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

5.3.5 Estimation for the multivariate Poisson finite mixture models
5.3.5.1 The EM algorithm
The EM algorithm is a popular algorithm for the ML estimation in statistical
applications (Dempster et al., 1977; McLachlan and Krishnan, 1997). It is appropriate to
the problems with missing values or problems that can be seen as containing missing
values. Suppose that there are observed data Y obs
and unobservable/missing data Y mis
,
which are perhaps missing values or even non-observable latent variables. The idea is to
augment the observed and the unobserved data, taking the complete data
Y = Y , Y ). The key idea of this algorithm is to iterate between two steps. The
com
(
obs mis
first step, the E-step, computes the conditional expectation of the complete data
loglikelihood with respect to the missing data, while the second step, the M-step,
maximizes the complete data likelihood.
Consider the multivariate reduction proposed earlier (section 5.1) in this thesis. The
observed data are the q -dimensional vectors Y = Y , Y , Y ) . The standard data
i
(
1i
2i
3i
argumentation is used for finite mixture models by introducing as latent variables the
vectors Z = Z , Z ,..., Z ) that correspond to the component memberships with
i
(
i1 i2
ik
Z = 1 if the i -th observation belongs to the j th component, and 0 otherwise.
ij
Furthermore, some more latent variables are introduced as follows: The component
specific latent variables, i.e. for the j th
component are introduced using the
j j j j j j j
unobservable vectors X = X , X , X , X , X , X ) , where the superscript
i
(
1i
2i
3i
12i
13i
23i
87