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

Karlis and Meligkotsidou (2006) discussed the correlation structure of the multivariate
Poisson mixture models. These mixture models allow for both negative correlations and
overdispersion in addition to being computationally feasible.
The multivariate Poisson distribution is discussed again in section 7.2, followed by the
properties of the finite mixture models. In section 7.5, these properties were applied to
both multivariate Poisson finite mixture models and multivariate Poisson hidden
Markov models for several applications.
7.2 The multivariate Poisson distribution
Consider a vector X = ( X1, X2,..., X k
) where X
i
’s are independent and each follows a
Poisson distribution with parameter
λ
j
, j = 1,...,
k . Suppose that matrix A has
dimensions
n × k with zeros and ones. Then the vector Y = Y , Y ,..., Y ) defined as the
(
1 2 n
Y = AX follows a n-variate Poisson distribution. The most general form of a n-variate
n
Poisson distribution assumes that A is a matrix of size 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 ⎜ ⎟ columns. The matrix
⎝i
⎠
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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