A formal test for nonstationarity of spatial stochastic processes

Spatial statistics is one of the major methodologies of image analysis, field trials, remote sensing, and environmental statistics. The standard methodology in spatial statistics is essentially based on the assumption of stationary and isotropic random fields. Such assumptions might not hold in large heterogeneous fields. Thus, it is important to understand when stationarity and isotropy are reasonable assumptions. Most of the work that has been done so far to test the nonstationarity of a random process is in one dimension. Unfortunately, there is not much literature of formal procedures to test for stationarity of spatial stochastic processes. s manuscript, we consider the problem of testing a given spatial process for stationarity and isotropy. The approach is based on a spatial spectral analysis, this means spectral functions which are space dependent. The proposed method consists essentially in testing the homogeneity of a set of spatial spectra evaluated at different locations. In addition to testing stationarity and isotropy, the analysis provides also a method for testing whether the observed process fits a uniformly modulated model, and a test for randomness (white noise). Applications include modeling and testing for nonstationary of air pollution concentrations over different geo-political boundaries.