ABSTRACT

A good model for the covariance function of a stationary process in space and time should accurately describe the variances and correlations of all linear combinations of the process. In particular, it does not suffice to find a model that describes the purely temporal covariances and the purely spatial covariances accurately. Rather, it is critical to capture the spatial-temporal interactions as well. We consider a number of properties of spatial-temporal covariance functions and how these relate to the spatial-temporal interactions of the process. First, we examine how the smoothness away from the origin of a spatial-temporal covariance function affects, for example, temporal correlations of spatial differences. Second, we examine the implications of a Markov assumption in time on spatial-temporal covariance functions. Third, we consider models that are asymmetric in space-time: the correlation between site A at time t and site B at time s is different than the correlation between site A at time s and site B at time t. We apply some of these data to wind data in Ireland and show that some of the proposed models fit the data much better than previously suggested models.