Heavy rainfall modeling in high dimensions Philippe Naveau - - PowerPoint PPT Presentation

Intro Metric PAM Spectral Conclusions

Heavy rainfall modeling in high dimensions

Philippe Naveau naveau@lsce.ipsl.fr

Laboratoire des Sciences du Climat et l’Environnement (LSCE) Gif-sur-Yvette, France joint work with A. Sabourin, E. Bernard, M. Vrac and O. Mestre FP7-ACQWA, GIS-PEPER, MIRACLE & ANR-McSim, MOPERA

9 novembre 2012

Intro Metric PAM Spectral Conclusions

Hourly precipitation for 92 stations, 1992-2011 (Olivier Mestre)

−4 −2 2 4 6 8 42 44 46 48 50 Longitudes

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Intro Metric PAM Spectral Conclusions

Our game plan to handle extremes from this big rainfall dataset Spatial scale Large (country) Local (region) Problem Dimension reduction Spectral density in moderate dimension Data Weekly maxima Heavy hourly rainfall

• f hourly precipitation

excesses Method Clustering algorithms Mixture of for maxima Dirichlet Without imposing a given parametric structure

Intro Metric PAM Spectral Conclusions

Clustering of maxima (joint work with E. Bernard, M. Vrac and O. Mestre) Meteo-France subset data Weekly maxima of hourly precipitation 228 points = 19 years x 3 months x 4 weeks at each of the 92 locations Task 1 Clustering 92 grid points into around 10-20 climatologically homogeneous groups wrt spatial dependence

Intro Metric PAM Spectral Conclusions

Applying the kmeans algorithm to maxima (15 clusters)

−4 −2 2 4 6 8 42 44 46 48 50

PRECIP

Longitudes

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Kmeans Fall

−4 −2 2 4 6 8 42 44 46 48 50

log(PRECIP)

Longitudes

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Kmeans Fall

Intro Metric PAM Spectral Conclusions

The scale and shape GEV parameters

−4 −2 2 4 6 8 42 44 46 48 50

GEV scale

Longitudes

Fall

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6.1

−4 −2 2 4 6 8 42 44 46 48 50

GEV shape

Longitudes

0.01 0.06 0.11 0.16 0.21 0.26 0.31 0.35 0.4 0.45

Intro Metric PAM Spectral Conclusions

Kmeans clusterings Drawbacks Comparing apples and oranges An average of maxima (centroid of a cluster) is not a maximum variances have to be finite Clustering mixed intensity and dependence among maxima Difficult interpretation of clusters Questions How to find an appropriate metric for maxima ? How to create cluster centroids that are maxima ?

Intro Metric PAM Spectral Conclusions

A L1 marginal free distance (Cooley, Poncet and N., 2005, N. and al., 2007) d(x, y) = 1 2E |F y(M(y)) − F x(M(x))|

Intro Metric PAM Spectral Conclusions

A L1 marginal free distance (Cooley, Poncet and N., 2005, N. and al., 2007) d(x, y) = 1 2E |F y(M(y)) − F x(M(x))| If M(x) and M(y) bivariate GEV, then

extremal coefficient = 1 + 2d(x, y) 1 − 2d(x, y)

Intro Metric PAM Spectral Conclusions

Extension for the asymptotically independent case (Ramos and Ledford)

Guillou et al, 2012

η-Madogram

ν(η) = 1 2E

• Fη(M ∗1/η

X

) − Fη(M ∗1/η

Y

)

• =

1 2E [|F(M ∗

X) − F(M ∗ Y )|]

where Fη (resp. F) is the df of M ∗1/η

X

and M ∗1/η

Y

(resp. of M ∗

X and M ∗ Y )

ν(η) = Vη(1, 1)/Vη(1, ∞) 1 + Vη(1, 1)/Vη(1, ∞) − 1 2

Intro Metric PAM Spectral Conclusions

Extension for the asymptotically independent case (Ramos and Ledford)

Guillou et al, 2012

Estimation of the η−madogram

• FX, resp.

FY , be the empirical df of M ∗

Xi, resp. M ∗ Yi

• ν(η) =

1 2N

N

• i=1
• FX(M ∗

Xi) −

FY (M ∗

Yi)

• Theorem 1. Let
• M ∗

Xi, M ∗ Yi

• be a sample of N bivariate vectors such that

M ∗

Xi

bn , M ∗

Yi

bn

• converges in distribution to a bivariate extreme value distribution with an

η−extremal function. Then as n → ∞ and N → ∞ √ N

• ν(η) − 1

2E|F(M ∗

X) − F(M ∗ Y )|

d →

• [0,1]2 NC(u, v)dJ(u, v)

Intro Metric PAM Spectral Conclusions

Clusterings Questions How to find an appropriate metric for maxima ? How to create cluster centroids that are maxima ?

