arXiv subjects

Antonio Punzo

Publications and source records attributed to Antonio Punzo.

At least 37 records · Page 2Linked to original sources

arXiv2020

The multivariate tail-inflated normal distribution and its application in finance

This paper introduces the multivariate tail-inflated normal (MTIN) distribution, an elliptical heavy-tails generalization of the multivariate normal (MN). The MTIN belongs to the family of MN scale mixtures by choosing a convenient continuous uniform as mixing distribution. Moreover, it has a closed-form for the probability density function characterized by only one additional ``inflation'' parameter, with respect to the nested MN, governing the tail-weight. The first four moments are also computed; interestingly, they always exist and the excess kurtosis can assume any positive value. The method of moments and maximum likelihood (ML) are considered for estimation. As concerns the latter, a direct approach, as well as a variant of the EM algorithm, are illustrated. The existence of the ML estimates is also evaluated. Since the inflation parameter is estimated from the data, robust estimates of the mean vector and covariance matrix of the nested MN distribution are automatically obtained by down-weighting. Simulations are performed to compare the estimation methods/algorithms and to investigate the ability of AIC and BIC to select among a set of candidate elliptical models. For illustrative purposes, the MTIN distribution is finally fitted to multivariate financial data where its usefulness is also shown in comparison with other well-established multivariate elliptical distributions.

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arXiv2020

Mixtures of Contaminated Matrix Variate Normal Distributions

Analysis of three-way data is becoming ever more prevalent in the literature, especially in the area of clustering and classification. Real data, including real three-way data, are often contaminated by potential outlying observations. Their detection, as well as the development of robust models insensitive to their presence, is particularly important for this type of data because of the practical issues concerning their effective visualization. Herein, the contaminated matrix variate normal distribution is discussed and then utilized in the mixture model paradigm for clustering. One key advantage of the proposed model is the ability to automatically detect potential outlying matrices by computing their \textit{a posteriori} probability to be a "good" or "bad" point. Such detection is currently unavailable using existing matrix variate methods. An expectation conditional maximization algorithm is used for parameter estimation, and both simulated and real data are used for illustration.

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arXiv2019

High-dimensional unsupervised classification via parsimonious contaminated mixtures

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.

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arXiv2019

Unconstrained representation of orthogonal matrices with application to common principle components

Many statistical problems involve the estimation of a $\left(d\times d\right)$ orthogonal matrix $\textbf{Q}$. Such an estimation is often challenging due to the orthonormality constraints on $\textbf{Q}$. To cope with this problem, we propose a very simple decomposition for orthogonal matrices which we abbreviate as PLR decomposition. It produces a one-to-one correspondence between $\textbf{Q}$ and a $\left(d\times d\right)$ unit lower triangular matrix $\textbf{L}$ whose $d\left(d-1\right)/2$ entries below the diagonal are unconstrained real values. Once the decomposition is applied, regardless of the objective function under consideration, we can use any classical unconstrained optimization method to find the minimum (or maximum) of the objective function with respect to $\textbf{L}$. For illustrative purposes, we apply the PLR decomposition in common principle components analysis (CPCA) for the maximum likelihood estimation of the common orthogonal matrix when a multivariate leptokurtic-normal distribution is assumed in each group. Compared to the commonly used normal distribution, the leptokurtic-normal has an additional parameter governing the excess kurtosis; this makes the estimation of $\textbf{Q}$ in CPCA more robust against mild outliers. The usefulness of the PLR decomposition in leptokurtic-normal CPCA is illustrated by two biometric data analyses.

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arXiv2018

Multiple Scaled Contaminated Normal Distribution and Its Application in Clustering

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, referred to as "bad" points. The MCN can also automatically detect bad points. The price of these advantages is two additional parameters, both with specific and useful interpretations: proportion of good observations and degree of contamination. However, points may be bad in some dimensions but good in others. The use of an overall proportion of good observations and of an overall degree of contamination is limiting. To overcome this limitation, we propose a multiple scaled contaminated normal (MSCN) distribution with a proportion of good observations and a degree of contamination for each dimension. Once the model is fitted, each observation has a posterior probability of being good with respect to each dimension. Thanks to this probability, we have a method for simultaneous directional robust estimation of the parameters of the MN distribution based on down-weighting and for the automatic directional detection of bad points by means of maximum a posteriori probabilities. The term "directional" is added to specify that the method works separately for each dimension. Mixtures of MSCN distributions are also proposed as an application of the proposed model for robust clustering. An extension of the EM algorithm is used for parameter estimation based on the maximum likelihood approach. Real and simulated data are used to show the usefulness of our mixture with respect to well-established mixtures of symmetric distributions with heavy tails.

