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Extreme Value Theory (or how to go beyond the data range) - Cirm

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Motivation
Basics
Applic
Max-stable
Mado
Spectral
Extreme Value Theory
(or how to go beyond the data range)
Philippe Naveau
Laboratoire des Sciences du Climat et l’Environnement (LSCE)
Gif-sur-Yvette, France
17 novembre 2010
Conclusion
Motivation
Basics
Applic
Max-stable
Nous avons anticipeВґ dans la
mesure du possible mais on ne
Вґ
Вґ
peut pas prevoir
l’imprevisible”
Xynthia’s storm, 25th of Feb,
2010
“Il est impossible que l’improbable
n’arrive jamais”
Emil Julius Gumbel (1891-1966)
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Extreme quotes
1
“Man can believe the impossible, but man can never believe the
improbable”
Oscar Wilde (Intentions, 1891)
Extreme events ? ... a probabilistic
concept linked to the tail behavior :
low frequency of occurrence, large
uncertainty and sometimes strong
amplitude.
Region of interest
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Important issues in Extreme Value Theory
Applied statistics
An asymptotic probabilistic
concept
A statistical approach for
extrapolation of quantiles
A general framework with “weak”
assumptions (ie no model for the
full data set)
Assessing uncertainties
Non-stationarity
Multivariate
Univariate
Non-parametric
Parametric
Independence
Theoritical probability
Motivation
Basics
Applic
Max-stable
An active statistical and probabilistic field
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Historical perspective
Gumbel (1891-1966)
Weibull (1887-1979)
Вґ
Frechet
(1878-1973)
Emil Gumbel was born and trained as a statistician in Germany, forced to move to
France and then the U.S. because of his pacifist and socialist views. He was a
pioneer in the application of extreme value theory, particularly to climate and
hydrology.
Waloddi Weibull was a Swedish engineer famous for his pioneering work on
reliability, providing a statistical treatment of fatigue, strength, and lifetime.
Maurice Frechet was a French mathematician who made major contributions to
pure mathematics as well as probability and statistics. He also collected empirical
examples of heavy-tailed distributions.
Other important names : Fisher and Tippet (1928), Gnedenko (1943), etc
Motivation
Basics
Applic
Max-stable
Return levels and return periods
A return level with a return period of
T = 1/p years is a high threshold zp
whose probability of exceedance is p.
E.g., p = 0.01 в‡’ T = 100 years.
A Return level interpretation
Number of events : Average
number of events occurring within
a T -year time period is one
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Max-stability
Let Mn = max(X1 , . . . , Xn ) with Xi iid with distribution F .
Definition : F max-stable if
„
В«
M n в€’ bn
P
< x = F n (an x + bn ) = F (x)
an
Blackboard
`
Unit-Frechet
F (x) = exp(в€’1/x) for x > 0. Then an = n & bn = 0
Gumbel F (x) = exp(в€’ exp(в€’x)) for all real x. Then an = 1 & bn = log n
Weibull F (x) = exp(в€’(в€’x)О± ) for x < 0 (1 otherwise). Then an = nв€’1/О± ,
bn = 0
Conclusion
Motivation
Basics
Gumbel
Maxima Distribution
Applic
(1891-1966)
Max-stable
Mado
Spectral
Weibull (1887-1979)
Conclusion
FrВґ
echet (1878-1973)
Distribution du maximum
Normal density в‡’
в‡ђ Gumbel density
Uniform density в‡’
в‡ђ Weibull density
Cauchy density в‡’
в‡ђ FrВґ
echet density
n = 50
⇓
↓
↑
⇑
Extr^
emes? Mesurer
n = 100
Interpoler
RВґ
egionaliser
6
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Generalized Extreme Value (GEV) distribution
„
P
M n в€’ an
<x
bn
В«
пљѕ h
“ x − µ ”i−1/ξ ff
в€ј GEV(x) = exp в€’ 1 + Оѕ
Пѓ
+
0.2
0.0
0.1
density
0.3
0.4
--2
-1
1
2
3
4
m
0.0
0.1
0.2
0.3
0
x0.4
d
M
density
2ensity
1
.0
.1
.2
.3
.4
-2
-1
0
1
2
x
Home work : show that a GEV is max-stable
3
4
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
GEV and return levels
пљѕ h
“ x − µ ”i−1/ξ ff
GEV(x) = exp в€’ 1 + Оѕ
Пѓ
+
Computing the return level zp such that GEV(zp ) = 1 в€’ p
zp = GEVв€’1 (1 в€’ p)
`
Вґ
Hence, zp = Вµ + ПѓОѕ [в€’ ln(1 в€’ p)]в€’Оѕ в€’ 1]
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
GEV and return levels estimation
zp = Вµ +
”
σ“
[в€’ ln(1 в€’ p)]в€’Оѕ в€’ 1]
Оѕ
Estimating the return level zp
zˆp = µ
Л†+
Пѓ
Л†
ОѕЛ†
“
”
Л†
[в€’ ln(1 в€’ p)]в€’Оѕ в€’ 1]
Estimating the GEV parameters estimates
Maximum likelihood estimation
Methods of moments type (PWM and GPWM, Ribereau et al., 2010)
Exhaustive tail-index approaches
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
GEV and return levels estimation
zˆp = µ
Л†+
”
Л†
Пѓ
ˆ“
[в€’ ln(1 в€’ p)]в€’Оѕ в€’ 1]
ОѕЛ†
Л†t
Maximum likelihood estimates of (Л†
Вµ, Пѓ
Л† , Оѕ)
Asymptotically distributed as a multivariate Gaussian vector with mean
Л† t and covariance matrix that is the inverse of the expected
Оё = (Л†
Вµ, Пѓ
Л† , Оѕ)
information matrix whose elements are equal
„
В«
∂ 2 log l(θ)
E в€’
∂θi ∂θj
where l(Оё) is the likelihood function of the GEV distributed sample
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
2600
Daily maxima of CH4 at Gif-sur-Yvette (Toulemonde et al., 2009, Environmetrics)
!
!
!
!
!
!
!
2000
2200
!
!
1800
Daily maxima of CH4 (ppb)
2400
!
!
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2002
2003
2004
2005
t
2006
2007
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Maxima of CH4 at Gif-sur-Yvette (Toulemonde et al., 2009, Environmetrics)
QQplot of the weekly maxima of CH4 with a Gumbel
0.0035
Histogram of weekly maxima of CH4
0.0030
2600
!
!
2400
!
!
2000
0.0010
0.0015
2200
0.0020
!
1800
0.0005
0.0000
Density
0.0025
!
1800
2000
2200
2400
2600
!
!
!
!
!
!
!
!
! !
!
!!
!
!!
