Extreme Value Theory
(or how to go beyond of the range data)
Sept 2007, Romania
Philippe Naveau
Laboratoire des Sciences du Climat et l’Environnement (LSCE)
Gif-sur-Yvette, France
Katz et al., Statistics of extremes in hydrology,
Advances in Water Resources 25 (2002) 1287-1304
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Extreme quotes
1
“Man can believe the impossible, but man can never believe the
improbable”
Oscar Wilde (Intentions, 1891)
2
“Il est impossible que l’improbable n’arrive jamais”
Emil Julius Gumbel (1891-1966)
Extreme events ? ... a probabilistic
concept linked to the tail behavior :
low frequency of occurrence, large
uncertainty and sometimes strong
amplitude.
Region of interest
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Important issues in Extreme Value Theory
Applied statistics
An asymptotic probabilistic
concept
A statistical modeling approach
Identifying clearly assumptions
Assessing uncertainties
Goodness of fit and model
selection
Non-stationarity
Multivariate
Univariate
Non-parametric
Parametric
Independence
Theoritical probability
Motivation
Univariate EVT
Non-stationary extremes
Outline
1 Motivation
Heavy rainfalls
Three applications
2 Univariate EVT
Asymptotic result
Historical perspective
GPD Parameters estimation
Brief summary of univariate iid EVT
3 Non-stationary extremes
Spatial interpolation of return levels
Downscaling of heavy rainfalls
4 Spatial extremes
Assessing spatial dependences among maxima
5 Conclusions
II Main Part
Spatial extremes
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Why heavy rainfalls are important in geosciences ?
”It is very likely that hot extremes, heat waves, and heavy precipitation events
will continue to become more frequent” and that “precipitation is highly
variable spatially and temporally”
The policymakers summary of the 2007 Intergovernmental Panel on Climate Change
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Random variable types
Climate : maxima or mimina (daily, monthly, annually), dry spells, etc
Hydrology : return levels.
a quantile estimation pb : how to find zp such that P(Z > zp ) = p
=⇒ Exceedances
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
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 ⇒ T = 100 years.
Return level interpretations
Waiting time : Average waiting
time until next occurrence of
event is T years
Number of events : Average
number of events occurring within
a T -year time period is one
Spatial extremes
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Our 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
Kharin
ay!1
50
20
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
cccma3.1/t47[5]
er
and
Zwiers, giss
Journal
of
cccma3.1/t63
inm cm3.0 era15
cnrm cm3
ipsl cm4[2]
echo g[3]
gfdl cm2.0
gfdl cm2.1
giss aom
0E
Climate
30N
era40
2007,
P
ncep2 20
ncep1
era15
60N
miroc3.2/hires
(1981-2000)
miroc3.2/medres[3]
mpi echam5
mri cgcm2.3.2[5]
ncar ccsm3[6]
ncar pcm1[4]
Motivation
Univariate EVT
Non-stationary extremes
Heavy rainfall distributions
The problems at hand
Classical distributions (Gamma, Weibull,
Stretched-exponential, . . .) not satisfying for
extremes
EVT not adequate low and medium
precipitation
Our main question
How to go beyond the univariate
site-per-site modeling and to take into
account the spatial pairwise dependence
among sites ?
Spatial extremes
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Three applications
Measuring the spatial dependence among maxima (Max-stable
processes) :
Vannitsem & Naveau (2007), Schlather & Tawn (2003), de Haan &
Pereira (2005)
Spatial Interpolation of return levels in Colorado (Hierarchical
Bayesian models) :
Cooley, Nychka and Naveau (2007), Coles & Tawn (1996).
Downscaling extremes over Illinois (latent processes) :
Vrac and Naveau (2007)
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Two toy examples
Annual maximum peak flow
Crete ice core
R.W. Katz et al. / Advances in Water Resources 25 (2002) 1287–1304
Potomac river (cfs)
Greenland (ecm)
Data = crete
Fig. 3. Annual peak flow of Potomac River at Point of Rocks, MD,
USA, 1895–2000.
