Stationary space-time Gaussian fields and their time autoregressive representation

We compare two different modelling strategies for continuous space discrete time data. The first strategy is in the spirit of Gaussian kriging. The model is a general stationary space-time Gaussian field where the key point is the choice of a parametric form for the covariance function. In the main, covariance functions that are used are separable in space and time. Nonseparable covariance functions are useful in many applications, but construction of these is not easy. The second strategy is to model the time evolution of the process more directly. We consider models of the autoregressive type where the process at time t is obtained by convolving the process at time t - 1 and adding spatially correlated noise. Under specific conditions, the two strategies describe two different formulations of the same stochastic process. We show how the two representations look in different cases. Furthermore, by transforming time-dynamic convolution models to Gaussian fields we can obtain new covariance functions and by writing a Gaussian field as a time-dynamic convolution model, interesting properties are discovered. The computational aspects of the two strategies are discussed through experiments on a dataset of daily UK temperatures. Although algorithms for performing estimation, simulation, and so on are easy to do for the first strategy, more computer-efficient algorithms based on the second strategy can be constructed.