Maximum likelihood estimation in constrained parameter spaces for mixtures of factor analyzers

Mixtures of factor analyzers are becoming more and more popular in the area of model based clustering of multivariate data. According to the likelihood approach in data modeling, it is well known that the unconstrained likelihood function may present spurious maxima and singularities. To reduce such drawbacks, in this paper we introduce a procedure for parameter estimation of mixtures of factor analyzers, which maximizes the likelihood function under the mild requirement that the eigenvalues of the covariance matrices lie into some interval [a,b]. Moreover, we give a recipe on how to select appropriate bounds for the constrained EM algorithm, directly from the handled data. We then analyze and measure its performance, compared with the usual non-constrained approach, and also with other constrained models in the literature. Results show that the data-driven constraints improve the estimation and the subsequent classification, at the same time.