There is an increasingly rich literature about Bayesian nonparametric models for clustering functional observations. However, most of the recent proposals rely on infinite-dimensional characterizations that might lead to overly complex cluster solutions. In addition, while prior knowledge about the functional shapes is typically available, its practical exploitation might be a difficult modeling task. Motivated by an application in e-commerce, we propose a novel enriched Dirichlet mixture model for functional data. Our proposal accommodates the incorporation of functional constraints while bounding the model complexity. To clarify the underlying partition mechanism, we characterize the prior process through a Pólya urn scheme. These features lead to a very interpretable clustering method compared to available techniques. To overcome computational bottlenecks, we employ a variational Bayes approximation for tractable posterior inference.
The modelling of action potentials from extracellular recordings, or spike sorting, is a rich area of neuroscience research in which latent variable models are often used. Two such models, Overfitted Finite Mixture models (OFMs) and Dirichlet Process Mixture models (DPMs) are considered to provide insights for unsupervised clustering of complex, multivariate medical data when the number of clusters is unknown. OFM and DPM are structured in a similar hierarchical fashion but they are based on different philosophies with different underlying assumptions. This study investigates how these differe…
This chapter investigates the latent structure of bipartite networks via a model-based clustering approach which is able to capture both latent groups of sending nodes and latent variability of the propensity of sending nodes to create links with receiving nodes within each group. This modelling approach is very flexible and can be estimated by using fast inferential approaches such as variational inference. We apply this model to the analysis of a terrorist network in order to identify the main latent groups of terrorists and their latent trait scores based on their attendance to some events.
Motivated by the problem of accurately predicting gap times between successive blood donations, we present here a general class of Bayesian nonparametric models for clustering. These models allow for prediction of new recurrences, accommodating covariate information that describes the personal characteristics of the sample individuals. We introduce a prior for the random partition of the sample individuals which encourages two individuals to be co-clustered if they have similar covariate values. Our prior generalizes PPMx models in the literature, which are defined in terms of cohesion and sim…
Finite mixture models have become a popular tool for clustering. Amongst other uses, they have been applied for clustering longitudinal data and clustering high-dimensional data. In the latter case, a latent Gaussian mixture model is sometimes used. Although there has been much work on clustering using latent variables and on clustering longitudinal data, respectively, there has been a paucity of work that combines these features. An approach is developed for clustering longitudinal data with many time points based on an extension of the mixture of common factor analyzers model. A variation of…
In many modern applications, there is interest in analyzing enormous data sets that cannot be easily moved across computers or loaded into memory on a single computer. In such settings, it is very common to be interested in clustering. Existing distributed clustering algorithms are mostly distance or density based without a likelihood specification, precluding the possibility of formal statistical inference. Model-based clustering allows statistical inference, yet research on distributed inference has emphasized nonparametric Bayesian mixture models over finite mixture models. To fill this gap…
University of Warwick（华威大学）；University of Edinburgh（爱丁堡大学）
Mixtures of experts probabilistically divide the input space into regions, where the assumptions of each expert, or conditional model, need only hold locally. Combined with Gaussian process (GP) experts, this results in a powerful and highly flexible model. We focus on alternative mixtures of GP experts, which model the joint distribution of the inputs and targets explicitly. We highlight issues of this approach in multi-dimensional input spaces, namely, poor scalability and the need for an unnecessarily large number of experts, degrading the predictive performance and increasing uncertainty. …
The majority of finite mixture models suffer from not allowing asymmetric tail dependencies within components and not capturing non-elliptical clusters in clustering applications. Since vine copulas are very flexible in capturing these types of dependencies, we propose a novel vine copula mixture model for continuous data. We discuss the model selection and parameter estimation problems and further formulate a new model-based clustering algorithm. The use of vine copulas in clustering allows for a range of shapes and dependency structures for the clusters. Our simulation experiments illustrate…
We are concerned in clustering continuous data sets subject to non-ignorable missingness. We perform clustering with a specific semi-parametric mixture, under the assumption of conditional independence given the component. The mixture model isused for clustering and not for estimating the density of the full variables (observed and unobserved), thus we do not need other assumptions on the component distribution neither to specify the missingness mechanism. Estimation is performed by maximizing an extension of smoothed likelihood allowing missingness. This optimization is achieved by a Majoriza…
