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
Jan 19, 2016
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