Testing for separability of spatial–temporal covariance functions

Most applications in spatial statistics involve modeling of complex spatial–temporal dependency structures, and many of the problems of space and time modeling can be overcome by using separable processes. This subclass of spatial–temporal processes has several advantages, including rapid fitting and simple extensions of many techniques developed and successfully used in time series and classical geostatistics. In particular, a major advantage of these processes is that the covariance matrix for a realization can be expressed as the Kronecker product of two smaller matrices that arise separately from the temporal and purely spatial processes, and hence its determinant and inverse are easily determinable. However, these separable models are not always realistic, and there are no formal tests for separability of general spatial–temporal processes. We present here a formal method to test for separability. Our approach can be also used to test for lack of stationarity of the process. The beauty of our approach is that by using spectral methods the mechanics of the test can be reduced to a simple two-factor analysis of variance (ANOVA) procedure. The approach we propose is based on only one realization of the spatial–temporal process.We apply the statistical methods proposed here to test for separability and stationarity of spatial–temporal ozone fields using data provided by the US Environmental Protection Agency (EPA).