An EM Algorithm for Singular Gaussian Mixture Models

In this paper, we introduce finite mixture models with singular multivariate normal components. These models are useful when the observed data involves collinearities, that is when the covariance matrices are singular. They are also useful when the covariance matrices are ill-conditioned. In the latter case, the classical approaches may lead to numerical instabilities and give inaccurate estimations. Hence, an extension of the Expectation Maximization algorithm, with complete proof, is proposed to derive the maximum likelihood estimators and cluster the data instances for mixtures of singular multivariate normal distributions. The accuracy of the proposed algorithm is then demonstrated on the grounds of several numerical experiments. Finally, we discuss the application of the proposed distribution to financial asset returns modeling and portfolio selection.