A weighted likelihood approach for robust fitting of a mixture of multivariate Gaussian components is developed in this work. Two approaches have been proposed that are driven by a suitable modification of the standard EM and CEM algorithms, respectively. In both techniques, the M-step is enhanced by the computation of weights aimed at downweighting outliers. The weights are based on Pearson residuals stemming from robust Mahalanobis-type distances. Formal rules for robust clustering and outlier detection can be also defined based on the fitted mixture model. The behavior of the proposed metho…
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 …
We introduce a copula mixture model to perform dependency-seeking clustering when co-occurring samples from different data sources are available. The model takes advantage of the great flexibility offered by the copulas framework to extend mixtures of Canonical Correlation Analysis to multivariate data with arbitrary continuous marginal densities. We formulate our model as a non-parametric Bayesian mixture, while providing efficient MCMC inference. Experiments on synthetic and real data demonstrate that the increased flexibility of the copula mixture significantly improves the clustering and t…
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…
The two most extended density-based approaches to clustering are surely mixture model clustering and modal clustering. In the mixture model approach, the density is represented as a mixture and clusters are associated to the different mixture components. In modal clustering, clusters are understood as regions of high density separated from each other by zones of lower density, so that they are closely related to certain regions around the density modes. If the true density is indeed in the assumed class of mixture densities, then mixture model clustering allows to scrutinize more subtle situat…
A mixture of joint generalized hyperbolic distributions (MJGHD) is introduced for asymmetric clustering for high-dimensional data. The MJGHD approach takes into account the cluster-specific subspace, thereby limiting the number of parameters to estimate while also facilitating visualization of results. Identifiability is discussed, and a multi-cycle ECM algorithm is outlined for parameter estimation. The MJGHD approach is illustrated on two real data sets, where the Bayesian information criterion is used for model selection.
In model-based-clustering mixture models are used to group data points into clusters. A useful concept introduced for Gaussian mixtures by Malsiner Walli et al (2016) are sparse finite mixtures, where the prior distribution on the weight distribution of a mixture with $K$ components is chosen in such a way that a priori the number of clusters in the data is random and is allowed to be smaller than $K$ with high probability. The number of cluster is then inferred a posteriori from the data. The present paper makes the following contributions in the context of sparse finite mixture modelling. Fi…
Clustering time series into similar groups can improve models by combining information across like time series. While there is a well developed body of literature for clustering of time series, these approaches tend to generate clusters independently of model training which can lead to poor model fit. We propose a novel distributed approach that simultaneously clusters and fits autoregression models for groups of similar individuals. We apply a Wishart mixture model so as to cluster individuals while modeling the corresponding autocovariance matrices at the same time. The fitted Wishart scale …
Time-course gene expression data such as yeast cell cycle data may be periodically expressed. To cluster such data, currently used Fourier series approximations of periodic gene expressions have been found not to be sufficiently adequate to model the complexity of the time-course data, partly due to their ignoring the dependence between the expression measurements over time and the correlation among gene expression profiles. We further investigate the advantages and limitations of available models in the literature and propose a new mixture model with AR(1) random effects for the clustering of…
Analysis of a Bayesian mixture model for the Matrix Langevin distribution on the Stiefel manifold is presented. The model exploits a particular parametrization of the Matrix Langevin distribution, various aspects of which are elaborated on. A general, and novel, family of conjugate priors, and an efficient Markov chain Monte Carlo (MCMC) sampling scheme for the corresponding posteriors is then developed for the mixture model. Theoretical properties of the prior and posterior distributions, including posterior consistency, are explored in detail. Extensive simulation experiments are presented t…
In some contexts, mixture models can fit certain variables well at the expense of others in ways beyond the analyst's control. For example, when the data include some variables with non-trivial amounts of missing values, the mixture model may fit the marginal distributions of the nearly and fully complete variables at the expense of the variables with high fractions of missing data. Motivated by this setting, we present a mixture model for mixed ordinal and nominal data that splits variables into two groups, focus variables and remainder variables. The model allows the analyst to specify a ric…
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 the framework of Bayesian model-based clustering based on a finite mixture of Gaussian distributions, we present a joint approach to estimate the number of mixture components and identify cluster-relevant variables simultaneously as well as to obtain an identified model. Our approach consists in specifying sparse hierarchical priors on the mixture weights and component means. In a deliberately overfitting mixture model the sparse prior on the weights empties superfluous components during MCMC. A straightforward estimator for the true number of components is given by the most frequent number…
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.
