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A spectral domain test for stationarity of spatial and spatio-temporal processes.

Many random phenomena in the environmental and geophysical sciences are functions of space or both space and time and one important aim is to develop statistical models based on what is observed. While doing so a commonly used assumption is that the underlying process is second order 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 tests for spatial and spatio-temporal stationarity. The tests are based on the dichotomy that Fourier transforms of stochastic processes are near uncorrelated if the process is second order stationary but correlated if the process is second order nonstationary. Using this as motivation, a Discrete Fourier transform for irregularly spaced spatial data as well as for spatio-temporal data observed over discrete equidistant times but on irregularly spaced spatial locations are defined. The test statistics are based on measuring the degree of correlation in the transformed data. The asymptotic sampling properties of听 the test statistics are derived under both stationarity and nonstationarity of the random fields. These results motivate a graphical tool which allows a visual representation of the nonstationary features. The methods are also illustrated with simulations and real data examples.