Hybrid Parametric Classes of Isotropic Covariance Functions for Spatial Random Fields

Covariance functions are the core of spatial statistics, stochastic processes, machine learning as well as many other theoretical and applied disciplines. The properties of the covariance function at small and large distances determine the geometric attributes of the associated Gaussian random field. Having covariance functions that allow to specify both local and global properties is certainly on demand. This paper provides a method to find new classes of covariance functions having such properties. We term these models hybrid as they are obtained as scale mixtures of piecewise covariance kernels against measures that are also defined as piecewise linear combination of parametric families of measures. In order to illustrate our methodology, we provide new families of covariance functions that are proved to be richer with respect to other well known families that have been proposed by earlier literature. More precisely, we derive a hybrid Cauchy-Mat\'ern model, which allows us to index both long memory and mean square differentiability of the random field, and a hybrid Hole-Effect-Mat\'ern model, which is capable of attaining negative values (hole effect), while preserving the local attributes of the traditional Mat\'ern model. Our findings are illustrated through numerical studies with both simulated and real data.