Parametric nonstationary covariance functions on spheres

Abstract

Gaussian Processes are powerful tools for modelling spatial data. In this context, a significant amount of modelling focus is placed on specifying the covariance function, which is required to be symmetric and positive definite. Covariance functions have classically been defined and used in Euclidean space. However, as data collected from the globe becomes more prevalent, accounting for Earth's geometry becomes increasingly important. Using Euclidean distance can be suboptimal for these data. We survey the literature for recent developments related to construction of nonstationary covariance functions on spheres, which historically has been a challenging area. We present contributions in this effort by providing three general forms for families of parametric nonstationary covariance functions.

Large‐Domain Multisite Precipitation Generation: Operational Blueprint and Demonstration for 1,000 Sites

“…A main limitation of using a stationary STCS over large regions is that it cannot reproduce variations in the correlation behavior at different locations. However, over the last years there has been intense research on developing nonstationary STCS (see e.g., Blake et al., 2022; Kolovos et al., 2004; Paciorek & Schervish, 2006). Thus, the mixed‐Uniform framework developed here can be extended by using such formulas when applied to large regions.…”

Novel whitening approaches in functional settings

“…Other alternatives such as kernel regressors can certainly be considered. The proposed whitening procedures might have further potential applications for spatio-temporal functional data when stationary and isotropic assumptions are not satisfied; see, for example, Mateu and Giraldo (2021) and Blake et al (2022) for a review of current spherical approaches. A spatio-temporal stochastic process is a particular case of a functional variable with values in a Hilbert space of three-argument functions defined on the three-dimensional spatio-temporal domain.…”