Uncertainty quantification: a minimum variance unbiased (joint) estimator of the non-normalized Sobol’ indices

Abstract

Often, uncertainty quantification is followed by the computation of sensitivity indices of input factors. Variance-based sensitivity analysis and multivariate sensitivity analysis (MSA) aim to apportion the variability of the model output(s) into input factors and their interactions. Sobol’ indices (first-order and total indices), which quantify the effects of input factor(s), serve as a practical tool to assess interactions among input factors, the order of interactions, and the magnitude of interactions. In this paper, we investigate a novel way of estimating both the first-order and total indices based on U-statistics, including the statistical properties of the new estimator. First, we provide a minimum variance unbiased estimator of the non-normalized Sobol’ indices as well as its optimal rate of convergence and its asymptotic distribution. Second, we derive a joint estimator of Sobol’ indices, its consistency and its asymptotic distribution, and third, we demonstrate the applicability of these results by means of numerical tests. The new estimator allows for improving the estimation of Sobol’ indices for some degrees of the kernel.