The identification of unlabelled neuronal electric signals is one of the most challenging open problems in neuroscience, widely known as Spike Sorting. Motivated to solve this problem, we propose a model-based approach within the mixture modeling framework for clustering oscillatory functional data called MixFMM. The core of the approach is the FMM waves, which are non-linear parametric time functions, flexible enough to describe different oscillatory patterns and simple enough to be estimated efficiently. In particular, specific model parameters describe the waveforms' phase, amplitude, and s…
Growth mixture models are an important tool for detecting group structure in repeated measures data. Unlike traditional clustering methods, they explicitly model the repeat measurements on observations, and the statistical framework they are based on allows for model selection methods to be used to select the number of clusters. However, the basic growth mixture model makes the assumption that all of the measurements in the data have grouping information/separate the clusters. In other clustering contexts, it has been shown that including non-clustering variables in clustering procedures can l…
Finite mixture models have been used for unsupervised learning for some time, and their use within the semi-supervised paradigm is becoming more commonplace. Clickstream data is one of the various emerging data types that demands particular attention because there is a notable paucity of statistical learning approaches currently available. A mixture of first-order continuous time Markov models is introduced for unsupervised and semi-supervised learning of clickstream data. This approach assumes continuous time, which distinguishes it from existing mixture model-based approaches; practically, t…
The learning of mixture models can be viewed as a clustering problem. Indeed, given data samples independently generated from a mixture of distributions, we often would like to find the {\it correct target clustering} of the samples according to which component distribution they were generated from. For a clustering problem, practitioners often choose to use the simple $k$-means algorithm. $k$-means attempts to find an {\it optimal clustering} that minimizes the sum-of-squares distance between each point and its cluster center. In this paper, we consider fundamental (i.e., information-theoreti…
Motivated by problems in data clustering, we establish general conditions under which families of nonparametric mixture models are identifiable, by introducing a novel framework involving clustering overfitted \emph{parametric} (i.e. misspecified) mixture models. These identifiability conditions generalize existing conditions in the literature, and are flexible enough to include for example mixtures of Gaussian mixtures. In contrast to the recent literature on estimating nonparametric mixtures, we allow for general nonparametric mixture components, and instead impose regularity assumptions on …
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…
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…
Dirichlet process mixtures are flexible non-parametric models, particularly suited to density estimation and probabilistic clustering. In this work we study the posterior distribution induced by Dirichlet process mixtures as the sample size increases, and more specifically focus on consistency for the unknown number of clusters when the observed data are generated from a finite mixture. Crucially, we consider the situation where a prior is placed on the concentration parameter of the underlying Dirichlet process. Previous findings in the literature suggest that Dirichlet process mixtures are t…
University of Hasan 1st（哈桑一世大学）；University of Abdelmalek Essaâdi（阿卜杜勒马利克·埃萨迪大学）
The mixture models have become widely used in clustering, given its probabilistic framework in which its based, however, for modern databases that are characterized by their large size, these models behave disappointingly in setting out the model, making essential the selection of relevant variables for this type of clustering. After recalling the basics of clustering based on a model, this article will examine the variable selection methods for model-based clustering, as well as presenting opportunities for improvement of these methods.
Korea Institute of Science and Technology Information（韩国科学技术信息研究院）；University of Lyon 2（里昂第二大学）
Evolutionary clustering aims at capturing the temporal evolution of clusters. This issue is particularly important in the context of social media data that are naturally temporally driven. In this paper, we propose a new probabilistic model-based evolutionary clustering technique. The Temporal Multinomial Mixture (TMM) is an extension of classical mixture model that optimizes feature co-occurrences in the trade-off with temporal smoothness. Our model is evaluated for two recent case studies on opinion aggregation over time. We compare four different probabilistic clustering models and we show …
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 …
In model-based clustering using finite mixture models, it is a significant challenge to determine the number of clusters (cluster size). It used to be equal to the number of mixture components (mixture size); however, this may not be valid in the presence of overlaps or weight biases. In this study, we propose to continuously measure the cluster size in a mixture model by a new concept called mixture complexity (MC). It is formally defined from the viewpoint of information theory and can be seen as a natural extension of the cluster size considering overlap and weight bias. Subsequently, we ap…
We consider clustering based on significance tests for Gaussian Mixture Models (GMMs). Our starting point is the SigClust method developed by Liu et al. (2008), which introduces a test based on the k-means objective (with k = 2) to decide whether the data should be split into two clusters. When applied recursively, this test yields a method for hierarchical clustering that is equipped with a significance guarantee. We study the limiting distribution and power of this approach in some examples and show that there are large regions of the parameter space where the power is low. We then introduce…
