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The multivariate contaminated normal (MCN) distribution represents a simple heavy-tailed generalization of the multivariate normal (MN) distribution to model elliptical contoured scatters in the presence of mild outliers (also referred to as 'bad' points herein) and automatically detect bad points. The price of these advantages is two additional parameters: proportion of good observations and degree of contamination. However, in a multivariate setting, only one proportion of good observations and only one degree of contamination may be limiting. To overcome this limitation, we propose a multiple scaled contaminated normal (MSCN) distribution. Among its parameters, we have an orthogonal matrix Γ. In the space spanned by the vectors (principal components) of Γ, there is a proportion of good observations and a degree of contamination for each component. Moreover, each observation has a posterior probability of being good with respect to each principal component. Thanks to this probability, the method provides directional robust estimates of the parameters of the nested MN and automatic directional detection of bad points. The term 'directional' is added to specify that the method works separately for each principal component. Mixtures of MSCN distributions are also proposed, and an expectation-maximization algorithm is used for parameter estimation. Real and simulated data are considered to show the usefulness of our mixture with respect to well-established mixtures of symmetric distributions with heavy tails. [ABSTRACT FROM AUTHOR]

A mixture of multivariate contaminated normal distributions is developed for model-based clustering. In addition to the parameters of the classical normal mixture, our contaminated mixture has, for each cluster, a parameter controlling the proportion of mild outliers and one specifying the degree of contamination. Crucially, these parameters do not have to be specified a priori, adding a flexibility to our approach. Parsimony is introduced via eigen-decomposition of the component covariance matrices, and sufficient conditions for the identifiability of all the members of the resulting family are provided. An expectation-conditional maximization algorithm is outlined for parameter estimation and various implementation issues are discussed. Using a large-scale simulation study, the behavior of the proposed approach is investigated and comparison with well-established finite mixtures is provided. The performance of this novel family of models is also illustrated on artificial and real data. [ABSTRACT FROM AUTHOR]

The analysis of the decision boundaries plays an important role in understanding the characteristics of a classifier in the framework of model-based clustering and discriminant analysis. The wider is the family of decision boundaries generated by a classifier the larger is its flexibility for classification purposes. In this paper, we present rigorous results concerning the decision boundaries of mixtures of (linear) regressions under Gaussian assumptions. In particular, three types of mixtures of regressions are considered: with fixed covariates, with concomitant variables, and with random covariates. The obtained decision boundaries have a geometrical interpretation in terms hyperquadrics and define a taxonomy of the considered models. Beyond Gaussian assumptions, decision boundaries can be investigated numerically; as an example, we illustrate the case of the t distribution. [ABSTRACT FROM AUTHOR]

In the context of mixture models with random covariates, this article presents the polynomial Gaussian cluster-weighted model (CWM). It extends the linear Gaussian CWM, for bivariate data, in a twofold way. First, it allows for possible nonlinear dependencies in the mixture components by considering a polynomial regression. Second, it is not restricted to be used for model-based clustering only being contextualized in the most general model-based classification framework. Maximum likelihood parameter estimates are derived using the EM algorithm and model selection is carried out using the Bayesian information criterion (BIC) and the integrated completed likelihood (ICL). The article also investigates the conditions under which the posterior probabilities of component-membership from a polynomial Gaussian CWM coincide with those of other well-established mixture-models which are related to it. When applied to artificial and real data, the polynomial Gaussian CWM has shown to outperform the mixture of polynomial Gaussian regressions, which is its natural competitor in the class of mixture models with fixed covariates. [ABSTRACT FROM AUTHOR]

• We propose a robust method for simultaneous unsupervised classification and dimensionality reduction for high-dimensional data. • The proposed approach is effective for identifying mild outliers in high-dimensional unsupervised classification problems. • The proportion of mild outliers is learned and so does not need to be pre-specified. The contaminated Gaussian distribution represents a simple heavy-tailed elliptical generalization of the Gaussian distribution; unlike the often-considered t -distribution, it also allows for automatic detection of mild outlying or "bad" points in the same way that observations are typically assigned to the groups in the finite mixture model context. Starting from this distribution, we propose the contaminated factor analysis model as a method for dimensionality reduction and detection of bad points in higher dimensions. A mixture of contaminated Gaussian factor analyzers (MCGFA) model follows therefrom, and extends the recently proposed mixture of contaminated Gaussian distributions to high-dimensional data. We introduce a family of 32 parsimonious models formed by introducing constraints on the covariance and contamination structures of the general MCGFA model. We outline a variant of the expectation-maximization algorithm for parameter estimation. Various implementation issues are discussed, and the novel family of models is compared to well-established approaches on both simulated and real data. [ABSTRACT FROM AUTHOR]