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

E[ X | Y, u ( i) = 1, Φ ] = d = y −d −d
j j j j
1i j 1i 1i 12i 13i
E[ X | Y, u ( i) = 1, Φ ] = d = y −d −d
j j j j
2i j 2i 2i 12i 23i
E[ X | Y, u ( i) = 1, Φ ] = d = y −d −d
.
j j j j
3i j 3i 3i 13i 23i
Let denote d j ( j j j j j j
i
= d
1i,d 2i,d 3i,d 12i,d 13i,d23i),
i= 1,..., n,
j = 1,..., m.
Then M-step computes the posteriori probabilities using the following equation.
α
j() i β
j() i α
j() i β
j()
i
uj() i = P[ Si = j| Y= y]
= =
m m
α ( n) α ( i) β ( i)
∑
∑
l j j
l= 1 j=
1
(5.31)
and then re-estimate the rates as follows:
n
j
∑uj()
i di
ˆ i=
1
λ
j
= ,
n
∑
i=
1
u () i
j
j = 1,..., m.
(5.32)
The M-step will give the parameter estimates
λ
1
,...,λ
m
for the k th iteration and then go
back, and repeat the algorithm until the convergence criterion is met.
We extended the univariate Markov-dependent Poisson mixture model to a multivariate
Poisson model (bivariate and trivariate). We carried out Splus/R codes for the analysis
of the multivariate Poisson hidden Markov model according to sections 5.4.2, 5.4.2.1
and 5.4.2.2.
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