A randomized orthogonal array-based procedure for the estimation of first- and second-order Sobol' indices

Abstract

In variance-based sensitivity analysis, the method of Sobol' [Sensitivity analysis for nonlinear mathematical models. Math Model Comput Exp. 1993;1:407–414] allows one to compute Sobol' indices (SI) using the Monte Carlo integration. One of the main drawbacks of this approach is that estimating SI requires a number of simulations which is dependent on the dimension of the model of interest. For example, estimating all the first- or second-order SI of a d-dimensional function basically requires or independent input vectors, respectively. Some interesting combinatorial results have been introduced to weaken this defect, in particular by Saltelli [Making best use of model evaluations to compute sensitivity indices. Comput Phys Commun. 2002;145:280–297] and more recently by Owen [Variance components and generalized Sobol' indices. SIAM/ASA J Uncertain Quantification. 2013;1:19–41], but the quantities they estimate still depend linearly on the dimension d. In this paper, we introduce a new approach to estimate all the first- and second-order SI by using only two input vectors. We establish theoretical properties of such a method for the estimation of first-order SI and discuss the generalization to higher order indices. In particular, we prove on numerical examples that this procedure is tractable and competitive for the estimation of all first- and second-order SI. 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 – that is, valid without strong regularity assumptions – sensitivity analysis, as the method of Sobol' provides, could be of great help. This article has supplementary material online.

Keywords:

• global sensitivity analysis
• variance-based sensitivity indices
• numerical integration
• orthogonal arrays

Acknowledgments

The authors are grateful to Eric Blayo, Jean-Michel Brankart and Pierre Brasseur for valuable discussions on the simulator MODECOGeL and more generally on marine ecosystem models. They also thank Art Owen for his helpful comments.

Supplementary data

Supplemental data for this article can be accessed 10.1080/00949655.2014.971799.

Notes

1. GHER: GeoHydrodynamics and Environment Research, Université de Liège, Belgium. LOBEPM: Laboratoire d'Océanologie Biologique et d'Écologie du Plancton Marin, Université Pierre et Marie Curie, France

Additional information

Funding

This work has been partially supported by French National Research Agency (ANR) through COSINUS program (project COSTA-BRAVA no. ANR-09-COSI-015).