A new interpretation and validation of variance based importance measures for models with correlated inputs

In order to explore the contributions by correlated input variables to the variance of the output, a novel interpretation framework of importance measure indices is proposed for a model with correlated inputs, which includes the indices of the total correlated contribution and the total uncorrelated contribution. The proposed indices accurately describe the connotations of the contributions by the correlated input to the variance of output, and they can be viewed as the complement and correction of the interpretation about the contributions by the correlated inputs presented in "Estimation of global sensitivity indices for models with dependent variables, Computer Physics Communications, 183 (2012) 937-946". Both of them contain the independent contribution by an individual input. Taking the general form of quadratic polynomial as an illustration, the total correlated contribution and the independent contribution by an individual input are derived analytically, from which the components and their origins of both contributions of correlated input can be clarified without any ambiguity. In the special case that no square term is included in the quadratic polynomial model, the total correlated contribution by the input can be further decomposed into the variance contribution related to the correlation of the input with other inputs and the independent contribution by the input itself, and the total uncorrelated contribution can be further decomposed into the independent part by interaction between the input and others and the independent part by the input itself. Numerical examples are employed and their results demonstrate that the derived analytical expressions of the variance-based importance measure are correct, and the clarification of the correlated input contribution to model output by the analytical derivation is very important for expanding the theory and solutions of uncorrelated input to those of the correlated one.