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

The estimated covariance matrix and correlation coefficient are given below:
⎡ 5.3742 − 0.9564⎤
⎢
⎥ r = - 0.1669.
⎣−
0.9564 6.1084 ⎦
Table 7.3 gives the logliklihood, the AIC and the BIC together with the number of
components for the bivariate independent Poisson finite mixture model (7.17).
k
px ( , x) = ∑ pPox ( , x | λ , λ ),
1 2 j 1 2 1j 2j
j=
1
x1
x2
min( x1
, x2
)
−(
λ1 + λ ) λ1
λ2
⎛ x1
⎞⎛
x
2
2 ⎞
where Po(
x1,
x2
| λ1,
λ2
) = e ∑ i!
x1!
x2!
⎜
i 0 i
⎟
⎜
i
⎟ . (7.17)
= ⎝ ⎠⎝
⎠
In this case, the AIC and the BIC criterion select different component models: the AIC
selects the four-component model and the BIC selects the three-component model. The
method described in section 7.2 is used to calculate the covariance matrices.
Table 7.3: Loglikelihood, AIC and BIC together with the number of components for
the local independence multivariate Poisson finite mixture Model
Number of Number of free Loglikelihood AIC BIC
components ( k ) parameters
1 2 -450.6038 -452.6038 -455.2089
2 5 -433.5880 -438.5881 -445.1009
3 8 -423.6535 -431.6536 -442.0742
4 11 -420.2611 -431.2615 -445.5895
5 14 -419.2967 -433.2967 -451.5329
6 17 -419.2967 -436.2967 -458.4406
The estimated covariance matrix (AIC selection) and the correlation coefficient are
⎡ 5.7531
⎢
⎣−1.0169
−1.0169⎤
6.3774
⎥
⎦
and r = - 0.1679 respectively.
The estimated covariance matrix (BIC selection) and the correlation coefficient are
⎡ 5.2607
⎢
⎣−1.3547
−1.3547⎤
6.0806
⎥ and r = - 0.2395 respectively.
⎦
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