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

m
∑
α ( i)
= α ( i −1)
P f ( y ; λ )
(5.27)
j
k = 1
k
(1)
[ α 1) = P f ( y ; λ ), j 1,..., m]
, and
j
(
j 1 j
=
m
∑
k = 1
kj
i+ 1
; λ
k
)
k
( i + 1)
i
j
β ( i)
= P f ( y β
(5.28)
j
jk
[ β ( n)
= 1, j 1,..., m]
.
j
=
Note that the α'
s are computed by a forward pass through the observations and the β ' s
by a backward pass after evaluating the α'
s. The likelihood is then simply calculated by
the expression∑ α
j
( n)
.
m
j=
1
The calculations of X
1,
X
2,
X
3,
X
12,
X
13,
and X 23
can be carried out using the same
formulas explained in section 5.2.2. The multivariate Poisson model is defined as
Y = X + X + X
1 1 12 13
Y = X + X + X
2 2 12 23
Y = X + X + X
3 3 13 23
.
(5.29)
E-step: Using the current values of the parameters calculate
E[ X | Y, u ( i) = 1, Φ ] = d
j
j
12i j 12i
min( y1i, y2i)
rPo( y1 i
−r | λ1j) Po( y2i −r | λ2j) Po( r | λ12j)
= ∑ . (5.30)
f ( y | λ )
r=
0
i
j
j
j
The corresponding expressions for E[ X13 i
| Y, uj( i) = 1, Φ ] = d13
i
and
E[ X | Y, u ( i) = 1, Φ ] = d follow by analogy. Then
j
j
23i j 23i
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