Weak derivative-based expansion of functions: ANOVA and some inequalities

Abstract

Functional analysis of variance (FANOVA) is widely used in statistical modeling and sensitivity analysis. Knowing the important role of derivatives in models’ analysis and/or development such as variational formulation of phenomena, we develop new expansions of functions using their weak derivatives, the density and distribution functions of input variables. We also derive the expressions of ANOVA components based on weak cross-partial derivatives. We investigate its application in uncertainty quantification by proposing the derivative-based multivariate sensitivity analysis. Our approach allows for better assessing the main, interaction and total effects of inputs using weak derivatives, and we provide minimum variance unbiased estimators of the cross-covariances of ANOVA functionals for computing such effects in presence of multivariate and/or functional response models.