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

⎡1 0 0⎤
A =
⎢
0 1 0
⎥
⎢ ⎥
⎢⎣
0 0 1⎥⎦
⎡1 0 0⎤
where A
1
=
⎢
0 1 0
⎥
⎢ ⎥
.
⎢⎣0 0 1⎥⎦
In this situation, the likelihood function for the general k component mixture model for
q weed species takes a very simple form:
y
( ) il
exp( −
lj
)
L( Θ ; y ) = f( y | Θ ) = ∑ p
,
!
n n k q
lj
li li j
i= 1 i= 1 j=
1 l=
1 yli
∏ ∏ ∏ θ θ where
j
p are mixing
proportions.
Moreover, for the general k component mixture model for q weed species, we have
k − 1 different p ’s, and k different θ ’s per weed species. The number of parameters
need to be estimated is
( k − 1)
+ k × q . Details of the multivariate Poisson finite mixture
model is given in section 5.3. The loglikelihood is then expressed as:
n
y
⎡ k q
( ) li
lj
exp( − ) ⎤
lj
LL( ∀ p, θ | data) = ∑ln⎢∑ p
j∏ θ θ
⎥.
i= 1 ⎢⎣
j= 1 l=
1 yli
! ⎥⎦
5.1.4 The multivariate Poisson model with restricted covariance
The multivariate Poisson models presented in section 5.1.1 and 5.1.2 represent two
extreme approaches to model the interdependent count rates. From a theoretical aspect,
the fully structured model is preferable to the model with common covariance structure
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