Bayesian Inference for Dynamic Spatio-temporal Models

Spatio-temporal processes are phenomena evolving in space, either by being a point, a field or a map and also they vary in time. A stochastic process may be proposed as a vehicle to infer and hence offer predictions of the future. In this era high dimensional datasets can be available where measurements are observed daily or even hourly at more than one locations along with many predictors. Therefore, what we would like to infer is high dimensional and the analysis is difficult to come through due to high complexity of calculations or efficiency from a computational aspect. The first Reduced-dimension Dynamic Spatio Temporal Models (DSTMs) were developed to jointly describe the spatial and temporal evolution of a function observed subject to noise. A basic state space model is adopted for the discrete temporal variation, while a continuous autoregressive structure describes the continuous spatial evolution. Application of DTSMs rely upon the pre-selection of a suitable reduced set of basis functions and this can present a challenge in practice. In this thesis we propose a Hierarchical Bayesian framework for high dimensional spatio-temporal data based upon DTSMs which attempts to resolve this issue allowing the basis to adapt to the observed data. Specifically, we present a wavelet decomposition for the spatial evolution but where one would typically expect parsimony. This believed parsimony can be achieved by placing a Spike and Slab prior distribution on the wavelet coefficients. The aim of using the Spike and Slab prior, is to filter wavelet coefficients with low contribution, and thus achieve the dimension reduction with significant computational savings. We then propose an Hierarchical Bayesian State-space model, for the estimation of which we offer an appropriate Forward Filtering Backward Sampling algorithm under an MCMC procedure. Then, we extend this model for estimating Poisson counts and Multinomial cell probabilities through proposing a Conditional Particle Filtering framework.