scholarly journals One-dimensional Markov random fields, Markov chains and topological Markov fields

2013 ◽  
Vol 142 (1) ◽  
pp. 227-242 ◽  
Author(s):  
Nishant Chandgotia ◽  
Guangyue Han ◽  
Brian Marcus ◽  
Tom Meyerovitch ◽  
Ronnie Pavlov
Keyword(s):  
Markov Chains ◽  
Random Fields ◽  
Markov Random Fields ◽  
One Dimensional ◽  
Markov Random ◽  
Markov Fields

Markov Chains in Many Dimensions

1994 ◽  
Vol 26 (3) ◽  
pp. 756-774 ◽  
Author(s):  
Dimitris N. Politis
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Maximum Entropy ◽  
Stationary Distribution ◽  
Random Fields ◽  
Special Class ◽  
Markov Random Fields ◽  
One Dimension ◽  
Stationary Markov Chain ◽  
Markov Random

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

An example of phase transition in countable one-dimensional Markov random fields

1977 ◽  
Vol 14 (1) ◽  
pp. 205-211 ◽  
Author(s):  
Ted Cox
Keyword(s):  
Phase Transition ◽  
Random Fields ◽  
Markov Random Fields ◽  
Conditional Probabilities ◽  
Positive Matrix ◽  
One Dimensional ◽  
Markov Random

Let S be a countable set, Q a strictly positive matrix on S × S. The set 𝒢(Q) of one-dimensional Markov random fields taking values in S with conditional probabilities determined by Q has been investigated by Spitzer [4], Föllmer [1] and Kesten [3]. In this paper a new result of Spitzer's is stated and proved, and used to present a specific example (the only one known) of a matrix Q which exhibits phase transition and admits a complete description of 𝒢 (Q).

A central limit theorem for conditionally centred random fields with an application to Markov fields

1998 ◽  
Vol 35 (03) ◽  
pp. 608-621
Author(s):  
Francis Comets ◽  
Martin Janžura
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Markov Random Fields ◽  
Moment Condition ◽  
Markov Random ◽  
Pseudo Likelihood ◽  
Strict Positivity ◽  
Markov Fields

We prove a central limit theorem for conditionally centred random fields, under a moment condition and strict positivity of the empirical variance per observation. We use a random normalization, which fits non-stationary situations. The theorem applies directly to Markov random fields, including the cases of phase transition and lack of stationarity. One consequence is the asymptotic normality of the maximum pseudo-likelihood estimator for Markov fields in complete generality.

scholarly journals On a Class of Tensor Markov Fields

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 451
Author(s):  
Enrique Hernández-Lemus
Keyword(s):  
Graphical Models ◽  
Random Fields ◽  
Markov Random Fields ◽  
Random Variables ◽  
Probability Measures ◽  
Gibbs Fields ◽  
Markov Random ◽  
Statistical Dependency ◽  
Markov Fields ◽  
Dependency Structures

Here, we introduce a class of Tensor Markov Fields intended as probabilistic graphical models from random variables spanned over multiplexed contexts. These fields are an extension of Markov Random Fields for tensor-valued random variables. By extending the results of Dobruschin, Hammersley and Clifford to such tensor valued fields, we proved that tensor Markov fields are indeed Gibbs fields, whenever strictly positive probability measures are considered. Hence, there is a direct relationship with many results from theoretical statistical mechanics. We showed how this class of Markov fields it can be built based on a statistical dependency structures inferred on information theoretical grounds over empirical data. Thus, aside from purely theoretical interest, the Tensor Markov Fields described here may be useful for mathematical modeling and data analysis due to their intrinsic simplicity and generality.

Mathematics and Innovation in Engineering

2008 ◽  
Vol 380 ◽  
pp. 3-14
Author(s):  
Elena Beretta ◽  
Alberto Gandolfi ◽  
C.C.A. Sastri
Keyword(s):  
Markov Chains ◽  
Large Deviations ◽  
Random Fields ◽  
Spin Glasses ◽  
Markov Random Fields ◽  
Imaging Technologies ◽  
Inverse Conductivity Problem ◽  
Markov Random ◽  
The Way ◽  
Conductivity Problem

We present some examples of mathematical discoveries whose original import was mainly theoretical but which later ended up triggering extraordinary ad- vances in engineering, sometimes all the way down to technological realizations and market products. The examples we cite include Markov chains and Markov random fields, spin glasses, large deviations and the inverse conductivity problem, and their effects in various areas such as communication and imaging technologies.

An example of phase transition in countable one-dimensional Markov random fields

1977 ◽  
Vol 14 (01) ◽  
pp. 205-211 ◽  
Author(s):  
Ted Cox
Keyword(s):  
Phase Transition ◽  
Random Fields ◽  
Markov Random Fields ◽  
Conditional Probabilities ◽  
Positive Matrix ◽  
One Dimensional ◽  
Markov Random

Let S be a countable set, Q a strictly positive matrix on S × S. The set 𝒢(Q) of one-dimensional Markov random fields taking values in S with conditional probabilities determined by Q has been investigated by Spitzer [4], Föllmer [1] and Kesten [3]. In this paper a new result of Spitzer's is stated and proved, and used to present a specific example (the only one known) of a matrix Q which exhibits phase transition and admits a complete description of 𝒢 (Q).

Markov Chains in Many Dimensions

1994 ◽  
Vol 26 (03) ◽  
pp. 756-774 ◽  
Author(s):  
Dimitris N. Politis
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Maximum Entropy ◽  
Stationary Distribution ◽  
Random Fields ◽  
Special Class ◽  
Markov Random Fields ◽  
One Dimension ◽  
Stationary Markov Chain ◽  
Markov Random

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

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