scholarly journals Random Fields in Physics, Biology and Data Science

2021 ◽  
Vol 9 ◽  
Author(s):  
Enrique Hernández-Lemus
Keyword(s):  
Random Field ◽  
Random Fields ◽  
Markov Random Fields ◽  
Statistical Physics ◽  
Data Science ◽  
Joint Probability ◽  
Theoretical Aspect ◽  
Joint Probability Distribution ◽  
Computational Molecular Biology ◽  
Markov Random

A random field is the representation of the joint probability distribution for a set of random variables. Markov fields, in particular, have a long standing tradition as the theoretical foundation of many applications in statistical physics and probability. For strictly positive probability densities, a Markov random field is also a Gibbs field, i.e., a random field supplemented with a measure that implies the existence of a regular conditional distribution. Markov random fields have been used in statistical physics, dating back as far as the Ehrenfests. However, their measure theoretical foundations were developed much later by Dobruschin, Lanford and Ruelle, as well as by Hammersley and Clifford. Aside from its enormous theoretical relevance, due to its generality and simplicity, Markov random fields have been used in a broad range of applications in equilibrium and non-equilibrium statistical physics, in non-linear dynamics and ergodic theory. Also in computational molecular biology, ecology, structural biology, computer vision, control theory, complex networks and data science, to name but a few. Often these applications have been inspired by the original statistical physics approaches. Here, we will briefly present a modern introduction to the theory of random fields, later we will explore and discuss some of the recent applications of random fields in physics, biology and data science. Our aim is to highlight the relevance of this powerful theoretical aspect of statistical physics and its relation to the broad success of its many interdisciplinary applications.

New models for Markov random fields

1992 ◽  
Vol 29 (04) ◽  
pp. 877-884 ◽  
Author(s):  
Noel Cressie ◽  
Subhash Lele
Keyword(s):  
Random Field ◽  
Conditional Probability ◽  
Markov Random Field ◽  
Random Fields ◽  
Markov Random Fields ◽  
Sufficient Conditions ◽  
Joint Probability ◽  
Probability Models ◽  
Markov Random ◽  
New Models

The Hammersley–Clifford theorem gives the form that the joint probability density (or mass) function of a Markov random field must take. Its exponent must be a sum of functions of variables, where each function in the summand involves only those variables whose sites form a clique. From a statistical modeling point of view, it is important to establish the converse result, namely, to give the conditional probability specifications that yield a Markov random field. Besag (1974) addressed this question by developing a one-parameter exponential family of conditional probability models. In this article, we develop new models for Markov random fields by establishing sufficient conditions for the conditional probability specifications to yield a Markov random field.

New models for Markov random fields

1992 ◽  
Vol 29 (4) ◽  
pp. 877-884 ◽  
Author(s):  
Noel Cressie ◽  
Subhash Lele
Keyword(s):  
Random Field ◽  
Conditional Probability ◽  
Markov Random Field ◽  
Random Fields ◽  
Markov Random Fields ◽  
Sufficient Conditions ◽  
Joint Probability ◽  
Probability Models ◽  
Markov Random ◽  
New Models

The Hammersley–Clifford theorem gives the form that the joint probability density (or mass) function of a Markov random field must take. Its exponent must be a sum of functions of variables, where each function in the summand involves only those variables whose sites form a clique. From a statistical modeling point of view, it is important to establish the converse result, namely, to give the conditional probability specifications that yield a Markov random field. Besag (1974) addressed this question by developing a one-parameter exponential family of conditional probability models. In this article, we develop new models for Markov random fields by establishing sufficient conditions for the conditional probability specifications to yield a Markov random field.

Conditional cyclic Markov random fields

1996 ◽  
Vol 28 (1) ◽  
pp. 1-12 ◽  
Author(s):  
John T. Kent ◽  
Kanti V. Mardia ◽  
Alistair N. Walder
Keyword(s):  
Random Field ◽  
Markov Random Field ◽  
Random Fields ◽  
Markov Random Fields ◽  
Field Model ◽  
Random Field Model ◽  
Markov Random Field Model ◽  
Gaussian Markov Random Field ◽  
Markov Random ◽  
Limiting Behaviour

Grenander et al. (1991) proposed a conditional cyclic Gaussian Markov random field model for the edges of a closed outline in the plane. In this paper the model is recast as an improper cyclic Gaussian Markov random field for the vertices. The limiting behaviour of this model when the vertices become closely spaced is also described and in particular its relationship with the theory of ‘snakes' (Kass et al. 1987) is established. Applications are given in Grenander et al. (1991), Mardia et al. (1991) and Kent et al. (1992).

