Notebooks

Random Fields

Last update: 21 Apr 2025 21:17
First version:

Stochastic processes where the index variable is space, or something space-like. (Formally, one-dimensional space works a lot like time; and space-time works a lot like a higher-dimensional space, though not always.) This is of course important for modeling spatial and spatio-temporal data, but also data on networks, and statistical mechanics.

    Recommended, general:
  • Pierre Brémaud, Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues
  • Carlo Gaetan and Xavier Guyon, Spatial Statistics and Modeling
  • Geoffrey Grimmett, Probability on Graphs: Random Processes on Graphs and Lattices
  • Xavier Guyon, Random Fields on a Network
  • Peter Guttorp, Stochastic Modeling of Scientific Data
  • Brian D. Ripley, Statistical Inference for Spatial Processes
  • Rinaldo B. Schinazi, Classical and Spatial Stochastic Processes
    Recommended, of more specialized interest:
  • Ole E. Barndorff-Nielsen, Fred Espen Benth and Almut E. D. Veraart, Ambit Stochastics
  • J.-R. Chazottes, P. Collet, C. Kuelske and F. Redig, "Deviation inequalities via coupling for stochastic processes and random fields", math.PR/0503483
  • Jérôme Dedecker, Paul Doukhan, Gabriel Lang, José Rafael León R., Sana Louhichi and Clémentine Prieur, Weak Dependence: With Examples and Applications
  • David Griffeath, "Introduction to Markov Random Fields", ch. 12 in Kemeny, Knapp and Snell, Denumerable Markov Chains [One of the proofs of the equivalence between the Markov property and having a Gibbs distribution, conventionally but misleadingly called the Hammersley-Clifford Theorem. Pollard, below, provides an on-line summary.]
  • Mark Kaiser, "Statistical Dependence in Markov Random Field Models" [abstract, preprint]
  • Andee Kaplan, Mark S. Kaiser, Soumendra N. Lahiri, Daniel J. Nordman, "Simulating Markov random fields with a conclique-based Gibbs sampler", arxiv:1808.04739 [Presentation by Dr. Kaplan]
  • David Pollard, "Markov random fields and Gibbs distributions" [Online PDF. A proof of the theorem linking Markov random fields to Gibbs distributions, following the approach of David Griffeath.]
  • Jeffrey E. Steif, "Consistent estimation of joint distributions for sufficiently mixing random fields", Annals of Statistics 25 (1997): 293--304
    To read:
  • Jan Ambjorn et al., Quantum Geometry: A Statistical Field Theory Approach [I am interested in the stuff about random surfaces.]
  • K. Bahlali, M. Eddahbi and M. Mellouk, "Stability and genericity for SPDEs driven by spatially correlated noise", math.PR/0610174
  • Raluca M. Balan, "A strong invariance principle for associated random fields", Annals of Probability 33 (2005): 823--840 = math.OR/0503661
  • M. S. Bartlett, "Physical Nearest-Neighbour Models and Non-Linear Time Series", Journal of Applied Probability 8 (1971): 222--232
  • Michel Bauer, Denis Bernard, "2D growth processes: SLE and Loewner chains", math-ph/0602049
  • Denis Belomestny, Vladimir Spokoiny, "Concentration inequalities for smooth random fields", arxiv:1307.1565
  • Anton Bovier, Statistical Mechanics of Disordered Systems
  • Alexander Bulinski and Alexey Shashkin, "Strong invariance principle for dependent random fields", math.PR/0608237
  • M. Cassandro, A. Galves and E. Löcherbach, "Partially Observed Markov Random Fields Are Variable Neighborhood Random Fields", Journal of Statistical Physics 147 (2012): 795--807, arxiv:1111.1177
  • Ruslan K. Chornei, Hans Daduna, and Pavel S. Knopov
  • Giuseppe Da Prato, Arnaud Debussche and Luciano Tubaro, "Coupling for some partial differential equations driven by white noise", math.AP/0410441
  • Jean-Dominique Deuschel and Andreas Greven (eds.), Interacting Stochastic Systems [This looks deeply cool]