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1800
2000
2200
2400
2600
Motivation
Annual Maxima
Basics
Applic
Peak over Threshold (POT)
Max-stable
Time series в‡’ 1 obs/yea
Mado
Spectral
Conclusion
POT
Time series в‡’ О» obs/yea
Markovian
Time series в‡’ all excee
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Thresholding : the Generalized Pareto Distribution (GPD)
P{Rв€’u > y |R > u} =
„
В«в€’1/Оѕ
Оѕy
1+
Пѓu +
Vilfredo Pareto : 1848-1923
Born in France and trained as an
engineer in Italy, he turned to the
social sciences and ended his
career in Switzerland. He
formulated the power-law
distribution (or ”Pareto’s Law”), as
a model for how income or wealth
is distributed across society.
Motivation
Basics
Applic
Max-stable
Mado
Spectral
1.0
1.5
GPD : “From Bounded to Heavy tails”
7
0.0
0.5
!=-0.5
1
2
3
4
5
6
Index
20
0
!=0.0
5
10
15
Index
0
!=0.5
0
50
100
150
Index
200
250
300
Conclusion
Motivation
Basics
Applic
Max-stable
GPD
Two GPD examples
Exponential F (x) = 1 в€’ exp(в€’x) for x > 0.
Uniform F (x) = x for 0 < x < 1.
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
GPD
GPD return level zp
Пѓu
zp = u +
Оѕ
В»
p
P(R > u)
–−ξ
!
в€’1
Estimating the return level zp
Пѓ
Л†u
zˆp = u +
ОѕЛ†
В»
pГ—n
Nu
–−ξˆ
!
в€’1
GPD parameters estimation
Maximum likelihood estimation (Smith, 1985)
Methods of moments type (PWM and GPWM, Ribereau et al., 2010)
Exhaustive tail-index approaches
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
в€’1.0
в€’0.8
в€’0.6
correlation
в€’0.4
в€’0.2
0.0
GPD estimators
в€’0.5
0.0
0.5
Оі
1.0
1.5
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Generalized Pareto Distribution (GPD)
“
P{R в€’ u > y |R > u} = 1 +
Оѕy
Пѓu
”−1/ξ
Parameters
u = predetermined threshold
Пѓu = scale parameter to be estimated
Оѕ = shape parameter to be estimated
Advantages & Practical issues
Flexibility to describe three different types of tail behavior
More data are kept for the statistical inference
Problem of threshold selection
Stability property
If the exceedance (R в€’ u|R > u) follows a GPD(Пѓu , Оѕ) then the higher
exceedance (R в€’ v |R > v ) also follows GPD(Пѓu + (v в€’ u)Оѕ, Оѕ)
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Crete ice core Greenland (ecm)
0.8 0
0.4
Time
0.0
Posterior proba
200
400
600
800
1000
Data = crete
600
800
1000
1200
Time
1400
1600
1800
2000
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
GPD diagnostics & models selection for a Crete data
400
Probability plot
!
!!
!
!!
!
50
0.2
160
600
!!
200
0.0
100
!!
empirical
350
0.8
250
300
model
200
0.4
!
0.6
170
180
1.0
190
empirical
u
ОѕЛ† = 0.56 (0.37)
!
!
!
150
!
!!
!!
150
Mean Excess
!!
Quantile Plot
!
!
!!
!!!!
!
!
!
!
!
!
!
!
!
!
200
400
600
200
model
800
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Intro summary
Modeling maxima : GEV
Stability for the max operator and X0 , X1 , . . . Xn idd GEV
a max(X1 , . . . , Xn ) + b = X
Modeling excedances : GPD
If exceedances (R в€’ u|R > u) follows a GPD(Пѓu , Оѕ) then higher exceedances
(R в€’ v |R > v ) also follows GPD(Пѓu + (v в€’ u)Оѕ, Оѕ)
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
A few studies dealing with geophysical extremes
Casson and Coles (1999) a Bayesian hierarchical model for wind speeds
exceedances
Stephenson and Tawn (2005) Bayesian modeling of sea-level and
rainfall extremes
Chavez and Davison (2005) GAM for extreme temperatures (NAO)
Wang et al. (2004) Wave heights with covariates
Turkman et al. (2007), Spatial extremes of wildfire sizes
Ribatet et al. (2010), Spatial R package for extremes
Bel, Bacro, Lantujenoul (2010), Spatial extremes
Extreme snow , Blanchet et al., 2010
Special issue of the journal Extremes, 2010
Вґ
Pratique du calcul bayesien,
JJ. Boreux, E. Parent et J. Bernier, 2010
Biodiversity and extreme temperatures, Sang and Gelfand, 2009
Lichenometry, Jomelli et al., 2007
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Three applications
Spatial Interpolation of precipitation return levels in Colorado
(Hierarchical Bayesian models) :
Cooley, Nychka and Naveau (2007, JASA).
Measuring the spatial dependence among rainfall maxima in
Bourgonge (Max-stable processes) :
Naveau & et al., (2009, Biometrika)
Modeling multivariate dependence among pollutants (spectral EVT
measures) :
Cooley, Davis and Naveau (2009, JMVA)
Motivation
Basics
Applic
Max-stable
Mado
Spectral
41
Daily precipitation (April-October, 1948-2001, 56 stations)
40
Ft. Collins
в—Џ
Denver
в—Џ
39
Grand
Junction
в—Џ
Limon
в—Џ
Colo в—ЏSpgs
38
Pueblo
в—Џ
37
latitude
Breckenridge
в—Џ
в€’109
в€’108
в€’107
в€’106
в€’105
longitude
в€’104
в€’103
в€’102
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Precipitation in Colorado’s front range
Data
56 weather stations in Colorado (semi-arid and mountainous region)
Daily precipitation for the months April-October
Time span = 1948-2001
Not all stations have the same number of data points
Precision : 1971 from 1/10th of an inche to 1/100
D. Cooley, D. Nychka and P. Naveau, (2007). Bayesian
Spatial Modeling of Extreme Precipitation Return Levels.
Journal of The American Statistical Association.
Motivation
Basics
Applic
Pierre Simon Laplace (1749-1827)
Вґ
“L’analyse des probabilites
assigne la probabiliteВґ de ces
causes, et elle indique les
moyens d’accroitre de plus
Вґ
en plus cette probabilite.”
“Essai Philosophiques sur
Вґ (1774)
les probabilites”
Max-stable
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Pierre Simon Laplace (1749-1827)
“If an event can be produced by a number of n different causes, then the
probabilities of the causes given the event ... are equal to the probability of
the event given that cause, divided by the sum of all the probabilities of the
event given each of the causes.”
