Symmetry and Separability in Spatial-Temporal Processes
Man Sik Park, PhD
Postdoctoral Fellow
Department of Statistics
Colorado State University
Wednesday, February 1, 2006
3:10 p.m.
E202 Engineering Building

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 $36km \times 36 km$.



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