Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent Extreme Value Theory and Dimension GARDES Inference on reduction for the study of hyperspectral images Weibull tail distributions Extreme conditional quantile estimation Laurent GARDES Dimension reduction and regression INRIA Rhˆ one-Alpes, LJK, Team MISTIS Further works http://mistis.inrialpes.fr/people/gardes/ Habilitation ` a diriger des recherches 1
Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail Outline distributions Extreme conditional I. Inference on Weibull tail distributions. quantile estimation II. Extreme conditional quantile estimation. Dimension reduction and regression III. Dimension reduction and regression. Further works IV. Further works. 2
Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent GARDES 1 Inference on Weibull tail distributions Inference on Weibull tail distributions 2 Extreme conditional quantile estimation Extreme conditional quantile 3 Dimension reduction and regression estimation Dimension reduction and regression 4 Further works Further works 3
Extreme Value Theory and Dimension reduction for the study of hyperspectral images Laurent In collaboration with GARDES J. Diebolt (D.R. CNRS) Inference on Weibull tail S. Girard (C.R. INRIA) distributions Extreme A. Guillou (Professeur, Universit´ e de Strasbourg) conditional quantile estimation Associated publications Dimension TEST (2008) reduction and regression JSPI (two in 2008) Further works REVStat (2006) CIS (2005) 4
Extreme Value Theory and Dimension Recalls on Extreme Value Theory reduction for the study of hyperspectral images Laurent GARDES Inference on Let X 1 , . . . , X n be n independent random variables with the same Weibull tail distributions cumulative distribution function F . The order statistics are denoted Extreme by X 1 , n ≤ . . . ≤ X n , n . conditional quantile estimation Goal: Extreme quantile estimation i.e. for α n → 0 as n → ∞ , Dimension estimation of reduction and regression q ( α n ) = ¯ F ← ( α n ) . Further works Main difficulty: If α n is small ( n α n → 0), P ( q ( α n ) > X n , n ) → 1 . An extrapolation is thus needed! 5
Extreme Value Theory and Dimension Recalls on Extreme Value Theory reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works 6
Extreme Value Theory and Dimension Recalls on Extreme Value Theory reduction for the study of hyperspectral images Laurent GARDES The main result is the extreme value theorem: Inference on Weibull tail distributions Theorem If there exist two sequences ( a n > 0), ( b n ) and γ ∈ R such Extreme conditional that quantile X n , n − b n ff estimation ≤ x → H γ ( x ) , P a n Dimension reduction and then regression exp[ − (1 + γ x ) − 1 /γ ] if γ � = 0 , Further works H γ ( x ) = + exp( − e − x ) if γ = 0 , • H γ ( . ) is the extreme value distribution. • γ is the extreme value index. 7
Extreme Value Theory and Dimension Recalls on Extreme Value Theory reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions Extreme Three maximum domains of attraction (MDA) conditional quantile estimation • γ > 0: Fr´ echet MDA (Pareto, student, Cauchy, . . . ) Dimension • γ < 0: Weibull MDA (uniform) reduction and regression • γ = 0: Gumbel MDA (normal, Weibull, exponential, log-normal, Further works etc . . . ) 8
Extreme Value Theory and Dimension Weibull tail distributions reduction for the study of hyperspectral images Laurent GARDES • Sub family of the Gumbel MDA. Inference on • The survival function is given by: Weibull tail distributions n o Extreme ¯ − x 1 /θ L ( x ) F ( x ) = exp , θ > 0 . conditional quantile estimation L ( . ) is a slowly varying function: for all λ > 0, Dimension reduction and L ( λ x ) regression lim L ( x ) = 1 . x →∞ Further works • θ is the Weibull tail-coefficient. • Examples of Weibull tail distributions: Weibull, normal, gamma, exponential, etc . . . • Log-normal is not a Weibull tail distribution. 9
Extreme Value Theory and Dimension Weibull tail distributions reduction for the study of hyperspectral images Laurent GARDES • An estimator of θ was proposed by Beirlant et al. (1996) Inference on Weibull tail distributions k n − 1 , k n − 1 ˆ X X Extreme θ B n = (log( X n − i +1 , n ) − log( X n − k n +1 , n )) (log 2 ( n / i ) − log 2 ( n / k n )) , conditional quantile i =1 i =1 estimation where log 2 ( . ) = log log( . ) and ( k n ) is a sequence of integers such Dimension reduction and that 1 < k n < n . regression • The corresponding extreme quantile estimator is defined by: Further works „ log(1 /α n ) « ˆ θ B n q B ( α n ) = X n − k n +1 , n ˆ . log( n / k n ) • The asymptotic properties of these estimators are not established by the authors. 10
