Sensitivity Analysis on the Sphere and a Spherical ANOVA Decomposition
It provides a theoretical framework for sensitivity analysis on spherical domains, which is foundational for applications in geophysics, astrophysics, and directional statistics.
This paper establishes sensitivity analysis on the sphere by decomposing functions into terms with different variable dependencies and parity, enabling modeling of high-dimensional functions with low-dimensional interactions.
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 $f_{\boldsymbol u,\boldsymbol ξ}$. The index $\boldsymbol u$ is a subset of $\{1,2,\ldots,d+1\}$, where each term $f_{\boldsymbol u,\boldsymbol ξ}$ 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 $\boldsymbol ξ$. 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.