Criteria and characterizations for spatially isotropic and temporally symmetric matrix-valued covariance functions

Abstract

We consider spatial matrix-valued isotropic covariance functions in Euclidean spaces and provide a very short proof of a celebrated characterization result proposed by earlier literature. We then provide a characterization theorem to create a bridge between a class of matrix-valued functions and the class of matrix-valued positive semidefinite functions in finite-dimensional Euclidean spaces. We culminate with criteria of the Pólya type for matrix-valued isotropic covariance functions, and with a generalization of Schlather's class of multivariate spatial covariance functions.We then challenge the problem of matrix-valued space-time covariance functions, and provide a general class that encompasses all the proposals on the Gneiting nonseparable class provided by earlier literature.

The Matérn Model: A Journey through Statistics, Numerical Analysis and Machine Learning

“…M νij,αijψ(u 2 ) ( x ) for x ∈ R d , u ∈ R and a suitable positive valued and continuous function ψ. This setting has been recently generalised by [6] and through a technical approach by [125]: for both contribution, the idea is to replace (pointwise) the mapping ψ with the mapping ψ having continuous and strictly positive elements ψ ij . c) Multivariate Nonstationary.…”

“…This last remark has motivated alternative approaches, and the reader is referred to Apanasovich et al [11] and more recently to Emery et al [54]. Extensions to multivariate space-time Matérn structures have been provided by Allard et al [6] and through a technical approach by Porcu et al [125]. Multivariate nonstationary Matérn functions have been proposed by Kleiber and Nychka [88].…”

Covariance Models for Multivariate Random Fields resulting from Pseudo Cross-Variograms

“…This intimate connection adds another dimension to the range of applications of pseudo cross-variograms, that is, the construction of valid covariance models for multivariate random fields. In this regard, pseudo crossvariograms have already been used in Dörr and Schlather (2021), Allard et al (2022) and Porcu et al (2022) to propose several extensions of Gneiting's popular univariate space-time covariance model (Gneiting, 2002b), thereby meeting one of the requests of Chen et al (2021) for flexible space-time cross-covariance models.…”