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

j
j
E[ X
12 i
| Yi
, Z
ij
= 1, Φ ] = d12i
min( , )
y1i
y2i
∑
r=
0
j
12i 1i 2i ij
= rP[ X = r| y , y , Z = 1, Φ ]
=
y1i y
j
2i r X
i
r y
i
y
i
Zij
min( , )
∑
r=
0
P[ = , , | = 1, Φ]
12 1 2
P[ y , y | Z = 1, Φ]
1i 2i ij
rPoy ( −r| θ ) Poy ( −r| θ ) Por ( | θ )
min( y1i, y2i)
1i 1j 2i 2 j 12 j
= ∑ . (5.18)
py ( | θ )
r=
0
i j
The corresponding expressions for
j
j
E[ X
13 i
| Yi
, Zij
1, ] = d13
i
= Φ and
j
j
E[ X
23 i
| Yi
, Zij
1, ] = d23
i
= Φ follow by analogy. Then
E[ X | Y, Z = 1, Φ ] = d = y −d −d
j j j j
1i i ij 1i 1i 12i 13i
EX [ | Y, Z = 1, Φ ] = d = y −d −d
j j j j
2i i ij 2i 2i 12i 23i
EX [ | Y, Z = 1, Φ ] = d = y −d −d
.
j j j j
3i i ij 3i 3i 13i 23i
M-step: Update the parameters
p
j
n
∑
wij
wd
ij
i=
= 1 i=
1
and θ
tj
=
n
n
w
n
∑
∑
i=
1
ij
j
ti
, for j = 1,...,
k,
t ∈ Ω . (5.19)
If some convergence criterion is satisfied, stop iterating; otherwise go back to the E-
step. Here the following stopping criterion is used.
L
( k + 1) − L(
k)
−12
L(
k)
< 10
, where
L (k) is the loglikelihood at the k th iteration. The similarities with the standard EM
algorithm for the finite mixture are straightforward. The quantities
w ij
at the
termination of the algorithm are the posterior probabilities that the i th observation
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