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

6.4.1 Results for the different multivariate Poisson finite mixture models
All three models, i.e. the local independence model, the common covariance model, and
the model with the restricted covariance structure, were fitted sequentially for 1 to 7
components ( k =1,…,7). Furthermore, in order to overcome the famous shortcomings of
the EM algorithm, i.e. the dependence on the initial starting values for the model
parameters, 10 different sets of starting values were chosen at random. In fact, the
mixing proportions ( p ) were uniform random numbers. These proportions were
rescaled so that the summation of all p ’s is equal to 1. The λ ’s were generated from a
uniform distribution over the range of the data points. For each set of initial values, the
algorithm was run for 150 iterations without considering any convergence criterion.
Then, the set of initial starting values with the largest loglikelihood was selected. The
EM iterations were continued with these selected initial values until the convergence
criterion is satisfied, i.e., until the relative change of the loglikelihood between two
successive iterations was smaller than
12
10 − . This procedure is repeated 7 times for each
value of k . The number of cluster selection was based on the most well-known
information criterion (section 5.3.4), i.e., the Akaike Information Criterion (AIC) and
the Bayesian Information Criterion (BIC). For the restricted covariance, the independent
and the common covariance models d k
is d = 7k− 1, d = 4k− 1 and d = 5k−
1
respectively. The AIC and BIC criterions serve as a guide for the researcher to select the
optimal number of components in the data.
k
k
k
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