Flexible Integro-Differential Equations for Bayesian Modeling of Spatio-Temporal Data

Integro-Differential Equations (IDEs) are a novel way of dynamically modeling spatio-temporal data. IDEs are characterized by a kernel which controls the spatial and temporal associations. The ubiquitous choice for kernel has been Gaussian. We explore advantages of more flexible kernel choices. One-dimensional space is considered initially, replacing the Gaussian IDE kernel with more flexible parametric families of distributions. The kernels are chosen based on stochastic partial differential equation approximations which connect characteristics of the kernel with interpretable physical properties of the underlying process controlling the data. Next, Dirichlet process mixtures of normal distributions are used to model non-parametrically the IDE kernel. Computational issues arise using non-parametric kernels which are solved using Hermite polynomials and Hamiltonian Monte Carlo sampling. To develop flexible modeling in two-dimensional space, we propose bivariate stable distributions as IDE kernels. By using Bernstein polynomials as a prior for the measure defining the bivariate stable, a wide variety of shapes can be achieved. Bivariate stable kernels will be shown to outperform the Gaussian kernel by comparing K-step ahead predictions for Pacific sea surface temperature anomalies. Through study of properties for the proposed models, and empirical investigation with synthetic and real data, we demonstrate that the methodology has the potential to significantly improve the inference and forecasting capacity of IDE models based on Gaussian kernels.