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Clustering is an important technique in exploratory data analysis, with applications in image processing, object classification, target recognition, data mining etc. The aim is to partition data according to natural classes present in it, assigning data points that are more similar to the same cluster. We solved this ill-posed problem by adopting a fuzzy clustering method, based on mixtures of skew Gaussian, endowed by the joint usage of trimming and constrained estimation of scatter matrices. A set of membership values are used to fuzzy partition the data and to contribute to the robust estimates of the mixture parameters. The purpose is to adopt the basic skew Gaussian component for the mixture and apply impartial trimming to the data, to model the skew core of the clusters and to adapt to any type of tail behaviour. The choice of the skew Gaussian components is motivated by the fact that, with the increased availability of multivariate datasets, often underlying asymmetric structures appear. In these cases, the extremely useful paradigm for clustering given by the mixtures of Gaussian distributions appeared somehow unrealistic. Moreover, impartial trimming provides robust ML estimation, even in presence of outliers in the data. Finally, synthetic and real data are analyzed, to show how intermediate membership values are estimated for observations lying at cluster overlap, while cluster cores are composed by observations that are assigned to a cluster in a crisp way.

With the increasing availability of multivariate datasets, asymmetric structures in the data ask for more realistic assumptions, with respect to the incredibly useful paradigm given by the Gaussian distribution. Moreover, in performing ML estimation we know that a few outliers in the data can affect the estimation, hence providing unreliable inference. Challenged by such issues, more flexible and solid tools for modeling heterogeneous skew data are needed. Our fuzzy clustering method is based on mixtures of Skew Gaussian components, endowed by the joint usage of impartial trimming and constrained estimation of scatter matrices, in a modified maximum likelihood approach. The algorithm generates a set of membership values, that are used to fuzzy partition the data set and to contribute to the robust estimates of the mixture parameters. The new methodology has been shown to be resistant to different types of contamination, by applying it on artificial data. A brief discussion on the tuning parameters has been developed, also with the help of some heuristic tools for their choice. Finally, synthetic and real dataset are analyzed, to show how intermediate membership values are estimated for observations lying at cluster overlap, while cluster cores are composed by observations that are assigned to a cluster in a crisp way.

A clustering algorithm that combines the advantages of fuzzy clustering and robust statistical estimators is presented. It is based on mixtures of Factor Analyzers, endowed by the joint usage of trimming and the constrained estimation of scatter matrices, in a modified maximum likelihood approach. The algorithm generates a set of membership values, that are used to fuzzy partition the data set and to contribute to the robust estimates of the mixture parameters. The adoption of clusters modeled by Gaussian Factor Analysis allows for dimension reduction and for discovering local linear structures in the data. The new methodology has been shown to be resistant to different types of contamination, by applying it on artificial data. A brief discussion on the tuning parameters, such as the trimming level, the fuzzifier parameter, the number of clusters and the value of the scatter matrices constraint, has been developed, also with the help of some heuristic tools for their choice. Finally, a real data set has been analyzed, to show how intermediate membership values are estimated for observations lying at cluster overlap, while cluster cores are composed by observations that are assigned to a cluster in a crisp way

This paper presents a review about the usage of eigenvalues restrictions for constrained parameter estimation in mixtures of elliptical distributions according to the likelihood approach. The restrictions serve a twofold purpose: to avoid convergence to degenerate solutions and to reduce the onset of non interesting (spurious) local maximizers, related to complex likelihood surfaces. The paper shows how the constraints may play a key role in the theory of Euclidean data clustering. The aim here is to provide a reasoned survey of the constraints and their applications, considering the contributions of many authors and spanning the literature of the last 30 years.

The existing robust methods for model-based classification and clustering deal with elliptically contoured components. Here we introduce robust estimation for mixtures of skew-normal, by the joint usage of trimming and constraints. The model allows to fit heterogeneous skew data with great flexibility.