Classes of nonseparable, spatio-temporal stationary covariance functions

Suppose that a random process Z(s; t), indexed in space and time, has spatio-temporal stationary covariance C(h; u), where h epsilon R-d (d greater than or equal to 1) is a spatial lag and u epsilon R is a temporal lag. Separable spatio-temporal covariances have the property that they can be written as a product of a purely spatial covariance and a purely temporal covariance. Their ease of definition is counterbalanced by the rather limited class of random processes to which they correspond. In this article we derive a new approach that allows one to obtain many classes of nonseparable, spatio-temporal stationary covariance functions and fit several such classes to spatio-temporal data on wind speed over a region in the tropical western Pacific ocean.