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

ecause the former captures more of the existing variance in the data. However, from a
practical aspect, the model with common covariance structure is preferable to the fully
structured model because it requires fewer parameters to be estimated.
Therefore, the important question is whether a model somewhere in between the two
presented extreme models can be found, that is, both a) theoretically good enough to
describe most of the existing covariances, and b) practically suitable in terms of the
number of parameters to be estimated.
The model introduced in this section was tried to address the above mentioned problem.
The main idea is to simplify the variance/covariance structure as much as possible by
including only statistically significant m-fold interactions. For this trivariate model, the
statistical significance of the weed count interactions between Wild Buckwheat ( Y 1
),
Dandelion ( Y 2
) and Wild Oats ( Y 3
) will study by using of loglinear analysis (see section
5.7 and section 6.3). The loglinear analysis is particularly appropriate for the
development of a simpler multivariate Poisson mixture model for clustering since it
facilitates to discover, which interaction terms in the variance/covariance matrix can be
set equal to zero. We will show that (section 6.3) the p values of the goodness of fit
statistics of the models with some of the two-fold interactions and without any
interaction do not differ very much. Therefore, the two-fold interactions were kept in
the model. The latent variables, X = X , X , X , X , X , ) and the vector of
(
1 2 3 12 13
X
23
parameters, θ = ( θ1, θ2, θ3, θ12, θ13, θ23)
, are used to present the model and thus the
restricted covariance model can be defined as:
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