Testing for Directional Symmetry in Spatial Dependence Using the Periodogram

The characterization of spatial dependence is an important component of a spatial modeling exercise. For reasons of convenience, model parsimony, or computational efficiency, the spatial covariance structure is often assumed to be isotropic (direction-invariant), completely symmetric, or reflection symmetric (the latter two being forms of directional invariance somewhat weaker than isotropy). We propose some diagnostic tests of the latter two properties that are based on the two-dimensional periodogram. An advantage of basing tests on the periodogram rather than the sample semivariogram or covariance function is that the periodogram ordinates corresponding to different frequencies are asymptotically independent, leading to simpler distribution theory. A simulation study and two examples illustrate the usefulness and limitations of the proposed tests.