Finite mixture of skew distributions have emerged as an effective tool in modelling heterogeneous data with asymmetric features. With various proposals appearing rapidly in the recent years, which are similar but not identical, the connections between them and their relative performance becomes rather unclear. This paper aims to provide a concise overview of these developments by presenting a systematic classification of the existing skew distributions into four types, thereby clarifying their close relationships. This also aids in understanding the link between some of the proposed expectation-maximization (EM) based algorithms for the computation of the maximum likelihood estimates of the parameters of the models. The final part of this paper presents an illustration of the performance of these mixture models in clustering a real dataset, relative to other non-elliptically contoured clustering methods and associated algorithms for their implementation.
In this paper, we introduce a mixture of skew-t factor analyzers as well as a family of mixture models based thereon. The mixture of skew-t distributions model that we use arises as a limiting case of the mixture of generalized hyperbolic distributions. Like their Gaussian and t-distribution analogues, our mixture of skew-t factor analyzers are very well-suited to the model-based clustering of high-dimensional data. Imposing constraints on components of the decomposed covariance parameter results in the development of eight flexible models. The alternating expectation-conditional maximization …
Mixtures of factor analyzers (MFA) provide a powerful tool for modelling high-dimensional datasets. In recent years, several generalizations of MFA have been developed where the normality assumption of the factors and/or of the errors was relaxed to allow for skewness in the data. However, due to the form of the adopted component densities, the distribution of the factors/errors in most of these models is typically limited to modelling skewness oncentrated in a single direction. Here, we introduce a more flexible finite mixture of factor analyzers based on the class of scale mixtures of canoni…
In recent work, robust mixture modelling approaches using skewed distributions have been explored to accommodate asymmetric data. We introduce parsimony by developing skew-t and skew-normal analogues of the popular GPCM family that employ an eigenvalue decomposition of a positive-semidefinite matrix. The methods developed in this paper are compared to existing models in both an unsupervised and semi-supervised classification framework. Parameter estimation is carried out using the expectation-maximization algorithm and models are selected using the Bayesian information criterion. The efficacy …
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 c…
We present a proposal to deal with the non-normality issue in the context of regression models with measurement errors when both the response and the explanatory variable are observed with error. We extend the normal model by jointly modeling the unobserved covariate and the random errors by a finite mixture of scale mixture of skew-normal distributions. This approach allows us to model data with great flexibility, accommodating skewness, heavy tails, and multi-modality.
Clustering is the process of finding underlying group structures in data. Although mixture model-based clustering is firmly established in the multivariate case, there is a relative paucity of work on matrix variate distributions and none for clustering with mixtures of skewed matrix variate distributions. Four finite mixtures of skewed matrix variate distributions are considered. Parameter estimation is carried out using an expectation-conditional maximization algorithm, and both simulated and real data are used for illustration.
Mixtures of multivariate normal inverse Gaussian (MNIG) distributions can be used to cluster data that exhibit features such as skewness and heavy tails. However, for cluster analysis, using a traditional finite mixture model framework, either the number of components needs to be known $a$-$priori$ or needs to be estimated $a$-$posteriori$ using some model selection criterion after deriving results for a range of possible number of components. However, different model selection criteria can sometimes result in different number of components yielding uncertainty. Here, an infinite mixture model…
Most Item Response Theory (IRT) models for dichotomous responses are based on probit or logit link functions which assume a symmetric relationship between the probability of a correct response and the latent traits of individuals submitted to a test. This assumption restricts the use of those models to the case in which all items have a symmetric behaviour. On the other hand, asymmetric models proposed in the literature impose that all the items in a test have an asymmetric behaviour. This assumption is inappropriate for great part of the tests which are, in general, composed by both symmetric…
In recent years, data have become increasingly higher dimensional and, therefore, an increased need has arisen for dimension reduction techniques for clustering. Although such techniques are firmly established in the literature for multivariate data, there is a relative paucity in the area of matrix variate, or three-way, data. Furthermore, the few methods that are available all assume matrix variate normality, which is not always sensible if cluster skewness or excess kurtosis is present. Mixtures of bilinear factor analyzers using skewed matrix variate distributions are proposed. In all, fou…
