Shapley effect estimation in reliability-oriented sensitivity analysis with correlated inputs by importance sampling

TL;DR

This paper develops efficient importance sampling methods for estimating Shapley effects in reliability-oriented sensitivity analysis with correlated inputs, enabling accurate failure probability assessment with fewer model evaluations.

Contribution

It introduces two importance-sampling-based estimators for Shapley effects that are more efficient for small failure probabilities and extends the approach to use only existing data from importance sampling.

Findings

The proposed estimators reduce variance compared to existing methods.

The methods are unbiased and benefit from importance sampling.

Numerical tests demonstrate practical efficiency and accuracy.

Abstract

Reliability-oriented sensitivity analysis aims at combining both reliability and sensitivity analyses by quantifying the influence of each input variable of a numerical model on a quantity of interest related to its failure. In particular, target sensitivity analysis focuses on the occurrence of the failure, and more precisely aims to determine which inputs are more likely to lead to the failure of the system. The Shapley effects are quantitative global sensitivity indices which are able to deal with correlated input variables. They have been recently adapted to the target sensitivity analysis framework. In this article, we investigate two importance-sampling-based estimation schemes of these indices which are more efficient than the existing ones when the failure probability is small. Moreover, an extension to the case where only an i.i.d. input/output N-sample distributed according to the importance sampling auxiliary distribution is proposed. This extension allows to estimate the Shapley effects only with a data set distributed according to the importance sampling auxiliary distribution stemming from a reliability analysis without additional calls to the numerical model. In addition, we study theoretically the absence of bias of some estimators as well as the benefit of importance sampling. We also provide numerical guidelines and finally, realistic test cases show the practical interest of the proposed methods.