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Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Extreme Value Theory and Dimension reduction for the study of hyperspectral images

Laurent GARDES

INRIA Rhˆ

• ne-Alpes, LJK, Team MISTIS

http://mistis.inrialpes.fr/people/gardes/

Habilitation ` a diriger des recherches

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Outline

• I. Inference on Weibull tail distributions.
• II. Extreme conditional quantile estimation.
• III. Dimension reduction and regression.
• IV. Further works.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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1 Inference on Weibull tail distributions 2 Extreme conditional quantile estimation 3 Dimension reduction and regression 4 Further works

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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In collaboration with

• J. Diebolt (D.R. CNRS)
• S. Girard (C.R. INRIA)
• A. Guillou (Professeur, Universit´

e de Strasbourg) Associated publications TEST (2008) JSPI (two in 2008) REVStat (2006) CIS (2005)

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Recalls on Extreme Value Theory

Let X1, . . . , Xn be n independent random variables with the same cumulative distribution function F. The order statistics are denoted by X1,n ≤ . . . ≤ Xn,n. Goal: Extreme quantile estimation i.e. for αn → 0 as n → ∞, estimation of

q(αn) = ¯ F ←(αn).

Main difficulty: If αn is small (nαn → 0),

P(q(αn) > Xn,n) → 1.

An extrapolation is thus needed!

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Recalls on Extreme Value Theory

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Recalls on Extreme Value Theory

The main result is the extreme value theorem: Theorem If there exist two sequences (an > 0), (bn) and γ ∈ R such that

P Xn,n − bn an ≤ x ff → Hγ(x),

then

Hγ(x) =  exp[−(1 + γx)−1/γ

+

] if γ = 0, exp(−e−x) if γ = 0,

• Hγ(.) is the extreme value distribution.
• γ is the extreme value index.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Recalls on Extreme Value Theory

Three maximum domains of attraction (MDA)

• γ > 0: Fr´

echet MDA (Pareto, student, Cauchy, . . .)

• γ < 0: Weibull MDA (uniform)
• γ = 0: Gumbel MDA (normal, Weibull, exponential, log-normal,

etc . . .)

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Weibull tail distributions

• Sub family of the Gumbel MDA.
• The survival function is given by:

¯ F(x) = exp n −x1/θL(x)

• , θ > 0.

L(.) is a slowly varying function: for all λ > 0,

lim

x→∞

L(λx) L(x) = 1.

• θ is the Weibull tail-coefficient.
• Examples of Weibull tail distributions: Weibull, normal, gamma,

exponential, etc . . .

• Log-normal is not a Weibull tail distribution.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Weibull tail distributions

• An estimator of θ was proposed by Beirlant et al. (1996)

ˆ θB

n = kn−1

X

i=1

(log(Xn−i+1,n) − log(Xn−kn+1,n)) ,kn−1 X

i=1

(log2(n/i) − log2(n/kn)) ,

where log2(.) = log log(.) and (kn) is a sequence of integers such that 1 < kn < n.

• The corresponding extreme quantile estimator is defined by:

ˆ qB(αn) = Xn−kn+1,n „ log(1/αn) log(n/kn) «ˆ

θB

n

.

• The asymptotic properties of these estimators are not

established by the authors.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Contributions

• Generalizations of the estimator ˆ

θB

n .

− Introducing weighted estimators.

kn−1

X

i=1

W (i/kn)(log(Xn−i+1,n)−log(Xn−kn+1,n)) ,kn−1 X

i=1

W (i/kn)(log2(n/i) − log2(n/kn))

− Using others normalizing sequences.

1 Tn

kn−1

X

i=1

(log(Xn−i+1,n) − log(Xn−kn+1,n)), Tn ∼ kn log(n/kn) .

− Bias corrected estimator of θ.

• Generalization of ˆ

qB(αn) by replacing ˆ θB

n by any other estimator

• f θ.
• Bias corrected estimator of q(αn).

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Bias corrected estimator of θ

The estimators are based on the following approximation: for α and β small enough:

q(α) q(β) = „ − log(α) − log(β) «θ ℓ(− log(α)) ℓ(− log(β)) ≈ „ − log(α) − log(β) «θ .

