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

Now consider a random field { X ( s) : s∈S } defined on S , that is, a collection X (s)
of
random variables indexed by sites in S . These random variables are assumed to take
their values in a finite set χ , the state space. Some examples of χ are χ = { −1,
+ 1}
and
χ = { 1,2,..., r}.
The set χ S is the set of elements of the form x= { x( s) : s∈S}
with
x(
s)
∈χ for each s. An element of
S
χ will often called as a configuration (of the random
field). Also often we can simply write this as X for { X ( s) : s∈S } and think of X as a
random variable with values in χ S , the set of configurations. Letting | S | denote the
number of elements of S and similarly for χ , the number of elements of the
configuration space χ S is
|
| S|
χ | and it is hence often extremely large. For example, if
χ = { −1,
+ 1} and S is a lattice of size 128× 128, its size is
2
128
2 . If A is a subset of S ,
write X ( A ) for { X ( s) : s∈
A } , that is the collection of random variables on A , and
similarly for a particular configuration x= { x( s) : s∈S }. The symbol \ denotes setdifference;
for example, S \ { s}
is the set of sites in S except s , and write this difference
as
S \ s . Now the random field { X ( s) : s∈S}
is a Markov random field (MRF) on
S (with respect to the given neighbourhood structure) if
PX [ ( s) = x( s)| X( S\ s) = x( S\ s)] = PX [ ( s) = x( s)| X( Ν ) = x( Ν )] for all sites s ∈ S
S
and all configurations x ∈χ . In other words, the distribution of X ( s ) , given all other
sites, depends at the realized values in its neighbourhood only. These conditional
distributions are often called the local specification of the MRF. Two examples are
presented to get an idea of different MRF:
s
s
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