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

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.