P(event|causei ) Г— P(causei )
P(causei |event) = Pn
j=1 P(event|causej ) Г— P(causej )
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Bayes’ formula = calculating conditional probability
[x|y] в€ќ [y|x] Г— [x]
1701( ?)- 1761 “An essay
towards solving a Problem in
the Doctrine of Chances”
(1764)
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Bayesian approach
[x|y] в€ќ [y|x] Г— [x]
Drawbacks
Advantages
Integration of expert
information via prior [x]
Integration of expert information via prior
[x]
densities are needed
Deals with the full
distribution
Complex algorithmic techniques (MCMC,
particle-filtering)
Non-Gaussian
Can be slow and not adapted for large
data sets
Non-linear
Adequacy with EVT
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Hierarchical Bayesian Model with three levels
P(process, parameters|data)
в€ќ
P(data|process, parameters)
Г—P(process|parameters)
Г—P(parameters)
Process level : the scale and shape GPD parameters (Оѕ(x), Пѓ(x)) are hidden
random fields
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Our main assumptions
Process layer : The scale and shape GPD parameters (Оѕ(x), Пѓ(x)) are
random fields and result from an unobservable latent spatial process
Conditional independence : precipitation are independent given the GPD
parameters
Our main variable change
Пѓ(x) = exp(П†(x))
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Our three levels
A) Data layer := P(data|process, parameters) =
„
PОё {R(xi ) в€’ u > y |R(xi ) > u} =
1+
Оѕi y
exp П†i
В«в€’1/Оѕi
B) Process layer := P(process|parameters) :
e.g. П†i = О±0 + О±1 Г— elevationi + MVN (0, ОІ 0 exp(в€’ОІ 1 ||x k в€’ x j ||))
пљѕ
and Оѕ i
=
Оѕ moutains , if x i в€€ mountains
Оѕ plains , if x i в€€ plains
C) Parameters layer (priors) := P(parameters) :
e.g. (Оѕ moutains , Оѕ plains ) Gaussian distribution with non-informative mean and
variance
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Notre mod`
ele Bayesien hiВґ
erarchiq
Bayesian hierarchical modeling
Priors вњІ
О±0 + О±1 elev
Пѓ
В Priors вњІ ОІ 0 exp(в€’ОІ 1||.||)
вњ›
Priors
Оѕ plains
вњ›
Priors
В В вњ вќ…
вќ…
вќ�
вњ’
В Оѕ moutains
вњІ
zx
вњ›
Оѕ
вњ»
P (R(x) > u)
вњ»
Priors
вќ…
в– вќ…
Motivation
Basics
Applic
Max-stable
Spectral
Journal of the American
Statistical Mado
Association, ????
0
selection Table 1. Several of the Different GPD Hierarchical Models Tested and
ally.Model
The exceedance
model. Each simulafirst 2,000 iterations
he remaining iterauce dependence. We
an (1996) to test for
w the suggested critall parameters of all
s otherwise noted in
bution for the return
From (3), zr (x) is a
), and the (indepenus, it is sufficient to
Our method allows
s, which in turn can
, consider the logceedance model. We
m which we need to
umed that the parae the mean and co-
Their Corresponding DIC Scores
Baseline model
DВЇ
Model 0: П† = П†
Оѕ=Оѕ
73,595.5
Models in latitude/longitude space
Model 1: П† = О±0 + П†
Оѕ=Оѕ
Model 2: П† = О±0 + О±1 (msp) + П†
Оѕ=Оѕ
Model 3: П† = О±0 + О±1 (elev) + П†
Оѕ=Оѕ
Model 4: П† = О±0 + О±1 (elev)+ О±2 (msp)+ П†
Оѕ=Оѕ
Models in climate space
Model 5: П† = О±0 + П†
Оѕ=Оѕ
Model 6: П† = О±0 + О±1 (elev) + П†
Оѕ=Оѕ
Model 7: П† = О±0 + П†
Оѕ = Оѕ mtn , Оѕ plains
Model 8: П† = О±0 + О±1 (elev) + П†
Оѕ = Оѕ mtn , Оѕ plains
Model 9: П† = О±0 + П†
Оѕ=Оѕ+ Оѕ
DВЇ
Conclusion
60
61
pD
DIC
2.0 73,597.2
pD
DIC
62
63
64
65
66
73,442.0 40.9 73,482.9
67
73,441.6 40.8 73,482.4
68
69
73,443.0 35.5 73,478.5
70
73,443.7 35.0 73,478.6
71
72
DВЇ
pD
DIC
73,437.1 30.4 73,467.5
73,438.8 28.3 73,467.1
73,437.5 28.8 73,466.3
73,436.7 30.3 73,467.0
73,433.9 54.6 73,488.5
NOTE: Models in the climate space had better scores than models in the longitude/latitude
space. В· в€ј MVN(0, ), where [Пѓ ]i, j = ОІВ·, 0 exp(в€’ОІВ·, 1 xi в€’ xj ).
73
74
75
76
77
78
79
80
81
82
83
84
Motivation
Basics
Applic
Max-stable
Mado
Return levels posterior mean
Ft. Collins
! Greeley
Ft. Collins
! Greeley
!
!
oulder
Boulder
40
8
!
8
!
Denver
Denver
!
!
Colo Spgs
!
39
7
latitude
7
Colo Spgs
!
6
6
Pueblo
!
!
38
Pueblo
5
37
5
!105.0
longitude
!106.0
!105.0
longitude
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Posterior quantiles of return levels (.025, .975)
Ft. Collins
Ft. Collins
Greeley
!
!
Greeley
!
10
!
9
!
Denver
Denver
!
!
!!
!
!
!
!
!
8
4.0
!
!
40
9
!
!
!
!
Boulder
40
40
Boulder
!
!
10
!
! !
!
!
!!
!
8
3.5
!
!
!
!
!
!
3.0
!
7
!
39
Colo Spgs
latitude
39
latitude
39
7
!
!
!
!
!
!!
!
!
!
!
!
!
!
!
!!
2.5
!
6
6
Pueblo
Pueblo
!
!
!
!
38
38
5
38
2.0
5
!
!
!
4
4
!
!
1.5
!
!106.0
!105.5
!105.0
longitude
!104.5
!106.0
37
37
!
37
latitude
!
Colo Spgs
!105.5
!105.0
longitude
!104.5
!106.0
!105.5
!105.0
longitude
!104.5
Motivation
Basics
Applic
Max-stable
Patrick Galois (Meteo France) “Les
Л†
Вґ
`
tempetes
sont des phenom
enes
que
l’on observe tous les cinq a` dix ans en
Вґ climatiques.
France en raison d’aleas
Вґ
`
Si elle presente
un caractere
remarquable, Xynthia n’est pas pour
Вґ
`
`
autant le phenom
ene
du siecle.
Elle
est ainsi moins exceptionnelle que
celles de 1999 et ses vents sont moins
intenses qu’en 2009. Mais son issue
Вґ
dramatique reside
dans sa
conjonction a` un fort coefficient de
Вґ sur la cote
Л† atlantique, au
maree
Л†
´ haute.”
moment meme
de la maree
Xynthia’s storm, 25th of Feb, 2010
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
An illustration : fitting multivariate maxima
1000
Air pollutants (Leeds, UK, winter 94-98, daily max) NO vs. PM10 (left), SO2 vs. PM10
(center), and SO2 vs. NO (right) (Heffernan& Tawn 2004, Boldi & Davison, 2007)
!
