On ANOVA-Type Decompositions of Functions with Non-independent Variables: Sensitivity Analysis

Abstract. ANOVA decompositions are widely used in uncertainty quantification with interesting properties when the input variables are independent. This study proposes the unique decomposition of every complex, computational model evaluated at non-independent variables so that the main and interaction effects sum to the output variance. The proposed dependent ANOVA (DANOVA) relies on the minimal, symmetric equivalent representations of the model output, and an algorithm is provided for selecting such representations. New sensitivity indices that are interpretable in terms of the percentages of the output variance are derived. The main and total indices enable the identification of model structures, which are relevant for emulations. Such indices boil down to Shapley effects of Gaussian inputs for linear models: minimum variance and unbiased estimators of DANOVA components, estimators of indices, and a real application to autoregressive models are provided.