Reliable error estimation for Sobol' indices

Lluís Antoni Jiménez Rugama (1) , Laurent Gilquin (2)

1 Illinois Institute of Technology - Department of Applied Mathematics
2 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows

In the field of sensitivity analysis, Sobol’ indices are sensitivity measures widely used to assess the importance of inputs of a model to its output. The estimation of these indices is often performed trough Monte Carlo or quasi-Monte Carlo methods. A notable method is the replication procedure that estimates first-order indices at a reduced cost in terms of number of model evaluations. An inherent practical problem of this estimation is how to quantify the number of model evaluations needed to ensure that estimates satisfy a desired error tolerance. This paper addresses this challenge by proposing a reliable error bound for first-order and total effect Sobol’ indices. Starting from the integral formula of the indices, the error bound is defined in terms of the discrete Walsh coefficients of the different integrands. We propose a sequential estimation procedure of Sobol’ indices using the error bound as a stopping criterion. The sequential procedure combines Sobol’ sequences with either Saltelli’s strategy to estimate both first-order and total effect indices, or the replication procedure to estimate only first-order indices.

Mots clés: Walsh coefficients; sequential procedure; Sobol' sequences; error bound; Sobol' indices

Domaines: Méthodologie [stat.ME]; Analyse numérique [math.NA]; Statistiques [math.ST]

Lluís Antoni Jiménez Rugama, Laurent Gilquin. Reliable error estimation for Sobol' indices. Statistics and Computing, 2018, 28 (4), pp.725-738. ⟨10.1007/s11222-017-9759-1⟩. ⟨hal-01358067v2⟩