Theory of Probability and Mathematical Statistics
Boltzmann–Gibbs Random Fields with Mesh-free Precision Operators Based on Smoothed Particle Hydrodynamics
Dionissios T. Hristopulos
Link
Abstract: Boltzmann–Gibbs random fields are defined in terms of the exponential expression exp(-H), where H is a suitably defined energy functional of the field states x(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 v=1 are established.
Keywords: Random fields, kernel functions, precision matrix, smoothed particle hydrodynamics
Bibliography: D. Allard, D. T. Hristopulos, and T. Opitz, Linking physics and spatial statistics: a new family of Boltzmann-Gibbs random fields, Electron. J. Stat. 15 (2021), no. 2, 4085–4116. MR 4308768
N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437
S. Bochner, Lectures on Fourier integrals, Annals of Mathematics Studies, No. 42, Princeton University Press, Princeton, N.J., 1959, Translated by Morris Tenenbaum and Harry Pollard. MR 0107124
A. Chorti and D. T. Hristopulos, Nonparametric identification of anisotropic (elliptic) correlations in spatially distributed data sets, IEEE Trans. Signal Process. 56 (2008), no. 10, part 1, 4738–4751. MR 2517209
N. Cressie, Spatial statistics, John Wiley and Sons, New York, 1993. MR 624436
L. C. Evans, Partial differential equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943
R. W. Franke, Scattered data interpolation: tests of some methods, Math. Comp. 38 (1982), no. 157, 181–200. MR 637296
J. Glimm and A. Jaffe, Quantum physics: A functional integral point of view, second ed., Springer-Verlag, New York, 1987. MR 887102
N. Goldenfeld, Lectures on phase transitions and the renormalization group, Frontiers in Physics, Addison-Wesley, Reading, MA, 1993.
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. i. Gaussian interface fluctuations, Phys. Rev. E 47 (1993), no. 6, 4289–4300.
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. ii. Monte Carlo simulations, Phys. Rev. E 47 (1993), no. 6, 4301–4314.
T. Hayden, The extension of bilinear functionals, Pacific J. Math. 22 (1967), 99–108. MR 227736
D. Higdon, Space and space-time modeling using process convolutions, Quantitative methods for current environmental issues, Springer, London, 2002, pp. 37–56. MR 2059819
D. T. Hristopulos, Spartan Gibbs random field models for geostatistical applications, SIAM J. Sci. Comput. 24 (2003), no. 6, 2125–2162. MR 2005624
D. T. Hristopulos, Covariance functions motivated by spatial random field models with local interactions, Stoch. Environ. Res. Risk Assess. 29 (2015), no. 3, 739–754.
D. T. Hristopulos, Stochastic local interaction (SLI) model: Bridging machine learning and geostatistics, Comput. Geosci. 85 (2015), 26–37.
D. T. Hristopulos,Random fields for spatial data modeling: A primer for scientists and engineers, Springer, Dordrecht, the Netherlands, 2020.
D. T. Hristopulos and V. D. Agou, Stochastic local interaction model with sparse precision matrix for space–time interpolation, Spat. Stat. 40 (2020), 100403, 22. MR 4181140
D. T. Hristopulos and S. Elogne, Analytic properties and covariance functions for a new class of generalized Gibbs random fields, IEEE Trans. Inform. Theory 53 (2007), no. 12, 4667–4679. MR 2446930
E. Ising, Contribution to the theory of ferromagnetism, Zeitschrift für Physik 31 (1925), no. 1, 253–258.
M. Kardar, Statistical physics of fields, Cambridge University Press, Cambridge, 2007. MR 2374147
F. Lindgren, D. Bolin, and H. Rue, The SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running, Spat. Stat. 50 (2022), Paper No. 100599. MR 4439328
F. Lindgren, H. Rue, and J. Lindström, An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach, J. R. Stat. Soc. Ser. B Stat. Methodol. 73 (2011), no. 4, 423–498, With discussion and a reply by the authors. MR 2853727
J. J. Monaghan, Smoothed particle hydrodynamics, Annu. Rev. Astron. Astrophys. 30 (1992), no. 1, 543–574.
J. J Monaghan, Smoothed particle hydrodynamics, Rep. Progr. Phys. 68 (2005), no. 8, 1703–1759. MR 2158506
G. Mussardo, Statistical field theory: An introduction to exactly solved models in statistical physics, Oxford Graduate Texts, Oxford University Press, Oxford, 2010. MR 2559725
Néel, M. L., Propriétés magnétiques des ferrites; ferrimagnétisme et antiferromagnétisme, Annales de Physique 12 (1948), no. 3, 137–198.
M. Nica, Eigenvalues and eigenfunctions of the Laplacian, The Waterloo Mathematics Review 1 (2011), no. 2, 23–34.
M. P. Petrakis and D. T. Hristopulos, Non-parametric approximations for anisotropy estimation in two-dimensional differentiable Gaussian random fields, Stoch. Environ. Res. Risk Assess. 31 (2017), no. 7, 1853–1870.
C. E. Rasmussen and C. K. I. Williams, Gaussian processes for machine learning, Adaptive Computation and Machine Learning, MIT Press, Cambridge, MA, 2006. MR 2514435
J. A. Rozanov, Markov random fields and stochastic differential equations, Mathematics of the USSR-Sbornik 32 (1977), no. 4, 515–534. MR 505113
Y. A. Rozanov, Markov random fields, Applications of Mathematics, Springer-Verlag, New York-Berlin, 1982, Translated from the Russian by Constance M. Elson. MR 676644
H. Rue and L. Held, Gaussian Markov random fields: Theory and applications, Monographs on Statistics and Applied Probability, vol. 104, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2130347
B. Schölkopf, R. Herbrich, and A. J. Smola, A generalized representer theorem, Computational learning theory (Amsterdam, 2001), Lecture Notes in Comput. Sci., vol. 2111, Springer, Berlin, 2001, pp. 416–426. MR 2042050
B. Schölkopf and A. J. Smola, Learning with kernels: Support vector machines, regularization, optimization, and beyond, MIT Press, Cambridge, MA, USA, 2002.
S. Sheffield, Gaussian free fields for mathematicians, Probab. Theory Related Fields 139 (2007), no. 3-4, 521–541. MR 2322706
L. N. Trefethen and M. Embree, Spectra and pseudospectra: The behavior of nonnormal matrices and operators, Princeton University Press, Princeton, NJ, 2005. MR 2155029
H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), no. 4, 441–479. MR 1511670
M. I. Yadrenko, Spectral theory of random fields, Translation Series in Mathematics and Engineering, Optimization Software, Inc., Publications Division, New York, 1983, Translated from the Russian. MR 697386
A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. II, Springer Series in Statistics, Springer-Verlag, New York, 1987, Supplementary notes and references. MR 915557
M. Yaremchuk and S. Smith, On the correlation functions associated with polynomials of the diffusion operator, Q.J.R. Meteorol. Soc. 137 (2011), no. 660, 1927–1932.
