An approximation theoretic perspective of the Sobol' indices with dependent variables

TL;DR

This paper develops an approximation theoretic approach to interpret Sobol' indices when variables are dependent, enhancing their application in dimension reduction and sensitivity analysis.

Contribution

It introduces a new perspective to interpret Sobol' indices with dependent variables, addressing a key challenge in global sensitivity analysis.

Findings

Provides a theoretical framework for dependent variables

Demonstrates improved interpretation in dimension reduction

Includes illustrative examples and analysis

Abstract

The Sobol' indices are a recognized tool in global sensitivity analysis. When the uncertain variables in a model are statistically independent, the Sobol' indices may be easily interpreted and utilized. However, their interpretation and utility is more challenging with statistically dependent variables. This article develops an approximation theoretic perspective to interpret Sobol' indices in the presence of variable dependencies. The value of this perspective is demonstrated in the context of dimension reduction, a common application of the Sobol' indices. Theoretical analysis and illustrative examples are provided.