0.8 0
0.4
0.0
Posterior proba
200
400
600
800
1000
1292
3.1.1. Poisson–GP model
Arising as an approximation for the distribut
excesses above a high threshold, the cumulative
bution and quantile functions for the GP are giv
F ðx; r" ; cÞ ¼ 1 % ½1 þ cðx=r" Þ(
r" > 0; 1 þ cðx=r" Þ > 0;
%1=c
;
F %1 ð1 % p; r" ; cÞ ¼ ðr" =cÞðp%c % 1Þ;
0 < p < 1:
Here r" and c are the scale and shape paramete
spectively. The interpretation of the shape parame
Time for the GEV distribution (
equivalent to that
c > 0, then the GP distribution is heavy taile
convention,
0 refers
the 1800
limiting
600
800
1000c ¼1200
1400 to
1600
2000 case obtai
c ! 0 in Eq. (4) Time
of the exponential distribution (
unbounded, thin tail).
Let X1 ; X2 ; . . . ; Xn , denote a time series (assum
now, to be independent and identically distri
whose high extreme values are of interest. The Po
GP model consists of two components (Chapte
Motivation
Univariate EVT
Gumbel
Maxima Distribution
Non-stationary extremes
(1891-1966)
Spatial extremes
Weibull (1887-1979)
Conclusions
Fréchet (1878-1973)
Distribution du maximum
Normal density ⇒
⇐ Gumbel density
Uniform density ⇒
⇐ Weibull density
Cauchy density ⇒
⇐ Fréchet density
n = 50
⇓
↓
↑
⇑
Extr^
emes? Mesurer
n = 100
Interpoler
Régionaliser
6
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Max-stability
Let Mn = max(X1 , . . . , Xn ) with Xi iid with distribution F .
Problem : find an and bn for a given F such that
„
P
M n − an
<x
bn
«
= F n (an x + bn ) = F (x)
Home work A
Unit-Frèchet F (x) = exp(−1/x) for x > 0. Then an = 1 & bn = 0
Gumbel F (x) = exp(− exp(−x)) for all real x. Then an = 1 & bn = − log n
Weibull F (x) = exp(−|x|α ) for x < 0 (1 otherwise). Then an = n1/α & bn = 0
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Max-stability
Let Mn = max(X1 , . . . , Xn ) with Xi iid with distribution F .
Problem : find an and bn for a given F such that
„
lim P
n→∞
M n − an
<x
bn
«
= lim F n (an x + bn ) = F (x)
n→∞
Home work B
Exponential F (x) = 1 − exp(−x) for x > 0. Then an = 1 & bn = log n
Uniform F (x) = x for 0 < x < 1. Then an = 1/n & bn = 1
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Generalized Extreme Value (GEV) distribution
„
P
M n − an
<x
bn
«
 h
“ x − µ ”i−1/ξ ff
∼ GEV(x) = exp − 1 + ξ
σ
+
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
3
x
Home work C : show that a GEV is max-stable
4
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Historical perspective
Gumbel (1891-1966)
Weibull (1887-1979)
Fréchet (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
Univariate EVT
Non-stationary extremes
An active statistical and probabilistic field
Spatial extremes
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
GEV and return levels
 h
“ x − µ ”i−1/ξ ff
GEV(x) = exp − 1 + ξ
σ
+
Computing the return level zp such that GEV(zp ) = 1 − p
zp = GEV−1 (1 − p)
Hence, zp = µ +
σ
ξ
`
´
[− ln(1 − p)]−ξ − 1]
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
GEV and return levels estimation
zp = µ +
”
σ“
[− ln(1 − p)]−ξ − 1]
ξ
Estimating the return level zp
ẑp = µ̂ +
σ̂
ξ̂
“
”
[− ln(1 − p)]−ξ̂ − 1]
ˆ
Estimating the GEV parameters estimates (µ̂, σ̂, ξ)
Maximum likelihood estimation
Methods of moments type (PWM and GPWM)
Exhaustive tail-index approaches
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
GEV and return levels estimation
ẑp = µ̂ +
”
σ̂ “
[− ln(1 − p)]−ξ̂ − 1]
ξˆ
ˆt
Maximum likelihood estimates of (µ̂, σ̂, ξ)
Asymptotically distributed as a multivariate Gaussian vector with mean
ˆ t and covariance matrix that is the inverse of the expected
θ = (µ̂, σ̂, ξ)
information matrix whose elements are equal
„
«
∂ 2 log l(θ)
E −
∂θi ∂θj
where l(θ) is the likelihood function of the GEV distributed sample
Conclusions
Motivation
Univariate EVT
Our first toy example
1292
r"
spective
equivale
c > 0, t
convent
c ! 0 in
Fig. 3. Annual peak flow of Potomac River at Point of Rocks, MD,
unboun
USA, 1895–2000.