  • Rick Durrett, Stochastic Spatial Models: A Hyper-Tutorial
  • Vlad Elgart and Alex Kamenev, "Rare Events Statistics in Reaction--Diffusion Systems", cond-mat/0404241
  • Mohamed El Machkouri, Dalibor Volny, Wei Biao Wu, "A central limit theorem for stationary random fields", arxiv:1109.0838
  • H. Follmer, "On entropy and information gain in random fields", Z. Wahrsh. verw. Geb. 26 (1973): 207--217
  • T. Funaki, D. Surgailis and W. A. Woyczynski, "Gibbs-Cox Random Fields and Burgers Turbulence", Annals of Applied Probability 5 (1995): 461--492
  • L. Garcia-Ojalvo and J. Sancho, Noise in Spatially Extended Systems
  • B. M. Gurevich and A. A. Tempelman, "Markov approximation of homogeneous lattice random fields", Probability Theory and Related Fields 131 (2005): 519--527
  • Allan Gut and Ulrich Stadtmuller, "Cesaro Summation for Random Fields", Journal of Theoretical Probability 23 (2010): 715--728
  • Peter Hall, Introduction to the Theory of Coverage Processes [= point process with a random shape attached to each point]
  • Reza Hosseini, "Conditional information and definition of neighbor in categorical random fields", arxiv:1101.0255 ["Who then is my neighbor?" (Not an actual quote from the paper.)]
  • Xiangping Hu, Daniel Simpson, Finn Lindgren, Havard Rue, "Multivariate Gaussian Random Fields Using Systems of Stochastic Partial Differential Equations", arxiv:1307.1379
  • Niels Jacob and Alexander Potrykus, "Some thoughts on multiparameter stochastic processes", math.PR/0607744
  • Wolfgang Karcher, Elena Shmileva, Evgeny Spodarev, "Extrapolation of stable random fields", arxiv:1107.1654
  • M. Kerscher, "Constructing, characterizing, and simulating Gaussian and higher-order point distributions," astro-ph/0102153
  • Ross Kindermann and J. Laurie Snell, Markov Random Fields and Their Applications [Free online!]
  • P. Kotelenez, Stochastic Space-Time Models and Limit Theorems
  • Michael A. Kouritzin and Hongwei Long, "Convergence of Markov chain approximations to stochastic reaction-diffusion equations", Annals of Applied Probability 12 (2002): 1039--1070
  • Jean-Francois Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations
  • Atul Mallik, Michael Woodroofe, "A Central Limit Theorem For Linear Random Fields", arxiv:1007.1490
  • Jonathan C. Mattingly, "On Recent Progress for the Stochastic Navier Stokes Equations", math.PR/0409194
  • A. I. Olemskoi, D. O. Kahrchenko and I. A. Knyaz', "Phase transitions induced by noise cross-correlations", cond-mat/0403583
  • Rupert Paget, "Strong Markov Random Field Model", IEEE Transactions on Pattern Analysis and Machine Intelligence 26 (2004): 408--413
  • Marcelo Pereyra, Nicolas Dobigeon, Hadj Batatia, Jean-Yves Tourneret, "Computing the Cramer-Rao bound of Markov random field parameters: Application to the Ising and the Potts models", arxiv:1206.3985
  • Liang Qiao, Radek Erban, C. T. Kelley and Ioannis G. Kevrekidis, "Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation", q-bio.QM/0606006
  • Havard Rue and Leonhard Held, Gaussian Markov Random Fields: Theory and Applications
  • Andre Toom, "Law of Large Numbers for Non-Local Functions of Probabilistic Cellular Automata", Journal of Statistical Physics 133 (2008): 883--897
  • M. N. M. van Lieshout, "Markovianity in space and time", math.PR/0608242
  • Divyanshu Vats and Jose M. F. Moura, "Telescoping Recursive Representations and Estimation of Gauss-Markov Random Fields", arxiv:0907.5397
  • Benjamin Yakir, Extremes in Random Fields: A Theory and its Applications
  • Eunho Yang, Pradeep K. Ravikumar, Genevera I. Allen, Zhandong Liu, "Conditional Random Fields via Univariate Exponential Families", NIPS 2013

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