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Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)



Boltzmann–Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics

Author: Dionissios T. Hristopulos
Journal: Theor. Probability and Math. Statist. 107 (2022), 37-60
MSC (2020): Primary 60G15, 62P12; Secondary 62P30, 62M40
Published electronically: November 8, 2022
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Abstract: Boltzmann–Gibbs random fields are defined in terms of the exponential expression $\exp \left (-\mathcal {H}\right )$, where $\mathcal {H}$ is a suitably defined energy functional of the field states $x(\mathbf {s})$. This paper presents a new Boltzmann–Gibbs model which features local interactions in the energy functional. The interactions are embodied in a spatial coupling function which uses smoothed kernel-function approximations of spatial derivatives inspired from the theory of smoothed particle hydrodynamics. A specific model for the interactions based on a second-degree polynomial of the Laplace operator is studied. An explicit, mesh-free expression of the spatial coupling function (precision function) is derived for the case of the squared exponential (Gaussian) smoothing kernel. This coupling function allows the model to seamlessly extend from discrete data vectors to continuum fields. Connections with Gaussian Markov random fields and the Matérn field with $\nu =1$ are established.

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Additional Information

Dionissios T. Hristopulos
Affiliation: School of Electrical and Computer Engineering, Technical University of Crete, Chania, 73100 Greece

Keywords: Random fields, kernel functions, precision matrix, smoothed particle hydrodynamics
Received by editor(s): July 30, 2021
Accepted for publication: January 20, 2022
Published electronically: November 8, 2022
Article copyright: © Copyright 2022 Taras Shevchenko National University of Kyiv