On ANOVA decompositions of kernels and Gaussian random field paths

TL;DR

This paper introduces KANOVA, a kernel decomposition method for Gaussian random fields that enhances high-dimensional modeling and sensitivity analysis by controlling sparsity and dependence structures.

Contribution

It presents a novel kernel decomposition called KANOVA for Gaussian fields, linking tensor projections to model sparsity and dependence, aiding high-dimensional kriging.

Findings

KANOVA controls Gaussian field sparsity.

Projected kernels influence dependence structures.

Application to simulated data demonstrates effectiveness.

Abstract

The FANOVA (or "Sobol'-Hoeffding") decomposition of multivariate functions has been used for high-dimensional model representation and global sensitivity analysis. When the objective function f has no simple analytic form and is costly to evaluate, a practical limitation is that computing FANOVA terms may be unaffordable due to numerical integration costs. Several approximate approaches relying on random field models have been proposed to alleviate these costs, where f is substituted by a (kriging) predictor or by conditional simulations. In the present work, we focus on FANOVA decompositions of Gaussian random field sample paths, and we notably introduce an associated kernel decomposition (into 2^{2d} terms) called KANOVA. An interpretation in terms of tensor product projections is obtained, and it is shown that projected kernels control both the sparsity of Gaussian random field sample paths and the dependence structure between FANOVA effects. Applications on simulated data show the relevance of the approach for designing new classes of covariance kernels dedicated to high-dimensional kriging.