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

can be interpreted as the expected number of transitions from state S
i
to state S j
. That
is
T
∑ − 1
t=
1
γ ( i)
=
t
Expected number of transition from S
i
and
T
∑ − 1
t=
1
ξ ( i,
j)
=
t
Expected number of transitions from S
i
to
S
j
.
Now assuming a starting model λ = ( A,
B,
π ) , we use the model to calculate the α'
s,
β ' s using equations (3.3) to (3.8) then we use the
γ ' s using equations (3.14) to (3.17).
α'
s and
β ' s to calculate the ξ ' s and
The next step is to define re-estimated model as ˆ λ = ( Aˆ,
Bˆ,
ˆ π ) . The re-estimation
formulas for
A ˆ,
B ˆ, πˆ are
πˆ
i
=
Expected frequency in state Si
at time t = 1
= γ 1(
i ), 1 ≤ i ≤ K . (3.18)
Pˆ
ij
=
Expected Number of transitions from state Sto
i
state
Expected Number of transitions from state S
i
S
j
T −1
∑
t=
1
= T −
∑
t=
1
ξt
( i,
j)
,
1
γ ( i)
t
1 ≤ i,
j ≤ K . (3.19)
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