Extreme Value Theory and Dimension Contributions reduction for the study of hyperspectral images Laurent • Generalizations of the estimator ˆ θ B GARDES n . − Introducing weighted estimators. Inference on Weibull tail distributions kn − 1 , kn − 1 Extreme X X W ( i / k n )(log( X n − i +1 , n ) − log( X n − kn +1 , n )) W ( i / k n )(log 2 ( n / i ) − log 2 ( n / k n )) conditional i =1 i =1 quantile estimation Dimension − Using others normalizing sequences. reduction and regression k n − 1 Further works 1 k n X (log( X n − i +1 , n ) − log( X n − k n +1 , n )) , T n ∼ log( n / k n ) . T n i =1 − Bias corrected estimator of θ . q B ( α n ) by replacing ˆ θ B • Generalization of ˆ n by any other estimator of θ . • Bias corrected estimator of q ( α n ). 11
Extreme Value Theory and Dimension Bias corrected estimator of θ reduction for the study of hyperspectral images The estimators are based on the following approximation: for α and Laurent GARDES β small enough: „ − log( α ) „ − log( α ) Inference on « θ ℓ ( − log( α )) « θ q ( α ) Weibull tail ℓ ( − log( β )) ≈ q ( β ) = . distributions − log( β ) − log( β ) Extreme conditional A second order condition is required in order to specify the bias term: quantile estimation Dimension (H.1) There exist ρ < 0 and a function b ( . ) satisfying b ( x ) → 0 as reduction and regression x → ∞ such that locally uniformaly, Further works log( ℓ ( λ x ) /ℓ ( x )) lim = 1 , x →∞ b ( x ) K ρ ( λ ) where K ρ ( λ ) = ( λ ρ − 1) /ρ . The function b ( . ) ∈ RV ρ i.e. b ( x ) = x ρ ℓ ∗ ( x ) where ℓ ∗ ( . ) is a slowly varying function. 12
Extreme Value Theory and Dimension Bias corrected estimator of θ reduction for the study of hyperspectral images Laurent GARDES Under (H.1) , we have approximately: Inference on Weibull tail Z j ≈ θ + b (log( n / k n )) x j + η j , j = 1 , . . . , k n , distributions Extreme where Z j = j log( n / j )(log( X n − j +1 , n ) − log( X n − j , n )), conditional quantile x j = log( n / k n ) / log( n / j ) and η j is an error term. estimation Dimension reduction and � k n Ignoring the bias term, leads to the estimator k − 1 j =1 Z j . regression n Further works Estimating θ and b (log( n / k n )) by the method of least-squares leads to the bias corrected estimator: k n k n ˆ n = 1 b (log( n / k n )) θ D ˆ X X Z j − x j , k n k n j =1 j =1 13
Extreme Value Theory and Dimension Bias corrected estimator of θ reduction for the study of hyperspectral images Laurent GARDES Inference on Theorem Under (H.1) , if x | b ( x ) | → ∞ as x → ∞ and Weibull tail distributions k 1 / 2 Extreme n log( n / k n ) b (log( n / k n )) → Λ � = 0 , conditional quantile estimation then, Dimension k 1 / 2 reduction and n n − θ ) d log( n / k n ) (ˆ θ D → N (0 , θ 2 ) . regression Further works • Condition x | b ( x ) | → ∞ implies that ρ ≥ − 1. • ˆ n converges to θ with the same rate of convergence as ˆ θ D θ B n but without asymptotic bias. 14
Extreme Value Theory and Dimension Bias corrected estimator of θ reduction for the study of hyperspectral images Laurent GARDES Illustration with a simulation of N = 100 samples of size n = 500 Inference on from a N (0 , 1) distribution ( θ = 1 / 2). Weibull tail distributions Extreme conditional quantile estimation Dimension reduction and regression Further works Horizontal axis: k n . Vertical axis: mean of the estimator (left) and MSE of the estimator (right). In black: ˆ n and in grey: ˆ θ B θ D n . 15
Extreme Value Theory and Dimension Further works reduction for the study of hyperspectral images Laurent GARDES Inference on Weibull tail distributions • Propose a new model that incompasses Weibull tail and Heavy Extreme conditional tail distributions (work accepted in JSPI). quantile estimation • Use the previous model to construct statistical hypothesis test Dimension reduction and on the tail distribution. regression • Prove asymptotic results on the estimators when the Further works observations are not independent. • Estimation of the second order parameter ρ . 16
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