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

the estimated transition probability matrix for the restricted covariance model. This
model had the highest probability of 1 when moving from state four to state three
indicating that the distribution of state four almost surely moved to state three. The next
highest probability was 0.5383 when moving from state two to state three. It can be seen
that the average rates of weed distributions were different in different states. In the
restricted covariance model, there is only one important covariance term between
Dandelion ( Y 2
) and Wild Oats ( Y 3
). Also we see that the distribution of state one only
consist of Wild Oats with very high rate ( λ 3
=41.8286). The state three and state four do
not have any correlation between species. The interpretation of other parameters was
the same as for the independence case.
Table 6.13: Parameter estimates (bootstrapped standard errors) of the four states hidden
Markov restricted covariance model
State λ
1
λ
2
λ
3
λ
12
λ
13
λ
23
1 0.0000
(0.0549)
0.0000
(0.0000)
41.8286
(1.5246)
0.0000
(0.0000)
0.0000
(0.0000)
0.0000
(0.0000)
2 1.6504
(0.1101)
0.2749
(0.0473)
9.6423
(1.6981)
0.0000
(0.0000)
0.0000
(0.0000)
0.7309
(0.1452)
3 1.8772 0.4052 1.8312 0.0000 0.0000 0.0000
(0.0284)
4 0.6173
(0.0188)
(0.0204)
0.0941
(0.0115)
(0.3436)
0.0689
(0.0221)
(0.0000)
0.0000
(0.0000)
(0.0000)
0.0000
(0.0000)
(0.0000)
0.0000
(0.0000)
Table 6.14: Transition probability matrix of the hidden Markov restricted covariance
model
⎡0.6733
⎢
0.1438
⎢
⎢0.2883
⎢
⎣0.0000
0.1444
0.3179
0.1544
0.0000
0.1663
0.5383
0.5573
1.0000
0.0160⎤
0.0000
⎥
⎥
0.0000⎥
⎥
0.0000⎦
130