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-- doctoral thesis --
Bayesian Cluster Analysis - Some Extensions to Non-standard Situations
The Bayesian approach to cluster analysis is presented. We assume that all data
stem from a finite mixture model, where each component corresponds to one cluster
and is given by a multivariate normal distribution with unknown mean and
variance. The method produces posterior distributions of all cluster parameters
and proportions as well as associated cluster probabilities for all objects. We
extend this method in several directions to some common but non-standard situations.
The first extension covers the case with a few deviant observations not
belonging to one of the normal clusters. An extra component/cluster is created for
them, which has a larger variance or a different distribution, e.g. is uniform over
the whole range. The second extension is clustering of longitudinal data. All units
are clustered at all time points separately and the movements between time points
are modeled by Markov transition matrices. This means that the clustering at
one time point will be affected by what happens at the neighbouring time points.
The third extension handles datasets with missing data, e.g. item non-response.
We impute the missing values iteratively in an extra step of the Gibbs sampler
estimation algorithm. The Bayesian inference of mixture models has many advantages
over the classical approach. However, it is not without computational
difficulties. A software package, written in Matlab for Bayesian inference of mixture
models, is introduced. The programs of the package handle the basic cases
of clustering data that are assumed to arise from mixture models of multivariate
normal distributions, as well as the non-standard situations.
Keywords: Cluster analysis, Clustering, Classification, Mixture model, Gaussian,
Bayesian inference, MCMC, Gibbs sampler, Deviant group, Longitudinal, Missing
data, Multiple imputation.
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