Further advances on isotropic correlation models with negative values

Most of the covariance models utilized in spatial and in spatio-temporal data analysis are non negative; at this purpose, covariance functions characterized by negative values need to be explored not only to analyze some particular case studies characterized by negative correlation, but also and most importantly to provide richer families of correlation models. In this paper new families of correlation functions will be proposed, among the others, a generalization of the Cauchy correlation family as well as a generalization of a subclass of Matérn family. Moreover, apart from the Bochner’s spectral representation, in order to construct further families of correlation functions, a technique based on the differentiation of covariance models will be provided. Some of these new classes are characterized by a finite number of zeros as many as desired; at our knowledge, there seem not to exist correlation functions which present this relevant characteristic. Moreover, some of these new families can be built through differentiation by starting from covariance functions which are always positive in their domain. As it will be shown, the proposed classes are extremely flexible since they are able to model either correlation structures just characterized by positive values, either correlation structures which are negative in a subset of their domain: this is a relevant property of the proposed covariance models, essentially because most of the well known covariance functions, belonging to the same family, are not able to model both negative and positive correlation structures, according to their parameters values. A graphical representation and some general remarks on the proposed families of correlation models have been provided.