Multivariate sensitivity analysis: Minimum variance unbiased estimators of the first-order and total-effect covariance matrices

In uncertainty quantification, multivariate sensitivity analysis (MSA) extends variance-based sensitivity analysis to cope with the multivariate response, and it aims to apportion the variability of the multivariate response into input factors and their interactions. The first-order and total-effect covariance matrices from MSA, which assess the effects of input factors, provide useful information about interactions among input factors, the order of interactions, and the magnitude of interactions over all model outputs. In this paper, first, we propose and study generalized sensitivity indices (GSIs) using the first-order and total-effect covariance matrices. The new GSIs make use of matrix norms when partial orders such as the Loewner ordering on covariance matrices is not possible, and we obtain the classical GSIs using the Frobenius norm. Second, we propose minimum variance unbiased estimators (MVUEs) of the first-order and total-effect covariance matrices, and third, we provide an efficient estimator of the first-order and total (classical) GSIs. We also derive the consistency, the asymptotic normality, and the asymptotic confidence regions of these estimators. Our estimator allows for improving the GSIs estimates.