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Paul D. McNicholas

Publications and source records attributed to Paul D. McNicholas.

At least 73 records · Page 4Linked to original sources
arXiv2017

A Mixture of SDB Skew-t Factor Analyzers

Mixtures of skew-t distributions offer a flexible choice for model-based clustering. A mixture model of this sort can be implemented using a variety of formulations of the skew-t distribution. Herein we develop a mixture of skew-t factor analyzers model for clustering of high-dimensional data using a flexible formulation of the skew-t distribution. Methodological details of our approach, which represents an extension of the mixture of factor analyzers model to a flexible skew-t distribution, are outlined and details of parameter estimation are provided. Clustering results are illustrated and compared to an alternative formulation of the mixture of skew-t factor analyzers model as well as the mixture of factor analyzers model.

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arXiv2017

A Matrix Variate Skew-t Distribution

Although there is ample work in the literature dealing with skewness in the multivariate setting, there is a relative paucity of work in the matrix variate paradigm. Such work is, for example, useful for modelling three-way data. A matrix variate skew-t distribution is derived based on a mean-variance matrix normal mixture. An expectation-conditional maximization algorithm is developed for parameter estimation. Simulated data are used for illustration.

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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

Fractionally-Supervised Classification

Traditionally, there are three species of classification: unsupervised, supervised, and semi-supervised. Supervised and semi-supervised classification differ by whether or not weight is given to unlabelled observations in the classification procedure. In unsupervised classification, or clustering, all observations are unlabeled and hence full weight is given to unlabelled observations. When some observations are unlabelled, it can be very difficult to \textit{a~priori} choose the optimal level of supervision, and the consequences of a sub-optimal choice can be non-trivial. A flexible fractionally-supervised approach to classification is introduced, where any level of supervision --- ranging from unsupervised to supervised --- can be attained. Our approach uses a weighted likelihood, wherein weights control the relative role that labelled and unlabelled data have in building a classifier. A comparison between our approach and the traditional species is presented using simulated and real data. Gaussian mixture models are used as a vehicle to illustrate our fractionally-supervised classification approach; however, it is broadly applicable and variations on the postulated model can be easily made.

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arXiv2015

Clustering, Classification, Discriminant Analysis, and Dimension Reduction via Generalized Hyperbolic Mixtures

A method for dimension reduction with clustering, classification, or discriminant analysis is introduced. This mixture model-based approach is based on fitting generalized hyperbolic mixtures on a reduced subspace within the paradigm of model-based clustering, classification, or discriminant analysis. A reduced subspace of the data is derived by considering the extent to which group means and group covariances vary. The members of the subspace arise through linear combinations of the original data, and are ordered by importance via the associated eigenvalues. The observations can be projected onto the subspace, resulting in a set of variables that captures most of the clustering information available. The use of generalized hyperbolic mixtures gives a robust framework capable of dealing with skewed clusters. Although dimension reduction is increasingly in demand across many application areas, the authors are most familiar with biological applications and so two of the five real data examples are within that sphere. Simulated data are also used for illustration. The approach introduced herein can be considered the most general such approach available, and so we compare results to three special and limiting cases. Comparisons with several well established techniques illustrate its promising performance.

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arXiv2015

Mixtures of Multivariate Power Exponential Distributions

An expanded family of mixtures of multivariate power exponential distributions is introduced. While fitting heavy-tails and skewness has received much attention in the model-based clustering literature recently, we investigate the use of a distribution that can deal with both varying tail-weight and peakedness of data. A family of parsimonious models is proposed using an eigen-decomposition of the scale matrix. A generalized expectation-maximization algorithm is presented that combines convex optimization via a minorization-maximization approach and optimization based on accelerated line search algorithms on the Stiefel manifold. Lastly, the utility of this family of models is illustrated using both toy and benchmark data.

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arXiv2015

A Mixture of Generalized Hyperbolic Factor Analyzers

Model-based clustering imposes a finite mixture modelling structure on data for clustering. Finite mixture models assume that the population is a convex combination of a finite number of densities, the distribution within each population is a basic assumption of each particular model. Among all distributions that have been tried, the generalized hyperbolic distribution has the advantage that is a generalization of several other methods, such as the Gaussian distribution, the skew t-distribution, etc. With specific parameters, it can represent either a symmetric or a skewed distribution. While its inherent flexibility is an advantage in many ways, it means the estimation of more parameters than its special and limiting cases. The aim of this work is to propose a mixture of generalized hyperbolic factor analyzers to introduce parsimony and extend the method to high dimensional data. This work can be seen as an extension of the mixture of factor analyzers model to generalized hyperbolic mixtures. The performance of our generalized hyperbolic factor analyzers is illustrated on real data, where it performs favourably compared to its Gaussian analogue.

