Gibbs Markov Random Fields with Continuous Values based on the Modified Planar Rotator Model

Abstract

We introduce a novel Gibbs Markov random field for spatial data on Cartesian grids based on the modified planar rotator (MPR) model of statistical physics. The MPR captures spatial correlations using nearest-neighbor interactions of continuously-valued spins and does not rely on Gaussian assumptions. The only model parameter is the reduced temperature, which we estimate by means of an ergodic specific energy matching principle. We propose an efficient hybrid Monte Carlo simulation algorithm that leads to fast relaxation of the MPR model and allows vectorization. Consequently, the MPR computational time for inference and simulation scales approximately linearly with system size. This makes it more suitable for big data sets, such as satellite and radar images, than conventional geostatistical approaches. The performance (accuracy and computational speed) of the MPR model is validated with conditional simulation of Gaussian synthetic and non-Gaussian real data (atmospheric heat release measurements and Walker-lake DEM-based concentrations) and comparisons with standard gap-filling methods.