Linking physics and spatial statistics: A new family of boltzmann-gibbs random fields

Abstract

We investigate a connection between spatial statistics and statistical physics to obtain new covariance functions with direct physical in-terpretation for spatial random fields. These covariance functions are based on the exponential Boltzmann-Gibbs representation and use an energy functional to represent interactions between the values of the random field at different points in space. This formulation results in closed-form generalized covariance functions, which display infinite variance in Euclidean spaces of dimension larger than one. We propose regularization schemes in real and reciprocal (spectral) space that lead to well-behaved covariance structures. The real-space regularization parameter allows a continuous interpolation between the Boltzmann-Gibbs covariance and the exponential covariance. We also propose discretized approximations on regular grids, and we show that they represent reparametrized versions of the well-known Besag and Leroux lattice models. We then discuss parameter estimation and spatial prediction for the regularized Boltzmann-Gibbs covariance model in two di-mensions. We recommend using the pairwise difference likelihood that com-bines satisfactory estimation performance and good scalability with many observation points. The predictive performance of the regularized covari-ance function is assessed by means of cross-validation statistics. Irregularly-spaced samples from the Walker Lake dataset are used, and spatial prediction is conducted by means of ordinary kriging. The regularized Boltzmann-Gibbs covariance yields improved predictive performance compared to the exponential covariance model.