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

Since original HMMs were designed as one-dimensional Markov chains with firstorder
neighborhood systems, it cannot directly be used in two-dimensional problems
such as image segmentation. A special case of an HMM, in which the underlying
stochastic process is an MRF instead of a Markov chain, is referred to as a hidden
Markov random field model (Zhang et al., 2001). Mathematically, an HMRF model
is characterized by the following:
• Hidden Markov Random Field (HMRF)
The random field X = { X ( s)
: s ∈ S}
is an underlying HMRF assuming values in
a finite state space L = ( 1,...., )
with probability distribution π . The state of X
is unobservable.
• Observable Random Field
Y = { Y ( s)
: s ∈S}
is a random field with a finite state space D = ( 1,..., d)
. Given
any particular configuration
x ∈ χ , every Y () s follows a known conditional
probability distribution py ( ( s) | x( s )) of the same functional form f( y( s); θ
x( s)
),
where θ
x( s)
are the involved parameters. This distribution is called the emission
probability function and Y is also referred to as the emitted random field.
• Conditional Independence
For any
x ∈ χ , the random variables Y ( s ) are conditional independent
∑
p ( y | x)
= p(
y(
s)
| x(
s)).
Based on the above, the joint probability of ( X , Y)
can be written as
∑
s∈S
s∈S
p ( y , x)
= p(
y | x)
p(
x)
= p(
x)
p(
y(
s)
| x(
s)).
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