Models of Covariance Functions of Gaussian Random Fields Escaping From Isotropy, Stationarity and Non Negativity

Abstract

This paper represents a survey of recent advances in modeling of space or space-time Gaussian Random Fields (GRF), tools of Geostatistics at hand for the understanding of special cases of noise in image analysis. They can be used when stationarity or isotropy are unrealistic assumptions, or even when negative covariance between some couples of locations are evident. We show some strategies in order to escape from these restrictions, on the basis of rich classes of well known stationary or isotropic non negative covariance models, and through suitable operations, like linear combinations, generalized means, or with particular Fourier transforms.

On Some Characteristics of Gaussian Covariance Functions

“…Some of the results concerning isotropy, separability, as well as strict positive definiteness and asymmetry have never been highlighted in a homogeneous and compact way and have significant practical implications in modelling covariance functions. Note that the concept of Gaussian random field, as well as its properties and characterisations, is out of the aim of this paper; the readers can find wide literature on this subject (Adler, 1981; Doob, 1953; VanMarcke, 2010; Yaglom, 1987a) and recent reviews for spatial and spatiotemporal phenomena given by Gelfand & Schliep (2016) and Gregori et al (2014). Indeed, the extensive use of Gaussian random fields in statistical modelling is based on some properties of the normal distribution, which is completely defined from its second‐order statistics.…”