Limiting results for arrays of binary random variables on rectangular lattices under sparseness conditions

1979 ◽  
Vol 16 (03) ◽  
pp. 554-566 ◽  
Author(s):  
Roy Saunders ◽  
Richard J. Kryscio ◽  
Gerald M. Funk
Keyword(s):  
Simulation Study ◽  
Random Fields ◽  
Markov Random Fields ◽  
Random Variables ◽  
Poisson Approximation ◽  
Markov Random ◽  
Binary Random Variables ◽  
Inference Techniques ◽  
Image Position ◽  
Rectangular Lattices

In this article we give limiting results for arrays {Xij (m, n) (i, j) Dmn } of binary random variables distributed as particular types of Markov random fields over m x n rectangular lattices Dmn. Under some sparseness conditions which restrict the number of X ij (m, n)'s which are equal to one we show that the random variables (l = 1, ···, r) converge to independent Poisson random variables for 0 < d1 < d2 < · ·· < dr when m→∞ nd∞. The particular types of Markov random fields considered here provide clustering (or repulsion) alternatives to randomness and involve several parameters. The limiting results are used to consider statistical inference for these parameters. Finally, a simulation study is presented which examines the adequacy of the Poisson approximation and the inference techniques when the lattice dimensions are only moderately large.

Limiting results for arrays of binary random variables on rectangular lattices under sparseness conditions

1979 ◽  
Vol 16 (3) ◽  
pp. 554-566 ◽  
Author(s):  
Roy Saunders ◽  
Richard J. Kryscio ◽  
Gerald M. Funk
Keyword(s):  
Simulation Study ◽  
Random Fields ◽  
Markov Random Fields ◽  
Random Variables ◽  
Poisson Approximation ◽  
Markov Random ◽  
Binary Random Variables ◽  
Inference Techniques ◽  
Image Position ◽  
Rectangular Lattices

In this article we give limiting results for arrays {Xij (m, n) (i, j) Dmn} of binary random variables distributed as particular types of Markov random fields over m x n rectangular lattices Dmn. Under some sparseness conditions which restrict the number of Xij (m, n)'s which are equal to one we show that the random variables (l = 1, ···, r) converge to independent Poisson random variables for 0 < d1 < d2 < · ·· < dr when m→∞ nd∞. The particular types of Markov random fields considered here provide clustering (or repulsion) alternatives to randomness and involve several parameters. The limiting results are used to consider statistical inference for these parameters. Finally, a simulation study is presented which examines the adequacy of the Poisson approximation and the inference techniques when the lattice dimensions are only moderately large.

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.

scholarly journals Structure Learning of Gaussian Markov Random Fields with False Discovery Rate Control

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1311
Author(s):  
Sangkyun Lee ◽  
Piotr Sobczyk ◽  
Malgorzata Bogdan
Keyword(s):  
False Discovery Rate ◽  
Random Fields ◽  
Markov Random Fields ◽  
Structure Learning ◽  
Random Variables ◽  
Estimation Procedure ◽  
Complex Data ◽  
Gaussian Markov Random Fields ◽  
False Discovery ◽  
Markov Random

In this paper, we propose a new estimation procedure for discovering the structure of Gaussian Markov random fields (MRFs) with false discovery rate (FDR) control, making use of the sorted ℓ 1 -norm (SL1) regularization. A Gaussian MRF is an acyclic graph representing a multivariate Gaussian distribution, where nodes are random variables and edges represent the conditional dependence between the connected nodes. Since it is possible to learn the edge structure of Gaussian MRFs directly from data, Gaussian MRFs provide an excellent way to understand complex data by revealing the dependence structure among many inputs features, such as genes, sensors, users, documents, etc. In learning the graphical structure of Gaussian MRFs, it is desired to discover the actual edges of the underlying but unknown probabilistic graphical model—it becomes more complicated when the number of random variables (features) p increases, compared to the number of data points n. In particular, when p ≫ n , it is statistically unavoidable for any estimation procedure to include false edges. Therefore, there have been many trials to reduce the false detection of edges, in particular, using different types of regularization on the learning parameters. Our method makes use of the SL1 regularization, introduced recently for model selection in linear regression. We focus on the benefit of SL1 regularization that it can be used to control the FDR of detecting important random variables. Adapting SL1 for probabilistic graphical models, we show that SL1 can be used for the structure learning of Gaussian MRFs using our suggested procedure nsSLOPE (neighborhood selection Sorted L-One Penalized Estimation), controlling the FDR of detecting edges.

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

scholarly journals A Methodology for Redesigning Networks by Using Markov Random Fields

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1389
Author(s):  
Julia García Cabello ◽  
Pedro A. Castillo ◽  
Maria-del-Carmen Aguilar-Luzon ◽  
Francisco Chiclana ◽  
Enrique Herrera-Viedma
Keyword(s):  
Decision Making ◽  
Random Fields ◽  
Markov Random Fields ◽  
Application Performance ◽  
Cross Border ◽  
Markov Random ◽  
User Groups ◽  
Universal Definition ◽  
Definition Of ◽  
Decision Making Model

Standard methodologies for redesigning physical networks rely on Geographic Information Systems (GIS), which strongly depend on local demographic specifications. The absence of a universal definition of demography makes its use for cross-border purposes much more difficult. This paper presents a Decision Making Model (DMM) for redesigning networks that works without geographical constraints. There are multiple advantages of this approach: on one hand, it can be used in any country of the world; on the other hand, the absence of geographical constraints widens the application scope of our approach, meaning that it can be successfully implemented either in physical (ATM networks) or non-physical networks such as in group decision making, social networks, e-commerce, e-governance and all fields in which user groups make decisions collectively. Case studies involving both types of situations are conducted in order to illustrate the methodology. The model has been designed under a data reduction strategy in order to improve application performance.

Sign in / Sign up

Export Citation Format

Share Document