A second order condition is required in order to specify the bias term: (H.1) There exist ρ < 0 and a function b(.) satisfying b(x) → 0 as x → ∞ such that locally uniformaly,

lim

x→∞

log(ℓ(λx)/ℓ(x)) b(x)Kρ(λ) = 1,

where Kρ(λ) = (λρ − 1)/ρ. The function b(.) ∈ RVρ i.e. b(x) = xρℓ∗(x) where ℓ∗(.) is a slowly varying function.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Bias corrected estimator of θ

Under (H.1), we have approximately:

Zj ≈ θ + b(log(n/kn))xj + ηj, j = 1, . . . , kn,

where Zj = j log(n/j)(log(Xn−j+1,n) − log(Xn−j,n)), xj = log(n/kn)/ log(n/j) and ηj is an error term. Ignoring the bias term, leads to the estimator k−1

n

kn

j=1 Zj.

Estimating θ and b(log(n/kn)) by the method of least-squares leads to the bias corrected estimator:

ˆ θD

n = 1

kn

kn

X

j=1

Zj − ˆ b(log(n/kn)) kn

kn

X

j=1

xj,

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Bias corrected estimator of θ

Theorem Under (H.1), if x|b(x)| → ∞ as x → ∞ and

k1/2

n

log(n/kn) b(log(n/kn)) → Λ = 0,

then,

k1/2

n

log(n/kn) (ˆ θD

n − θ) d

→ N(0, θ2).

• Condition x|b(x)| → ∞ implies that ρ ≥ −1.
• ˆ

θD

n converges to θ with the same rate of convergence as ˆ

θB

n but

without asymptotic bias.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Bias corrected estimator of θ

Illustration with a simulation of N = 100 samples of size n = 500 from a N(0, 1) distribution (θ = 1/2). Horizontal axis: kn. Vertical axis: mean of the estimator (left) and MSE of the estimator (right). In black: ˆ θB

n and in grey: ˆ

θD

n .

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Further works

• Propose a new model that incompasses Weibull tail and Heavy

tail distributions (work accepted in JSPI).

• Use the previous model to construct statistical hypothesis test
• n the tail distribution.
• Prove asymptotic results on the estimators when the
• bservations are not independent.
• Estimation of the second order parameter ρ.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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1 Inference on Weibull tail distributions 2 Extreme conditional quantile estimation 3 Dimension reduction and regression 4 Further works

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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In collaboration with

• A. Daouia (MCF, Universit´

e Toulouse 1)

• S. Girard (C.R. INRIA)

Supervision research activities

• E. Ursu (Postdoctoral fellow)
• A. Lekina (past PhD student)
• J. el-Methni (new PhD student)

Contracts and grants ANR (French Research Agency), program VMC CEA (Atomic Energies Commission) of Cadarache Associated publications TEST (to appear) Extremes (2010) JMVA (2008 and 2010)

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Motivation

Rainfalls data in the C´ evennes-Vivarias region Data provided by Laboratoire d’´ etude des Transferts en Hydrologie et Environnement (LTHE). Work supported by the ANR.

Horizontal axis: longitude. Vertical axis: latitude. Color scale: altitude.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Motivation

• 142 rain gauge stations.
• Hourly rainfalls measured from 1993 to 2000.
• Total number of observations: n = 264056.
• Y is the variable of interest (hourly rainfall).
• X is the covariate recorded with Y (longitude + latitude +

altitude).

• Aim: estimate the amount of rain expected to be exceeded once

every T years i.e estimate the T-year return level.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Framework

• The conditional survival distribution function of Y when the

covariate is equal to x is:

¯ F(y, x) = y−1/γ(x)L(y, x).

• γ(.) is an unknown positive function called the conditional tail

index.

• For x fixed, L(., x) is a slowly varying function.
• In early studies, the conditional distribution is assumed to be

Gumbel: unrealistic assumption.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Main contributions

Two situations were investigated:

• Random covariate

− Estimation of the conditional tail index γ(x). − Estimation of conditional extreme quantiles. − Estimation of small tail probabilities.

• Deterministic covariate

− Estimation of the conditional tail index γ(x). − Estimation of conditional extreme quantiles. − Application to the estimation of return levels.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Methodology

• Let (Y1, x1), . . . , (Yn, xn) be a sample of independent
• bservations
• To estimate the conditional quantile q(αn, x), only the
• bservations for which the associated covariates are ”close” to x

are required.

• We consider the observations Yi for which the associated

covariates belong to the ball B(x, hn,x) (ball centered at point x with radius hn,x → 0 as n → ∞).