Skewed generalizations of the normal distribution have been a topic of great interest in the statistics community due to their diverse applications across several domains. One of the most popular skew normal distributions, due to its intuitive appeal, is the Azzalini's skew normal distribution. However, due to the nature of the distribution it suffers from serious inferential problems. Interestingly, the Bayesian approach has been shown to mitigate these issues. Recently, another skew normal distribution, the Geometric skew normal distribution, which is structurally different from Azzalini's s…
Many approximate Bayesian inference methods assume a particular parametric form for approximating the posterior distribution. A multivariate Gaussian distribution provides a convenient density for such approaches; examples include the Laplace, penalized quasi-likelihood, Gaussian variational, and expectation propagation methods. Unfortunately, these all ignore the potential skewness of the posterior distribution. We propose a modification that accounts for skewness, where key statistics of the posterior distribution are matched instead to a multivariate skew-normal distribution. A combination …
The skew-normal and the skew-$t$ distributions are parametric families which are currently under intense investigation since they provide a more flexible formulation compared to the classical normal and $t$ distributions by introducing a parameter which regulates their skewness. While these families enjoy attractive formal properties from the probability viewpoint, a practical problem with their usage in applications is the possibility that the maximum likelihood estimate of the parameter which regulates skewness diverges. This situation has vanishing probability for increasing sample size, bu…
The multivariate extended skew-normal distribution allows for accommodating raw data which are skewed and heavy tailed, and has at least three appealing statistical properties, namely closure under conditioning, affine transformations, and marginalization. In this paper we propose a Bayesian computational approach based on a sequential Monte Carlo (SMC) sampler to estimate such distributions. The practical implementation of each step of the algorithm is discussed and the elicitation of prior distributions takes into consideration some unusual behaviour of the likelihood function and the corres…
Discrete mixture models are one of the most successful approaches for density estimation. Under a Bayesian nonparametric framework, Dirichlet process location-scale mixture of Gaussian kernels is the golden standard, both having nice theoretical properties and computational tractability. In this paper we explore the use of the skew-normal kernel, which can naturally accommodate several degrees of skewness by the use of a third parameter. The choice of this kernel function allows us to formulate nonparametric location-scale-shape mixture prior with large support and good performance in differen…
Motivated by the analysis of the distribution of university grades, which is usually asymmetric, we discuss two informative priors for the shape parameter of the skew-normal distribution, showing that they lead to closed-form full-conditional posterior distributions, particularly useful in MCMC computation. Gibbs sampling algorithms are discussed for the joint vector of parameters, given independent prior distributions for the location and scale parameters. Simulation studies are performed to assess the performance of Gibbs samplers and to compare the choice of informative priors against a non…
We study the Jeffreys prior of the skewness parameter of a general class of scalar skew--symmetric models. It is shown that this prior is symmetric about 0, proper, and with tails $O(λ^{-3/2})$ under mild regularity conditions. We also calculate the independence Jeffreys prior for the case with unknown location and scale parameters. Sufficient conditions for the existence of the corresponding posterior distribution are investigated for the case when the sampling model belongs to the family of skew--symmetric scale mixtures of normal distributions. The usefulness of these results is illustrated…
Non-Gaussian mixture models are gaining increasing attention for mixture model-based clustering particularly when dealing with data that exhibit features such as skewness and heavy tails. Here, such a mixture distribution is presented, based on the multivariate normal inverse Gaussian (MNIG) distribution. For parameter estimation of the mixture, a Bayesian approach via Gibbs sampler is used; for this, a novel approach to simulate univariate generalized inverse Gaussian random variables and matrix generalized inverse Gaussian random matrices is provided. The proposed algorithm will be applied t…
The contribution of this work is the introduction of a multivariate circular-linear (or poly- cylindrical) distribution obtained by combining the projected and the skew-normal. We show the flexibility of our proposal, its property of closure under marginalization and how to quantify multivariate dependence. Due to a non-identifiability issue that our proposal inherits from the projected normal, a compu- tational problem arises. We overcome it in a Bayesian framework, adding suitable latent variables and showing that posterior samples can be obtained with a post-processing of the estimation alg…
