The thesis tackles the problem of uncovering hidden structures in high-dimensional data
in the presence of noise and non informative variables. It proposes a supervised and
an unsupervised mixture models that select the relevant variables and are robust to
measurement errors and outliers.
Within the class of unsupervised clustering models we extend variable selection to
the family of Student's t mixture models. While t distributions are naturally robust to
noise and extreme events, sparsity is achieved by imposing regularization on the location
and dispersion parameters. An EM algorithm is implemented to return the maximum
likelihood estimate of the model parameters given the added penalty term. To further
asses the contribution of each variable we propose a resampling procedure that ranks the
variables according to their selection probability.
Supervised clustering is implemented in a Bayesian framework. The model assumes
a mixture of Lasso type regressions with t-distributed errors. While the Lasso representation of the normal linear model imposes regularization on the regression coefficient,
variable selection is explicitly modelled by a latent binary indicator variable. The model
relies on particle Markov chain Monte Carlo algorithm to approximate the posterior
distribution of the parameters of interest.
To highlight the properties and advantages of the proposed models, two real life
problems are considered. The first one requires us to identify subtypes of breast cancer
tumors by grouping patients based only on their gene expression levels when only few of
the thousands genes are informative. In the second case our aim is to cluster different
financial markets spanning several macro sectors and explain their trading performance
only on the basis of the observed statistical features of their price dynamics.