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Methodology

• These observations are denoted by Z1, . . . , Zmn,x. The number of

covariates in the ball B(x, hn,x) is denoted by mn,x.

• Corresponding order statistics: Z1,mn,x ≤ . . . ≤ Zmn,x,mn,x.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Estimation of q(αn, x)

We propose the following estimator:

ˆ q(αn, x) = Zmn,x −kn,x +1,mn,x „ kn,x αnmn,x «ˆ

γn(x)

.

• kn,x is a positive sequence such that 1 < kn,x < mn,x.
• Estimator in the same spirit as the Weissman (1978) estimator.
• ˆ

γn(x) estimator of γ(x) such that:

k1/2

n,x (ˆ

γn(x) − γ(x)) d → N(0, AV(x)).

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Estimation of q(αn, x)

Two sources of bias

• The bias introduced by the slowly varying function. It is

controlled via the function b(., x) ∈ RVρ(x) (that appears in a second order condition).

• The bias due to the conditional framework. It is controlled by

ωn(a) = sup ˛ ˛ ˛ ˛log q(α, t) q(α, t′) ˛ ˛ ˛ ˛ , α ∈]a, 1 − a[, (t, t′) ∈ B(x, hn,x)2 ff .

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Asymptotic normality

Under a second order condition, if kn,x → ∞, mn,x/kn,x → ∞, if there exists δ2 > 0 such that

k2

n,xωn(m−(1+δ2) n,x

) → 0 and k1/2

n,x b(mn,x/kn,x, x) → 0,

then, if αn is such that αn → 0 and ⌊mn,xαn⌋ → c ∈ N,

k1/2

n,x

log(kn,x/(mn,xαmn,x )) „ ˆ q(αn, x) q(αn, x) − 1 «

d

→ N(0, AV(x)).

• The asymptotic normality is provided by ˆ

γn(x).

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Return levels estimation

• Estimation in the whole region of the 10-years return level.
• Conditional extreme quantile of order 1/(365 × 24 × 10).
• ”Extreme” since there are only 7 years of observations.

Montpellier Nimes Ales Privas Valence Le Puy Mende Millau 75 80 85 90 95 100 105 110 115

• Return level is globally decreasing with the altitude.
• Altitude is not the unique factor.
• Result validated by the LTHE.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Further works

• Estimation of frontier functions (work accepted in the book

”Festschrift in honor of L. Simar, Springer”)

• Estimation of conditional extreme quantile under more general

assumptions on the tail distribution.

• Choice of the sequences (hn,t) and (kn,t).
• In a short simulation study, we remark that:

− temporal dependence does not affect the bias (but increases the variance). − spatial dependence slightly affects the bias.

Make a theoretical study to confirm these points.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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1 Inference on Weibull tail distributions 2 Extreme conditional quantile estimation 3 Dimension reduction and regression 4 Further works

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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In collaboration with

• S. Dout´

e (C.R. CNRS, LPG)

• S. Girard (C.R. INRIA)

Supervision research activities

• C. Bernard-Michel (postdoctoral fellow)
• M. Fauvel (postdoctoral fellow)

Contracts and grants ANR MDC0 Associated publications Journal of Geophysical Research (2009) Statistics and Computing (2009) Biometrics (2008)

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Motivation

• Study of hyperspectral images from south polar region of Mars

(work supported by the ANR, in collaboration with the Laboratoire de Plan´ etologie de Grenoble (LPG)).

• Spectra (of dimension d ≈ 184) collected by the imaging

spectrometer OMEGA aboard MARS express.

• Aim: estimate some ground properties (proportion of water,

CO2, etc . . .)

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Motivation

• Learning data-set (Xi, Yi), i = 1, . . . , n (n ≈ 30000)
• Yi ∈ R: given value of a physical parameter (for example

proportion of water).

• Xi ∈ Rd (d = 184): the associated spectra obtained by the

radiative transfert algorithm.

• Aim: estimate the link function G defined by:

Yi = G(Xi, ηi), i = 1, . . . , n,

where ηi is a random error.

• Estimation of G is difficult (curse of dimensionality) when d is

large.

• Existing methods:

− Nearest neighbors. − Partial Least Square regression (PLS). − Support Vector Machine regression (SVM). − Sliced Inverse Regression.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Recall on SIR method

• Single index model: (Xi, Yi) independent observations from the

random vector (X, Y ) ∈ Rd × R satisfying

Y = g(βtX, η), η and X independent.

