Indices de Sobol généralisés aux variables dépendantes : tests de performance de l’algorithme HOGS couplé à plusieurs estimateurs paramétriques

Sainte-Marie, Julien; Viaud, Gautier; Cournède, Paul-Henry

Journal de la société française de statistique, Tome 158 (2017) no. 1, pp. 68-89.

L’algorithme “Hierarchically Orthogonal Gram–Schmidt” (HOGS) ( Chastaing et al., 2015 ) estime des indices de Sobol généralisés aux modèles à entrées dépendantes, quantifiant explicitement la sensibilité du modèle due aux corrélations. HOGS construit un méta-modèle pour chaque variable d’intérêt par projection sur une base fonctionnelle bien choisie pour le calcul des indices. Les coefficients de projection sont obtenus par l’estimateur des moindres carrés (OLS) ou les régressions pénalisées lasso, ridge et Elastic Net (EN). Quatre cas d’étude sont proposés : trois modèles simples permettent d’appréhender le fonctionnement de HOGS, et le modèle LNAS (Log-Normal Allocation and Senescence) dédié à la dynamique complexe de la croissance des plantes. Plusieurs configurations de HOGS et la pertinence du méta-modèle sont étudiées grâce à un indice de consistance. L’interprétation des indices de Sobol est illustrée grâce à LNAS. En conclusion, HOGS-OLS est la méthode la plus performante lorsque les ressources informatiques ne sont pas limitantes. Dans le cas contraire, la question de l’estimation paramétrique avec hypothèse de sparsité met en évidence que : i) EN est plus robuste mais plus coûteux numériquement que le Lasso, ii) HOGS génère une base trop grande, créant de la sparsité artificielle. Un amendement de HOGS a été proposé pour réduire la dimension de la base.

The algorithm “Hierarchically Orthogonal Gram-Schmidt” (HOGS) ( Chastaing et al., 2015 ) estimates generalized Sobol indices dedicated to models with dependent inputs, quantifying explicitly the model sensitivity due to correlations. HOGS constructs a meta-model for each variable of interest by projection on a functional basis suited to indices calculation. Regression coefficients are obtained with the ordinary least-square estimator (OLS) or penalized regression methods Lasso, Ridge and Elastic Net (EN). Four study cases are proposed: three toy models allowing to investigate HOGS functioning and numerical properties, and the LNAS (Log-Normal Allocation and Senescence) model dedicated to the complex dynamics of plant growth. Several HOGS configurations and meta-model accuracy are tested by means of a consistency index. An interpretation of Sobol indices is given for LNAS. It appears that HOGS-OLS is the most efficient method when simulation resources are not limited. Otherwise, considering the issue of parameter estimation with sparsity highlights that: i) EN is more robust but more costly than Lasso, ii) the basis constructed by HOGS is too large which creates artificial sparsity. A modification of HOGS has been proposed to reduce the dimension of the basis.

Mot clés : analyse de sensibilité, HDMR, HOGS, régression pénalisée
Keywords: sensitivity analysis, HDMR, HOGS, penalized regression

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