Risk of estimators for Sobol' sensitivity indices based on metamodels

TL;DR

This paper develops a new theoretical framework for assessing the accuracy of Sobol' sensitivity indices estimated via metamodels, providing risk bounds and convergence rates, supported by numerical experiments.

Contribution

It introduces a novel method for quality control of Sobol' indices estimates and derives risk bounds for a broad class of metamodels, linking model approximation quality to sensitivity index estimation accuracy.

Findings

Risk bounds for Sobol' indices estimates are established.

Fast convergence rates are possible with noiseless data.

Numerical experiments validate theoretical results.

Abstract

Sobol' sensitivity indices allow to quantify the respective effects of random input variables and their combinations on the variance of mathematical model output. We focus on the problem of Sobol' indices estimation via a metamodeling approach where we replace the true mathematical model with a sample-based approximation to compute sensitivity indices. We propose a new method for indices quality control and obtain asymptotic and non-asymptotic risk bounds for Sobol' indices estimates based on a general class of metamodels. Our analysis is closely connected with the problem of nonparametric function fitting using the orthogonal system of functions in the random design setting. It considers the relation between the metamodel quality and the error of the corresponding estimator for Sobol' indices and shows the possibility of fast convergence rates in the case of noiseless observations. The theoretical results are complemented with numerical experiments for the approximations based on multivariate Legendre and Trigonometric polynomials.