Variance-based sensitivity analysis in the presence of correlated input variables

TL;DR

Variance-based sensitivity analysis with correlated inputs is addressed by extending Sobol' estimators through a linear-correlation decomposition in standard normal space and a Nataf transformation for non-normal marginals, yielding sampling matrices that preserve the original joint distribution. The authors introduce a matrix-recombination approach and a regression-based alternative to estimate first-order and total-effect indices under correlation, along with separate measures for correlated and uncorrelated contributions $S_i^C$, $S_i^U$, $S_{T_i}^C$, and $S_{T_i}^U$. Numerical experiments on additive, coupled nonlinear, and Ishigami functions validate accuracy and highlight interpretability gains, showing how correlation reshapes index contributions and suggests robust factor fixing when using both correlated and uncorrelated perspectives. The method enables model-independent evaluation under dependency, with practical impact for uncertainty quantification and model simplification in systems with dependent inputs.

Abstract

In this paper we propose an extension of the classical Sobol' estimator for the estimation of variance based sensitivity indices. The approach assumes a linear correlation model between the input variables which is used to decompose the contribution of an input variable into a correlated and an uncorrelated part. This method provides sampling matrices following the original joint probability distribution which are used directly to compute the model output without any assumptions or approximations of the model response function.

Figures (4)

• Figure 1: Scatter plot of a single input variable with the model output including subset averages of sorted sample subsets
• Figure 2: Original and modified joint probability distributions of two correlated variables using the original and the extended matrix combination approach
• Figure 3: Standard deviation of the regression coefficients (left) and the estimated total effect sensitivity indices of $X_2$ (right) dependent on the correlation for the additive model including random noise
• Figure 4: Anthill plots of the response of the coupled nonlinear function with respect to $X_1$ and $X_2$ by assuming $\rho=0.0$ (left) and $\rho=0.8$ (right)