Sobol' indices and variance reduction diagram estimation from samples used for uncertainty propagation
Résumé
In this paper we present an efficient algorithm to estimate first order and second order Sobol' indices (SI) and its relationships with the inputs parameters variation range width. In the variance-based global sensitivity measure, SI are quantities defined by normalizing parts of variance in ANOVA decomposition (Sobol’, 1993). They are estimated from the ratio between the variance of the conditional expectation of the output given the input and the unconditional variance of the output (eq .1). Many techniques have been proposed to estimate these indices. A recent review of these methods can be found in (Borgonovo & Plischke, 2015). Among others Monte-Carlo-based algorithm (Sobol’ 1993, Saltelli et al., 1999, Jansen 1999, Monod et al., 2006, etc.) require a special random sampling scheme i.e. they cannot directly use samples used for the uncertainty propagation. Building on a similar idea than Plischkle (Plischke, 2010) we propose a methodology that allows the computation of SI form a set of given data. We propose to start from the local conditional variance to derive the global sensitivity indicator by estimating the variance of the conditional expectation (Eq. 2). The local information on sensitivity is summarized under the form of a Diagram of Expected Variance Reduction (DEVR), which relates the local reduction in uncertainty with the domain of variation of the considered input parameter with reduced width.
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