• All the information on Y contained in X is also contained in

βtX.

• β ∈ Rd is the effective dimension reduction (edr) direction.
• Estimation of G : Rd → R replaced by the estimation of

g : R → R

• Estimation of β:
• divide the support of Y into H slices S1, . . . , SH.
• maximize the estimated variance between slices.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Illustration of SIR method

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Recall on SIR method

• The estimator ˆ

β of the e.d.r. direction is solution of the problem:

arg max

β

βtˆ Γβ under the constraint βt ˆ Σβ = 1.

where

ˆ Σ = 1 n

n

X

i=1

(Xi − ¯ X)(Xi − ¯ X)t, with ¯ X = 1 n

n

X

i=1

Xi,

and

ˆ Γ = 1 n

H

X

j=1

nj( ¯ Xj − ¯ X)( ¯ Xj − ¯ X)t, with ¯ Xj = 1 nj X

i:Yi ∈Sj

Xi.

• ˆ

β is the eigenvector of the matrix ˆ Σ−1ˆ Γ associated to the largest eigenvalue.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Recall on SIR method

Other way to introduce the SIR estimator ˆ β

• Inverse regression model (proposed by Cook (2007)):

X = µ + V β

H−1

X

j=1

cjsj(Y ) + ζ,

where µ ∈ Rd, V is a d × d covariance matrix, ζ a Nd(0, V ) random vector independent of Y , cj ∈ R and sj : R → R, for j = 1, . . . , H − 1.

• Taking

sj(.) = I{. ∈ Sj} − nj n , where nj =

n

X

i=1

I{Yi ∈ Sj}, j = 1, . . . , H − 1,

the maximum likelihood estimator of β is (up to a scale parameter) the SIR estimator ˆ β.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Contribution

SIR requires the inversion of ˆ Σ. When d is large, ˆ Σ is ill-conditioned.

• Proposition of a regularized SIR version in order to avoid the

inversion of ˆ Σ.

• Application to hyperspectral images from Mars.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Regularized SIR method

• Introduction of a Gaussian prior in Cook’s model: conditionally

to (˜ ρ, Y ),

Θ = @˜ ρ−1/2

H−1

X

j=1

cjsj(Y ) 1 A β ∼ Nd(0, Ω),

where ˜ ρ = (βt ˆ Σβ)/(βtV β).

• The prior covariance matrix Ω is known.
• conditionally to (˜

ρ, Y ), Θ is proportional to β.

• Ω describes which directions are most likely to contain β.
• The ”MAP” estimator of β is the eigenvector associated to the

largest eigenvalue of:

(Ωˆ Σ + Id)−1Ωˆ Γ,

• The inversion of ˆ

Σ is not required.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Regularized SIR method

Links with existing methods

• Taking Ω = ˆ

Σ−1 leads to SIR. Directions corresponding to small variance are most likely.

• Regularized SIR also incompasses the ”Ridge-regularization” and

the ”PCA+SIR” regularization. New method

• Taking Ω = τ −1 ˆ

Σ leads to ”Tikhonov-SIR”. τ is a regularization

• parameter. Directions corresponding to large variance are most

likely.

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Application to hyperspectral images from Mars

Estimation of the proportion of CO2 by SIR (left) and Regularized SIR (right).

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Further works

• Asymptotic results for the regularized SIR estimator of β (joint

work with A.F. Yao, Universit´ e Aix-Marseille II)

• Add a spatial regularization.
• Propose regularization methods for other regression tools (PLS,

Nearest Neighbors, etc . . .).

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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1 Inference on Weibull tail distributions 2 Extreme conditional quantile estimation 3 Dimension reduction and regression 4 Further works

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Further works

• Study of conditional Weibull tail distributions (”Weibull tail” +

”conditional extremes”).

• Dimension reduction for the estimation of conditional extreme

quantiles (”conditional extremes” + ”Dimension reduction”).

• Use other statistical tools for the study of extreme values (for

example, use nonparametric estimation methods to estimate the slowly varying function).

Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works

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Summary (Period 2004-2010)

• 13 articles in international scientific journals.
• 2 book chapters.
• 2 PhD students.
• 3 postdoctoral fellows.
• 1 contract with the CEA.
• 2 financial supports by the ANR.