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arXiv2018

Robust Model-Based Clustering of Voting Records

We explore the possibility of discovering extreme voting patterns in the U.S. Congressional voting records by drawing ideas from the mixture of contaminated normal distributions. A mixture of latent trait models via contaminated normal distributions is proposed. We assume that the low dimensional continuous latent variable comes from a contaminated normal distribution and, therefore, picks up extreme patterns in the observed binary data while clustering. We consider in particular such model for the analysis of voting records. The model is applied to a U.S. Congressional Voting data set on 16 issues. Note this approach is the first instance within the literature of a mixture model handling binary data with possible extreme patterns.

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arXiv2018

Asymmetric Clusters and Outliers: Mixtures of Multivariate Contaminated Shifted Asymmetric Laplace Distributions

Mixtures of multivariate contaminated shifted asymmetric Laplace distributions are developed for handling asymmetric clusters in the presence of outliers (also referred to as bad points herein). In addition to the parameters of the related non-contaminated mixture, for each (asymmetric) cluster, our model has one parameter controlling the proportion of 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 that is absent from other approaches such as trimming. Moreover, each observation is given a posterior probability of belonging to a particular cluster, and of being an outlier or not; advantageously, this allows for the automatic detection of outliers. An expectation-conditional maximization algorithm is outlined for parameter estimation and various implementation issues are discussed. The behaviour of the proposed model is investigated, and compared with well-established finite mixtures, on artificial and real data.

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arXiv2018

Cluster-weighted latent class modeling

Usually in Latent Class Analysis (LCA), external predictors are taken to be cluster conditional probability predictors (LC models with covariates), and/or score conditional probability predictors (LC regression models). In such cases, their distribution is not of interest. Class specific distribution is of interest in the distal outcome model, when the distribution of the external variable(s) is assumed to dependent on LC membership. In this paper, we consider a more general formulation, typical in cluster-weighted models, which embeds both the latent class regression and the distal outcome models. This allows us to test simultaneously both whether the distribution of the covariate(s) differs across classes, and whether there are significant direct effects of the covariate(s) on the indicators, by including most of the information about the covariate(s) - latent variable relationship. We show the advantages of the proposed modeling approach through a set of population studies and an empirical application on assets ownership of Italian households.

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arXiv2017

A new look at the inverse Gaussian distribution

The inverse Gaussian (IG) is one of the most famous and considered distributions with positive support. We propose a convenient mode-based parameterization yielding the reparametrized IG (rIG) distribution; it allows/simplifies the use of the IG distribution in various statistical fields, and we give some examples in nonparametric statistics, robust statistics, and model-based clustering. In nonparametric statistics, we define a smoother based on rIG kernels. By construction, the estimator is well-defined and free of boundary bias. We adopt likelihood cross-validation to select the smoothing parameter. In robust statistics, we propose the contaminated IG distribution, a heavy-tailed generalization of the rIG distribution to accommodate mild outliers; they can be automatically detected by the model via maximum a posteriori probabilities. To obtain maximum likelihood estimates of the parameters, we illustrate an expectation-maximization (EM) algorithm. Finally, for model-based clustering and semiparametric density estimation, we present finite mixtures of rIG distributions. We use the EM algorithm to obtain ML estimates of the parameters of the mixture model. Applications to economic and insurance data are finally illustrated to exemplify and enhance the use of the proposed models.

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arXiv2016

ContaminatedMixt: An R Package for Fitting Parsimonious Mixtures of Multivariate Contaminated Normal Distributions

We introduce the R package ContaminatedMixt, conceived to disseminate the use of mixtures of multivariate contaminated normal distributions as a tool for robust clustering and classification under the common assumption of elliptically contoured groups. Thirteen variants of the model are also implemented to introduce parsimony. The expectation-conditional maximization algorithm is adopted to obtain maximum likelihood parameter estimates, and likelihood-based model selection criteria are used to select the model and the number of groups. Parallel computation can be used on multicore PCs and computer clusters, when several models have to be fitted. Differently from the more popular mixtures of multivariate normal and t distributions, this approach also allows for automatic detection of mild outliers via the maximum a posteriori probabilities procedure. To exemplify the use of the package, applications to artificial and real data are presented.

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arXiv2016

Parsimonious mixtures of multivariate contaminated normal distributions

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 behaviour 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.

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arXiv2016

Multivariate response and parsimony for Gaussian cluster-weighted models

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.

