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

Table 7.1 gives the counts of surface and interior faults in 100 lenses presented in
Aitchison and Ho (1989). The observed covariance matrix and the correlation between
surface ( x 1
) and interior ( x 2
) counts are given below:
⎡ 5.2227
⎢
⎣−1.0227
−1.0227⎤
6.2072
⎥
⎦
r = - 0.1796.
The correlation coefficient indicates that data have a negative correlation. Table 7.2
gives the loglikelihood, the AIC and the BIC together with the number of components
for the bivariate common covariance Poisson finite mixture model (7.16).
k
px ( , x) = ∑ pPox ( , x | λ , λ , λ ),
1 2 j 1 2 1j 2j 3j
j=
1
where
Po(
x
⎛ x ⎞⎛
x ⎞
x1
x2
min( x1
, x2
)
−(
λ1
+ λ2
+ λ3
) 1 2
1 2
1
, x2
| λ1,
λ2
, λ3)
= e ∑ i
x x
⎜
i i
⎟
⎜
i
⎟ !
1!
2!
= 0
λ
λ
⎝
⎠⎝
⎠
i
⎛ λ3
⎟ ⎞
⎜ . (7.16)
⎝ λ1λ2
⎠
According to the AIC and the BIC criterion (section 5.3.4), the larger the criterion, the
better the model in comparison with another. Therefore, the three-component model
with loglikelihood –420.6121 was selected as the best model (Table 7.2). The
covariance matrix and the correlation between x
1
and x
2
were estimated.
Table 7.2: Loglikelihood, AIC and BIC together with the number of components for
the common covariance multivariate Poisson finite mixture Model
Number of Number of free Loglikelihood AIC BIC
components ( k ) parameters
1 3 -450.6038 -453.6038 -457.5115
2 7 -432.6901 -439.6901 -448.8082
3 11 -420.6121 -431.6121 -445.9405
4 15 -419.8284 -434.8284 -454.3672
5 19 -419.7168 -438.7168 -463.4659
6 23 -419.3221 -442.3221 -472.2815
7 27 -419.3221 -446.3221 -481.4919
148