Covariance functions that are stationary or nonstationary in space and stationary in time

There is a great demand for statistical modelling of phenomena that evolve in both space and time, and thus, there is a growing literature on covariance function models for spatio-temporal processes. Although several nonseparable space-time covariance models are available in the literature, very few of them can be used for spatially anisotropic data. In this paper, we propose a new class of stationary nonseparable covariance functions that can be used for both geometrically and zonally anistropic data. In addition, we show some desirable mathematical features of this class. Another important aspect, only partially covered by the literature, is that of spatial nonstationarity. We show a very simple criteria allowing for the construction of space-time covariance functions that are nonseparable, nonstationary in space and stationary in time. Part of the theoretical results proposed in the paper will then be used for the analysis of Irish wind speed data as in HASLETT and RAFTERY (Applied Statistics, 38, 1989, 1).