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

indicates the component, and the variables are the latent variables used to construct the
model in section 5.1. Thus, the complete data are the vectors Y , X , Z ) . The vector of
(
i i i
parameters is defined by Φ, and then the complete loglikelihood takes the following
form:
n k
j
∑∑ ij j ∏ ti tj
i= 1 j= 1
t∈Ω
L( Φ ) = Z (log p + log f( X | θ ))
n k n k
j
j
Zij pj Zij θtj Xti θtj Xti
i= 1 j= 1 i= 1 j= 1 t∈Ω
∑∑ ∑∑ ∑ , (5.16)
= log + ( − + log −log !)
where Ω = {1,2,3,12,13,23}
. The relevant part of the complete likelihood is give by
n
k
j
∑∑ ( − Zijθtj + Zij Xti log θtj
) and hence, one needs the expectations E ( Z ij
) and
i= 1 j=
1
E ( X j Z ti ij
) . However for the latter, since Z
ij
is a binary random variable, when
j
X
ti
is 0
if the observation does not belong to the j th component and takes the value
j
X
ti
if
j
j
Z =1. Thus E [ X Z ] p(
Z ) E[
X | Z = 1]
.
ij
ti
ij
=
ij ti ij
j
The E [ X ti
| Z = 1]
is the expectation of the latent variable
ij
j
X
ti
given that it belongs to
the j th component. Thus, at the E-step one needs the expectations E [ Z ij
| Y i
, Φ]
for
j
i = 1,...,n
, j = 1,...,
k and E [ X | Y , Z = 1, Φ]
for i = 1,...,
n , j = 1,...,
k and t ∈ Ω.
ti
i
ij
More formally, the procedure can be described as follows:
E-step: Using the current values of the parameters calculate
py (
i
| θ
j)
wij = E[ Zij | Yi , Φ ] = pj
, i= 1,..., n, j = 1,...,
k . (5.17)
py ( )
i
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