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arXiv2015

Model-based clustering via linear cluster-weighted models

A novel family of twelve mixture models with random covariates, nested in the linear $t$ cluster-weighted model (CWM), is introduced for model-based clustering. The linear $t$ CWM was recently presented as a robust alternative to the better known linear Gaussian CWM. The proposed family of models provides a unified framework that also includes the linear Gaussian CWM as a special case. Maximum likelihood parameter estimation is carried out within the EM framework, and both the BIC and the ICL are used for model selection. A simple and effective hierarchical random initialization is also proposed for the EM algorithm. The novel model-based clustering technique is illustrated in some applications to real data. Finally, a simulation study for evaluating the performance of the BIC and the ICL is presented.

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arXiv2015

Improving the autodependogram using the Kulback-Leibler divergence

The autodependogram is a graphical device recently proposed in the literature to analyze autodependencies. It is defined computing the classical Pearson chi-square statistics of independence at various lags in order to point out the presence lag-depedencies. This paper proposes an improvement of this diagram obtained by substituting the chi-square statistics with an estimator of the Kulback-Leibler divergence between the bivariate density of two delayed variables and the product of their marginal distributions. A simulation study, on well-established time series models, shows that this new autodependogram is more powerful than the previous one. An application to financial data is also shown.

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arXiv2014

Robust Clustering in Regression Analysis via the Contaminated Gaussian Cluster-Weighted Model

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, it adopts a Gaussian distribution for both the covariates and the responses given the covariates. To robustify the approach with respect to possible elliptical heavy tailed departures from normality, due to the presence of atypical observations, the contaminated Gaussian CWM is here introduced. In addition to the parameters of the Gaussian CWM, each mixture component of our contaminated CWM 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 one specifying the degree of contamination with respect to the covariates. Crucially, these parameters do not have to be specified a priori, adding flexibility to our approach. Furthermore, once the model is estimated and the observations are assigned to the groups, a finer intra-group classification in typical points, 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 to the estimators from the Gaussian CWM. A sensitivity study is also conducted based on a real data set.

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arXiv2014

Hypothesis Testing for Parsimonious Gaussian Mixture Models

Gaussian mixture models with eigen-decomposed covariance structures make up the most popular family of mixture models for clustering and classification, i.e., the Gaussian parsimonious clustering models (GPCM). Although the GPCM family has been used for almost 20 years, selecting the best member of the family in a given situation remains a troublesome problem. Likelihood ratio tests are developed to tackle this problems. These likelihood ratio tests use the heteroscedastic model under the alternative hypothesis but provide much more flexibility and real-world applicability than previous approaches that compare the homoscedastic Gaussian mixture versus the heteroscedastic one. Along the way, a novel maximum likelihood estimation procedure is developed for two members of the GPCM family. Simulations show that the $χ^2$ reference distribution gives reasonable approximation for the LR statistics only when the sample size is considerable and when the mixture components are well separated; accordingly, following Lo (2008), a parametric bootstrap is adopted. Furthermore, by generalizing the idea of Greselin and Punzo (2013) to the clustering context, a closed testing procedure, having the defined likelihood ratio tests as local tests, is introduced to assess a unique model in the general family. The advantages of this likelihood ratio testing procedure are illustrated via an application to the well-known Iris data set.

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arXiv2014

KernSmoothIRT: An R Package for Kernel Smoothing in Item Response Theory

Item response theory (IRT) models are a class of statistical models used to describe the response behaviors of individuals to a set of items having a certain number of options. They are adopted by researchers in social science, particularly in the analysis of performance or attitudinal data, in psychology, education, medicine, marketing and other fields where the aim is to measure latent constructs. Most IRT analyses use parametric models that rely on assumptions that often are not satisfied. In such cases, a nonparametric approach might be preferable; nevertheless, there are not many software applications allowing to use that. To address this gap, this paper presents the R package KernSmoothIRT. It implements kernel smoothing for the estimation of option characteristic curves, and adds several plotting and analytical tools to evaluate the whole test/questionnaire, the items, and the subjects. In order to show the package's capabilities, two real datasets are used, one employing multiple-choice responses, and the other scaled responses.

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arXiv2013

Fitting Bivariate Mixed-Type Data via the Generalized Linear Exponential Cluster-Weighted Model

The cluster-weighted model (CWM) is a mixture model with random covariates which allows for flexible clustering and density estimation of a random vector composed by a response variable and by a set of covariates. In this class of models, the generalized linear exponential CWM is here introduced especially for modeling bivariate data of mixed-type. Its natural counterpart, in the family of latent class models, is also defined. Maximum likelihood parameter estimates are derived using the EM algorithm and model selection is carried out using the Bayesian information criterion (BIC). Artificial and real data are finally considered to exemplify and appreciate the proposed model.

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