Uncertainty importance measure for models with correlated normal variables

In order to explore the contributions by correlated input variables to the variance of the model output, the contribution decomposition of the correlated input variables based on Mara's definition is investigated in detail. By taking the quadratic polynomial output without cross term as an illustration, the solution of the contribution decomposition is derived analytically using the statistical inference theory. After the correction of the analytical solution is validated by the numerical examples, they are employed to two engineering examples to show their wide application. The derived analytical solutions can directly be used to recognize the contributions by the correlated input variables in case of the quadratic or linear polynomial output without cross term, and the analytical inference method can be extended to the case of higher order polynomial output. Additionally, the origins of the interaction contribution of the correlated inputs are analyzed, and the comparisons of the existing contribution indices are completed, on which the engineer can select the suitable indices to know the necessary information. At last, the degeneration of the correlated inputs to the uncorrelated ones and some computational issues are discussed in concept.