The Gauss Hypergeometric Covariance Kernel for Modeling Second-Order Stationary Random Fields in Euclidean Spaces: its Compact Support, Properties and Spectral Representation

TL;DR

This paper introduces a new family of compactly-supported covariance kernels based on hypergeometric functions, capable of modeling diverse second-order stationary random fields in Euclidean spaces with flexible properties.

Contribution

It develops a hypergeometric covariance kernel family with analytic spectral expressions, encompassing many classical covariances and enabling versatile modeling of multivariate spatial fields.

Findings

Includes spherical, Matérn, Gaussian, and other covariances as special cases.

Provides conditions for valid multivariate covariance kernels with different component behaviors.

Analyzes properties like continuity, smoothness, and scale transformations of the hypergeometric kernels.

Abstract

This paper presents a parametric family of compactly-supported positive semidefinite kernels aimed to model the covariance structure of second-order stationary isotropic random fields defined in the $d$-dimensional Euclidean space. Both the covariance and its spectral density have an analytic expression involving the hypergeometric functions ${}_2F_1$ and ${}_1F_2$, respectively, and four real-valued parameters related to the correlation range, smoothness and shape of the covariance. The presented hypergeometric kernel family contains, as special cases, the spherical, cubic, penta, Askey, generalized Wendland and truncated power covariances and, as asymptotic cases, the Mat\'ern, Laguerre, Tricomi, incomplete gamma and Gaussian covariances, among others. The parameter space of the univariate hypergeometric kernel is identified and its functional properties -- continuity, smoothness, transitive upscaling (mont\'ee) and downscaling (descente) -- are examined. Several sets of sufficient conditions are also derived to obtain valid stationary bivariate and multivariate covariance kernels, characterized by four matrix-valued parameters. Such kernels turn out to be versatile, insofar as the direct and cross-covariances do not necessarily have the same shapes, correlation ranges or behaviors at short scale, thus associated with vector random fields whose components are cross-correlated but have different spatial structures.