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5.3.2 Model-based cluster estimation
The purpose of model-based clustering, described in previous section, is to estimate the
parameter vector Φ. The maximum likelihood (ML) and the maximum a posterior
(MAP) estimation (Vermunt et al., 2002) are the two popular methods to estimate this
parameter vector. Of these two, maximum likelihood estimation is used in this thesis.
5.3.3 ML estimation with the EM algorithm
One purpose of model-based clustering approach is to estimate the parameters
( p1,...,
p k − 1,
θ1,...,
θ
k
) . Following the maximum likelihood (ML) estimation approach,
the estimation involves maximizing the loglikelihood (5.14), as stated earlier. In other
words, the idea is to find the optimal values for the parameter vector, say Φ opt , such that
the observations y i
( i = 1,..., n)
are more likely came from f ( y i
| Φ opt ) than from
f ( y i
| Φ) for any other value of Φ (McLachlan and Peel, 2000).
To maximize this loglikelihood, different approaches such as Newton-Raphson
algorithm (McHugh, 1956), expectation-maximization (EM) (Dempster et al., 1977;
McLachlan and Krishnan, 1997) algorithm etc. can be used. Most of software either
uses Newton-Raphson algorithm or expectation-maximization (EM) algorithm, or a
combination of both. Most recent techniques increasing in popularity are the stochastic
EM method (Diebolt, 1996) and MCMC (Markov Chain Monte Carlo) (Robert, 1996).
Moreover, since the EM is relatively slow, recent research efforts focus on modifying
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