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arXiv2014

Mixtures of Variance-Gamma Distributions

A mixture of variance-gamma distributions is introduced and developed for model-based clustering and classification. The latest in a growing line of non-Gaussian mixture approaches to clustering and classification, the proposed mixture of variance-gamma distributions is a special case of the recently developed mixture of generalized hyperbolic distributions, and a restriction is required to ensure identifiability. Our mixture of variance-gamma distributions is perhaps the most useful such special case and, we will contend, may be more useful than the mixture of generalized hyperbolic distributions in some cases. In addition to being an alternative to the mixture of generalized hyperbolic distributions, our mixture of variance-gamma distributions serves as an alternative to the ubiquitous mixture of Gaussian distributions, which is a special case, as well as several non-Gaussian approaches, some of which are special cases. The mathematical development of our mixture of variance-gamma distributions model relies on its relationship with the generalized inverse Gaussian distribution; accordingly, the latter is reviewed before our mixture of variance-gamma distributions is presented. Parameter estimation carried out within the expectation-maximization framework.

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arXiv2014

A Mixture of Generalized Hyperbolic Distributions

We introduce a mixture of generalized hyperbolic distributions as an alternative to the ubiquitous mixture of Gaussian distributions as well as their near relatives of which the mixture of multivariate t and skew-t distributions are predominant. The mathematical development of our mixture of generalized hyperbolic distributions model relies on its relationship with the generalized inverse Gaussian distribution. The latter is reviewed before our mixture models are presented along with details of the aforesaid reliance. Parameter estimation is outlined within the expectation-maximization framework before the clustering performance of our mixture models is illustrated via applications on simulated and real data. In particular, the ability of our models to recover parameters for data from underlying Gaussian and skew-t distributions is demonstrated. Finally, the role of Generalized hyperbolic mixtures within the wider model-based clustering, classification, and density estimation literature is discussed.

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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

Unsupervised Learning via Mixtures of Skewed Distributions with Hypercube Contours

Mixture models whose components have skewed hypercube contours are developed via a generalization of the multivariate shifted asymmetric Laplace density. Specifically, we develop mixtures of multiple scaled shifted asymmetric Laplace distributions. The component densities have two unique features: they include a multivariate weight function, and the marginal distributions are also asymmetric Laplace. We use these mixtures of multiple scaled shifted asymmetric Laplace distributions for clustering applications, but they could equally well be used in the supervised or semi-supervised paradigms. The expectation-maximization algorithm is used for parameter estimation and the Bayesian information criterion is used for model selection. Simulated and real data sets are used to illustrate the approach and, in some cases, to visualize the skewed hypercube structure of the components.

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arXiv2014

Mixture Model Averaging for Clustering

In mixture model-based clustering applications, it is common to fit several models from a family and report clustering results from only the `best' one. In such circumstances, selection of this best model is achieved using a model selection criterion, most often the Bayesian information criterion. Rather than throw away all but the best model, we average multiple models that are in some sense close to the best one, thereby producing a weighted average of clustering results. Two (weighted) averaging approaches are considered: averaging the component membership probabilities and averaging models. In both cases, Occam's window is used to determine closeness to the best model and weights are computed within a Bayesian model averaging paradigm. In some cases, we need to merge components before averaging; we introduce a method for merging mixture components based on the adjusted Rand index. The effectiveness of our model-based clustering averaging approaches is illustrated using a family of Gaussian mixture models on real and simulated data.

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arXiv2014

An Adaptive LASSO-Penalized BIC

Mixture models are becoming a popular tool for the clustering and classification of high-dimensional data. In such high dimensional applications, model selection is problematic. The Bayesian information criterion, which is popular in lower dimensional applications, tends to underestimate the true number of components in high dimensions. We introduce an adaptive LASSO-penalized BIC (ALPBIC) to mitigate this problem. This efficacy of the ALPBIC is illustrated via applications of parsimonious mixtures of factor analyzers. The selection of the best model by ALPBIC is shown to be consistent with increasing numbers of observations based on simulated and real data analyses.

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arXiv2014

Modelling Receiver Operating Characteristic Curves Using Gaussian Mixtures

The receiver operating characteristic curve is widely applied in measuring the performance of diagnostic tests. Many direct and indirect approaches have been proposed for modelling the ROC curve, and because of its tractability, the Gaussian distribution has typically been used to model both populations. We propose using a Gaussian mixture model, leading to a more flexible approach that better accounts for atypical data. Monte Carlo simulation is used to circumvent the issue of absence of a closed-form. We show that our method performs favourably when compared to the crude binormal curve and to the semi-parametric frequentist binormal ROC using the famous LABROC procedure.

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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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