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

focus of this section. Random fields, which possess a Markov property, are called
Markov random fields (Elliott et al., 1995 and 1996).
Let us generalize this idea in a two-dimensional setting. Let Z be the set of integers, and
let
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S ⊂ Z be a finite rectangular two-dimensional lattice of integer points. Typically, it
will take S = { 0,1,..., n −1}
× {0,1,..., m −1}
, for some n and m . S is a two-dimensional
lattice containing
n × m points. The points in S are often called sites. To define a
Markov structure on the set S , we define what is meant by two points being
neighbours. Different definitions may suit different purposes, or applications. However,
the following two general conditions should include in the definition.
(i)
A site must be a neighbour of itself.
(ii) If t is neighbour of s , then s is a neighbour of t .
The second condition is a symmetry requirement. It can be written
s ~ t if the sites
s , t
∈
S
are neighbours. Two common neighbourhood structures are given in
Figure 2.6. If s is a site, the neighbourhood
Νs
of s can be defined as the set of all its
neighbours; Ν = { t∈S : t ~ s}.
Hence, Figure 2.6 illustrates the neighbourhood of the
s
middle site, for two different structures. In these structures, special care must be taken at
the edge of the lattice S , since sites located there have smaller neighbourhoods. One
way of defining the neighbourhood structure is “wrapping around” the lattice and define
sites at the other end of the lattice as neighbours.
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