multivariate poisson hidden markov models for analysis of spatial ...

concept of Markov models to include the case where the observation is a probabilistic
function of the states.
The concept of hidden Markov Model has been the object of considerable study since
the basic theory of hidden Markov models was initially introduced and studied during
the late 1960’s and early 1970’s by Baum and his colleagues (Baum et al., 1966, 1967
and 1970). The primary concern in the hidden Markov modeling technique is the
estimation of the model parameters from the observed sequences. One method of
estimating the parameters of the hidden Markov models is to use the well-known Baum-
Welch re-estimation method (Baum and Petrie, 1966). Baum and Eagon first proposed
the algorithm in 1967 for the estimation problem of hidden Markov models with
discrete observation densities. Baum and others (1970) later extended this algorithm to
continuous density hidden Markov models with some limitations.
1.2.3 Hidden Markov model and hidden Markov random field model
Hidden Markov models are well known models in modeling the unknown state
sequence given the observation sequence. As mentioned in the previous section, this has
been successfully applied in the fields of speech recognition, biological modeling
(protein sequences and DNA sequences) and many other fields. The hidden Markov
models presented in section 1.2.2 are one-dimensional models, and they cannot take
spatial dependencies into account. To overcome this drawback, Markov random fields
and hidden Markov random fields (HMRF) can be used in more than one dimension
when considering the spatial dependencies. For example, when the state space or
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