Gaussian linear approximation for the estimation of the Shapley effects

TL;DR

This paper proposes a Gaussian linear approximation method for estimating Shapley effects in sensitivity analysis, providing theoretical convergence results and demonstrating computational efficiency and accuracy through numerical experiments.

Contribution

It introduces a Gaussian linear approximation framework for estimating Shapley effects in non-linear models, with proven convergence and practical validation.

Findings

The Gaussian linear approximation accurately estimates Shapley effects for small input variances.

Convergence of the approximation improves as the number of terms in the empirical mean increases.

The method significantly reduces computational time while maintaining estimation accuracy.

Abstract

In this paper, we address the estimation of the sensitivity indices called "Shapley eects". These sensitivity indices enable to handle dependent input variables. The Shapley eects are generally dicult to estimate, but they are easily computable in the Gaussian linear framework. The aim of this work is to use the values of the Shapley eects in an approximated Gaussian linear framework as estimators of the true Shapley eects corresponding to a non-linear model. First, we assume that the input variables are Gaussian with small variances. We provide rates of convergence of the estimated Shapley eects to the true Shapley eects. Then, we focus on the case where the inputs are given by an non-Gaussian empirical mean. We prove that, under some mild assumptions, when the number of terms in the empirical mean increases, the dierence between the true Shapley eects and the estimated Shapley eects given by the Gaussian linear approximation converges to 0. Our theoretical results are supported by numerical studies, showing that the Gaussian linear approximation is accurate and enables to decrease the computational time signicantly.