Conditional cyclic Markov random fields

1996 ◽  
Vol 28 (01) ◽  
pp. 1-12 ◽  
Author(s):  
John T. Kent ◽  
Kanti V. Mardia ◽  
Alistair N. Walder
Keyword(s):  
Random Field ◽  
Markov Random Field ◽  
Random Fields ◽  
Markov Random Fields ◽  
Field Model ◽  
Random Field Model ◽  
Markov Random Field Model ◽  
Gaussian Markov Random Field ◽  
Markov Random ◽  
Limiting Behaviour

Grenander et al. (1991) proposed a conditional cyclic Gaussian Markov random field model for the edges of a closed outline in the plane. In this paper the model is recast as an improper cyclic Gaussian Markov random field for the vertices. The limiting behaviour of this model when the vertices become closely spaced is also described and in particular its relationship with the theory of ‘snakes' (Kass et al. 1987) is established. Applications are given in Grenander et al. (1991), Mardia et al. (1991) and Kent et al. (1992).

scholarly journals Modeling of the Markiv random field for the purpose of its further optimization and application

Connectivity ◽  
2020 ◽  
Vol 146 (5) ◽  
Author(s):  
V. V. Zhebka ◽  
Keyword(s):  
Random Field ◽  
Markov Random Field ◽  
Random Fields ◽  
Markov Random Fields ◽  
Markov Models ◽  
Scale Model ◽  
Small Scale ◽  
Advantages And Disadvantages ◽  
Markov Random ◽  
Different Types

Models of the Markov random field are investigated. The main improvements of the Markov random field model are investigated. If we consider Markov models of random fields with binary conditional distributions, which include stochastic evolution in time, which is based on the autoregression structure for a large-scale model, these models retain the flexibility of static Markov random field models to reproduce the representation of spatial dependence in a small-scale model. Bayesian estimation in this case is achieved through the use of a so-called algorithm that requires the generation of auxiliary random fields, but does not require the use of ideal samples. Markov random fields are a powerful tool in machine learning. It is often necessary to model such fields between dissimilar objects, which leads to the fact that the nodes in the graph belong to different types of data. To model inhomogeneous areas using graphical models, it is necessary to assign different types of distributions (binary, Gaussian, Poisson, exponent, exponential, etc.) to the model nodes. The concept of conditional random fields is considered in the article, their features, advantages and disadvantages are established. The application of binary data in Markov models of random fields is considered, which generates a class of models of binary Markov random fields. It is established that the discrete nature of Markov random fields allows a wider range of possible values of dependence, ie negative dependence. The model, loss function and distribution of the Markov random field function are investigated. Strengthening of Markov random fields is proposed. The pairwise exponential Markov random field is considered.

Poisson and extreme value limit theorems for Markov random fields

1987 ◽  
Vol 19 (01) ◽  
pp. 106-122 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Random Field ◽  
Markov Random Field ◽  
Random Fields ◽  
Conditional Distribution ◽  
Markov Random Fields ◽  
Limit Theorems ◽  
Nearest Neighbor ◽  
Extreme Value ◽  
Markov Random ◽  
Corner Points

Let Xt , be a Markov random field assuming values in RM. Let In be a rectangular box in Zm with its center at 0 and corner points with coordinates ±n. Let (An ) be a sequence of measurable subsets of RM such that neighborhood of t) → 0, for n → ∞; and let fn (x) be the indicator of An. Under appropriate conditions on the nearest-neighbor distributions of (Xt ), the conditional distribution of given the values of Xs , for s on the boundary of In , converges to the Poisson distribution. An immediate application is an extreme value limit theorem for a real-valued Markov random field.

Poisson and extreme value limit theorems for Markov random fields

1987 ◽  
Vol 19 (1) ◽  
pp. 106-122 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Random Field ◽  
Markov Random Field ◽  
Random Fields ◽  
Conditional Distribution ◽  
Markov Random Fields ◽  
Limit Theorems ◽  
Nearest Neighbor ◽  
Extreme Value ◽  
Markov Random ◽  
Corner Points

Let Xt, be a Markov random field assuming values in RM. Let In be a rectangular box in Zm with its center at 0 and corner points with coordinates ±n. Let (An) be a sequence of measurable subsets of RM such that neighborhood of t) → 0, for n → ∞; and let fn(x) be the indicator of An. Under appropriate conditions on the nearest-neighbor distributions of (Xt), the conditional distribution of given the values of Xs, for s on the boundary of In, converges to the Poisson distribution. An immediate application is an extreme value limit theorem for a real-valued Markov random field.

Sampling from Gaussian Markov random fields conditioned on linear constraints

2008 ◽  
Vol 48 ◽  
pp. 1041 ◽  
Author(s):  
Daniel Peter Simpson ◽  
Ian W. Turner ◽  
A. N. Pettitt
Keyword(s):  
Random Fields ◽  
Markov Random Fields ◽  
Linear Constraints ◽  
Gaussian Markov Random Fields ◽  
Markov Random
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