A test for stationarity for spatio-temporal data

Many random phenomena in the environmental and geophysical sciences are functions of both space and time; these are usually called spatio-temporal processes. Typically, the spatio-temporal process is observed over discrete equidistant time and at irregularly spaced locations in space. One important aim is to develop statistical models based on what is observed. While doing so a commonly used assumption is that the underlying spatial-temporal process is stationary. If this assumption does not hold, then either the mean or the covariance function is misspecified. This can, for example, lead to inaccurate predictions. In this paper we propose a test for spatio-temporal stationarity. The test is based on the dichotomy that Fourier transforms of stochastic processes are near uncorrelated if the process is second order stationarity but correlated if the process is second order nonstationary. Using this as motivation, a Discrete Fourier transform for spatio-temporal data over discrete equidistant times but on irregularly spaced spatial locations is defined. Two statistics which measure the degree of correlation in the Discrete Fourier transforms are proposed. These statistics are used to test for spatio-temporal stationarity. It is shown that the same statistics can also be adapted to test for the one-way stationarity (either spatial or temporal stationarity). The proposed methodology is illustrated with a small simulation study.