Let X
now, to
R.W. Katz et al. / Advances in Water Resources 25 (2002) 1287–1304
whose h
GP mo
3.1.1. Poisson–GP model
[15,25],
Arising as an approximation for the distributio
ceedanc
excesses above a high threshold, the cumulative
d
i) are ge
bution and quantile functions for the GP arek);given
and
some i)
%1=c
F ðx; r" ; cÞ ¼ 1 % ½1 þ cðx=r" Þ( ;
paramet
distribu
"
"
r > 0; 1 þ cðx=r Þ > 0;
dependi
F %1 ð1 % p; r" ; cÞ ¼ ðr" =cÞðp%c % 1Þ; 0 < p <A).
1: As
pendenc
"
Here r and c are the scale and shape parameters
instead
sumptio
spectively. The interpretation of the shape paramete
rameter
equivalent to that for the GEV distribution
(e.g
(e.g., an
Non-stationary extremes
Spatial extremes
Here
Conclusions
c > 0, then the GP distribution is heavy tailed)
obtaine
3.1.2. P
c ! 0 in Eq. (4) of the exponential distributionAmo
(i.e
Fig. 3. Annual peak flow of Potomac River at Point of Rocks, MD,
unbounded, thin tail).
theory n
(Fig.
4)
indicates
that
the
fit
is
reasonably
adequate,
ˆ1895–2000.
USA,ξ
= 0.191 with a P-value of 0.002 for likelihood
ratio
of
0 a timeannual
the stat
X1tail.
;test
X2 ;In. .Section
. ; Xξn , =
denote
series (assumed
even in the Let
upper
5.2.2, another
volves r
now,
to be
distribu
peak flow
time series
willindependent
be analyzed for and
which identically
the fit of
GP mod
the GEVwhose
distribution
not appear
be acceptable.
high does
extreme
valuestoare
of interest. The
Pois
excesses
GP model consists of two components (Chapter
sional n
[15,25], Chapter 5 in [77]): (i) the occurrences
of
is time,
3. Methodological
ceedancesdevelopments
of some high threshold u (i.e., Xi > of
u, the
forGs
i) are generated by a Poisson process (with rateapproac
param
Fig. 4. Q–Q plot for fit of GEV distribution to annual peak flow of
convention, c ¼ 0 refers to the limiting case
Potomac River (line of equality indicates perfect fit).
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
!
Annual Maxima
!
Time series ⇒ 1 obs/yea
!
POT
!
Time series ⇒ λ obs/yea
!
Markovian
!
