Theoretical generalization of Markov chain random field from potential function perspective

Abstract

The inner relationship between Markov random field (MRF) and Markov chain random field (MCRF) is discussed. MCRF is a special MRF for dealing with high-order interactions of sparse data. It consists of a single spatial Markov chain (SMC) that can move in the whole space. Generally, the theoretical backbone of MCRF is conditional independence assumption, which is a way around the problem of knowing joint probabilities of multi-points. This so-called Naive Bayes assumption should not be taken lightly and should be checked whenever possible because it is mathematically difficult to prove. Rather than trap in this independence proving, an appropriate potential function in MRF theory is chosen instead. The MCRF formulas are well deduced and the joint probability of MRF is presented by localization approach, so that the complicated parameter estimation algorithm and iteration process can be avoided. The MCRF model is then applied to the lithofacies identification of a region and compared with triplex Markov chain (TMC) simulation. Analyses show that the MCRF model will not cause underestimation problem and can better reflect the geological sedimentation process.