Elliptical Anisotropy Statistics of Two-Dimensional Differentiable Gaussian Random Fields: Joint Probability Density Function and Confidence Regions

Two-dimensional data often have autocovariance functions with elliptical equipotential contours, a property known as statistical anisotropy. The anisotropy parameters include the tilt of the ellipse (orientation angle) $\theta$ with respect to the coordinate system and the ratio $R$ of the principal correlation lengths. Sample estimates of anisotropy parameters are needed for defining suitable spatial models and for interpolation of incomplete data. The sampling joint probability density function characterizes the distribution of anisotropy statistics $(\hat{R}, \hat{\theta})$. By means of analytical calculations, we derive an explicit expression for the joint probability density function, which is valid for Gaussian, stationary and differentiable random fields. Based on it, we derive an approximation of the joint probability density function that is independent of the autocovariance function and provides conservative confidence regions for the sample-based estimates $(\hat{R},\hat{\theta})$. We also formulate a statistical test for isotropy based on the approximation of the joint probability density function. The latter provides (i) a stand-alone approximate estimate of the $(\hR, \htheta)$ distribution (ii) computationally efficient initial values for maximum likelihood estimation, and (iii) a useful prior for Bayesian anisotropy inference. We validate the theoretical analysis by means of simulations, and we illustrate the use of confidence regions with a real-data case study.