Hybrid Parametric Classes of Isotropic Covariance Functions for Spatial Random Fields

Abstract

Covariance functions are the core of spatial statistics, stochastic processes, machine learning, and 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. Covariance functions that allow one to specify both local and global properties are certainly in demand. This paper provides a method for finding new classes of covariance functions having such properties. We refer to these models as hybrid, as they are obtained as scale mixtures of piecewise covariance kernels against measures that are also defined as piecewise linear combinations of parametric families of measures. To illustrate our methodology, we provide new families of covariance functions that are proved to be richer than other well-known families proposed in earlier literature. More precisely, we derive a hybrid Cauchy–Matérn model, which allows us to index both long memory and mean square differentiability of the random field, and a hybrid hole-effect–Matérn model which is capable of attaining negative values (hole effect) while preserving the local attributes of the traditional Matérn model. Our findings are illustrated through numerical studies with both simulated and real data.