Intro Metric PAM Spectral Conclusions

Partitioning Around Medoids (PAM) (Kaufman, L. and Rousseeuw, P.J. (1987))

Intro Metric PAM Spectral Conclusions

PAM : Choose K initial mediods

Intro Metric PAM Spectral Conclusions

PAM : Assign each point to each closest mediod

Intro Metric PAM Spectral Conclusions

PAM : Recompute each mediod as the gravity center of each cluster

Intro Metric PAM Spectral Conclusions

PAM : continue if a mediod has been moved

Intro Metric PAM Spectral Conclusions

PAM : Assign each point to each closest mediod

Intro Metric PAM Spectral Conclusions

PAM : Recompute each mediod as the gravity center of each cluster

Intro Metric PAM Spectral Conclusions

−4 −2 2 4 6 8 42 44 46 48 50

PAM with K= 15

Longitudes Latitudes

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Fall

Intro Metric PAM Spectral Conclusions

−4 −2 2 4 6 8 42 44 46 48 50

PAM with K= 20

Longitudes Latitudes

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Fall

Intro Metric PAM Spectral Conclusions

!! !"#$%&'()*+,-"(.-/0)+

1234567887+!57992!27:8+ " +

si = bi − ai max(ai, bi) ai bi i

ai bi, si ≈ 1 → Well classified ai ∼ bi, si ≈ 0 → Neutral ai bi, si ≈ −1 → Badly classified

Intro Metric PAM Spectral Conclusions

Silhouette width 0.0 0.2 0.4

• Sil. coeff. for K= 15

Average silhouette width : 0.09 median −4 −2 2 4 6 8 42 44 46 48 50

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Fall

Intro Metric PAM Spectral Conclusions

Summary on clustering of maxima Classical clustering algorithms (kmeans) are not in compliance with EVT Madogram provides a convenient distance that is marginal free and very fast to compute PAM applied with mado preserves maxima and gives interpretable results

Intro Metric PAM Spectral Conclusions

Our game plan to handle extremes from this rainfall dataset Spatial scale Large (country) Local (region) Problem Dimension reduction Spectral density in moderate dimension Data Weekly maxima Heavy hourly rainfall

• f hourly precipitation

excesses Method Clustering algorithms Mixture of for maxima Dirichlet

Intro Metric PAM Spectral Conclusions

Bayesian Dirichlet mixture model for multivariate excesses (joint work with A. Sabourin) Meteo-France data Wet hourly events at the regional scale (temporally declustered)

• f moderate dimensions (from 2 to 5)

Task 2 Assessing the dependence among rainfall excesses

Intro Metric PAM Spectral Conclusions

Focusing on the “Lyon” cluster

−4 −2 2 4 6 8 42 44 46 48 50

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Intro Metric PAM Spectral Conclusions

Marginal laws : extended Generalized Pareto

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

10 20 30 10 20 30

Station1

Observed Expected Fit with n=174 delta=11.68 sigma=2.37 xi=0.191

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

2 4 6 8 10 2 4 6 8 10

Station21

Observed Expected Fit with n=145 delta=12.18 sigma=1.15 xi=0.115

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

5 10 15 20 5 10 15 20

Station38

Observed Expected Fit with n=153 delta=34.05 sigma=2.24 xi=0.12

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! !! ! !! ! !!!! ! ! ! !!! !!! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

5 10 15 5 10 15

Station42

Observed Expected Fit with n=152 delta=22.28 sigma=1.88 xi=0.146

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! !! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

5 10 15 20 25 5 10 15 20 25

Station68

Observed Expected Fit with n=166 delta=11.76 sigma=2.51 xi=0.195

! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! !! ! ! !! ! ! ! ! ! !! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! !!! ! ! !! ! !!! ! !! !! ! !! !! ! ! ! ! ! ! ! ! ! ! ! !

2 4 6 8 10 12 14 2 4 6 8 10 12 14

Station70

Observed Expected Fit with n=167 delta=5.1 sigma=1.84 xi=0.082

Intro Metric PAM Spectral Conclusions

Defining radius and angular points Example with d = 3 and X = (X1, X2, X3) such that P(Xi < x) = e

−1 x

Simplex S3 = ˘ w = (w1, w2, w3) :

3

X

i=1

wi = 1, wi ≥ 0 ¯ .

