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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]

Two families of parsimonious mixture models are introduced for model-based clustering. They are based on two multivariate distributions-the shifted exponential normal and the tail-inflated normal-recently introduced in the literature as heavy-tailed generalizations of the multivariate normal. Parsimony is attained by the eigen-decomposition of the component scale matrices, as well as by the imposition of a constraint on the tailedness parameters. Identifiability conditions are also provided. Two variants of the expectation-maximization algorithm are presented for maximum likelihood parameter estimation. Parameter recovery and clustering performance are investigated via a simulation study. Comparisons with the unconstrained mixture models are obtained as by-product. A further simulated analysis is conducted to assess how sensitive our and some well-established parsimonious competitors are to their own generative scheme. Lastly, our and the competing models are evaluated in terms of fitting and clustering on three real datasets. [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]

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]

When considering a mixed probability measure on [0, 1] with atoms in zero and one, the frontier points have to be treated (and 'weighted') differently and separately with respect to the interior points. In order to avoid the troublesome consequences of a mixed model, an easily interpretable discretization of [0, 1] on uniformly spaced atoms is here proposed. This 'homogeneous support' is used to define two discrete models, a parametric and a nonparametric one. An application to real data on recovery risk of the Bank of Italy's loans is here considered to exemplify both discretization and proposed models. Finally, a simulation study is performed to analyze the behavior of the nonparametric proposal in terms of goodness-of-fit. [ABSTRACT FROM AUTHOR]