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Theory of Probability and Mathematical Statistics

Gaussian random fields: with and without covariances

N. H. Bingham and Tasmin L. Symons


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 Matérn 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.

Keywords: Gaussian random field, covariance function, Bessel potential, stochastic partial differential equation, Matérn process, Gaussian Markov random field, precision matrix, sparseness, numerical linear algebra

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