Gaussian random fields: with and without covariances

TL;DR

This paper explores the structure and properties of Gaussian random fields, including isotropic and Matérn processes, and discusses their discrete approximations via Gaussian Markov random fields for efficient computation.

Contribution

It provides a unified framework for understanding Gaussian random fields with and without covariances, linking continuous models to discrete GMRFs for scalable data analysis.

Findings

Covariance structures described via Bochner-Godement theorem.

Matérn processes on various manifolds analyzed using SPDEs.

GMRFs offer computational advantages through sparse precision matrices.

Abstract

We begin with isotropic Gaussian random fields, and show how the Bochner-Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Mat\'ern processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs). We then turn from this continuous setting to approximating discrete settings, Gaussian Markov random fields (GMRFs), and the computational advantages they bring in handling large data sets, by exploiting the sparseness properties of the relevant precision (concentration matrices).