Several classes of space-time correlation models have been proposed by various authors in the last years. However, most of these families utilize non negative covariance functions to be adapted to different case studies: indeed, the traditional classes of covariances, such as the Whittle-Matern class and the several families constructed by applying the classical properties are not so flexible to describe covariance functions characterized by negative values. A recent analysis, regarding the difference between two isotropic covariance functions, has underlined that these new families of models are more flexible than the traditional ones because the same models 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 different behaviours in proximity of the origin. In this paper several families of isotropic space-time covariance functions, among the ones proposed in the literature, have been reviewed in order to enrich the same families including models characterized by negative values in a subset of their domain. Furthermore, the definition of separability has been revised in order to enlarge the classical definition. Apart from the theoretical importance related to the new aspects, these new classes of covariance models are characterized by an extremely simple formalism and can be easily adapted to several case studies.
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