Generalized Functional ANOVA Diagnostics for High-Dimensional Functions of Dependent Variables

This article studies the problem of providing diagnostics for high-dimensional functions when the input variables are known to be dependent. In such situations, commonly used diagnostics can place an unduly large emphasis on functional behavior that occurs in regions of very low probability. Instead, a generalized functional ANOVA decomposition provides a natural representation of the function in terms of low-order components. This article details a weighted functional ANOVA that controls for the effect of dependence between input variables. The construction involves high-dimensional functions as nuisance parameters and suggests a novel estimation scheme for it. The methodology is demonstrated in the context of machine learning in which the possibility of poor extrapolation makes it important to restrict attention to regions of high data density.