!
400
400
800
500
!
500
!
!
!
!
!
200
0
0
50
100
PM10
!
!
!
!
!
!!!
!
!
!
! ! ! !!
!
! !
!
! ! !!
!
!
!
!
!
! !!
! !
! !
! ! !
!
! !
!
! !! ! ! ! ! !
! !!
!
!
! ! !
!
!! !! !!!!
!!
! ! ! ! !!!! ! !
! !
! !!
!
!
!
!!
!!!!
!
!
!
! !! !
!!
!
! !!
!!!
!!
!
!
!!
!
!
! !!
!!! !!
! !!
!!
!!
!
!!
!!
!!!!!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!!
! !!! !
! !!
!
!
!
!!
!
! !! !
!
!
!!
!!
! !
!
!
!!!
!
!!!
!
!!
!
! !
!
!
!
!!!
!
!
! !
!
!!
!
!
!!
!!!!
! !
! !
!!
!
!
!
!
!
!!
!
!
!!
! ! !
!!
!!
!!
!
!!
!!!
!!!
!!
!
!
!!
!!
!!
!
!
!
!
!
!
!!
! !!
!!
!
!!!
!!
!!
!
!!
!!
!! !
!
!
!!!
!
!
!!
!
!
!
!!
!
!
!
!
!!!!
!
!
!
!
!
!
!!!
!
!!
!
!
!
!
!
!
!!
!!
!
!!
!
!
!
!!
!!
!
!!
!
!
!!
!
!
!
!
!
!
!!
!
!!
!!
!
!! !
!!
!!
!! !
!!!
!!
!
!
!
!!
!
!!
!!
!
!
!!
!
!
!!
!
!
! !!
!!
SO2
!!
!
! !
!
!!
!
!
!
!
!
!
!
!
!
!
150
200
!
!
! !!
!! ! !
!
!
0
!
!
50
100
PM10
!!
!!
!
!
!
!
!
!!
!
!
!
!
!
!! !
!
!!
!
!
!
!
! !
!
!! !
!
!
!
! !
!
!
! !
!
!! ! ! ! ! ! !!
!
! ! !! !
!
!
!
! !
!
!! ! ! !
! ! ! ! !!
! !!! ! !
!
!
! !!
! !!
!! !!
!
!
!
! ! !
!
!!
! !!! !! !
!
! !!! !!
!
!
!!! !!
!!
!!
!! ! ! !
!
!!
!
!
! !
!!!
!!
!!
!
!!
!!!
!!
!!!!
!
!
!
!!! ! ! !
!! ! !
!!
!!!!
!!!
!
!
!
! !
!
!! !
! !
!!
!
!!
!
!
!
!!!!
!
!
!
!!
!!
!
!!
!
!
!
!
!
!
!
!
!
!
!!
!
!
!
!
!! !!
!!!
!!
!!
!
!
!
!
!
!
!
!!
!!
!
!!
!
!!
!
!!
!
!
! !!
!
!
!
!
!
!
!
!
!
!
!!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!!
!!
!
!
!
!!
!!!
!
!
!!
!
!
!!
!!
!! ! ! !
!!
!
!!
!
!
!
!
!
!
!
!
!
! !
!!!
!!
!!
!
!!
!
!
!
!!!
!
!
!
!
!
!
!
!
!
!
!
!
!!! !
!
!
!
!
!
!
!!
!!
!
!
!
!
!
!
!
!!
!
! !
!
!
!
!!
!
! !!
!
!!
!
!
!! !!
!!
!!
!
!
300
300
!
!
!
!
!
SO2
!
!
200
!
!
!
!
!
!
!!
!
!
!! !
!
!
!
!
150
!
!!
!
!
!
100
!
!
!
! !
!!
!
!!
! !
! !
!
!
!
!
0
400
!
!
!
!
200
NO
!
!
!!
100
! !
!
0
600
!
!
200
!
! !
! ! !!
!! ! !
!
!
!
!
!
!
!
!
!
!
!
!
!! ! !
!! !
!
! ! !!!
!!
! !
!
!
!
!
!
! ! ! !
!!
!
! !!! !
! !
!!! !
! ! !! ! ! ! !
!
!!!
!! ! ! ! ! !
! !
!! ! !!!
!!
!
!!
! ! !!
!!
! ! ! !
!
!!
!! ! ! !! !
!!
!
!
!!
!!!!!!
!
!
!!
!
!!
!
!!!
!
!
!!
!!
!
!
!
!
!! !
!!!
!
!
!
!
!
!
!
!
!
!
!
!!
! !
!
!!
!
!!
!!
!
! !!!
!!
!!
!
!!
!!
!
!
!
!
!
!
!
!!!
!
!
!!
! !
!
!
!
!
!
!!
! ! !!
!!!
!
!
!
!!
!
!
!
!
!
!
!
!
!
!
!
!
!
!!
!
!!
!
!
!!
!
!!
!
!
!
!
!!
!!!
!!
!
!
!
!!
!
!
!
!
!
!
!!!! ! !!! ! !!
!
!
!!
!
!
!
!
!
!
!!
!
!!!
!!
!
!
!
!
!
!
!
!
!!
! !
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
! !
!
!
!
!
!
!!
!
!
!
!
!
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!
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!
!
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!
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!!
!
!
!
!
!
!
!!
! !!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!!
!!
!
!
!
!
!!
!
! !!
!
!
!
!!
!
!
!
!!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!! !!
!!
!
!
!
!
!
!!!
!
!
!
!
0
200
400
!
!
!!!
!
!
!!
!
!!
!!
600
800
1000
NO
Figure 1: Scatterplots of NO vs. PM10 (left), SO2 vs. PM10 (center), and SO2 vs. NO (right).
The extremes of PM10 and NO appear to have relatively strong dependence, while the extremes of
SO2 and the other two pollutants appear to have much weaker dependence.
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Mutivariate extremes
A few Approaches for modeling multivariate extremes
Max-stable processes : Adapting asymptotic results for multivariate
extremes
Schlather & Tawn (2003), de Haan & Pereira (2005)
Latent models : structure indirectly modeled via the EVT parameters
distribution
Ribatet et al., 2010, Vrac et al., 2008, Garcon et al., 2010,
Complete modeling : Auto-Regressive spatio-temporal heavy tailed
processes, Davis and Mikosch (2007), AR-Gumbel Toulemonde et al.
(2009)
Copula approach : uniform marginals with extreme copulas,
Genest et al., Arthur
Conditional simulations : combining conditional simulations and MEVT,
The Bacro ANR
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
1
0
20
в€’1
10
y
2
30
3
40
Spatial Statistics for Maxima
10
20
x
30
40
How to describe the spatial
dependence as a function of
the distance between two
points ?
