On Some Characteristics of Gaussian Covariance Functions

Abstract

Summary The concepts of isotropy/anisotropy and separability/non‐separability of a covariance function are strictly related. If a covariance function is separable, it cannot be isotropic or geometrically anisotropic, except for the Gaussian covariance function, which is the only model both separable and isotropic. In this paper, some interesting results concerning the Gaussian covariance model and its properties related to isotropy and separability are given, and moreover, some examples are provided. Finally, a discussion on asymmetric models, with Gaussian marginals, is furnished and the strictly positive definiteness condition is discussed.

Blind recovery of sources for multivariate space-time random fields

“…Note that in the above case the concept of isotropy, which has in general no meaning for spatio-temporal random fields, is not recalled, and the assumption of anisotropy is used instead. It is worth pointing out that the Gauss kernel is the only function which can be separable and isotropic (De Iaco et al 2020), thus defining a spatio-temporal metric or keeping separate the two distances is essentially equivalent. In the following, for a given second order stationary space-time random process, it is assumed that it is spatially isotropic in the weak sense (alternatively called second order isotropic), that is the covariance is a function of the spatial and temporal distances khk and jsj, respectively.…”