Isotropy, symmetry, separability and strict positive definiteness for covariance functions: A critical review

Abstract

Although isotropy, symmetry and separability are commonly assumed for practical reasons, anisotropic, asymmetric and non-separable covariance functions are often more realistic; in addition, strict positive definiteness is also desirable, since it ensures the invertibility of the kriging coefficient matrix. In this paper, a critical review of these concepts is proposed and it is shown how these aspects are strictly related. In particular, separable covariance models represent a simple way to construct component-wise anisotropic models which, under suitable conditions, are strictly positive definite. Similarly, some other results on strict positive definiteness can be used to obtain non-separable anisotropic models. Covariance functions defined on partially overlapped domains are used to construct non-geometric spatial anisotropic covariance functions, also characterized by non-separability and strict positive definiteness. Moreover, anisotropic and asymmetric covariance functions that are also strictly positive definite are presented.