In Bayesian nonparametrics there exists a rich variety of discrete priors, including the Dirichlet process and its generalizations, which are nowadays well-established tools. Despite the remarkable advances, few proposals are tailored for modeling observations lying on product spaces, such as $\mathbb{R}^p$. Indeed, for multivariate random measures, most available priors lack flexibility and do not allow for separate partition structures among the spaces. We introduce a discrete nonparametric prior, termed enriched Pitman-Yor process (EPY), aimed at addressing these issues. Theoretical propert…
The dissimilarity mixture autoencoder (DMAE) is a neural network model for feature-based clustering that incorporates a flexible dissimilarity function and can be integrated into any kind of deep learning architecture. It internally represents a dissimilarity mixture model (DMM) that extends classical methods like K-Means, Gaussian mixture models, or Bregman clustering to any convex and differentiable dissimilarity function through the reinterpretation of probabilities as neural network representations. DMAE can be integrated with deep learning architectures into end-to-end models, allowing th…
Mixture models are a popular tool in model-based clustering. Such a model is often fitted by a procedure that maximizes the likelihood, such as the EM algorithm. At convergence, the maximum likelihood parameter estimates are typically reported, but in most cases little emphasis is placed on the variability associated with these estimates. In part this may be due to the fact that standard errors are not directly calculated in the model-fitting algorithm, either because they are not required to fit the model, or because they are difficult to compute. The examination of standard errors in model-b…
We propose an extension of non-parametric multivariate finite mixture models by dropping the standard conditional independence assumption and incorporating the independent component analysis (ICA) structure instead. We formulate an objective function in terms of penalized smoothed Kullback Leibler distance and introduce the nonlinear smoothed majorization-minimization independent component analysis (NSMM-ICA) algorithm for optimizing this function and estimating the model parameters. We have implemented a practical version of this algorithm, which utilizes the FastICA algorithm, in the R packa…
Air Force Institute of Technology（空军理工学院）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
We introduce a model-free relax-and-round algorithm for k-means clustering based on a semidefinite relaxation due to Peng and Wei. The algorithm interprets the SDP output as a denoised version of the original data and then rounds this output to a hard clustering. We provide a generic method for proving performance guarantees for this algorithm, and we analyze the algorithm in the context of subgaussian mixture models. We also study the fundamental limits of estimating Gaussian centers by k-means clustering in order to compare our approximation guarantee to the theoretically optimal k-means clu…
Normandie Univ（诺曼底大学）；UNICAEN（卡昂大学）；CNRS（法国国家科学研究中心）；Aix Marseille Université（艾克斯-马赛大学）；ENSAM（法国国立高等工程技术学校）；Université de Toulon（土伦大学）
The parsimonious Gaussian mixture models, which exploit an eigenvalue decomposition of the group covariance matrices of the Gaussian mixture, have shown their success in particular in cluster analysis. Their estimation is in general performed by maximum likelihood estimation and has also been considered from a parametric Bayesian prospective. We propose new Dirichlet Process Parsimonious mixtures (DPPM) which represent a Bayesian nonparametric formulation of these parsimonious Gaussian mixture models. The proposed DPPM models are Bayesian nonparametric parsimonious mixture models that allow to…
University of Ljubljana（卢布尔雅那大学）；Slovenian Environment Agency（斯洛文尼亚环境署）；Tampere University（坦佩雷大学）
We investigate a novel non-parametric regression-based clustering algorithm for longitudinal data analysis. Combining natural cubic splines with Gaussian mixture models (GMM), the algorithm can produce smooth cluster means that describe the underlying data well. However, there are some shortcomings in the algorithm: high computational complexity in the parameter estimation procedure and a numerically unstable variance estimator. Therefore, to further increase the usability of the method, we incorporated approaches to reduce its computational complexity, we developed a new, more stable variance…
Recent work on overfitting Bayesian mixtures of distributions offers a powerful framework for clustering multivariate data using a latent Gaussian model which resembles the factor analysis model. The flexibility provided by overfitting mixture models yields a simple and efficient way in order to estimate the unknown number of clusters and model parameters by Markov chain Monte Carlo (MCMC) sampling. The present study extends this approach by considering a set of eight parameterizations, giving rise to parsimonious representations of the covariance matrix per cluster. A Gibbs sampler combined w…
ETH Zürich（苏黎世联邦理工学院）；University of Pennsylvania（宾夕法尼亚大学）；Yale University（耶鲁大学）
Spectral clustering is one of the most popular algorithms to group high dimensional data. It is easy to implement and computationally efficient. Despite its popularity and successful applications, its theoretical properties have not been fully understood. In this paper, we show that spectral clustering is minimax optimal in the Gaussian Mixture Model with isotropic covariance matrix, when the number of clusters is fixed and the signal-to-noise ratio is large enough. Spectral gap conditions are widely assumed in the literature to analyze spectral clustering. On the contrary, these conditions ar…
