A parameter transformation of the anisotropic Matérn covariance function

Abstract

We describe a polar coordinate transformation of the anisotropy parameters of the Matérn covariance function, which provides two benefits over the standard parameterization. First, it identifies a single point (the origin) with the special case of isotropy. Second, the posterior distribution of the transformed anisotropic angle and ratio is approximately bell-shaped and unimodal even in the case of isotropy. This has advantages for parameter inference and density estimation. We also apply a transformation to the standard deviation and range such that they are approximately orthogonal. We demonstrate this parameter transformation through two simulated and two real data sets, and conclude by considering possible extensions, such as implementing this transformation for approximate Bayesian inference methods.