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

The parameters
θ ( j ∈ Rm , m = 2, 3) correspond to a m-way covariance in a similar
j
way to the m-way interactive terms, and thus, they impose structure on the data.
Mardia (1970) introduced the multivariate reduction technique to create the multivariate
Poisson distribution. This reduction technique has been used extensively for the
construction of multivariate models. The idea of the method is to start with some
independent random variables and to create new variables by considering some
functions of the original variables. Then, since the new variables contain jointly some of
the original ones, a kind of structure is imposed creating multivariate models
(Tsiamyrtzis et al., 2004).
We can represent this model using following matrix notations. Assume that
X i
,
i = 1,...,k
are independent Poisson random variables and A is a n× k matrix with zeros
and ones. Then the vector Y = Y , Y ,..., Y ) defined as Y = AX follows a n -variate
(
1 2 n
Poisson distribution. The most general form assumes that A is a matrix of size
n
n× (2 − 1) of the form
A=[A 1 , A 2 , A 3 ,…,A n ]
where
A
i
,
⎛n⎞
i = 1,...,
n are matrices with n rows and ⎜ ⎟
⎝i
⎠
columns. The matrix
A i
contains columns with exactly i ones and n− i zeros, with no duplicate columns, for
i = 1,...,n
. Thus A
n
is the column vector of 1’s while A
1
becomes the identity matrix of
size n× n. For example, the fully structured multivariate Poisson model for three
variables can be represented as follows:
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