Karhunen-Lo\`eve Expansions for Axially Symmetric Gaussian Processes: Modeling Strategies and $L^2$ Approximations

TL;DR

This paper develops flexible Karhunen-Loève expansions for axially symmetric Gaussian processes on spheres, allowing for diverse covariance structures and asymmetries, with theoretical error bounds and numerical validation.

Contribution

It introduces a parametric family of Karhunen-Loève coefficients for versatile covariance modeling and a method to induce asymmetries in longitudinal processes.

Findings

Versatile covariance functions including isotropic and longitudinally independent cases.

A strategy to create asymmetric covariance functions along longitudes.

Error bounds for $L^2$-approximations of truncated expansions.

Abstract

Axially symmetric processes on spheres, for which the second-order dependency structure may substantially vary with shifts in latitude, are a prominent alternative to model the spatial uncertainty of natural variables located over large portions of the Earth. In this paper, we focus on Karhunen-Lo\`eve expansions of axially symmetric Gaussian processes. First, we investigate a parametric family of Karhunen-Lo\`eve coefficients that allows for versatile spatial covariance functions. The isotropy as well as the longitudinal independence can be obtained as limit cases of our proposal. Second, we introduce a strategy to render any longitudinally reversible process irreversible, which means that its covariance function could admit certain types of asymmetries along longitudes. Then, finitely truncated Karhunen-Lo\`eve expansions are used to approximate axially symmetric processes. For such approximations, bounds for the $L^2$-error are provided. Numerical experiments are conducted to illustrate our findings.