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Summary: The expectation–maximization (EM) algorithm is a familiar tool for computing the maximum likelihood estimate of the parameters in hidden Markov and semi‐Markov models. This paper carries out a detailed study on the influence that the initial values of the parameters impose on the results produced by the algorithm. We compare random starts and partitional and model‐based strategies for choosing the initial values for the EM algorithm in the case of multivariate Gaussian emission distributions (EDs) and assess the performance of each strategy with different assessment criteria. Several data generation settings are considered with varying number of latent states, of variables as well as of the level of fuzziness in the data, and discussion on how each factor influences the obtained results is provided. Simulation results show that different initialization strategies may lead to different log‐likelihood values and, accordingly, to different estimated partitions. A clear indication of which strategies should be preferred is given. We further include two real‐data examples, widely analysed in the hidden semi‐Markov model literature. [ABSTRACT FROM AUTHOR]

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]

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]

We revisit the generalized hyperbolic (GH) distribution and its nested models. These include widely used parametric choices like the multivariate normal, skew‐t$$ t $$, Laplace, and several others. We also introduce the multiple‐choice LASSO, a novel penalized method for choosing among alternative constraints on the same parameter. A hierarchical multiple‐choice Least Absolute Shrinkage and Selection Operator (LASSO) penalized likelihood is optimized to perform simultaneous model selection and inference within the GH family. We illustrate our approach through a simulation study and a real data example. The methodology proposed in this paper has been implemented in R functions which are available as supplementary material. [ABSTRACT FROM AUTHOR]

A family of parsimonious Gaussian cluster-weighted models is presented. This family concerns a multivariate extension to cluster-weighted modelling that can account for correlations between multivariate responses. Parsimony is attained by constraining parts of an eigen-decomposition imposed on the component covariance matrices. A sufficient condition for identifiability is provided and an expectation-maximization algorithm is presented for parameter estimation. Model performance is investigated on both synthetic and classical real data sets and compared with some popular approaches. Finally, accounting for linear dependencies in the presence of a linear regression structure is shown to offer better performance, vis-à-vis clustering, over existing methodologies. [ABSTRACT FROM AUTHOR]

This article proposes the elliptical multivariate leptokurtic-normal (MLN) distribution to fit data with excess kurtosis. The MLN distribution is a multivariate Gram-Charlier expansion of the multivariate normal (MN) distribution and has a closed-form representation characterized by one additional parameter denoting the excess kurtosis. It is obtained from the elliptical representation of the MN distribution, by reshaping its generating variate with the associated orthogonal polynomials. The strength of this approach for obtaining the MLN distribution lies in its general applicability as it can be applied to any multivariate elliptical law to get a suitable distribution to fit data. Maximum likelihood is discussed as a parameter estimation technique for the MLN distribution. Mixtures of MLN distributions are also proposed for robust model-based clustering. An EM algorithm is presented to obtain estimates of the mixture parameters. Benchmark real data are used to show the usefulness of mixtures of MLN distributions. The Canadian Journal of Statistics 45: 95-119; 2017 © 2016 Statistical Society of Canada [ABSTRACT FROM AUTHOR]

We propose the family of dimension-wise scaled normal mixtures (DSNMs) to model the joint distribution of a d -variate random variable with real-valued components. Each member of the family generalizes the multivariate normal (MN) distribution in two directions. Firstly, the DSNM has a more general type of symmetry with respect to the elliptical symmetry of the MN distribution. Secondly, the univariate marginals have similar heavy-tailed normal scale mixture distributions with (possibly) different tailedness parameters; as a consequence of practical interest, the DSNM allows for a different excess kurtosis on each dimension. These peculiarities are obtained in an MN scale mixture framework by introducing a d -variate mixing random variable with independent and similar components acting separately for each dimension. We examine several properties of DSNMs, such as the joint density function, hierarchical and stochastic representations, relations with other families of symmetric distributions, type of symmetry, marginal and conditional distributions, no correlation implying independence, moment generating function, and moments of practical interest. For illustrative purposes, we describe two members of the DSNM family obtained in the case of components of the mixing random vector being either uniform or shifted exponential; these are examples of mixing distributions that guarantee a closed-form expression for the joint density of the DSNM. For the two DSNMs analyzed in detail, we introduce parsimony by allowing the d tailedness parameters to be tied across dimensions, and describe algorithms, based on the expectation–maximization (EM) principle, to estimate the parameters by maximum likelihood. We use real data from the financial and biometrical fields to appreciate the advantages of our DSNMs over other symmetric heavy-tailed distributions available in the literature. [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]