A vine copula–based method for analyzing the moment-independent importance measure of the multivariate output

Abstract

The moment-independent importance measure technique for exploring how uncertainty allocates from output to inputs has been widely used to help engineers estimate the degree of confidence of decision results and assess risks. Solving the Borgonovo moment-independent importance measure in the presence of the multivariate output is still a challenging problem due to “curse of dimensionality,” and it is investigated in this contribution. For easily estimating the moment-independent importance measure, a novel method based on the vine copula is proposed. In the proposed method for estimating moment-independent importance measure, three steps are included. First, the moment-independent importance measure is expressed as a product of bivariate copula density functions through the vine copula trees. Second, the marginal probability density functions are obtained by the maximum entropy under the constraint of the fractional moments. Finally, the post-processed is executed to directly estimate the moment-independent importance measure by estimated copula density functions. The proposed method can handle multivariate output easily. The results of several examples indicate the validity and benefits of the proposed method.

Estimation of high-order moment-independent importance measures for Shapley value analysis

“…As far as Borgonovo's δ-sensitivity measures are concerned, copula estimation was conducted with Gaussian kernels in the first copula-focused paper [10] and by resorting to a maximum entropy approach in [11]. More recently, and in a slightly different context where Y is multivariate, one can find in [14] a first attempt to connect vine copula models and Borgonovo's indices. Before going further, let us recall the procedure leading toδ C u : 1.…”

“…Firstly, the authors developed an estimation scheme for high-order indices where the copula-based expression of Borgonovo's indices is coupled with numerical procedures for copula density estimation in high dimension [13]. Taking the most of copula distribution models in order to refine the estimation of Borgonovo's δ-sensitivity measures has been partly addressed in [14], but only within a very particular context where Y is multidimensional. In addition, this previous work focuses only on the specific case of parametric regular vine copulas whereas the present paper includes a comparative study of the estimators resulting from different copula distribution models.…”