Motivation
Basics
Applic
Max-stable
Mado
Spectral
40
Spatial Statistics for Maxima
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
30
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
10
в—Џ
в—Џ
в—Џ
в—Џ
в—Џв—Џ
в—Џ
в—Џ
в—Џв—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
0
y
в—Џ
20
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
10
20
x
30
40
How to perform
spatial interpolation of
extreme events ?
Conclusion
Motivation
Basics
Applic
Max-stable
Main question
How to model dependencies among maxima ?
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Вґ
Choice of marginals : unit-Frechet
F (x) = exp(в€’1/x), for x > 0
Вґ
Frechet
(1878-1973)
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Max-stable processes
Вґ
Max-stability in the univariate case with an unit-Frechet
margin
F t (tx) = F (x), for F (x) = exp(в€’1/x)
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Max-stable processes
Вґ
Max-stability in the univariate case with an unit-Frechet
margin
F t (tx) = F (x), for F (x) = exp(в€’1/x)
Вґ
Max-stability in the multivariate case with unit-Frechet
margins
F t (tu, tv ) = F (u, v )
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
A central question
P [M(x) < u, M(x + h) < v ] =??
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Asymptotic theory
Вґ
If one assumes that we have unit Frechet
margins then
В»
Mn (x) в€’ an
lim P
n→∞
bn
–
Mn (x + h) в€’ an
u,
bn
where
1
Z
Vh (u, v ) = 2
„
max
0
w 1в€’w
,
u
v
v = exp [в€’Vh (u, v )]
В«
with Hh (.) a distribution function on [0, 1] such that
dHh (w)
R1
0
w dHh (w) = 0.5.
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Bivariate case (M(x), M(x + h))
Complex non-parametric structure
В«
„
Z 1
w 1в€’w
dHh (w)
Vh (u, v ) = 2
max
,
u
v
0
Special case u = v
Note Vh (u, u) = Vh (1, 1)/u and define
Оё(h) := Vh (1, 1)
P [M(x) < u, M(x + h) < u]
because F (u) = exp(в€’1/u)
=
exp(в€’Оё(h)/u)
=
F (u)Оё(h)
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Оё(h) = Extremal coefficient
P [M(x) < u, M(x + h) < u] = F (u)Оё(h)
Interpretation
Independence в‡’ Оё(h) = 2
M(x) = M(x + h) в‡’ Оё(h) = 1
Similar to correlation coefficients for Gaussian but ...
No characterization of the full bivariate dependence
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Geostatistics : Variograms
Complex non-parametric
structure
в—Џ
1.0
в—Џ
в—Џ
в—Џ
в—Џ
Оі(h) =
0.6
в—Џ
в—Џ
Finite if light tails
0.4
0.2
в—Џ
Capture all spatial
structure if {Z (x)}
Gaussian fields
в—Џ
0.0
1
E|Z (x + h) в€’ Z (x)|2
2
в—Џ
в—Џ
0.0
semivariance
0.8
в—Џ
в—Џ
0.2
0.4
distance
0.6
0.8
but not well adapted for
extremes
Motivation
Basics
Applic
Max-stable
Mado
Spectral
A Different Variogram
ОЅh =
1
E |F (M(x + h)) в€’ F (M(x))|
2
Why does it work ?
1
1
|a в€’ b| = max(a, b) в€’ (a + b)
2
2
a = F (M(x + h)) and b = F (M(x))
Ea = Eb = 1/2
Оё(h)
E max(a, b) = EF (max(M(x + h), M(x))) =
1 + Оё(h)
|
{z
}
max-stable
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Madogram ОЅh в‡’ Extremal coeff Оё(h)
Оё(h) =
1 + 2ОЅh
1 в€’ 2ОЅh
The madogram ОЅh gives the extremal coefficient Оё(h)
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Comparisons with other estimators
Gumbel (1960)
( "„ « 1
„ « 1 #α )
1 О±
1 О±
P (X ≤ x, Y ≤ y ) = exp −
+
x
y
Four estimators
- Pickands’ estimator (1975)
- Deheuvels’ estimator (1991)
- Hall and Tajvidi’s estimator (2000)
Вґ a` et al. (1997) estimator
- Capera
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Comparisons with other estimators
О± = 0.3
О± = 0.7
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
1
0
20
в€’1
10
y
2
30
3
40
Schlather’s models (2003)
10
20
30
40
x
r
Оё(h) = 1 +
1в€’
1
(exp(в€’h/40) + 1)
2
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Madogram ОЅh в‡’ Extremal coeff Оё(h)
2.0
0.8
Schlather’s fields
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
1.8
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
1.6
в—Џ
в—Џ
1.4
0.4
в—Џ
thetaHat
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
16
18
20
в—Џ
в—Џ
0.2
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
1.2
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
1.0
в—Џ
0.0
estimated madogram
0.6
в—Џ
1
4
6
8
10
12
distance
14
16
18
20
1
4
6
8
10
12
distance
14
Motivation
Basics
Applic
Max-stable
Mado
Spectral
1
0
10
y
20
2
30
3
40
Smith’s models (2003)
10
20
30
40
x
Оё(h) = 2О¦
“√
”
hT ОЈв€’1 h/2
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Madogram ОЅh в‡’ Extremal coeff Оё(h)
2.0
1.0
Smith’s fields
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
1.8
0.8
в—Џ
в—Џ
в—Џ
thetaHat
в—Џ
в—Џ
в—Џ
0.4
в—Џ
в—Џ
1.6
в—Џ
в—Џ
в—Џ
1.4
0.6
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
1.2
0.2
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
в—Џ
1.0
в—Џ
0.0
estimated madogram
в—Џ
в—Џ
1
4
6
8
10
12
distance
14
16
18
20
1
4
6
8
10
12
distance
14
16
18
20
Motivation
Basics
Applic
Max-stable
Mado
Spectral
The О»в€’madogram
ОЅh (О») =
1
E F О» (M(x + h)) в€’ F 1в€’О» (M(x))
2
Properties
Defined for light & heavy tails
Called a О»-Madogram
Nice links with extreme value theory
ОЅh (0) = ОЅh (1) = 0.25
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
A fundamental relationship
ОЅh (О») =
Vh (О», 1 в€’ О»)
3
в€’ c(О»), with c(О») =
1 + Vh (О», 1 в€’ О»)
2(1 + О»)(2 в€’ О»)
Conversely,
Vh (О», 1 в€’ О») =
c(О») + ОЅh (О»)
1 в€’ c(О») в€’ ОЅh (О»)
Conclusion
Motivation
Basics
The О»в€’madogram
Applic
Max-stable
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
30-year maxima of daily precipitation in Bourgogne
146 stations of maxima of daily precipitation over 1970-1999 in Bourgogne
Conclusion
Basics
Applic . Figure
Max-stable
Spectral
Conclusion
us Motivation
now turn to the
5 displays Mado
the -madogram
as a function
ent distances h. The continuous line corresponds to the case for which the extrem
54-year maxima of daily precipitation in Belgium
pendent, while the dashed line to the full dependence. The empirical evaluations are
een these two asymptotic solutions and progressively converge to the independent s
creasing distances.