Time series ⇒ all excee
Peak over Threshold (POT)
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Thresholding : the Generalized Pareto Distribution (GPD)
P{R−u > y |R > u} =
„
«−1/ξ
ξy
1+
σu +
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
Univariate EVT
Non-stationary extremes
Spatial extremes
Generalized Pareto Distribution (GPD)
“
P{R − u > y|R > u} = 1 +
ξy
σu
”−1/ξ
Parameters
u = predetermined threshold
σu = scale parameter to be estimated
ξ = 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
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
GPD
P{R − u > y |R > u} =
„
«−1/ξ
ξy
1+
σu +
Special cases (home work D)
Unit-Frèchet F (x) = exp(−1/x) for x > 0. Then σ u = 1 and ξ = −1/α
Exponential F (x) = 1 − exp(−x) for x > 0. Then σ u = 1 and ξ = 0
Uniform F (x) = x for 0 < x < 1. Then σ u = 1 and ξ = −1
Stability property (home work E)
If the exceedance (R − u|R > u) follows a GPD(σu , ξ) then the higher
exceedance (R − v |R > v ) also follows GPD(σu + (v − u)ξ, ξ)
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
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
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
ˆ
Estimating the GPD parameters estimates (σ̂u , ξ)
Maximum likelihood estimation
Methods of moments type (PWM and GPWM)
Exhaustive tail-index approaches
Taking advantages of the stability property
Mean Excess function
E(R − u|R > u) =
σu + uξ
1−ξ
the scale parameter varies linearly in the threshold u
the shape parameter ξ is fixed wrt the threshold u
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
GPD diagnostics & models selection for our Crete data
400
170
180
u
ξˆ = 0.56 (0.37)
190
200
!
!!
!!
!
160
600
!!
!
0.2
150
!!
200
0.0
!
100
160
!
!
50
150
!
!!
empirical
!
0.4
250
300
model
350
0.8
!!
Quantile Plot
!!
150
200
Mean Excess
250
200
50
100
150
Mean Excess
300
350
400
Probability plot
170
0.6
180
1.0
190
u
empirical
!
!
!!
!!!!
!
!
!
!
!
!
!
!
!
!
200
400
600
200
model
800
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
GPD
GPD return level zp
zp = u +
σu
ξ
»
p
P(R > u)
!
–−ξ
−1
Estimating the return level zp
σ̂ u
ẑp = u +
ξˆ
»
p×n
Nu
–−ξ̂
!
−1
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
A first summary
Applied statistics
So far, we have assumed an idd
model
Asymptotic probability suggests
GEV and GPD
Maximum likelihood approach
provides asymptotic parameters
and return levels uncertainties
Goodness of fit and model
selection
Non-stationarity
Multivariate
Univariate
Non-parametric
Parametric
Independence
Theoritical probability
Motivation
Univariate EVT
Non-stationary extremes
Outline
1 Motivation
Heavy rainfalls
Three applications
2 Univariate EVT
Asymptotic result
Historical perspective
GPD Parameters estimation
Brief summary of univariate iid EVT
3 Non-stationary extremes
Spatial interpolation of return levels
Downscaling of heavy rainfalls
4 Spatial extremes
Assessing spatial dependences among maxima
5 Conclusions
II Main Part
Spatial extremes
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
41
Daily precipitation (April-October, 1948-2001, 56 stations)
40
Ft. Collins
●
Denver
●
39
Grand
Junction
●
Limon
●
Colo ●Spgs
38
Pueblo
●
37
latitude
Breckenridge
●
−109
−108
−107
−106
−105
longitude
−104
−103
−102
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
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 (in
press).
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Our main assumptions
Process layer : The scale and shape GPD parameters (ξ(x), σ(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
σ(x) = exp(φ(x))
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Hierarchical Bayesian Model with three levels
P(process, parameters|data)
∝
P(data|process, parameters)
×P(process|parameters)
×P(parameters)
Process level : the scale and shape GPD parameters (ξ(x), σ(x)) are hidden
random fields
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Our three levels
A) Data layer := P(data|process, parameters) =
„
Pθ {R(xi ) − u > y |R(xi ) > u} =
1+
ξi y
exp φi
«−1/ξi
B) Process layer := P(process|parameters) :
e.g. φi = α0 + α1 × elevationi + MVN (0, β 0 exp(−β 1 ||x k − x j ||))

and ξ i
=
ξ moutains , if x i ∈ mountains
ξ plains , if x i ∈ plains
C) Parameters layer (priors) := P(parameters) :
e.g. (ξ moutains , ξ plains ) Gaussian distribution with non-informative mean and
variance
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Notre modèle Bayesien hiérarchiq
Bayesian hierarchical modeling
Priors %
α0 + α1 elev
σ
$
#
Priors % β 0 exp(−β 1||.||)
&
Priors
ξ plains
&
Priors
#
#
'
!
