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

CHAPTER 5
MULTIVARIATE POISSON DISTRIBUTION, MULTIVARIATE POISSON
FINITE MIXTURE MODEL AND MULTIVARIATE POISSON HIDDEN
MARKOV MODEL
5.1 The multivariate Poisson distribution: general description
Without loss of generality, the explanation in this thesis is restricted to three variables.
Following the notation of Marshall and Olkin (1985), Johnson et al. (1997), Brijs (2002)
and Brijs et al. (2004), the sets R = {1,2,3
1
} , R = {12,13,23}
, }
2
R
3
= {123 are defined.
Let ∪ 3 R =
= 1
i
R i
. Now consider the independent variables X j
, which follow Poisson
distributions with parameters θ j
with
j ∈ R respectively. Furthermore, the observed
variables of interest Y i
, with i = 1,2, 3 are defined as Yi = ∑X j
where j ∈ R and j
contains the subscript i. For example, the general, fully saturated covariance model for
j
the case with three observed variables, where ∪ 3 R =
= 1
Y = X
Y
Y
1
2
3
1
= X
= X
2
3
+ X
+ X
12
+ X
12
13
+ X
+ X
13
+ X
23
23
i
R i
+ X
+ X
+ X
123
, is written as:
123
123.
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