55 stations of the Climatological network
0.25
l-madogram
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
l
0-10 km
30-50 km
70-90 km
130-150 km
0.6
0.7
0.8
0.9
1
independence
full dependence
full dependence
Figure 5:
55 stations of precipitation maxima over 1951-2005 in Belgium
*********
supported by the NSF-GMC (ATM-0327936) grant and the European commission project NEST No 1
ferences
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Examples : fitting multivariate maxima
1000
Air pollutants (Leeds, UK, winter 94-98, daily max) NO vs. PM10 (left), SO2 vs. PM10
(center), and SO2 vs. NO (right) (Heffernan& Tawn 2004, Boldi & Davison, 2007)
!
!
400
400
800
500
!
500
!
!
!
!
!
0
50
100
PM10
150
SO2
!
!
!
!
200
!
!
0
!
!
!
100
!
!
! !
! !!!!
!
!
!
!
!
!
50
!!
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400
200
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NO
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200
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100
600
!
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0
200
PM10
400
!
!
!!!
!
!
!!
!
!!
!!
600
800
1000
NO
Our
strategy
Figure
1: Scatterplots of NO vs. PM10 (left), SO2 vs. PM10 (center), and SO2 vs. NO (right).
The extremes of PM10 and NO appear to have relatively strong dependence, while the extremes of
1 Assume observations arise from a max-stable process
SO2 and the other two pollutants appear to have much weaker dependence.
2
Find and fit a flexible parametric model for the spectral density
3
Two desiderata : (A) interpretable parameters & (B) going beyond the
bivariate case
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Asymptotic theory (De Haan, Resnick)
Вґ
Assuming unit Frechet
margins then
F (u, v ) = exp [в€’V (u, v )]
where
1
„
w 1в€’w
,
u
v
В«
with H(.) a distribution function on [0, 1] such that
R1
Z
V (u, v ) = 2
max
0
dH(w)
0
w dH(w) = 0.5.
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Models for Multivariate MSD’s
Exponent measure function
V (z)
Logistic
Asymmetric Logistic
(Tawn, 88)
Negative Logistic
(Joe, 90)
+ Can obtain G(z)
в€’ Overparametrized ?
в€’ Less flexible ?
Spectral density
h(w )
Dirichlet
(Coles & Tawn, 91)
Dirichlet mixture
(Boldi & Davison, 2006)
Pairwise Beta (Cooley, Davis
and Naveau)
+ More flexibility ?
в€’ Cannot directly get G(z)
Conclusion
w; Оё). They show that if h is a positive function on Spв€’1 with finite first moments
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
в€— defining
an
a parametric
model for V (z;
Оё)d(w),
then
p Оё), one can alternatively define parametr
i h (w;
R
mp wp
1
1 w1
в€’(p+1)
в€— m
gular
density
h(w;
and
Tawn
describe
Dirichlet
model
(Coles,
Tawn,
1991)
wi (1991)
dH(w
1/p
h(w;
Оё) Оё).
= Coles
(m
В· w)
m
, . . .one
, method
; Оё of obtainin
jh ) =
Sp
p
p
m
В·
w
m
В·wwfinite first mome
в€—
Оё). They show that if1h is a в€’(p+1)
positive
function
on
S
with
m
m
w
j=1
p p
1 1 pв€’1
Оё) = (m В· w)
mj hв€—
,...,
;Оё
в€— (w; Оё)d(w),h(w;
then
p
mВ·w
mВ·w
id angular density which has all its massj=1
on the interior of Spв€’1 . In effect, if one think
erhaps unnormalized) density on Spв€’1 , then
(11) alters the density so that it has cen
p the
id angular density which
all its mass on
interior
.m
Inp w
effect,
if one thinks
m1 (1991)
wof1 Spв€’1used
1 has
p
в€’(p+1)
в€—Tawn
t (1/p, . . . , h(w;
1/p) and
total
mass
of
1.
Coles
and
their
to
Оё) = density
(m В· w)
, . .density
.,
Оёtechnique
j h alters the
erhaps unnormalized)
on Spв€’1 , thenm(11)
so; that
it has cen
m В· w a wellm
В· w density
ivariate extreme value pmodel from the j=1
Dirichlet density,
known
on th
t (1/p, . . . , 1/p) and total mass of 1. Coles and Tawn (1991) used their technique to
x which in p-dimensions is parameterized by О± = (О±1 , . . . , О±p )T and whose pdf is give
ivariate extreme value model from the Dirichlet density, a well known density on th
angular density
which has
allmodel
its massp on
the interior of ST and
. Inwhose
effect,pdf
if is
one
th
case : Dirichlet
x whichAinspecial
p-dimensions
is parameterized
by О± = (О±1 , . . . , О±p )pв€’1
given
О“(О±
В·S1)
О±j в€’1
в€—
haps unnormalized)
density
on
,
then
(11)
alters
the
density
so
that
it
has
pв€’1
h (w; О±) = p
wj
, О±j > 0, j = 1, . . . , p.
p
О“(О±
1/p, . . . , 1/p) and
total mass
1.j )Coles
Tawn (1991) used their technique
j=1of
О“(О±
В· 1)
j=1 О±and
j в€’1
в€—
h
(w;
О±)
=
w
,
О±j > 0, j a= well
1, . . .known
, p.
p
j
riate extreme value model j=1
from
Dirichlet density,
density on
О“(О±the
j ) j=1
в€’1
which
in p-dimensions
is parameterized
О± О±=j (О±1,,applying
. . . , О±p )T(11)
andone
whose
pdfthe
is ag
Dirichlet
density has moments
mi = О±i / bypj=1
obtains
в€’1
yDirichlet density has moments m = О± / p p О±
i
p i
p, applying (11) one obtains the an
j=1 j
О± в€’1
О±j О“(О±wВ·О±1j в€’1
1О“(О± В· 1)
+ ,1) О± > О±
hв€— (w;
О±)
=
0,j w
j j= 1,j . . . , p.
j
p
h(w; О±) =
j p+1
О“(О±
)
p
p
j
p
О“(О±
)
(О±
В·
w)
О±
В·
w
1j=1
О“(О± В· 1 + 1) j=1 О±j wj О±j в€’1
j=1 О±j j j=1
h(w; О±) =
p
О“(О±j ) (О± В· w)p+1 в€’1 О± В· w
can be asymmetric.
j=1
j=1
p
richlet density has moments mi = О±i /
j=1 О±j
, applying (11) one obtains th
can be asymmetric.
