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

ypy ( ,0,0) = θ py ( − 1,0,0) y1 ≥ 1
1 1 1 1
yp(0, y,0) = θ p(0, y− 1,0) y2 ≥ 1
(5.11)
2 2 2 2
yp(0,0, y) = θ p(0,0, y− 1) y3 ≥ 1
3 3 3 3
p(0,0,0) = exp( − ( θ + θ + θ + θ + θ + θ )) . (5.12)
1 2 3 12 13 23
The above mentioned recurrence relations and the Flat algorithm (Tsiamyrtzis and
Karlis, 2004) are used to calculate the probabilities of the restricted covariance trivariate
Poisson model.
5.2.3 The Flat algorithm
Using the Flat Algorithm the calculation of py (
1, y2, y
3)
can be done in two stages. In
the first stage, one can move from ( y1, y2, y
3)
to the closest hyperplane using only one
of the recurrence relationships (5.7), and in the second stage, he can move down to the 0
point by the simplified recurrence relationships (5.10) and (5.11). Thus, starting from
( y1, y2, y
3)
and applying the recurrence relationship, we get three new points
( y1− 1, y2, y3)
, ( y1−1, y2− 1, y3)
and ( y 1
−1, y 2
−1, y 3
− 1) . Applying the same recurrence
relationship to these three points we get another six new points: ( y 1
− 2, y 2
, y 3
),
( y1−2, y2− 1, y3)
, ( y1−2, y2, y3− 1) , ( y1− 2, y2− 2, y3)
, ( y1−2, y2−1, y3− 1) and
( y −2, y , y − 2) . Figure 5.1 illustrates how coordinates can move to the closer plane
1 2 3
using the recurrence relationship (5.7) for the case y1 ≤ y2 ≤ y3
.
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