Sobol' Matrices For Multi-Output Models With Quantified Uncertainty

TL;DR

This paper introduces Sobol' matrices for multi-output models, extending variance-based sensitivity analysis to quantify input relevance across multiple outputs and their linkages, with methods for calculation and uncertainty quantification.

Contribution

It extends Sobol' indices to a matrix form for multi-output models, providing a natural and quantifiable measure of input relevance to output linkages.

Findings

Sobol' matrices can be numerically computed and benchmarked.

Standard errors of Sobol' matrices are related to model moments.

The approach is validated against analytical test functions.

Abstract

Variance based global sensitivity analysis measures the relevance of inputs to a single output using Sobol' indices. This paper extends the definition in a natural way to multiple outputs, directly measuring the relevance of inputs to the linkages between outputs in a correlation-like matrix of indices. The usual Sobol' indices constitute the diagonal of this matrix. Existence, uniqueness and uncertainty quantification are established by developing the indices from a putative multi-output model with quantified uncertainty. Sobol' matrices and their standard errors are related to the moments of the multi-output model, to enable calculation. These are benchmarked numerically against test functions (with added noise) whose Sobol' matrices are calculated analytically.