Sensitivity Analysis on the Sphere and a Spherical ANOVA Decomposition

TL;DR

This paper develops a sensitivity analysis framework on the sphere, introducing a spherical ANOVA decomposition that accounts for variable interactions and parity, enabling efficient modeling of high-dimensional functions.

Contribution

It presents a novel spherical ANOVA decomposition incorporating parity, extending classical sensitivity analysis to spherical domains with geometric dependencies.

Findings

Decomposition formulas for functions on the sphere are established.

The method models high-dimensional functions with low-dimensional interactions.

The approach accounts for dependencies induced by spherical geometry.

Abstract

We establish sensitivity analysis on the sphere. We present formulas that allow us to decompose a function $f\colon \mathbb S^d\rightarrow \mathbb R$ into a sum of terms $f_{\boldsymbol u,\boldsymbol \xi}$. The index $\boldsymbol u$ is a subset of $\{1,2,\ldots,d+1\}$, where each term $f_{\boldsymbol u,\boldsymbol \xi}$ depends only on the variables with indices in $\boldsymbol u$. In contrast to the classical analysis of variance (ANOVA) decomposition, we additionally use the decomposition of a function into functions with different parity, which adds the additional parameter $\boldsymbol \xi$. The natural geometry on the sphere naturally leads to the dependencies between the input variables. Using certain orthogonal basis functions for the function approximation, we are able to model high-dimensional functions with low-dimensional variable interactions.