Estimating Sobol' indices combining Monte Carlo estimators and Latin hypercube sampling
Résumé
In variance-based sensitivity analysis, the method of Sobol' (1993) allows to compute Sobol' indices using Monte Carlo integration. One of the main drawbacks of this approach is that the estimation of Sobol' indices requires the use of several samples. For example, in a d-dimensional space, the estimation of all the first-order Sobol' indices requires d+1 samples. Some interesting combinatorial results have been introduced to weaken this defect, in particular by Saltelli (2002) and more recently by Owen (2012) but the quantities they estimate still require O(d) samples. In this paper, we introduce a new approach to estimate for any k all the k-th order Sobol' indices by using only two samples. We establish theoretical properties of such a method for the first-order Sobol' indices and discuss the generalization to higher-order indices. As an illustration, we propose to apply this new approach to a marine ecosystem model of the Ligurian sea (northwestern Mediterranean) in order to study the relative importance of its several parameters. The calibration process of this kind of chemical simulators is well-known to be quite intricate, and a rigorous and robust --- i.e. valid without strong regularity assumptions --- sensitivity analysis, as the method of Sobol' provides, could be of great help.
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