Sobol' Matrices For Multi-Output Models With Quantified Uncertainty

TL;DR

This paper extends Sobol' sensitivity analysis from single outputs to multi-output settings by introducing Sobol' matrices that quantify how inputs influence not only individual outputs but the correlations between outputs. It builds a formal framework around multi-output models with quantified uncertainty (MQU), linking Sobol' matrices to the first two moments of the underlying stochastic process and providing analytic expressions for both the matrices and their standard errors. The authors develop a GP-based implementation to compute these matrices from MQU moments, validate them on a nine-output test function constructed from standard sensitivity benchmarks, and benchmark performance under varying data quantity and quality. The results show that Sobol' matrices derived from MQU moments can accurately capture input effects on output linkage for moderate dimensionality (up to about seven inputs) and offer a principled route to dimension reduction based on how inputs affect output correlations, with practical robustness to homoskedastic noise.

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.

Figures (6)

• Figure 1: Frequency of closed and first order Sobol' matrix element values across 9 outputs and 5 inputs.
• Figure 2: The Standard Deviation (SD) of prediction for the benchmark GPs. Each plot point is the mean over 2 folds and 9 outputs, so extreme outliers are visible as blemishes.
• Figure 3: The Root Mean Square Error (RMSE) of prediction for the benchmark GPs, obtained from 2-fold validation. Each plot point is the mean over 2 folds and 9 outputs, so extreme outliers are visible as blemishes.
• Figure 4: The absolute error $\lbrack{A_{\mathbf{m}}}\rbrack_{l\times l^{\prime}}$ of the closed Sobol' matrix element $\lbrack{S_{\mathbf{m}}}\rbrack_{l\times l^{\prime}}$ calculated from MQU moments. Ground truth is the MNU closed Sobol' matrix reported in \ref{['sec:Func']}. Each plot point is the median (top row) or 90% quantile (bottom row) $A$ over 2 folds, $L^{2}=81$ matrix elements and $M\in\left\lbrace{5,7}\right\rbrace\xspace$ closed matrices.The top row indicates accuracy, the bottom row reliability.
• Figure 5: The absolute error $\lbrack{A^{T}_{\mathbf{M-m}}}\rbrack_{l\times l^{\prime}}$ of the total Sobol' matrix element $\lbrack{S^{T}_{\mathbf{M-m}}}\rbrack_{l\times l^{\prime}}$ calculated from MQU moments. Ground truth is $S_{\mathbf{M}}-S_{\mathbf{m}}$ calculated from the MNU closed Sobol' matrices reported in \ref{['sec:Func']}. Each plot point is the median $A^{T}$ over 2 folds, $L^{2}=81$ matrix elements and one or two total matrices. Only MQUs with $M=7$ have been used.