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

ξ
α () iPβ
( jb ) ( y )
t ij t+ 1 j t+
1
t
(, i j)
=
K K
∑∑
i= 1 j=
1
α () iPβ
( jb ) ( y )
t ij t+ 1 j t+
1
(3.14)
() t () t
( t)
where α () i = P[ Y = y , S = i; λ]
Y = { Y 1
,...., Y },
t
t
t
*( t) *( t)
*( t)
Y y
t
λ
{ Y
t + 1,..., YT
}
β () i = P[ = | S = i; ]
t
Y = .
The second variable is defined as
γ () i = P[ S = i| Y=
y , λ]
t
t
PS [
t
= i, Y=
y; λ]
=
P[ Y=
y; λ]
() t () t *() t *() t
PS [
t
= i, Y = y ; λ] P[ Y = y | St
= i; λ]
=
P[ Y=
y; λ]
, (3.15)
which is the probability of being in state i at time t given the model and the
observation sequence. This can be expressed in forward and backward variables by
αt() i βt() i αt() i βt()
i
γ
t
() i = =
K
P[ Y=
y; λ]
α () i β () i
∑
i=
1
t
t
(3.16)
and one can see that the relationship between γ (i)
and ξ ( i,
j)
is given by
K
∑
γ ( i)
= ξ ( i,
j),
1 ≤ i ≤ K,
1 ≤ t ≤ T . (3.17)
t
j=
1
t
If we sum γ (i)
over the time index t , we get a quantity, which can be interpreted as the
t
expected (over time) number of times that state S i
is visited, or equivalently, the
expected number of transitions made from state S
i
(if we exclude the time slot t = T
from the summation). Similarly, summation of ξ ( i,
j)
over t (from t = 1 to t = T − 1)
t
t
t
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