An analytic solution for estimating two-dimensional hidden Markov models

Two-dimensional hidden Markov model (2-D HMM) is an extension of 1-D HMM to 2-D, it provides a reasonable statistical method to model matrix data. This paper presents some new strict definitions of 2-D HMM and proves the equivalence between them, and gives a study of the three basic problems for 2-D HMM, namely, probability evaluation, optimal state matrix and parameter estimation. By using the ideal that the sequences of states on columns or rows of a 2-D HMM can be seen as states of a 1-D HMM, several new formulae solving these problems are theoretically derived and further demonstrated by computer simulations.