We introduce two classes of multivariate log skewed distributions with normal kernel: the log canonical fundamental skew-normal (log-CFUSN) and the log unified skew-normal (log-SUN). We also discuss some properties of the log-CFUSN family of distributions. These new classes of log-skewed distributions include the log-normal and multivariate log-skew normal families as particular cases. We discuss some issues related to Bayesian inference in the log-CFUSN family of distributions, mainly we focus on how to model the prior uncertainty about the skewing parameter. Based on the stochastic represent…
Shannon and Renyi entropies are quantitative measures of uncertainty in a data set. They are developed by Renyi in the context of entropy theory. These measures have been studied in the case of the multivariate t-distributions. We extend these tools to the class of multivariate skew t-distributions and then to more families of finite mixture of multivariate skew t distributions. In particular, by using generalized Holder inequality and some properties of multinomial theorem, we find upper and lower bounds of entropies for these families. An approximate value of these entropies can be calculate…
Because of its mathematical tractability, the Gaussian mixture model holds a special place in the literature for clustering and classification. For all its benefits, however, the Gaussian mixture model poses problems when the data is skewed or contains outliers. Because of this, methods have been developed over the years for handling skewed data, and fall into two general categories. The first is to consider a mixture of more flexible skewed distributions, and the second is based on incorporating a transformation to near normality. Although these methods have been compared in their respective …
Frequentist and likelihood methods of inference based on the multivariate skew-normal model encounter several technical difficulties with this model. In spite of the popularity of this class of densities, there are no broadly satisfactory solutions for estimation and testing problems. A general population Monte Carlo algorithm is proposed which: 1) exploits the latent structure stochastic representation of skew-normal random variables to provide a full Bayesian analysis of the model and 2) accounts for the presence of constraints in the parameter space. The proposed approach can be defined as …
There has been growing interest on forecasting mortality. In this article, we propose a novel dynamic Bayesian approach for modeling and forecasting the age-at-death distribution, focusing on a three-components mixture of a Dirac mass, a Gaussian distribution and a Skew-Normal distribution. According to the specified model, the age-at-death distribution is characterized via seven parameters corresponding to the main aspects of infant, adult and old-age mortality. The proposed approach focuses on coherent modeling of multiple countries, and following a Bayesian approach to inference we allow to…
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 …
Mixture of factor analyzer (MFA) model is an efficient model for the analysis of high dimensional data through which the factor-analyzer technique based on the covariance matrices reducing the number of free parameters. The model also provides an important methodology to determine latent groups in data. There are several pieces of research to extend the model based on the asymmetrical and/or with outlier datasets with some known computational limitations that have been examined in frequentist cases. In this paper, an MFA model with a rich and flexible class of skew normal (unrestricted) genera…
Regression models for dichotomous data are ubiquitous in statistics. Besides being useful for inference on binary responses, these methods serve also as building blocks in more complex formulations, such as density regression, nonparametric classification and graphical models. Within the Bayesian framework, inference proceeds by updating the priors for the coefficients, typically set to be Gaussians, with the likelihood induced by probit or logit regressions for the responses. In this updating, the apparent absence of a tractable posterior has motivated a variety of computational methods, incl…
Growth mixture models (GMMs) incorporate both conventional random effects growth modeling and latent trajectory classes as in finite mixture modeling; therefore, they offer a way to handle the unobserved heterogeneity between subjects in their development. GMMs with Gaussian random effects dominate the literature. When the data are asymmetric and/or have heavier tails, more than one latent class is required to capture the observed variable distribution. Therefore, a GMM with continuous non-elliptical distributions is proposed to capture skewness and heavier tails in the data set. Specifically,…
Cluster-weighted models (CWMs) extend finite mixtures of regressions (FMRs) in order to allow the distribution of covariates to contribute to the clustering process. In a matrix-variate framework, the matrix-variate normal CWM has been recently introduced. However, problems may be encountered when data exhibit skewness or other deviations from normality in the responses, covariates or both. Thus, we introduce a family of 24 matrix-variate CWMs which are obtained by allowing both the responses and covariates to be modelled by using one of four existing skewed matrix-variate distributions or the…