ared to the parametric models for V (x, Оё), modeling h(w; Оё) directly allows for more flex
p
p
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Our Pairwise Beta Model
hp (w ; О±, ОІ)
=
Kp (О±)
X
hi,j (wi , wj ; О±, ОІi,j ), where
i=j
hi,j (wi , wj ; О±, ОІi,j )
=
(wi + wj )(pв€’1)(О±в€’1) (1 в€’ (wi + wj ))О±в€’1 Г—
„
«βi,j −1 „
В«ОІi,j в€’1
О“(2ОІi,j )
wj
wi
(О“(ОІi,j ))2 wi + wj
wi + wj
Advantages :
no
R adjustment necessary to get center of mass condition
wj dH(w) = 1/p
parameters have some interpretation : О± controls overall dependence,
βi,j ’s control pairwise dependence
largely specified by pairwise parameters
Middle ground between Coles & Tawn (1991) and Boldi & Davison (2007)
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Pairwise Beta Models
О± = 1, ОІ = (2, 4, 15)
О± = 4, ОІ = (2, 4, 15)
О± = 1, ОІ = (2, .5, .5)
О± = 1, ОІ = (2, 2, .5)
Conclusion
Motivationindicating Basics
that large
2,3
Applic
Max-stable
Madounlikely toSpectral
Conclusion
values
of components
two and three are
occur at the same
time.
Fitting the spectral density model
4
Estimation Procedure
Fitting a model for an angular density is a relatively straightforward exercise. Given a set of iid
observations y i , i = 1, . . . , n, one first fits distributions to the marginals, and then transforms
Beirlant
al., (2004), Coles & Tawn (2004), Boldi & Davison (2007)
z i = T (yat
i ) to have a common marginal with tail index О± = 1. One then makes a further transformation to pseudo polar coordinates yielding points (ri , wi ) where ri = z i and wi = z i z i в€’1 .
A high threshold t0 is selected and the points {(ri , wi ), i = 1, . . . , n : ri > t0 } are retained. Let
(r(i) , w(i) ), i = 1, . . . , Nt0 denote the reindexed threshold exceedances. Given that t0 is large enough,
we have
assumecommon
that the points
(r(i) , wwith
follow a Poisson process with intensity measure
(a)
marginals
unit tail index
(i) ) approximately
ОЅ given
in (3). into
Letting
A =coordinates
{(r, w) : r > and
t0 } the
approximate
likelihood
(Beirlant
(b)
transform
polar
select
exceedances
above
t0 et al., 2004, pp.
170-171)
of thethe
points
(r(i) , w(i) ), i = 1, . . . , Nt0 is given by
(c)
maximize
likelihood
Nt0
L(θ; (r(i) , w(i) ), i = 1, . . . , Nt0 ) ≈ exp(−ν(A))
Nt0
dОЅ(r(i) , w(i) ) = exp(в€’tв€’1
0 )
i=1
в€’2
r(i)
h(w(i) , Оё),
i=1
where h(w; Оё) is any parametric model for the angular measure. To find Оё which maximizes
Nt
this likelihood, we need to only note that L(Оё; (r(i) , w(i) ), i = 1, . . . , Nt0 ) в€ќ i=10 h(w(i) , Оё). The
estimate ОёЛ† can then be found via numerical optimization. This estimation procedure was used
by both Coles and Tawn (1994) and Boldi and Davison (2007). Since the pairwise beta angular
p
measure is a smooth function of О± в€€ (0, в€ћ) and ОІ в€€ (0, в€ћ)(2) and has bounded support on the
unit simplex Spв€’1 , if one assumes the marginals distribution is known, standard asymptotics hold
Л†
for the maximum likelihood estimators О±,
Л† ОІ.
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Table 1: Summary of the pairwise beta simulation results. Table gives the true values of the
Air pollutants
example
parameters, the mean of the MLE estimates for the 1000 simulations, the standard error of the
estimates suggested by the asymptotics, and the sample standard error of the estimates. The means
of the estimates are slightly larger than the actual parameter values and the sample standard errors
are slightly larger than the asymptotic estimates; both results are presumably due to the skewness
seen in Figure 3.
ОІ1,2
ОІ1,3
ОІ1,4
ОІ1,5
ОІ2,3
ОІ2,4
ОІ2,5
ОІ3,4
ОІ3,5
ОІ4,5
PM10, NO
PM10, NO2
PM10, O3
PM10, SO2
NO, NO2
NO, O3
NO, SO2
NO2, O3
NO2, SO2
O3, SO2
4.04
(0.139)
29.69
(1.222)
0.33
(0.006)
0.81
(0.026)
3.51
(0.119)
0.34
(0.006)
0.53
(0.014)
0.61
(0.013)
0.45
(0.011)
0.33
(0.006)
1000
О±
–
0.31
(0.002)
!
!
!
!
500
400
400
800
500
Table 2: Maximum likelihood parameter estimates for the five-dimensional air quality data for the
pairwise beta model. The largest ОІi,j values correspond to the PM10, NO, and NO2 components
as expected. Standard errors are in parentheses.
!
!
!
!
50
PM10
!!
!
! !
!
!!
!
!
!
!
!
!
!
!
!
!
150
200
!
!
! !!
!! ! !
!
!
!
!
0
!
50
100
PM10
!
!
Dirichlet
L(Оё)
k
AIC
2134.90 5.00 -4259.80
1367.97 5.00 -2725.94
640.95 5.00 -1271.89
!! !
!
!!
!
!
!!
!!
!
!
!
!
!
!!
!
!
! !
!
!! !
!
!
!
! !
!
!
! !
!
!! ! ! ! ! ! !!
!
! ! !! !
!
!
!
! !
!
!! ! ! !
! ! ! ! !!
! !!! ! !
!
!
! !!
! !!
!! !!
!
!
!
! ! !
!
!!
! !!! !! !
!
! !!! !!
!
!
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!!
!
!
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!!!!
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!
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!
!
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!
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!
!
!
!
!
!
!
!
!
!
! !!!!
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!!
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! ! ! !!!
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!
!
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!!!
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!!
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!! ! !
! !
!!!
!!
!!
!
!!
!
!
!
!!!
!
!
!
!
!
!
!
!
!
!
!
!!! !
!
!
!
!
!
!
!!
!!
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!
!
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!!
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!
!!
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!
!! !!
!!
!!
!
!
300
SO2
Pairwise Beta
L(Оё)
k
AIC
2430.53 11.00 -4839.06
1625.37 11.00 -3228.74
809.65 11.00 -1597.31
!
!
!
!
200
!
!
300
SO2
!
Nt0 = 750
Nt0 = 500
Nt0 = 250
100
!
!
!
!
150
! !
! !!!!
!
!
!
!
!
!
!
!!
!
!
!
100
200
0
0
!
!
!
!
!
!
!!
!
!
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!
!
!!!
!
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!
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!
0
400
!
!
!
!
!
200
NO
!
!
!
!!
100
! !
0
600
!
!
200
!!
