Maximum Likelihood Estimation and Model Comparison for Mixtures of Structural Equation Models with Ignorable Missing Data

The objective of this paper is to develop the maximum likelihood approach for analyzing a finite mixture of structural equation models with missing data that are missing at random. A Monte Carlo EM algorithm is proposed for obtaining the maximum likelihood estimates. A well-known statistic in model comparison, namely the Bayesian Information Criterion (BIC), is used for model comparison. With the presence of missing data, the computation of the observed-data likelihood function value involved in the BIC is not straightforward. A procedure based on path sampling is developed to compute this function value. It is shown by means of simulation studies that ignoring the incomplete data with missing entries gives less accurate ML estimates. An illustrative real example is also presented.     

A maximum likelihood methodology for clusterwise linear regression

This paper presents a conditional mixture, maximum likelihood methodology for performing clusterwise linear regression. This new methodology simultaneously estimates separate regression functions and membership inK clusters or groups. A review of related procedures is discussed with an associated critique. The conditional mixture, maximum likelihood methodology is introduced together with the E-M algorithm utilized for parameter estimation. A Monte Carlo analysis is performed via a fractional factorial design to examine the performance of the procedure. Next, a marketing application is presented concerning the evaluations of trade show performance by senior marketing executives. Finally, other potential applications and directions for future research are identified.     

A latent class procedure for the structural analysis of two-way compositional data

This paper develops a new procedure for simultaneously performing multidimensional scaling and cluster analysis on two-way compositional data of proportions. The objective of the proposed procedure is to delineate patterns of variability in compositions across subjects by simultaneously clustering subjects into latent classes or groups and estimating a joint space of stimulus coordinates and class-specific vectors in a multidimensional space. We use a conditional mixture, maximum likelihood framework with an E-M algorithm for parameter estimation. The proposed procedure is illustrated using a compositional data set reflecting proportions of viewing time across television networks for an area sample of households.     

The mixture method of clustering applied to three-way data

Clustering or classifying individuals into groups such that there is relative homogeneity within the groups and heterogeneity between the groups is a problem which has been considered for many years. Most available clustering techniques are applicable only to a two-way data set, where one of the modes is to be partitioned into groups on the basis of the other mode. Suppose, however, that the data set is three-way. Then what is needed is a multivariate technique which will cluster one of the modes on the basis of both of the other modes simultaneously. It is shown that by appropriate specification of the underlying model, the mixture maximum likelihood approach to clustering can be applied in the context of a three-way table. It is illustrated using a soybean data set which consists of multiattribute measurements on a number of genotypes each grown in several environments. Although the problem is set in the framework of clustering genotypes, the technique is applicable to other types of three-way data sets.     

A mixture likelihood approach for generalized linear models

A mixture model approach is developed that simultaneously estimates the posterior membership probabilities of observations to a number of unobservable groups or latent classes, and the parameters of a generalized linear model which relates the observations, distributed according to some member of the exponential family, to a set of specified covariates within each Class. We demonstrate how this approach handles many of the existing latent class regression procedures as special cases, as well as a host of other parametric specifications in the exponential family heretofore not mentioned in the latent class literature. As such we generalize the McCullagh and Nelder approach to a latent class framework. The parameters are estimated using maximum likelihood, and an EM algorithm for estimation is provided. A Monte Carlo study of the performance of the algorithm for several distributions is provided, and the model is illustrated in two empirical applications.     

Bayesian Regularization for Normal Mixture Estimation and Model-Based Clustering

Normal mixture models are widely used for statistical modeling of data, including cluster analysis. However maximum likelihood estimation (MLE) for normal mixtures using the EM algorithm may fail as the result of singularities or degeneracies. To avoid this, we propose replacing the MLE by a maximum a posteriori (MAP) estimator, also found by the EM algorithm. For choosing the number of components and the model parameterization, we propose a modified version of BIC, where the likelihood is evaluated at the MAP instead of the MLE. We use a highly dispersed proper conjugate prior, containing a small fraction of one observation's worth of information. The resulting method avoids degeneracies and singularities, but when these are not present it gives similar results to the standard method using MLE, EM and BIC.     

