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

1 1 2 2 3 3
(2) (3)
∑∑
PY [ = y, Y = y , Y = y]
= e
L
L
(2) (3)
x = 0x
= 0
−θ
∏
∏
θ
∑
∑
( y − x − x )
j k l
( j)
kR ∈ lR ∈ 3
2
∏
j
j∈R i∈R ∪R
1 2 3
∑
∑
θ
xi
i
∏
( y − x − x)! x!
j k l i
j∈R i∈R ∪R
( j)
1 k∈R
l∈R
2
3 2 3
with θ = θ + θ + θ + θ + θ + θ + θ .
1 2 3 12 13 23 123
To see why L
1
must be equal to min( y , y 1 2
) , one must look at the first two of the three
conditions on X
1,
X
2,
and X
3
specified in (5.2). Indeed, it is known that all x ’s should
be zero or a positive integer. For that reason, if all x ’s except for x 12
would be zero,
then the maximum acceptable value for the x
12
can be min( y , y 1 2
) to facilitate the first
two conditions. Similarly, the values for the other x ’s can be computed, based on the
preceding values (Mahamunulu, 1967), resulting in the admissible ranges for all L’s.
The above formulated trivariate Poisson model incorporates all possible interactions
(that is, two-way and three-way) that can exist between the counts of the three weed
species considered. In other words, this model can take into account for all possible
covariances between the weed counts.
The mixture variant of the multivariate Poisson model (details are given in section 5.3)
simply extends the multivariate Poisson model by assuming that k groups of species
have different parameter values for the θ ’s. Clearly, the number of parameters to be
optimized rapidly increase with the specification of different groups in the data.
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