Flow cytometry is a high-throughput technology used to quantify multiple surface and intracellular markers at the level of a single cell. This enables to identify cell sub-types, and to determine their relative proportions. Improvements of this technology allow to describe millions of individual cells from a blood sample using multiple markers. This results in high-dimensional datasets, whose manual analysis is highly time-consuming and poorly reproducible. While several methods have been developed to perform automatic recognition of cell populations, most of them treat and analyze each sample…
Flow cytometry (FCM) is the standard multi-parameter assay for measuring single cell phenotype and functionality. It is commonly used for quantifying the relative frequencies of cell subsets in blood and disaggregated tissues. A typical analysis of FCM data involves cell classification---that is, the identification of cell subgroups in the sample---and comparisons of the cell subgroups across samples or conditions. While modern experiments often necessitate the collection and processing of samples in multiple batches, analysis of FCM data across batches is challenging because differences acros…
In this paper, we present an algorithm for the fitting of a location-scale variant of the canonical fundamental skew t (CFUST) distribution, a superclass of the restricted and unrestricted skew t-distributions. In recent years, a few versions of the multivariate skew $t$ (MST) model have been put forward, together with various EM-type algorithms for parameter estimation. These formulations adopted either a restricted or unrestricted characterization for their MST densities. In this paper, we examine a natural generalization of these developments, employing the CFUST distribution as the paramet…
The canonical form of scale mixtures of multivariate skew-normal distribution is defined, emphasizing its role in summarizing some key properties of this class of distributions. It is also shown that the canonical form corresponds to an affine invariant co-ordinate system as defined in Tyler \emph{et} al. (2009), and a method for obtaining the linear transform that converts a scale mixture of multivariate skew-normal distribution into a canonical form is presented. Related results, where the particular case of the multivariate skew $t$ distribution is considered in greater detail, are the gene…
We consider Gaussian mixture models in high dimensions and concentrate on the twin tasks of detection and feature selection. Under sparsity assumptions on the difference in means, we derive information bounds and establish the performance of various procedures, including the top sparse eigenvalue of the sample covariance matrix and other projection tests based on moments, such as the skewness and kurtosis tests of Malkovich and Afifi (1973), and other variants which we were better able to control under the null.
It is shown that a simple Dirichlet process mixture of multivariate normals offers Bayesian density estimation with adaptive posterior convergence rates. Toward this, a novel sieve for non-parametric mixture densities is explored, and its rate adaptability to various smoothness classes of densities in arbitrary dimension is demonstrated. This sieve construction is expected to offer a substantial technical advancement in studying Bayesian non-parametric mixture models based on stick-breaking priors.
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. T…
Although there is much recent work developing flexible variational methods for Bayesian computation, Gaussian approximations with structured covariance matrices are often preferred computationally in high-dimensional settings. This paper considers approximate inference methods for complex latent variable models where the posterior is close to Gaussian, but with some skewness in the posterior marginals. We consider skew decomposable graphical models (SDGMs), which are based on the closed skew normal family of distributions, as variational approximations. These approximations can reflect the tru…
Multivariate scale mixtures of skew-normal (SMSN) variables are flexible models that account for non-normality in multivariate data scenarios by tail weight assessment and a shape vector representing the asymmetry of the model in a directional fashion. Its stochastic representation involves a skew-normal (SN) vector and a non negative mixing scalar variable, independent of the SN vector, that injects kurtosis into the SMSN model. We address the problem of finding the maximal skewness projection for vectors that follow a SMSN distribution; when simple conditions on the moments of the mixing var…
Skew normal mixture models provide a more flexible framework than the popular normal mixtures for modelling heterogeneous data with asymmetric behaviors. Due to the unboundedness of likelihood function and the divergency of shape parameters, the maximum likelihood estimators of the parameters of interest are often not well defined, leading to dissatisfactory inferential process. We put forward a proposal to deal with these issues simultaneously in the context of penalizing the likelihood function. The resulting penalized maximum likelihood estimator is proved to be strongly consistent when the…
The mixture of factor analyzers (MFA) model provides a powerful tool for analyzing high-dimensional data as it can reduce the number of free parameters through its factor-analytic representation of the component covariance matrices. This paper extends the MFA model to incorporate a restricted version of the multivariate skew-normal distribution to model the distribution of the latent component factors, called mixtures of skew-normal factor analyzers (MSNFA). The proposed MSNFA model allows us to relax the need for the normality assumption for the latent factors in order to accommodate skewness…