!
"
#
ξ moutains
%
zx
&
ξ
)
P (R(x) > u)
)
Priors
!
(
!
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
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-
Conclusions
Journal of the American Statistical Association, ???? 0
Their Corresponding DIC Scores
Baseline model
D̄
Model 0: φ = φ
ξ=ξ
73,595.5
Models in latitude/longitude space
Model 1: φ = α0 + %φ
ξ=ξ
Model 2: φ = α0 + α1 (msp) + %φ
ξ=ξ
Model 3: φ = α0 + α1 (elev) + %φ
ξ=ξ
Model 4: φ = α0 + α1 (elev)+ α2 (msp)+ %φ
ξ=ξ
Models in climate space
Model 5: φ = α0 + %φ
ξ=ξ
Model 6: φ = α0 + α1 (elev) + %φ
ξ=ξ
Model 7: φ = α0 + %φ
ξ = ξ mtn , ξ plains
Model 8: φ = α0 + α1 (elev) + %φ
ξ = ξ mtn , ξ plains
Model 9: φ = α0 + %φ
ξ = ξ + %ξ
D̄
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
D̄
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. %· ∼ MVN(0, & ), where [σ ]i, j = β·, 0 exp(−β·, 1 #xi − xj #).
73
74
75
76
77
78
79
80
81
82
83
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
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
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Posterior quantiles of return levels (.025, .975)
Ft. Collins
Ft. Collins
Greeley
!
!
Greeley
!
10
!
Denver
!
40
9
!
!
!!
!
!
!
!
!
8
4.0
!
!
Denver
!
!
!
Boulder
9
!
40
40
Boulder
!
!
10
!
! !
!
!
!!
!
8
3.5
!
!
!
!
!
!
3.0
!
!
39
7
latitude
39
latitude
39
Colo Spgs
!
!
!
!
!
!!
!
!
!
!
!
!
!
!
!!
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
7
!105.5
!105.0
longitude
!104.5
!106.0
!105.5
!105.0
longitude
!104.5
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Downscaling of rainfalls
Vrac and Naveau, (2007). Stochastic downscaling of
precipitation : From dry events to heavy rainfalls. Water
Resource Research (in press)
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Our data
Local scale : R t = Daily precipitation recorded at 37 stations 1980-1999
(DJF)
Large scale : X t = NCEP geopotential height, Q and DT at 850mb
Weather regimes : S t = Four regimes of precipitation
Our objective :
What is the precipitation probability distribution of R t given the large and
regional scale characteristics, X t and S t ?
Subsidiary questions :
What is the precipitation distribution at a given site ?
What are meaningful regional patterns ?
How to connect the different scales ?
Our strategy
A GPD latent process (hidden markov process + logistic model) that depends
on X t et de S t
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Illinois rainfall patterns
%$
%!
'#
'"
'(
Precipitation pattern 2
%&
%&
%$
%!
'#
'"
'(
Precipitation pattern 1
!!"
!!#
!$!
!$$
!!"
!$!
!$$
%$
%!
'#
'"
'(
Precipitation pattern 4
%&
%&
%$
%!
'#
'"
'(
Precipitation pattern 3
!!#
!!"
!!#
!$!
!$$
!!"
!!#
!$!
!$$
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
How to switch from one precipitation pattern to the other ?
