Generalized Functional ANOVA: A Complete Theoretical Framework

Abstract

Functional ANOVA separates a multivariate function into main effects and interactions. With dependent inputs, these components are hierarchically orthogonal, and the associated Hilbert subspaces are harder to characterize explicitly.

We study continuous inputs on a bounded hyperrectangle, assuming that the joint density is bounded above and away from zero. A distribution-adapted Riesz basis gives a stable representation of the function space and identifies the subspace associated with each group of variables. Under independence, this construction recovers the classical orthogonal decomposition.

This representation turns decomposition into coefficient estimation. A model-agnostic estimator illustrates the framework on synthetic and real data.

Version history

The September 2026 revision uses the title shown above. The original May 2026 version appeared as Generalized Functional ANOVA in Closed-Form: A Unified View of Additive Explanations.