Mathematical Geology

, Volume 39, Issue 3, pp 321–335

# Markov Chain Random Fields for Estimation of Categorical Variables

• Weidong Li
Article

## Abstract

Multi-dimensional Markov chain conditional simulation (or interpolation) models have potential for predicting and simulating categorical variables more accurately from sample data because they can incorporate interclass relationships. This paper introduces a Markov chain random field (MCRF) theory for building one to multi-dimensional Markov chain models for conditional simulation (or interpolation). A MCRF is defined as a single spatial Markov chain that moves (or jumps) in a space, with its conditional probability distribution at each location entirely depending on its nearest known neighbors in different directions. A general solution for conditional probability distribution of a random variable in a MCRF is derived explicitly based on the Bayes’ theorem and conditional independence assumption. One to multi-dimensional Markov chain models for prediction and conditional simulation of categorical variables can be drawn from the general solution and MCRF-based multi-dimensional Markov chain models are nonlinear.

## Keywords

Multi-dimensional Markov chain Markov random field Conditional simulation Interclass relationship Nonlinear Conditional independence

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© International Association for Mathematical Geology 2007

## Authors and Affiliations

1. 1.Department of GeographyKent State UniversityKentUSA