 3: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 4: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

PRUDENCE

European climate

Control 1961–1990

Scenarios 2071–2100

10 high-resolution, limited domain regional GCMs

6 driving global GCMs

Page 5: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 6: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 7: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 8: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 9: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

Temperature – Single-site Model

y100

Page 10: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 11: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 12: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

)

Page 13: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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)

Page 14: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 15: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

Temperature – Stochastic Links0 1

late

nt

(y 1

00)

glob

al (

y 100

)

Page 16: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 17: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

)

Page 18: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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}

Page 19: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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*

Page 20: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 21: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

Zurich Temperature (June – July)

Extremal Index

Threshold Percentile

Pr(cluster size > 1)

Threshold Percentile

Page 22: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 23: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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

Page 24: A Survey of Statistical Methods for Climate Extremes Chris Ferro Climate Analysis Group Department of Meteorology University of Reading, UK 9th International

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