Intro Metric PAM Spectral Conclusions

Six out of 18 simplex for the six stations of the “Lyon” cluster

S t a t i

• n

4 2 Station38

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

S t a t i

• n

1 Station42

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

S t a t i

• n

6 8 Station21

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

S t a t i

• n

3 8 Station42

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

S t a t i

• n

2 1 Station70

! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

S t a t i

• n

6 8 Station42

! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

Intro Metric PAM Spectral Conclusions

Mathematical constraints on the distribution of the angular points H P (W ∈ B, R > r) ∼

r→∞

1 r H(B) Features of H H can be non-parametric The gravity center of H has to be centered on the simplex ∀i ∈ {1, . . . , d}, R

Sd wi dH(w) = 1 d

Intro Metric PAM Spectral Conclusions

A few references on Bayesian non-parametric and semi-parametric spectral inference M.-O. Boldi and A. C. Davison. A mixture model for multivariate extremes. JRSS : Series B (Statistical Methodology), 69(2) :217–229, 2007.

• S. Guillotte, F. Perron, and J. Segers.

Non-parametric bayesian inference on bivariate extremes. JRSS : Series B (Statistical Methodology), 2011.

• A. Sabourin and P

. Naveau. Bayesian Drichlet mixture model for multivariate extremes. In revision. P .J. Green. Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82(4) :711, 1995. Roberts, G.O. and Rosenthal, J.S. Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains The Annals of Applied Probability,16,4,2123 :2139, 2006.

Intro Metric PAM Spectral Conclusions

Dirichlet distribution ∀w ∈

• Sd, diri(w | µ, ν) =

Γ(ν) Qd

i=1 Γ(νµi) d

Y

i=1

wνµi −1

i

.

0.00 0.35 0.71 1.06 1.41

w2

µ = (1/3, 1/3, 1/3) and ν = 9

Intro Metric PAM Spectral Conclusions

Dirichlet distribution ∀w ∈

• Sd, diri(w | µ, ν) =

Γ(ν) Qd

i=1 Γ(νµi) d

Y

i=1

wνµi −1

i

.

0.00 0.35 0.71 1.06 1.41

w2

µ = (1/3, 1/3, 1/3) and ν = 9

Intro Metric PAM Spectral Conclusions

Dirichlet distribution ∀w ∈

• Sd, diri(w | µ, ν) =

Γ(ν) Qd

i=1 Γ(νµi) d

Y

i=1

wνµi −1

i

.

0.00 0.35 0.71 1.06 1.41

w2

µ = (.15, .35, .05) and ν = 9 But this one is not centered ! !

Intro Metric PAM Spectral Conclusions

Mixture of Dirichlet distribution Boldi and Davision, 2007 h(µ,p,ν)(w) =

k

X

m=1

pm diri(w | µ · , m, νm) with µ = µ · ,1:k, ν = ν1:k, p = p1:k

Intro Metric PAM Spectral Conclusions

Mixture of Dirichlet distribution Boldi and Davision, 2007 h(µ,p,ν)(w) =

k

X

m=1

pm diri(w | µ · , m, νm) with µ = µ · ,1:k, ν = ν1:k, p = p1:k Constraint on (µ, p) p1 µ.,1 + · · · + pk µ.,k = ( 1

d , . . . , 1 d )

Intro Metric PAM Spectral Conclusions

Inference of Dirichlet density mixtures Boldi and Davison (2007) Prior of [µ|p ] defined on the set p1 µ.,1 + · · · + pk µ.,k = ( 1

d , . . . , 1 d )

Sequential inference : first p, then µ one coordinate after the other

• skewed, not interpretable, slow sampling
• Difficult inference in dimension > 3

Intro Metric PAM Spectral Conclusions

Inference of Dirichlet density mixtures How to build priors for (p, µ) such that p1 µ.,1 + · · · + pk µ.,k = ( 1

d , . . . , 1 d )