Markov chains
P(S t = j|S t−1 = i) ∝ γij
..., but these transition probabilities are independent of the atmospheric
variables X t like Q
Non-homogeneous Markov chains
»
–
1
P(S t = j|S t−1 = i, X t ) ∝ γij exp − (X t − µij )Σ−1 (X t − µij )0
2
where
Σ = atmospheric variables covariance
µij = atmospheric variables means
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Precipitation density : non-homogeneous Mixture Model
1.0
1.0
Mixture
Weibull
GPD
0.8
0.01
0.6
0.0001
0.4
1e-06
0.2
0.0
Mixture
Weibull
GPD
1e-08
0.0
1.0
2.0
3.0
4.0
5.0
0.0
1.0
2.0
3.0
4.0
Mixture Distribution
GPD and Gamma (Weibull after Frigessi et al. (2003))
”
´
1 “`
fmix (r ) =
1 − wµ,τ (r ) · fΓ(α,β) (r ) + wµ,τ (r ) · fG(σ,ξ) (r , u = 0)
Z |
{z
} | {z } | {z } |
{z
}
Gamma weight
Gamma pdf
GPD weight
GPD
5.0
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
0.8
0.6
GPD
0.2
0.4
GAMMA
0.0
Weight function [1]
1.0
Weight Function
0
1
2
3
4
Precipitation [cm]
5
6
Weight Function
Dynamic mixture model for unsupervised tail estimation without threshold
selection (Frigessi et al., 2002)
“r − µ”
1
1
wm,τ (r ) = + arctan
2
π
τ
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
QQ plots for the Spartan station
(a)
(b)
(c)
(d)
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Model selection
(0) gamma + GPD : parameters vary with location and precipitation pattern,
(i) only gamma : parameters vary with location and precipitation pattern,
(ii) gamma + GPD with one ξ parameter per pattern
(iii) same as (ii) with τ set to be equal to 0,
(iv) gamma + GPD with one common ξ for all stations and all patterns,
(v) same as (iv) with τ set to be equal to 0.
(iii)∗ same as model (iii) but only gamma distributions for pattern 1.
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Model selection
Model (0)
Model (i)
Model (ii)
Model (iii)
Model (iv)
Model (v)
Model (iii)∗
p = 24n
p = 8n
p = 20n + 4
p = 16n + 4
p = 20n + 1
p = 16n + 1
p = 12n + 5
Aledo
AIC=-796.52
AIC=-816.58
AIC=-795.76
AIC=-809.79
AIC=-819.46
AIC=-816.18
AIC=-816.79
Aurora
AIC=-1137.47 AIC=-1149.99 AIC=-1256.53 AIC=-1293.89 AIC=-1358.48
AIC=-1152.51
AIC=-1299.89
Station
37
Fairfield
AIC=14.36
AIC=103.07
AIC=22.45
AIC=22.37
AIC=-76.81
AIC=-10.21
AIC=16.37
Sparta
AIC=277.10
AIC=372.92
AIC=235.65
AIC=228.35
AIC=231.91
AIC=251.44
AIC=222.35
AIC=-1014.80
AIC=-920.68
AIC=-1028.91
AIC=-1023.59
Windsor
All five stations
AIC=-1016.25 AIC=-1017.59 AIC=-1069.99
AIC=-4433.18 AIC=-4422.27 AIC=-4479.50 AIC=-4515.13
AIC=-4425.06
AIC=-4423.78 AIC=-4553.13
Table 3: Akaike Information Criterion (AIC) values obtained for our five selected weather stations and for our seven models.
The bold values correspond to the optimal criterion per row. Below each model’s name, the number p of parameters for n
stations is provided.
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
1
0
20
−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
Univariate EVT
Non-stationary extremes
Spatial extremes
40
Spatial Statistics for Maxima
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●
30
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10
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●
0
y
●
20
●
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●
●
●
●
●
●
10
20
x
30
40
How to perform
spatial interpolation of
extreme events ?
