Nonparametric methods of assessing spatial isotropy

A common requirement for spatial analysis is the modeling of the second-order structure. While the assumption of isotropy is often made for this structure, it is not always appropriate. A conventional practice to check for isotropy is to informally assess plots of direction-specific sample second-order properties, e.g., sample variogram or sample second-order intensity function. While a useful diagnostic, these graphical techniques are difficult to assess and open to interpretation. Formal alternatives to graphical diagnostics are valuable, but have been applied to a limited class of models.

In this dissertation, we propose a formal approach testing for isotropy that is both objective and appropriate for a wide class of models. This approach, which is based on the asymptotic joint normality of the sample second-order properties, can be used to compare these properties in multiple directions. An L_2 consistent subsampling estimator for the asymptotic covariance matrix of the sample second-order properties is derived and used to construct the test statistic with a limiting $chi2$ distribution under the null hypothesis.

Our testing approach is purely nonparametric and can be applied to both quantitative spatial processes and spatial point processes. For quantitative processes, the results apply to both regularly spaced and irregularly spaced data when the point locations are generated by a homogeneous point process. In addition, the shape of the random field can be quite irregular. Examples and simulations demonstrate the efficacy of the approach.