Copula-based decomposition approach for the derivative-based sensitivity of variance contributions with dependent variables

Variance-based sensitivity analysis with dependent variables represents how the uncertainties and dependence of variables influence the output uncertainty. Since the distribution parameters of variables are difficult to be given precisely, this work defines the derivative-based sensitivity of variance contribution with respect to the distribution parameters, which reflects how small variation of distribution parameters influences the variance contributions. By introducing the copula functions to describe the dependence of variables, the derivative of variance contributions can be transformed into those of marginal PDF and copula function, which can be defined by kernel function and copula kernel function. Then the derivative-based sensitivity of variance contributions can be decomposed into the independent part and dependent part. Since the derivatives of marginal PDF and copula function can be given analytically, the proposed derivative-based sensitivity can be computed with no additional computational cost, which is seen as the ‘by-product’ of variance-based sensitivity analysis. To calculate the proposed sensitivity, two computational methods, numerical method and SDP (state dependent parameter) method are presented for comparison. Several examples are used to demonstrate the reasonability of the proposed sensitivity and the accuracy of the applied method.