Covariance functions and models for complex-valued random fields

Abstract

In Geostatistics, primary interest often lies in the study of the spatial, or spatial-temporal, correlation of real-valued random fields, anyway complexvalued random field theory is surely a natural extension of the real domain. In such a case, it is useful to consider complex covariance functions which are composed of an even real part and an odd imaginary part. Generating complex covariance functions is not simple at all, but the procedure, developed in this paper, allows generating permissible covariance functions for complex-valued random fields in a straightforward way. In particular, by recalling the spectral representation of the covariance and translating the spectral density function by using a shifting factor, complex covariances are obtained. Some general aspects and properties of complex-valued random fields and their moments are pointed out and some examples are given.In Geostatistics, one of the main problems consists in finding a function which can be used as a covariance, in order to describe the spatial or spatial-temporal correlation of the phenomenon under study. Practitioners usually study the correlation of a random field in the real domain and fit known permissible covariance models to the experimental covariances. The even covariance function of a real-valued random field could represent a limitation in some geostatistical applications. In the multivariate context, for example, the Linear Model of Coregionalization (Journel and Huijbregts, 1981;Wackernagel, 1998), in spite of its computational semplicity, has the limitation to the simple and cross-covariance to be even functions; Grzebyk (1993) proposed an extension of the Linear Model of Coregionalization to the complex domain in order to allow the use of not-even cross-covariance functions. Anyway the major contribution given by Grzebyk in the complex theory was oriented to a multivariate context.

Determinantal Point Process Models and Statistical Inference

“…()), and examples of stationary and anisotropic covariance functions are discussed in De laco et al . ().…”

Complex Gaussian Processes for Regression

“…3) Spectral kernels: Finally, there have also been some proposals to create an imaginary part from the real part of the covariance function in kriging [12], [13] for a multiple output learning framework. However, these proposals are for stationary random fields of just real inputs, i.e., k(x−x ), with x ∈ R d , and do not provide a pseudo-covariance function.…”

“…However, it is unclear what the function should be for the covariance matrix to have some given properties. In [13] the proposed covariance function exhibits a sinusoidal behavior in its real and imaginary parts that is in general not suitable for the application at hand.…”

New spatio-temporal complex covariance functions for vectorial data through positive mixtures

“…Recalling the theory of complex-valued random fields is justifiable for describing phenomena, which can be naturally decomposed in modulus and direction, as a wind speed, sea current or electric field. One of the first contributions on the complex formalism and on the methodology to model these phenomena in one-dimension can be found in Yaglom (1987); subsequently, for a spatial domain in two or more dimensions the pioneer works of Lajaunie and Be ´jaoui (1991), Grzebyk (1993), Wackernagel (2003) and De Iaco et al (2003) can be cited. Moreover, together with some advances focused on some parametric classes of covariance models defined on complex domains (Posa 2020(Posa , 2021, in the last decade practical aspects regarding the fitting procedure and the related computational difficulties were also discussed, as shown in De Iaco and Posa (2016).…”