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Depending on the selected hyper-parameters, cluster weighted modeling may produce a set of diverse solutions. Particularly, the user can manually specify the number of mixture components, the degree of heteroscedasticity of the clusters in the explanatory variables and of the errors around the regression lines. In addition, when performing robust inference, the level of impartial trimming enforced in the estimation needs to be selected. This flexibility gives rise to a variety of “legitimate” solutions. To mitigate the problem of model selection, we propose a two stage monitoring procedure to identify a set of “good models”. An application to the benchmark tone perception data showcases the benefits of the approach.

Recently, observed departures from the classical Gaussian mixture model in real datasets motivated the introduction of mixtures of skew t, and remarkably widened the application of model based clustering and classification to great many real datasets. Unfortunately,when data contamination occurs, classical inference for these models could be severely affected. In this paper we introduce robust estimation of mixtures of skew normal, to resist sparse outliers and even pointwise contamination that may arise in data collection. Hence, in each component, the skewed nature of the data is explicitly modeled, while any departure from it is dealt by the robust approach. Some applications on real data show the effectiveness of the proposal

Recently, observed departures from the classical Gaussian mixture model in real datasets have led to the introduction of more flexible tools for modeling heterogeneous skew data. Among the latest proposals in the literature, we consider mixtures of skew normal, to incorporate asymmetry in components, as well as mixtures of t, to down-weight the contribution of extremal observations. Clearly, mixtures of skew t have widened the application of model based clustering and classification to great many real datasets, as they can adapt to both asymmetry and leptokurtosis in the grouped data. Unfortunately, when data contamination occurs far from the bulk of the data, or even between the groups, classical inference for these models is not reliable. Our proposal is to address robust estimation of mixtures of skew normal, to resist sparse outliers and even pointwise contamination that could arise in data collection. We introduce a constructive way to obtain a robust estimator for the mixture of skew normal model, by incorporating impartial trimming and constraints in the EM algorithm. At each E-step, a low percentage of less plausible observations, under the estimated model, is tentatively trimmed; at the M-step, constraints on the scatter matrices are employed to avoid singularities and reduce spurious maximizers. Some applications on artificial and real data show the effectiveness of our proposal, and the joint role of trimming and constraints to achieve robustness

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., Ministerio de Economía y Competitividad grant MTM2017-86061-C2-1-P, y Consejería de Educación de la Junta de Castilla y León and FEDER grantVA005P17 y VA002G18

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.

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. These restrictions serve a twofold purpose: to avoid convergence to degenerate solutions and to reduce the onset of non interesting (spurious) 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 review of the constraints and their applications, along the contributions of many authors, spanning the literature of the last thirty years., Spanish Ministerio de Economía y Competitividad (grant MTM2017-86061-C2-1-P), Junta de Castilla y León - Fondo Europeo de Desarrollo Regional (grant VA005P17 and VA002G18)

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.

Mixtures of Gaussian factors are powerful tools for modeling an unobserved heterogeneous population, offering at the same time dimension reduction and model-based clustering. Unfortunately, the high prevalence of spurious solutions and the disturbing effects of outlying observations, along maximum likelihood estimation, open serious issues. We consider restrictions for the component covariances, to avoid spurious solutions, and trimming, to provide robustness against violations of normality assumptions of the underlying latent factors. A detailed AECM algorithm for this new approach is presented. Simulation results and an application to the AIS dataset show the aim and effectiveness of the proposed methodology