Asymptotic normality and efficiency of two Sobol index estimators

Abstract

Many mathematical models involve input parameters, which are not precisely known. Global sensitivity analysis aims to identify the parameters whose uncertainty has the largest impact on the variability of a quantity of interest (output of the model). One of the statistical tools used to quantify the influence of each input variable on the output is the Sobol sensitivity index. We consider the statistical estimation of this index from a finite sample of model outputs: we present two estimators and state a central limit theorem for each. We show that one of these estimators has an optimal asymptotic variance. We also generalize our results to the case where the true output is not observable, and is replaced by a noisy version.Résumé. De nombreux modèles mathématiques font intervenir plusieurs paramètres qui ne sont pas tous connus précisément. L'analyse de sensibilité globale se propose de sélectionner les paramètres d'entrée dont l'incertitude a le plus d'impact sur la variabilité d'une quantité d'intérêt (sortie du modèle). Un des outils statistiques pour quantifier l'influence de chacune des entrées sur la sortie est l'indice de sensibilité de Sobol. Nous considérons l'estimation statistique de cet indice à l'aide d'un nombre fini d'échantillons de sorties du modèle: nous présentons deux estimateurs de cet indice et énonçons un théorème central limite pour chacun d'eux. Nous démontrons que l'un de ces deux estimateurs est optimal en terme de variance asymptotique. Nous généralisons également nos résultats au cas où la vraie sortie du modèle n'est pas observée, mais où seule une version dégradée (bruitée) de la sortie est disponible.1991 Mathematics Subject Classification. 62G05, 62G20.The dates will be set by the publisher.

Sensitivity indices for multivariate outputs

“…Thanks to the simple form of this estimator, the following Proposition can be proved in a way similar to the one used to prove Proposition 2.2 and Proposition 2.5 of [1] (ie., by an application of the so-called Delta method). Proposition 4.1 Suppose Y ∈ L 4 (Ω, R k ), and that Σ is positive-definite.…”

“…In the scalar case (k = 1), it is customary to estimate S u,Scal (f ) by using a Monte-Carlo pick-freeze method [3,1], which uses a finite sample of evaluations of f . In this Section, we propose a pick-freeze estimator for the vector case which generalizes the T N estimator studied in [1]. We set: Y u = f (X u , X ′ ∼u ) where X ′ ∼u is an independent copy of X ∼u .…”

Sensitivity analysis of complex models: Coping with dynamic and static inputs

“…The proposed approach is an alternative to the Pick and Freeze method [27,28,29]. However, the Pick and Freeze method is computationally demanding because it requires N ⇥(N d +1) model runs where N d is the number of input factors and N is the sample size.…”

Estimation of quantile oriented sensitivity indices

“…Hence, it is natural to wonder how these indices could be estimated. In the case where the contrast function is given by mean-contrast functions which correspond to Sobol index, there exist a pretty large literature dedicated to the estimation of such index (see, e.g., [15,13,17,9,10]). In this section, we first describe how quantile contrast index can be estimated.…”

“…Following [10], in order to improve the efficiency of the estimator, CT E α (Y ) and E(Y ) could be estimated by using the complete sample X, X * . A further study is needed to get its asymptotic properties.…”