Special classes of isotropic covariance functions

Abstract In the literature classical properties concerning the class of covariance functions are well illustrated. The general problem which analyzes the requirements such that a covariance function can be obtained through the difference between two covariance functions can be retained a further property and a recent analysis regarding the difference between two covariance functions has been provided. In this paper the properties of these new classes of models have been explored; in particular, the present analysis has been given for isotropic covariance functions, because of their importance in many applied areas; moreover, isotropic covariance functions can be considered the starting point to construct anisotropic models. It has been pointed out that these new families of models are more flexible than the traditional ones because the same models, according to the values of their parameters, are able to select covariance functions which are always positive in their domain, as well as covariance functions which could be negative in a subset of their field of definition. Moreover, within the same class of models, it is possible to select covariance models which present a parabolic behaviour near the origin from covariance models which present a linear behaviour in proximity of the origin. Apart from the theoretical importance related to the new aspects presented throughout the paper, it is relevant to underline the practical aspects, since these new classes of isotropic co-variance models are characterized by an extremely simple formalism and can be easily adapted to several case studies, hence they result very useful for many practitioners.