Aucun titre de diapositive

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Philippe Naveau Paul Poncet Dan Cooley
Spatial Extremes
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Outline

• Geostatistics
• (2) Extreme value theory the bivariate case
• (3) Extreme value theory the multivariate case
• (5) Conclusions

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Classical Questions in Geostatistics
How to take into account the distance between
points when modeling spatial uncertainty? How to
perform spatial interpolation?
(Chiles and Delfiner 1999, Stein 1999, Cressie
1993, Wackernagel 2003)
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A Classical Tool
Z(x) a zero-mean spatial stationary process with
finite covariance
The VARIOGRAM
?(h) 1/2 ? EZ(xh)-Z(x)2
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New Questions in Geostatistics
How to take into account the distance between
points when modeling spatial extreme events? How
to perform spatial interpolation for extreme
events?
(Coles et al., Smith et al., Schlather et al.,
etc)
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Outline

• Geostatistics
• (2) Extreme value theory the bivariate case
• (3) Extreme value theory the multivariate case
• (4) Conclusions

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MAXIMA The Generalized Extreme Value
distribution
P(X lt x) exp(-1/u(x))
with u(x) 1? (x-?)/?1/?
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Bivariate extremes
What is the 2D-GEV?
Apply the principle of stability!!!
GEV case Maximum of 2 GEV vectors is still
GEV MAX-STABLE BIVECTORS
Find F such that F(u,v) Fm(mu,mv)
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Bivariate case with unit Frechet
margins P(Xltu)P(Yltu)exp(-1/u)
P(Xltu, Yltv)exp(-V(u,v))
with V(u,v)2 ?max(w/u,(1-w)/v)dH(w) and H
distribution on 0,1 with mean 0.5
Special case vu P(Xltu, Yltu) exp(-V(u,u))
exp(-V(1,1)/u) exp(-1/u)V(1,1)
P(Xltu)?
(Embrecht et al.,1997)
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What is ?? gt the extremal coefficient
P(Xlt u, Ylt u)P?(Xlt u)
? 1 (X,Y) complete dependence ? 2 (X,Y)
independence ? does not completely characterize
the bivariate dependence
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Madogram ? 1/2 ? EX-Y
X-Y2max(X,Y)-(XY)
(X,Y) bivariate extreme vectors if EX finite
? 1/2 ? EX-Y EXF?-1(X) - EX with F(t)
P(Xltt)P(Yltt)GEV or GPD
(Greenwood, Hosking Wallis)
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Estimating the extremal coefficient
Recall u(x) 1? (x-?)/?1/?
Bivariate-GEV case
?u(? ?/?(1- ?))
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Estimating the extremal coefficient in the I.I.D.
bivariate case
Recall u(x) 1? (x-?)/?1/?
Bivariate-GEV case
?nu(? ?n/?(1- ?))
with ?n ?Xi-Yi/2n
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Estimated bias and variance of ?n with ?1.5
Starting samples bi-GEV(?.5, ?.5, ?)
?-0.2
?0.2
?0.8
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Outline

• Geostatistics
• (2) Extreme value theory the bivariate case
• (3) Extreme value theory the multivariate case
• (4) Conclusions

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Simple assumptions in Geostatistics
Z(x) a spatial stationary process
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Geostatistics extremes
Extremes Z(x) a spatial stationary and
max-stable process
Example Maximum annual precipitations
(Schlather and Tawn 2002,2003)
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A example a field of extreme values
What is the spatial structure?
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The first-order variogram
The Madogram
?(h) 1/2 ? EZ(xh)-Z(x)
The natural estimator of ?(h)
?Z(xi)-Z(xj)/2N with N nb of all pairs s.t.
xi - xj h
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The first-order variogram
The Madogram
?(h) 1/2 ? EZ(xh)-Z(x)
0
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Constructing max-stable processes
Apply the principle of stability!!!
GEV case Maximum of GEV processes is still
GEV MAX-STABLE PROCESSES
Find F such that F(u1,, un) Fm(mu1,,mun)
(Fougeres,2002)
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-2logP(Z(0)ltt,Z(h)lts)(1/t1/s)1(1-21?(h)st/(s
t)20.5
Frechet Marginal
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Extremal coefficient ?(h)11-1?(h)/20.5

• - 75 locations
• 300 simulations
• ?(h)exp(- h/40)

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Extremal coefficient ?(h)11-1?(h)/20.5

• - 75 locations
• 300 simulations
• ?(h)exp(- h/40)

?(h)
h
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Extremal coefficient ?(h)11-1?(h)/20.5

• - 75 locations
• 300 simulations
• ?(h)exp(- h/40)

?(h)
h
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-logP(Z(0)ltt,Z(h)lts)?(a/2log(t/s)/a)/s?(a/2log
(s/t)/a)/t
with a ht?-1h
Frechet Marginal
Smiths field
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Extremal coefficient ?(h)2?(ht?-1h/2)

• - 75 locations
• 300 simulations
• ?(h)exp(- h/40)

