A Survey of Statistical Methods for Climate Extremes
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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
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Overview
Climate extremes– Aims and issues– PRUDENCE project
Extreme-value theory– Fundamental idea– Spatial modelling– Clustering
Concluding remarks
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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
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PRUDENCE
European climate
Control 1961–1990
Scenarios 2071–2100
10 high-resolution, limited domain regional GCMs
6 driving global GCMs
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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
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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
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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
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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
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Temperature – Single-site Model
y100
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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
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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
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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
)
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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)
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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
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Temperature – Stochastic Links0 1
late
nt
(y 1
00)
glob
al (
y 100
)
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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
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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
)
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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}
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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*
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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
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Zurich Temperature (June – July)
Extremal Index
Threshold Percentile
Pr(cluster size > 1)
Threshold Percentile
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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
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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
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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