Analytic Properties and Covariance Functions of a New Class of Generalized Gibbs Random Fields

Abstract

Spartan Spatial Random Fields (SSRFs) are generalized Gibbs random fields, equipped with a coarse-graining kernel that acts as a low-pass filter for the fluctuations. SSRFs are defined by means of physically motivated spatial interactions and a small set of free parameters (interaction couplings). This paper focuses on the FGC-SSRF model, which is defined on the Euclidean space ${\mathbb{R}}^d$ by means of interactions proportional to the squares of the field realizations, as well as their gradient and curvature. The permissibility criteria of FGC-SSRFs are extended by considering the impact of a finite-bandwidth kernel. It is proved that the FGC-SSRFs are almost surely differentiable in the case of finite bandwidth. Asymptotic explicit expressions for the Spartan covariance function are derived for $d=1$ and $d=3$; both known and new covariance functions are obtained depending on the value of the FGC-SSRF shape parameter. Nonlinear dependence of the covariance integral scale on the FGC-SSRF characteristic length is established, and it is shown that the relation becomes linear asymptotically. The results presented in this paper are useful in random field parameter inference, as well as in spatial interpolation of irregularly-spaced samples.

24 pages; 4 figures Submitted for publication to IEEE Transactions on Information Theory

References in corpus (1)

• Spatial Random Field Models Inspired from Statistical Physics with Applications in the Geosciences

Cited by in corpus (13)

• Stochastic Local Interaction (SLI) Model: Interfacing Machine Learning and Geostatistics
• Radial Covariance Functions Motivated by Spatial Random Field Models with Local Interactions
• Generalized Whittle-Mat$\acute{\text{E}}$rn random field as a model of correlated fluctuations
• Non-separable Covariance Kernels for Spatiotemporal Gaussian Processes based on a Hybrid Spectral Method and the Harmonic Oscillator
• Environmental Time Series Interpolation Based on Spartan Random Processes
• Space-Time Models based on Random Fields with Local Interactions
• Stochastic Local Interaction Model: Geostatistics without Kriging
• The Method of Normalized Correlations - A Fast Alternative to Maximum Likelihood Estimation for Random Processes and Isotropic Random Fields with Short-Range Dependence
• Gibbs Markov Random Fields with Continuous Values based on the Modified Planar Rotator Model
• Spartan Random Processes in Time Series Modeling
• Spatial data modeling by means of Gibbs Markov random fields based on a generalized planar rotator model
• Boltzmann-Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics
• A parsimonious, computationally efficient machine learning method for spatial regression