Bivariate covariance functions of Pólya type

Abstract

We provide sufficient conditions of Pólya type which guarantee the positive definiteness of an isotropic 2 × 2-matrix-valued function in R and R3. Several isotropic bivariate covariance models have been proposed in literature, where all components of the covariance matrix are of the same parametric family, such as the bivariate Matérn model. Based on the Pólya type conditions, we introduce two novel bivariate parametric covariance models of this class, the powered exponential (or stable) covariance model and the generalized Cauchy covariance model. Both models allow for flexible smoothness, variance, scale, and cross-correlation parameters. The smoothness parameters are in (0,1]. Additionally, the bivariate generalized Cauchy model allows for distinct long range parameters. We also show that the univariate spherical model can be generalized to the bivariate case within the above class only in a trivial way.