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

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.Comment: 39 pages; 8 figure

A Latent Variable Approach to Modelling Multivariate Geostatistical Skew-Normal Data

In this paper we propose a spatial latent factor model to deal with multivariate geostatistical skew-normal data. In this model we assume that the unobserved latent structure, responsible for the correlation among different variables as well as for the spatial autocorrelation among different sites is Gaussian, and that the observed variables are skew-normal. For this model we provide some of its properties like its spatial autocorrelation structure and its finite dimensional marginal distributions. Estimation of the unknown parameters of the model is carried out by employing a Monte Carlo Expectation Maximization algorithm, whereas prediction at unobserved sites is performed by using closed form formulas and Markov chain Monte Carlo algorithms. Simulation studies have been performed to evaluate the soundness of the proposed procedures