Sensitivity analysis of the variance contributions with respect to the distribution parameters by the kernel function

Variance based sensitivity indices represent how the input uncertainty influences the output uncertainty. In order to identify how the distribution parameters of inputs influence the variance contributions, this work proposes the sensitivity of the variance contributions, which is defined by the partial derivative of the first-order variance contribution with respect to the distribution parameter. The proposed sensitivity can reflect how small variation of the distribution parameter influences the first-order variance contribution. By simplifying the partial derivative of the first-order variance contribution into the form of expectation via the kernel function, the proposed sensitivity can be seen as a by-product of the variance based sensitivity analysis without any additional output evaluations. For the classical quadratic responses, the proposed sensitivity can be derived analytically based on the integral form, while for the complex responses, the state dependent parameter (SDP) based method, which has been applied in the variance sensitivity analysis, can be employed to compute the proposed sensitivity. Several examples are used to demonstrate the correctness of the analytical solutions and the efficiency of the SDP based method.