Sensitivity Analysis on the Sphere and a Spherical ANOVA Decomposition

Abstract

We establish sensitivity analysis on the sphere. We present formulas that allow us to decompose a function $f:{\mathbb{S}}^d\to \mathbb{R}$ into a sum of terms $f_{\mathbf{u},\hat{\mathbf{I}}¾}$. The index $\mathbf{u}$ is a subset of $\{1,2,\dots ,d+1\}$, where each term $f_{\mathbf{u},\hat{\mathbf{I}}¾}$ depends only on the variables with indices in $\mathbf{u}$. 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 $\hat{\mathbf{I}}¾$. 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.