General validation and decomposition of the variance-based measures for models with correlated inputs

To clearly explore the contributions by the correlated inputs to the variance of model output, the general expressions of the variance contributions of the correlated inputs are firstly derived in this paper, which provide a general validation for the accurate connotations of these variance contributions. The universal expressions of the components included in the variance contributions of the correlated inputs are also derived. These components not only allow the engineer to distinguish between the independent variance contribution and the correlated one of the input, but also provide information for differentiating the independent contribution by the input itself from that by the interaction between the input and the others. All the general expressions of the variance contributions in this paper are derived based on the high dimensional model representation (HDMR) of the output. Thus, they are independent of the form of the computational model. Compared with the existing interpretation and decomposing methods for the variance contributions of the correlated inputs, the proposed method has wider applicability. Based on the form of the general expressions of the various variance contributions, their efficient solutions are built by using the advantages of the sparse grid numerical integration (SGI). Numerical and engineering tests show the effectiveness and applicability of the derived variance contributions for the correlated inputs, as well as the efficiency and accuracy of the established SGI based method.