A COMPOSITE LIKELIHOOD APPROACH TO CLUSTER ORDINAL DATA Seminar

Roberto Rocci (in collaboration with Monia Ranalli)

A latent Gaussian mixture model to classify ordinal data is discussed. The observed categorical variables are considered as a discretization of an underlying finite mixture of Gaussians. The model is estimated within the expectation maximization (EM) framework maximizing a pairwise likelihood.

This allows us to overcome the computational problems arising in the full maximum likelihood approach due to the evaluation of multidimensional integrals that cannot be written in closed form. Moreover, a method to cluster the observations on the basis of the posterior probabilities in output of the pairwise EM algorithm is suggested.

The effectiveness of the proposal is shown comparing the pairwise likelihood approach with the full maximum likelihood and the maximum likelihood for continuous data ignoring the ordinal nature of the variables. The comparison is made on real and simulated data sets.

Some extensions of the model are also discussed: the case where some variables are continuous and the case where some noise variables/dimensions mask the clustering structure. In the latter, the noise dimensions are detected considering the variables underlying the ordinal data to be linear combinations of two independent sets of second-order latent variables where only one contains the information about the clustering structure. Applications to real and simulated data are discussed.