!!
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!
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!!
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!!!
!
!
!
!
!
!
0
200
400
!
!!!
!
!
!
!!
!
!!
!!
600
800
1000
NO
Table 3: The log-likelihood (L(Оё)), the number of parameters (k), and AIC values for the pairwise
Figure
1: Scatterplots
vs. the
PM10
SO2ofvs.
PM10 (center),
and SO2
vs. are
NO for
(right).
beta
and Dirichlet
modelsofasNO
fit to
five (left),
locations
simulated
spatial fields.
Results
The
extremes
PM10 and which
NO appear
to have
strong
dependence,
whilequantiles
the extremes
of
750,
500,
and 250ofexceedances
correspond
torelatively
the empirical
85%,
90%, and 95%
of
andcomponents
the other two
pollutants
appearoftothe
have
much
weakerfields.
dependence.
theSO2
radial
for the
five locations
5000
simulated
The AIC shows that the
pairwise beta model outperforms the Dirichlet model for this spatial data.
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Air pollutants example
Dirichlet model
1.0
1.0
Beta model
1.0
data
!
!
!
!
!
!
!
!
!
!
!
0.4
!
!
!
!
!
!
!
!
!
0.2
!
!
!
!
!
!
!
! !
!
!
! !!
!
!
!!
0.0
!
!
!
! !! !!
!
! !!
!!
!
!
!
!!
!
!
!
!
!
!
!
0.0
!
!!
!
!
!
!
0.6
!
!
0.4
!
!
!
!
! !
!
! !
0.2
0.6
!
0.6
!!
!
!
!
0.4
!
!
!
!
0.2
!
!
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.8
0.8
!
!
0.8
!
!
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
100 largest
observations.
Figure 4: Left plot shows the angular components of the largest 100 observations of the trivariate
corners=
PM10
(lower
right),
NO
(upper
left), SO2
(lower
air quality
data.
The lower
right
corner
corresponds
to large
valuesleft)
of PM10 and small values of
NO and SO2. The upper corner corresponds to large values of NO, and the lower left corresponds
to large values of SO2. As points tend to lie along the hypotenuse of the triangle and in the lower
left corner, this indicates that large values of PM10 and NO tend to occur together, while large
values of SO2 occur independently. The center plot shows the log density of the fitted pairwise
beta model, while right plot shows the log density of the fitted Dirichlet model.
1.0
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Summary of our spectral approach
“Simple” and flexible spectral density with interpretable parameters
Can be used for prediction or interpolation purposes
Can be generalized (Ballani, Schlather, 2010)
Can be extended to the asymptotic independent case (Qin, Smith, Ren,
2008)
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Take home messages
Multivariate EVT may help characterizing extremes dependencies in
space and time
Physical knowledge should be integrated into the statistical analysis
Computational issues can be arisen quickly
Modeling trade off between parametric and non-parametric approaches
Asymptotic independence can be an issue
Extremes here means very rare
Two advertisements
Extreme Value Analysis (EVA, Lyon June 27th to July 1st, 2011)
Environmental Risk and Extreme Events, Workshop, Ascona, July 10-15
2011
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Conditional independence assumption limits
40
BHM with CI assumption
40
Observed
50
30
30
60
50
40
30
30
y
20
y
20
40
20
10
10
20
10
10
30
40
10
30
Difficulty to work with explicit densities50
30
60
Ribatet, Cooley and Davison (2010)
20
(b)
30
40
40
20
(a)
40
10
60
30
y
20
y
20
40
10
0
20
10
10
20
40
0
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
An example with О± = 1, ОІ = (2, 4, 15)
1.0
0.0
1.0
0.8
0.8
0.6
0.6
0.4
0.8
0.2
0.6
1.0
0.0
0.2
0.4
0.6
0.
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Density
0.4
0.2
2
0
0.0
1
Density
0.6
3
0.8
4
An example with О± = 1, ОІ = (2, 4, 15) (. . . = asymptotic mle) 200 real * 1000
0.8
1.0
1.2
1.4
1.6
1.8
1
2
3
4
5
!_1,2
0.06
Density
0.04
0.20
0.15
0.02
0.10
0.00
0.05
0.00
Density
0.25
0.08
0.30
0.35
!
0
2
4
6
8
!_1,3
10
12
14
0
10
20
30
!_2,3
40
50
60
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
Multivariate Max-Stable Distributions
If Z = (Z (x 1 ), . . . , Z (x p ))T has a multivariate max-stable distribution with unit
Вґ
Frechet
margins (P(Z (x i ) ≤ z) = exp(−z −1 )) then :
G(z) = P(Z ≤ z) = exp[−V (z)], where
„ «
Z
wi
V (z) = p
max
dH(w ),
i
zi
Sp
H is a positive measure on Sp , s.t.
Z
wi dH(w ) = 1/p,
Sp
and Sp = {w в€€ Rp+ |w1 + . . . + wp = 1}.
Motivation
Basics
Applic
Going multivariate for max-stable
SEE YOU TOMORROW ! !
Max-stable
Mado
Spectral
Conclusion
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Conclusion
A main random variable of interest : precipitation
1
Relevant parameter in meteorology and climatology
2
Highly stochastic nature compared to other meteorological parameters
200
mm day!1
100
50
cccma3.1/t47[5]
cccma3.1/t63
cnrm cm3
echo g[3]
gfdl cm2.0
gfdl cm2.1
giss aom
giss er
inm cm3.0
ipsl cm4[2]
era40
ncep2
era15
20
ncep1
10
ncep2
era40
P20, 1981!2000
5
100
miroc3.2/hires
miroc3.2/medres[3]
mpi echam5
mri cgcm2.3.2[5]
ncar ccsm3[6]
ncar pcm1[4]
60S
30S
0E
30N
ncep1
era15
60N
era40
miroc3.2/hires
cccma3.1/t47[5]
er
Kharin and
Zwiers, giss
Journal
of Climate 2007,
P (1981-2000)
ncep2 20
ay!1
50
20
cccma3.1/t63
cnrm cm3
echo g[3]
gfdl cm2.0
gfdl cm2.1
giss aom
inm cm3.0
ipsl cm4[2]
era15
miroc3.2/medres[3]
mpi echam5
mri cgcm2.3.2[5]
ncar ccsm3[6]
ncar pcm1[4]
Motivation
Basics
Applic
Max-stable
Mado
Spectral
Л†
Estimating the GPD parameters estimates (Л†
Пѓu , Оѕ)
Maximum likelihood estimation
Methods of moments type (PWM and GPWM, Ribereau et al., 2010)
Exhaustive tail-index approaches
MCMC techniques
Taking advantages of the stability property
Mean Excess function
E(R в€’ u|R > u) =
Пѓu + uОѕ
1в€’Оѕ
the scale parameter varies linearly in the threshold u
the shape parameter Оѕ is fixed wrt the threshold u
Conclusion
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