Salvatore Ingrassia

Constrained monotone EM algorithms for finite mixture of multivariate Gaussians

The likelihood function for normal multivariate mixtures may present both local spurious maxima and also singularities and the latter may cause the failure of the optimization algorithms. Theoretical results assure that imposing some constraints on the eigenvalues of the covariance matrices of the multivariate normal components leads to a constrained parameter space with no singularities and at least a smaller number of local maxima of the likelihood function. Conditions assuring that an EM algorithm implementing such constraints maintains the monotonicity property of the usual EM algorithm are provided. Different approaches are presented and their performances are evaluated and compared using numerical experiments.

Local Statistical Modeling via a Cluster-Weighted Approach with Elliptical Distributions

Cluster-weighted modeling (CWM) is a mixture approach to modeling the joint probability of data coming from a heterogeneous population. Under Gaussian assumptions, we investigate statistical properties of CWM from both theoretical and numerical point of view; in particular, we show that Gaussian CWM includes mixtures of distributions and mixtures of regressions as special cases. Further, we introduce CWM based on Student-t distributions, which provides a more robust fit for groups of observations with longer than normal tails or noise data. Theoretical results are illustrated using some empirical studies, considering both simulated and real data. Some generalizations of such models are also outlined.

Model-based clustering via linear cluster-weighted models

A novel family of twelve mixture models with random covariates, nested in the linear t cluster-weighted model (CWM), is introduced for model-based clustering. The linear t CWM was recently presented as a robust alternative to the better known linear Gaussian CWM. The proposed family of models provides a unified framework that also includes the linear Gaussian CWM as a special case. Maximum likelihood parameter estimation is carried out within the EM framework, and both the BIC and the ICL are used for model selection. A simple and effective hierarchical-random initialization is also proposed for the EM algorithm. The novel model-based clustering technique is illustrated in some applications to real data. Finally, a simulation study for evaluating the performance of the BIC and the ICL is presented.

Constrained monotone EM algorithms for mixtures of multivariate t distributions

Mixtures of multivariate t distributions provide a robust parametric extension to the fitting of data with respect to normal mixtures. In presence of some noise component, potential outliers or data with longer-than-normal tails, one way to broaden the model can be provided by considering t distributions. In this framework, the degrees of freedom can act as a robustness parameter, tuning the heaviness of the tails, and downweighting the effect of the outliers on the parameters estimation. The aim of this paper is to extend to mixtures of multivariate elliptical distributions some theoretical results about the likelihood maximization on constrained parameter spaces. Further, a constrained monotone algorithm implementing maximum likelihood mixture decomposition of multivariate t distributions is proposed, to achieve improved convergence capabilities and robustness. Monte Carlo numerical simulations and a real data study illustrate the better performance of the algorithm, comparing it to earlier proposals.

Clustering and classification via cluster-weighted factor analyzers

In model-based clustering and classification, the cluster-weighted model is a convenient approach when the random vector of interest is constituted by a response variable

Cluster-weighted \(t\)-factor analyzers for robust model-based clustering and dimension reduction

Y

Assessing the pattern of covariance matrices via an augmentation multiple testing procedure

and by a vector