Robust fitting of mixture of factor analyzers using the trimmed likelihood estimator

Master of ScienceDepartment of StatisticsWeixin YaoMixtures of factor analyzers have been popularly used to cluster the high dimensional data. However, the traditional estimation method is based on the normality assumptions of random terms and thus is sensitive to outliers. In this article, we introduce a robust estimation procedure of mixtures of factor analyzers using the trimmed likelihood estimator (TLE). We use a simulation study and a real data application to demonstrate the robustness of the trimmed estimation procedure and compare it with the traditional normality based maximum likelihood estimate

Robust estimation for mixtures of Gaussian factor analyzers, based on trimming and constraints

Producción CientíficaMixtures 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. In this paper 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

The joint role of trimming and constraints in robust estimation for mixtures of Gaussian factor analyzers.

Producción CientíficaMixtures of Gaussian factors are powerful tools for modeling an unobserved heterogeneous population, offering – at the same time – dimension reduction and model-based clustering. The high prevalence of spurious solutions and the disturbing effects of outlying observations in maximum likelihood estimation may cause biased or misleading inferences. Restrictions for the component covariances are considered in order to avoid spurious solutions, and trimming is also adopted, 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.Ministerio de Economía y Competitividad and FEDER, grant MTM2014-56235-C2-1-P, and by Consejería de Educación de la Junta de Castilla y León, grant VA212U13, by grant FAR 2015 from the University of Milano-Bicocca and by grant FIR 2014 from the University of Catania

The joint role of trimming and constraints in robust estimation for mixtures of Gaussian factor analyzers.

Producción CientíficaMixtures of Gaussian factors are powerful tools for modeling an unobserved heterogeneous population, offering – at the same time – dimension reduction and model-based clustering. The high prevalence of spurious solutions and the disturbing effects of outlying observations in maximum likelihood estimation may cause biased or misleading inferences. Restrictions for the component covariances are considered in order to avoid spurious solutions, and trimming is also adopted, 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.Ministerio de Economía y Competitividad and FEDER, grant MTM2014-56235-C2-1-P, and by Consejería de Educación de la Junta de Castilla y León, grant VA212U13, by grant FAR 2015 from the University of Milano-Bicocca and by grant FIR 2014 from the University of Catania

Robust, fuzzy, and parsimonious clustering based on mixtures of Factor Analyzers

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 VA002G1

Graphical and computational tools to guide parameter choice for the cluster weighted robust model

The Cluster Weighted Robust Model (CWRM) is a recently introduced methodology to robustly estimate mixtures of regressions with random covariates. The CWRM allows users to flexibly perform regression clustering, safeguarding it against data contamination and spurious solutions. Nonetheless, the resulting solution depends on the chosen number of components in the mixture, the percentage of impartial trimming, the degree of heteroscedasticity of the errors around the regression lines and of the clusters in the explanatory variables. Therefore an appropriate model selection is crucially required. Such a complex modeling task may generate several “legitimate” solutions: each one derived from a distinct hyper-parameters specification. The present paper introduces a two step-monitoring procedure to help users effectively explore such a vast model space. The first phase uncovers the most appropriate percentages of trimming, whilst the second phase explores the whole set of solutions, conditioning on the outcome derived from the previous step. The final output singles out a set of “top” solutions, whose optimality, stability and validity is assessed. Novel graphical and computational tools - specifically tailored for the CWRM framework - will help the user make an educated choice among the optimal solutions. Three examples on real datasets showcase our proposal in action. Supplementary files for this article are available online