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

mixture model. The loglikelihood function for s , y ) , i = 1,...,
n (called the complete-
c
data loglikelihood) is log L ( Φ | s1,
y1,....,
sn
, y
n
)
n
m
m
∑∑∑
∑∑
jk
jk
i= 2 j= 1 k = 1 i= 1 j=
1
n
m
(
i i
(1)
= log P
1
+ v ( i)log
P + u ( i)log
f ( y ; λ ) , (5.21)
s
where u j
( i)
= 1if S i
= j and 0 otherwise, and
j
v jk
( i)
= 1, if a transition from j to k occurred at i (i.e; S
−
= j,
S ) and 0
otherwise. (Φ is suppressed for simplicity of notation).
i
j
i 1 i
= k
This loglikelihood function consists of two parts, the loglikelihood for a Markov chain,
depending merely on the transition probabilities
P jk
, and the loglikelihood for
independent observations, depending only on the parameters λ ,..., λ . Note that when
1 m
P
jk
is independent of j , (5.21) gives the complete-data likelihood for the independent
case, so that the independent model is nested in the hidden Markov model.
The M-step requires maximization of E[log L c ( Φ) | y1,...., y ] , which is obtained by
replacing the components of the missing data by their conditional means.
vˆ ( i) = E[ v ( i) | y ,..., y ] = P[ S = j, S = k| y ,..., y ]
(5.22)
jk jk 1 n i−1 i 1 n
n
and
uˆ ( i) = E[ u ( i) | y ,..., y ] = P[ S = j| y ,..., y ] . (5.23)
j j 1 n i 1 n
The transition probabilities are obtained using following formula:
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