?(h)
h
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Outline

• Geostatistics
• (2) Extreme value theory the univariate case
• 2-i Lichenometry
• 2-ii Ice core volcanic signals
• (3) Extreme value theory the bivariate case
• (4) Extreme value theory the multivariate case
• (5) Conclusions

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Conclusions

• EVT There exist mathematically sound tools to
  deal with extremes and exceedances
• (2) Spatial extremes Madogram captures some
  dependence in max-stable fields

Work in progress and future research

• Spatial extremes Developing spatial
  interpolation schemes apply to downscaling of
  extremes

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A special class of max-stable process
Z(x)max?j g(sj,x) jgt0
with ?j magnitude of the j-th storm sj type
of storm g(.,.) shape of the event (?j, sj)
points of a Poisson process with intensity d?
?(ds)/?2
(Schlather and Tawn, 2003, de Haan 1984, Gine et
al., Resnick et al.)
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Multivariate extremes
Recall Bivariate case
-log P(Xltu, Yltv)2 ?max(w/u,(1-w)/v)dH(w)
-log P(Z(x)ltz(x),?x)?maxg(s,x)/z(x) ?x?(ds)
-log P(Z(0)ltz, Z(h)ltz) ?g(s,0)?g(s,h)?(ds) /z
(Schlather and Tawn, 2003)
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Classical assumptions in Geostatistics
Z(x) a spatial stationary process with finite
covariances
Example Mean annual temperatures
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Unit Frechet Margins
P(Xltu, Yltv)exp(-V(u,v))
with V(u,v)2 ?max(w/u,(1-w)/v)dH(w) and H
distribution on 0,1 with mean 0.5
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Bivariate extremes
P(Xltx, Ylty)exp(-V(u(x),u(y)))
with V(u,v)2 ?max(w/u,(1-w)/v)dH(w) and H
distribution on 0,1 with mean 0.5
and u(x) 1? (x-?)/?1/?
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Limitations of the variogram

• Not adapted to heavy-tail distributions
• (2) Spatial dependence may not well be
• represented by the covariance for
• extreme fields

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Notations
The Madogram in the spatial case
?(h) 1/2 ? EZ(xh)-Z(x)
The Madogram in the bivariate case
? 1/2 ? EX-Y
with XY in distribution
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What is the relationship between madogram
extremes?
X-Y 2 max(X,Y) - (XY)
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Constructing valid madograms
?(h) 1/2 ? EZ(xh)-Z(x)2

• Variogram parametrization
• Exponential, Matern family, etc
• Semi-definitive positive, etc

Whats about the madogram?
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Extremal coefficient properties
?(h)EZ(xh)-Z(x)/2
If ?(h) is a valid extremal function, then 2-
?(h) is semi-definite positive ?(h) is not
differentiable at 0 If ?(h) such that 1-2(?(h)
-1)2 semi-definite positive Then ?(h) is a valid
extremal function
(Matheron 1987, Schlather and Tawn 2003)
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Ordinary kriging
A linear predictor for Z(x0) p?(Z(x0)) ? ?i
Z(xi)
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Finding the weights ?i
Minimizing EZ(x0)- p?(Z(x0))2 under the
constraint ? ?i 1
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Finding the weights ?i
Minimizing ?Z(x0),??iZ(xi) under the
constraint ? ?i 1
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Ordinary kriging for extremes
p?(Z(x0)) ??i Z(xi)
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Finding the weights ?i
Minimizing ?Z(x0), p?(Z(x0)) under the
constraint ? ??ig(s,xi)?i1,..,n ?(ds) 1
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Finding the weights ?i
Minimizing ?Z(x0), p?(Z(x0)) under the
constraint ? ??ig(s,xi)?i1,..,n ?(ds) 1
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Finding the weights ?i
Minimizing ?Z(x0), p?(Z(x0)) under the
constraint ? ??ig(s,xi)?i1,..,n ?(ds) 1
?Z(x0),??iZ(xi) ? max?(xs-xt) r,s 1,,n
Schlathers model ?(xs-xt)11-?(xs-xt)
(?s?t)2/?s?t0.5
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SUMMARY
GEV, GPD
Gaussian
1st order Variogram
2nd order Variogram
Correlation coefficient
Extremal coefficient
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Extremal coefficient ?(h)11-1?(h)/20.5
?(h)

• - 75 locations
• 300 simulations
• ?(h)exp(- h2/40)

h
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A Link between Ages and Diameters
GLOBAL MODEL!!
GEV Parameters Scale constant Shape
constant Location time dependent
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Estimated bias and variance of ?n
Starting samples bi-GEV(?.5, ?.5, ?)
?
n
?
GEV
Gumbel
Weibull
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Madogram Bivariate extremes
Bivariate-GEV case
If ??0, then ? ? ?(1- ?) (?? -1)/?
If ?0, then ? ? ln?
Bivariate-GP case
If ??0, then ? ?/? ?(1 ?) ?(1- ?) /?(1- ?
?)-1/(1- ?)
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Estimating the extremal coefficient in the I.I.D.
bivariate case
Our Approach
Step 1 Estimate (?,?,?) Step 2 Transform
variables to have GEV margins with ?lt0.5 Step
3 Estimate ? with ?