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

where P = Pr( S = k | S
− 1
j),
1 ≤ j,
k ≤ m denote the stationary transition
jk i i
=
probabilities of { i
}
The likelihood for Φ is
S and = [ λ λ λ λ λ λ ]
λ , where j = 1,...,
m .
j 11i 12i 13i 22i 23i 33i
m m
n
(1)
( | y , y
2,
...., y
n
) = ∑...
∑Ps
f ( y1;
λ
S
( Φ))
∏ PS
i−
S
( Φ)
f ( y
i
i
; λ
S
( Φ
1 1
1
i
S1
= 1 Sn
= 1
i=
2
L Φ
1
)) , (5.20)
(1)
where P j
= Pr( S 1
= j)
denote the initial probabilities of { S i
}. Leroux and Puterman
(1992) discussed in their paper that L Φ | y , y ...., y ) is a convex mixture of
(
1 2,
n
(1)
likelihood values obtained with a fixed initial state (i.e. with P = 1for some j),
(1) (1)
concurrently maximization of L Φ | y , y ...., y ) over Φ and ( P ,..., P ) can be
(
1 2,
n
j
1 m
accomplished by maximization over Φ with a fixed initial state. Thus, it follows that
the
(1)
P
j
are known. Cappé (2001) explained that with a single training sequence, the
initial distribution is a parameter that has not much effect and the initial distribution
cannot be estimates consistently. Taking the above reason into account, it is assumed
that the initial distribution is uniform (equal probabilities for all states of the model). In
this thesis, initial Uniform distribution is assumed.
5.4.2.1 The EM algorithm
The EM algorithm can be applied to determine the likelihood maximization for the
multivariate Poisson hidden Markov model, almost as simply as for the independent
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