This is an extended version of the paper Lee and McLachlan (2014b) with simulations and applications added. This paper introduces a finite mixture of canonical fundamental skew t (CFUST) distributions for a model-based approach to clustering where the clusters are asymmetric and possibly long-tailed (Lee and McLachlan, 2014b). The family of CFUST distributions includes the restricted multivariate skew t (rMST) and unrestricted multivariate skew t (uMST) distributions as special cases. In recent years, a few versions of the multivariate skew t (MST) model have been put forward, together with va…
Finite mixture models have been widely used to model and analyze data from a heterogeneous populations. Moreover, data of this kind can be missing or subject to some upper and/or lower detection limits because of the restriction of experimental apparatuses. Another complication arises when measures of each population depart significantly from normality, for instance, asymmetric behavior. For such data structures, we propose a robust model for censored and/or missing data based on finite mixtures of multivariate skew-normal distributions. This approach allows us to model data with great flexibi…
Trinity College（都柏林圣三一学院）；Dalle Molle Institute for Artificial Intelligence Research (IDSIA)（达尔莫尔人工智能研究所（IDSIA））
Skew-Gaussian processes (SkewGPs) extend the multivariate Unified Skew-Normal distributions over finite dimensional vectors to distribution over functions. SkewGPs are more general and flexible than Gaussian processes, as SkewGPs may also represent asymmetric distributions. In a recent contribution we showed that SkewGP and probit likelihood are conjugate, which allows us to compute the exact posterior for non-parametric binary classification and preference learning. In this paper, we generalize previous results and we prove that SkewGP is conjugate with both the normal and affine probit likel…
We show how the expectation-maximization (EM) algorithm can be applied exactly for the fitting of mixtures of general multivariate skew t (MST) distributions, eliminating the need for computationally expensive Monte Carlo estimation. Finite mixtures of MST distributions have proven to be useful in modelling heterogeneous data with asymmetric and heavy tail behaviour. Recently, they have been exploited as an effective tool for modelling flow cytometric data. However, without restrictions on the the characterizations of the component skew t-distributions, Monte Carlo methods have been used to fi…
Families of mixtures of multivariate power exponential (MPE) distributions have been previously introduced and shown to be competitive for cluster analysis in comparison to other elliptical mixtures including mixtures of Gaussian distributions. Herein, we propose a family of mixtures of multivariate skewed power exponential distributions to combine the flexibility of the MPE distribution with the ability to model skewness. These mixtures are more robust to variations from normality and can account for skewness, varying tail weight, and peakedness of data. A generalized expectation-maximization…
In the past few years, there have been a number of proposals for generalizing the factor analysis (FA) model and its mixture version (known as mixtures of factor analyzers (MFA)) using non-normal and asymmetric distributions. These models adopt various types of skew densities for either the factors or the errors. While the relationships between various choices of skew distributions have been discussed in the literature, the differences between placing the assumption of skewness on the factors or on the errors have not been closely studied. This paper examines these formulations and discusses t…
In this study, we propose a robust mixture regression procedure based on the skew t distribution to model heavy-tailed and/or skewed errors in a mixture regression setting. Using the scale mixture representation of the skew t distribution, we give an Expectation Maximization (EM) algorithm to compute the maximum likelihood (ML) estimates for the paramaters of interest. The performance of proposed estimators is demonstrated by a simulation study and a real data example.
Robust clustering from incomplete data is an important topic because, in many practical situations, real data sets are heavy-tailed, asymmetric, and/or have arbitrary patterns of missing observations. Flexible methods and algorithms for model-based clustering are presented via mixture of the generalized hyperbolic distributions and its limiting case, the mixture of multivariate skew-t distributions. An analytically feasible EM algorithm is formulated for parameter estimation and imputation of missing values for mixture models employing missing at random mechanisms. The proposed methodologies a…
Nonlinear mixed effects models have received a great deal of attention in the statistical literature in recent years because of their flexibility in handling longitudinal studies, including human immunodeficiency virus viral dynamics, pharmacokinetic analyses, and studies of growth and decay. A standard assumption in nonlinear mixed effects models for continuous responses is that the random effects and the within-subject errors are normally distributed, making the model sensitive to outliers. We present a novel class of asymmetric nonlinear mixed effects models that provides efficient paramete…
