Symmetry and Separability in Spatial-Temporal Processes

Abstract

Symmetry is one of most standard assumptions that are needed for a covariance function in spatial statistics. However, many studies in spatial research fields show that environmental data have complex spatial-temporal dependency structures that are difficult to model and estimate, due to the lack of symmetry and other standard assumptions of a covariance function. So, not much literature exists in statistics about asymmetric covariance functions and formal tests for lack of symmetry in spatial-temporal processes. In this study, we introduce certain types of symmetry in spatial-temporal processes and propose new classes of asymmetric spatial-temporal covariance models by using spectral representations. We also clarify the relationship between symmetry and separability and introduce nonseparable covariance models. Based on the proposed concept of symmetry in spatial-temporal processes, new formal tests for lack of symmetry are proposed in this study by employing spectral representations of the spatial-temporal covariance function. The advantage of the tests is that simple analysis of variance (ANOVA) approaches are employed for detecting lack of symmetry inherent in spatial-temporal processes. Our new classes of covariance models are applied to the methods for the fine particulate matters with a mass median diameter less than 2.5 $mu m$ ($mbox[PM]_[2.5]$) observed from U.S. Environmental Protection Agency (EPA). We evaluate the performance of the tests by a simulation study and, finally, apply to the $mbox[PM]_[2.5]$ daily concentration calculated by the Models-3/Community Multiscale Air Quality (CMAQ) modeling system with the spatial resolution of $36km imes 36 km$.