Intro Metric PAM Spectral Conclusions

New parametrisation Ex : k = 4 and d = 3 γm : ”Equilibrium” centers built from µ.,m+1, . . . , µ.,k. γm =

k

X

j=m+1

pj pm+1 + · · · + pk µ.,j

Intro Metric PAM Spectral Conclusions

New parametrisation Ex : k = 4 and d = 3

I1

µ1

1

µ.,1, e1 ⇒ γ1 : γ0 γ1 γ0 I1 = e1 ; ⇒ p1

Intro Metric PAM Spectral Conclusions

New parametrisation Ex : k = 4 and d = 3

I1 1 I2

µ µ

1 2

2

µ.,2, e2 ⇒ γ2 : γ1 γ2 γ1 I2 = e2 ; ⇒ p2

Intro Metric PAM Spectral Conclusions

New parametrisation Ex : k = 4 and d = 3

I1 1 I2 2 I3

µ µ µ

1 2 3

µ4

µ.,3, e3 ⇒ γ3 : γ2 γ3 γ2 I3 = e3 ; µ.,4 = γ3. ⇒ p3, p4

Intro Metric PAM Spectral Conclusions

New parametrisation Ex : k = 4 and d = 3

I1 1 I2 2 I3

µ µ µ

1 2 3

µ4

Parametrisation of h with θ = (µ.,1:k−1, e1:k−1, ν1:k) (µ.,1:k−1, e1:k−1) gives (µ.,1:k, p1:k)

Intro Metric PAM Spectral Conclusions

Unconstrained Bayesian modeling for Θ = ‘∞

k=1 Θk;

Θk = ˘ (Sd)k−1 × [0, 1)k−1 × (0, ∞]k−1¯ Prior k ∼ Truncated geometric µ.,m|(µ.,1:m−1, e1:m−1) ∼ Dirichlet em|(µ.,1:m, e1:m−1) ∼ Beta νm ∼ logN Posterior sampling : MCMC reversible jumps

Intro Metric PAM Spectral Conclusions

Summary of the Bayesian schemes Boldi and Davison (2012) Our approach

µ ν ν λ δ σ π w k ~

λ, kmax

πk

k χµ, emean.max, χe

πγ

mν, σν, νmin, νmax

πν

µ · , 1:k−1, e1:k−1

f

µ · , 1:k, p1:k ν1:k w ∈

• Sd

Intro Metric PAM Spectral Conclusions

Simulation example with d = 3 and k = 3 Simulated points with true density Predictive density

0.00 0.35 0.71 1.06 1.41

w3 w1 w2

• 0.00

0.35 0.71 1.06 1.41

w3 w1 w2

0.00 0.35 0.71 1.06 1.41

Intro Metric PAM Spectral Conclusions

Simulation example with d = 5 and k = 3

= =

Intro Metric PAM Spectral Conclusions

Back to our excesses of the “Lyon” cluster Stations 68, 70, 1

0.00 0.35 0.71 1.06 1.41

w3 w1 w2

Intro Metric PAM Spectral Conclusions

Different results from different Monte Carlo chains ? Stations 68, 70, 42

0.00 0.35 0.71 1.06 1.41

w3 w1 w2

• 0.00

0.35 0.71 1.06 1.41

w3 w1 w2

Intro Metric PAM Spectral Conclusions

Take home messages Heavy rainfall over France Clustering of weekly maxima with PAM is fast and gives spatially coherent structures Hourly heavy rainfall over Lyon region appear to be asymptotically independent Exploring high temperatures instead of rainfall data Statistical challenges Moving from bivariate (extremal coefficient) to truly multivariate based clustering algorithms Moving from parametric to truly semi or non parametric spectral models in high dimension (with uncertainty estimates) Handling asymptotically independence in geophysical data

Intro Metric PAM Spectral Conclusions

! ! ! ! ! ! ! ! !

K Silhouette coefficients 2 4 6 8 10 12 14 16 18 20 −0.1 0.0 0.1 0.2 0.3

Silhouette coefficients for different K

Intro Metric PAM Spectral Conclusions

Guillou et al, 2012

Dependence function Vη

Rε = {(x, y) : x > ε, y > ε} M•,n,ε componentwise maxima such that (Xi, Yi) occur within Rεbn lim

ε→0 lim n→∞ P

• MX,n,ε

bn ≤ x, MY,n,ε bn ≤ y

• = Gη(x, y) = exp
• − Vη(x, y)
• Vη(x, y) = η

1

• max

ω x , 1 − ω y 1

η

dHη(ω) ⇒ Vη homogeneous of order −1/η: Vη(tx, ty) = t−1/ηVη(x, y) ⇒ Gη max-stable: Gn

η(nηu, nηv) = G(x, y)

Intro Metric PAM Spectral Conclusions

Guillou et al, 2012

η-Madogram (cont’d)

Vη symmetric ⇒ extremal coefficient θ : 1 ≤ θ := Vη(1, 1) Vη(1, +∞) ≤ 2 ⇒ independence (θ → 2) between the marginal distributions ⇒ dependence (θ = 1) ν(η) = θ 1 + θ − 1 2