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Compositional data have peculiar characteristics that pose significant challenges to traditional statistical methods and models. Within this framework, we use a convenient mode parametrized Dirichlet distribution across multiple fields of statistics. In particular, we propose finite mixtures of unimodal Dirichlet (UD) distributions for model-based clustering and classification. Then, we introduce the contaminated UD (CUD) distribution, a heavy-tailed generalization of the UD distribution that allows for a more flexible tail behavior in the presence of atypical observations. Thirdly, we propose finite mixtures of CUD distributions to jointly account for the presence of clusters and atypical points in the data. Parameter estimation is carried out by directly maximizing the maximum likelihood or by using an expectation-maximization (EM) algorithm. Two analyses are conducted on simulated data to illustrate the effects of atypical observations on parameter estimation and data classification, and how our proposals address both aspects. Furthermore, two real datasets are investigated and the results obtained via our models are discussed. [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]

Two families of matrix-variate hidden Markov regression models (MV-HMRMs) are here introduced. The distinction between them relies on the role of the covariates, which can be treated as fixed or random. Parsimony is achieved by using the eigen-decomposition of the components’ covariance matrices. This generates a different number of parsimonious models between the two families: 98 MV-HMRMs with fixed covariates and 9604 MV-HMRMs with random covariates. ECM algorithms are discussed for parameter estimation and, because of the high number of parsimonious models, convenient initialization, and fitting strategies are proposed. Our models are first applied to simulated data, and then to a four-way real dataset concerning the relationship between unemployment and labor force participation in the Italian provinces. [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]

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]

In generalized linear models (GLMs), measures of lack of fit are typically defined as the deviance between two nested models, and a deviance-based R2 is commonly used to evaluate the fit. In this paper, we extend deviance measures to mixtures of GLMs, whose parameters are estimated by maximum likelihood (ML) via the EM algorithm. Such measures are defined both locally, i.e., at cluster-level, and globally, i.e., with reference to the whole sample. At the cluster-level, we propose a normalized two-term decomposition of the local deviance into explained, and unexplained local deviances. At the sample-level, we introduce an additive normalized decomposition of the total deviance into three terms, where each evaluates a different aspect of the fitted model: (1) the cluster separation on the dependent variable, (2) the proportion of the total deviance explained by the fitted model, and (3) the proportion of the total deviance which remains unexplained. We use both local and global decompositions to define, respectively, local and overall deviance R2 measures for mixtures of GLMs, which we illustrate—for Gaussian, Poisson and binomial responses—by means of a simulation study. The proposed fit measures are then used to assess, and interpret clusters of COVID-19 spread in Italy in two time points. [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]

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]

Finite mixtures of regressions with fixed covariates are a commonly used model-based clustering methodology to deal with regression data. However, they assume assignment independence, i.e., the allocation of data points to the clusters is made independently of the distribution of the covariates. To take into account the latter aspect, finite mixtures of regressions with random covariates, also known as cluster-weighted models (CWMs), have been proposed in the univariate and multivariate literature. In this paper, the CWM is extended to matrix data, e.g., those data where a set of variables are simultaneously observed at different time points or locations. Specifically, the cluster-specific marginal distribution of the covariates and the cluster-specific conditional distribution of the responses given the covariates are assumed to be matrix normal. Maximum likelihood parameter estimates are derived using an expectation-conditional maximization algorithm. Parameter recovery, classification assessment, and the capability of the Bayesian information criterion to detect the underlying groups are investigated using simulated data. Finally, two real data applications concerning educational indicators and the Italian non-life insurance market are presented. [ABSTRACT FROM AUTHOR]