Testing lack of symmetry in spatial-temporal processes

Symmetry and separability of a covariance function are common assumptions to simplify the modeling effort of spatial-temporal processes. However, many studies in environmental sciences show that real data have complex spatial-temporal dependency structures resulting from lack of symmetry or violation of other standard assumptions of the covariance function. In this study, we propose new formal tests for lack of symmetry by using spectral representations of the spatial-temporal covariance functions of regularly spaced spatial-temporal data. The advantage of the proposed tests is that classical analysis of variance (ANOVA) models can be used for detecting lack of symmetry inherent in spatial-temporal processes. We evaluate the performance of the tests with simulation studies and we apply them to air pollution data.