GAUSSIAN PARSIMONIOUS CLUSTERING MODELS

Gaussian clustering models are useful both for understanding and suggesting powerful criteria. BanfIeld and Raftery, Biometriks 49, 803-821 (1993), have considered a parameterization of the variance matrix Sigma(k) of a cluster P-k in terms of its eigenvalue decomposition, Sigma(k) = lambda(k)D(k)A(k)D'(k), where lambda(k) defines the volume of P-k, D-k is an orthogonal matrix which defines its orientation and A(k) is a diagonal matrix with determinant 1 which defines its shape. This parametrization allows us to propose many general clustering criteria from the simplest one (spherical clusters with equal volumes which leads to the classical k-means criterion) to the most complex one (unknown and different volumes, orientations and shapes for all clusters). Methods of optimization to derive the maximum likelihood estimates as well as the practical usefulness of these models are discussed. We especially analyse the influence of the volumes of clusters. We report Monte Carlo simulations and an application on stellar data which dramatically illustrated the relevance of allowing clusters to have different volumes.