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Spatial Statistics for Maxima
A few Approaches for modeling spatial extremes
Max-stable processes : Adapting asymptotic results for multivariate
extremes
Schlather & Tawn (2003), Naveau et al. (2007), de Haan & Pereira
(2005)
Bayesian or latent models : spatial structure indirectly modeled via
the EVT parameters distribution
Coles & Tawn (1996), Cooley et al. (2005)
Linear filtering : Auto-Regressive spatio-temporal heavy tailed
processes,
Davis and Mikosch (2007)
Gaussian anamorphosis : Transforming the field into a Gaussian one
Wackernagel (2003)
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Max-stable processes
Max-stability in the univariate case with an unit-Fréchet margin
F t (tx) = F (x), for F (x) = exp(−1/x)
Max-stability in the multivariate case with unit-Fréchet margins
F t (tx1 , . . . , txm ) = F (x1 , . . . , txm ), for Fi (xi ) = exp(−1/xi )
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
A central question
P [M(x) < u, M(x + h) < v ] =??
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Bivariate case for Maxima from an asymptotic point of view
If one assumes unit Fréchet margins then the distribution of the vector
(M(x), M(x + h)) goes to
F (u, v ) = exp [−Vh (u, v )]
where
1
Z
Vh (u, v ) = 2
„
max
0
w 1−w
,
u
v
«
with Hh (.) a distribution function on [0, 1] such that
dHh (w)
R1
0
w dHh (w) = 0.5.
Home work : check that F (u, v ) is bivariate max-stable
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Bivariate case (M(x), M(x + h))
Complex non-parametric structure
Z
Vh (u, v ) = 2
1
max
0
How to estimate Vh (u, v ) ?
w 1−w
,
u
v
dHh (w)
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Geostatistics : Variograms
Complex non-parametric
structure
●
1.0
●
●
●
●
●
0.6
●
●
Finite if light tails
0.4
0.2
●
Capture all spatial
structure if {Z (x)}
Gaussian fields
●
0.0
1
E|Z (x + h) − Z (x)|2
2
●
●
0.0
semivariance
0.8
γ(h) =
●
0.2
0.4
distance
0.6
0.8
but not well adapted for
extremes
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
A madogram type
νh =
1
E |F (M(x + h)) − F (M(x))|
2
Properties
Defined for light & heavy tails
nice link with EVT but only gives Vh (1, 1)
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
νh = 12 E |F (M(x + h)) − F (M(x))|
Madogram
0.8
40
simulated fields
●
●
3
●
0.4
●
●
●
●
●
●
●
●
●
●
0.2
estimated madogram
1
0
20
●
●
●
−1
10
●
●
0.0
y
2
0.6
30
●
10
20
x
30
40
1
4
6
8
10
12
distance
14
16
18
20
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
The λ−madogram
νh (λ) =
1 λ
E F (M(x + h)) − F 1−λ (M(x))
2
Properties
Defined for light & heavy tails
Called a λ-Madogram
Nice links with extreme value theory
νh (0) = νh (1) = 0.25
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
A fundamental relationship
Home work
νh (λ) =
Vh (λ, 1 − λ)
3
− c(λ), with c(λ) =
1 + Vh (λ, 1 − λ)
2(1 + λ)(2 − λ)
Conversely,
Vh (λ, 1 − λ) =
c(λ) + νh (λ)
1 − c(λ) − νh (λ)
Conclusions
Motivation
Univariate EVT
The λ−madogram
Non-stationary extremes
Spatial extremes
Conclusions
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
30-year maxima of daily precipitation in Bourgogne
146 stations of maxima of daily precipitation over 1970-1999 in Bourgogne
Conclusions
extremes
us Motivation
now turn to theUnivariate EVT
. Non-stationary
Figure 5extremes
displays the Spatial
-madogram
as a Conclusions
function
ent distances h. The continuous line corresponds to the case for which the extrem
54-yearwhile
maxima
of daily precipitation
Belgium
pendent,
the dashed
line to the fullindependence.
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
Motivation
Univariate EVT
Non-stationary extremes
Spatial extremes
Conclusions
Conclusions
Applied statistics
An asymptotic probabilistic
concept
A statistical modeling approach
Identifying clearly assumptions
Assessing uncertainties
Goodness of fit and model
selection
Non-stationarity
Multivariate
Univariate
Non-parametric
Parametric
Independence
Theoritical probability