Spatio-Temporal Modelling using Integro-Difference Equations with Bivariate Stable Kernels

Integro-difference equations can be represented as hierarchical spatio-temporal dynamic models using appropriate parameterizations. The dynamics of the process defined by an integro-difference equation depends on the choice of a bivariate kernel distribution, where more flexible shapes generally result in more flexible models. We consider the use of the stable family of distributions for the kernel, as they are infinitely divisible and offer a variety of tail behaviours, orientations and skewness. Many of the attributes of the bivariate stable distribution are controlled by a measure, which we model using a flexible Bernstein polynomial basis prior. This nonparametric prior specification, along with spatially dependent parameters for the stable kernel distribution, results in a model which is more general than existing integro-difference equation models. A Fourier basis representation with thresholding of the relevant basis functions and parallelization of certain parts of the posterior simulation algorithm allow the computational burden to be reduced. The method is shown to improve prediction over the state-of-the-art Gaussian kernel model in an example using Pacific sea surface temperature data.