Non-Parametric Approximations for Anisotropy Estimation in Two-dimensional Differentiable Gaussian Random Fields

TL;DR

This paper derives an explicit joint probability density function for anisotropy statistics in Gaussian random fields, enabling better estimation and testing of anisotropy parameters in spatial data analysis.

Contribution

It provides a novel analytical expression for the joint density of anisotropy statistics, independent of the autocovariance function, facilitating improved anisotropy estimation and hypothesis testing.

Findings

Derived explicit joint probability density function for anisotropy statistics.

Validated the theoretical results through simulations with synthetic data.

Demonstrated application in detecting anisotropy changes in real-world data.

Abstract

Spatially referenced data often have autocovariance functions with elliptical isolevel contours, a property known as geometric anisotropy. The anisotropy parameters include the tilt of the ellipse (orientation angle) with respect to a reference axis and the aspect ratio of the principal correlation lengths. Since these parameters are unknown a priori, sample estimates are needed to define suitable spatial models for the interpolation of incomplete data. The distribution of the anisotropy statistics is determined by a non-Gaussian sampling joint probability density. By means of analytical calculations, we derive an explicit expression for the joint probability density function of the anisotropy statistics for Gaussian, stationary and differentiable random fields. Based on this expression, we obtain an approximate joint density which we use to formulate a statistical test for isotropy. The approximate joint density is independent of the autocovariance function and provides conservative probability and confidence regions for the anisotropy parameters. We validate the theoretical analysis by means of simulations using synthetic data, and we illustrate the detection of anisotropy changes with a case study involving background radiation exposure data. The approximate joint density provides (i) a stand-alone approximate estimate of the anisotropy statistics distribution (ii) informed initial values for maximum likelihood estimation, and (iii) a useful prior for Bayesian anisotropy inference.