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One of the challenges in cluster analysis is the evaluation of the obtained clustering results without using auxiliary information. To this end, a common approach is to use internal validity criteria. For mixtures of linear regressions whose parameters are estimated by maximum likelihood, we propose a three-term decomposition of the total sum of squares as a starting point to define some internal validity criteria. In particular, three types of mixtures of regressions are considered: with fixed covariates, with concomitant variables, and with random covariates. A ternary diagram is also suggested for easier joint interpretation of the three terms of the proposed decomposition. Furthermore, local and overall coefficients of determination are respectively defined to judge how well the model fits the data group-by-group but also taken as a whole. Artificial data are considered to find out more about the proposed decomposition, including violations of the model assumptions. Finally, an application to real data illustrates the use and the usefulness of these proposals. [ABSTRACT FROM AUTHOR]

The Gaussian cluster-weighted model (CWM) is a mixture of regression models with random covariates that allows for flexible clustering of a random vector composed of response variables and covariates. In each mixture component, a Gaussian distribution is adopted for both the covariates and the responses given the covariates. To make the approach robust with respect to the presence of mildly atypical observations, the contaminated Gaussian CWM is introduced. In addition to the parameters of the Gaussian CWM, each mixture component has a parameter controlling the proportion of outliers, one controlling the proportion of leverage points, one specifying the degree of contamination with respect to the response variables, and another specifying the degree of contamination with respect to the covariates. Crucially, these parameters do not have to be specified a priori, adding flexibility to the approach. Furthermore, once the model is estimated and the observations are assigned to the components, a finer intra-group classification in typical points, (mild) outliers, good leverage points, and bad leverage points-concepts of primary importance in robust regression analysis-can be directly obtained. Relations with other mixture-based contaminated models are analyzed, identifiability conditions are provided, an expectation-conditional maximization algorithm is outlined for parameter estimation, and various implementation and operational issues are discussed. Properties of the estimators of the regression coefficients are evaluated through Monte Carlo experiments and compared with other procedures. A sensitivity study is also conducted based on a real data set. [ABSTRACT FROM AUTHOR]

Cluster-weighted models represent a convenient approach for model-based clustering, especially when the covariates contribute to defining the cluster-structure of the data. However, applicability may be limited when the number of covariates is high and performance may be affected by noise and outliers. To overcome these problems, common/uncommon $$t$$ -factor analyzers for the covariates, and a $$t$$ -distribution for the response variable, are here assumed in each mixture component. A family of twenty parsimonious variants of this model is also presented and the alternating expectation-conditional maximization algorithm, for maximum likelihood estimation of the parameters of all models in the family, is described. Artificial and real data show that these models have very good clustering performance and that the algorithm is able to recover the parameters very well. [ABSTRACT FROM AUTHOR]

Cluster-weighted models (CWMs) are a flexible family of mixture models for fitting the joint distribution of a random vector composed of a response variable and a set of covariates. CWMs act as a convex combination of the products of the marginal distribution of the covariates and the conditional distribution of the response given the covariates. In this paper, we introduce a broad family of CWMs in which the component conditional distributions are assumed to belong to the exponential family and the covariates are allowed to be of mixed-type. Under the assumption of Gaussian covariates, sufficient conditions for model identifiability are provided. Moreover, maximum likelihood parameter estimates are derived using the EM algorithm. Parameter recovery, classification assessment, and performance of some information criteria are investigated through a broad simulation design. An application to real data is finally presented, with the proposed model outperforming other well-established mixture-based approaches. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of estimating a density, with either a compact support or a support bounded at only one end, exploiting a general and natural form of a finite mixture of distributions. Due to the importance of the concept of multimodality in the mixture framework, unimodal beta and gamma densities are used as mixture components, leading to a flexible modeling approach. Accordingly, a mode-based parameterization of the components is provided. A partitional clustering method, named $$k$$-bumps, is also proposed; it is used as an ad hoc initialization strategy in the EM algorithm to obtain the maximum likelihood estimation of the mixture parameters. The performance of the $$k$$-bumps algorithm as an initialization tool, in comparison to other common initialization strategies, is evaluated through some simulation experiments. Finally, two real applications are presented. [ABSTRACT FROM AUTHOR]