Mixture Model Analysis of Complex Samples

We investigate the effects of a complex sampling design on the estimation of mixture models. An approximate or pseudo likelihood approach is proposed to obtain consistent estimates of class-specific parameters when the sample arises from such a complex design. The effects of ignoring the sample design are demonstrated empirically in the context of an international value segmentation study in which a multinomial mixture model is applied to identify segment-level value rankings. The analysis reveals that ignoring the sample design results in both an incorrect number of segments as identified by information criteria and biased estimates of segment-level parameters.     

Enhanced Model-Based Clustering,Density Estimation,and Discriminant Analysis Software: MCLUST

MCLUST is a software package for model-based clustering, density estimation and discriminant analysis interfaced to the S-PLUS commercial software and the R language. It implements parameterized Gaussian hierarchical clustering algorithms and the EM algorithm for parameterized Gaussian mixture models with the possible addition of a Poisson noise term. Also included are functions that combine hierarchical clustering, EM and the Bayesian Information Criterion (BIC) in comprehensive strategies for clustering, density estimation, and discriminant analysis. MCLUST provides functionality for displaying and visualizing clustering and classification results. A web page with related links can be found at .     

Comparing Several Parametric and Nonparametric Approaches to Time Series Clustering: A Simulation Study

One key point in cluster analysis is to determine a similarity or dissimilarity measure between data objects. When working with time series, the concept of similarity can be established in different ways. In this paper, several non-parametric statistics originally designed to test the equality of the log-spectra of two stochastic processes are proposed as dissimilarity measures between time series data. Their behavior in time series clustering is analyzed throughout a simulation study, and compared with the performance of several model-free and model-based dissimilarity measures. Up to three different classification settings were considered: (i) to distinguish between stationary and non-stationary time series, (ii) to classify different ARMA processes and (iii) to classify several non-linear time series models. As it was expected, the performance of a particular dissimilarity metric strongly depended on the type of processes subjected to clustering. Among all the measures studied, the nonparametric distances showed the most robust behavior.     

Piecewise Regression Mixture for Simultaneous Functional Data Clustering and Optimal Segmentation

This paper introduces a novel mixture model-based approach to the simultaneous clustering and optimal segmentation of functional data, which are curves presenting regime changes. The proposed model consists of a finite mixture of piecewise polynomial regression models. Each piecewise polynomial regression model is associated with a cluster, and within each cluster, each piecewise polynomial component is associated with a regime (i.e., a segment). We derive two approaches to learning the model parameters: the first is an estimation approach which maximizes the observed-data likelihood via a dedicated expectation-maximization (EM) algorithm, then yielding a fuzzy partition of the curves into K clusters obtained at convergence by maximizing the posterior cluster probabilities. The second is a classification approach and optimizes a specific classification likelihood criterion through a dedicated classification expectation-maximization (CEM) algorithm. The optimal curve segmentation is performed by using dynamic programming. In the classification approach, both the curve clustering and the optimal segmentation are performed simultaneously as the CEM learning proceeds. We show that the classification approach is a probabilistic version generalizing the deterministic K-means-like algorithm proposed in Hébrail, Hugueney, Lechevallier, and Rossi (2010). The proposed approach is evaluated using simulated curves and real-world curves. Comparisons with alternatives including regression mixture models and the K-means-like algorithm for piecewise regression demonstrate the effectiveness of the proposed approach.     

Variable Selection for Clustering and Classification

As data sets continue to grow in size and complexity, effective and efficient techniques are needed to target important features in the variable space. Many of the variable selection techniques that are commonly used alongside clustering algorithms are based upon determining the best variable subspace according to model fitting in a stepwise manner. These techniques are often computationally intensive and can require extended periods of time to run; in fact, some are prohibitively computationally expensive for high-dimensional data. In this paper, a novel variable selection technique is introduced for use in clustering and classification analyses that is both intuitive and computationally efficient. We focus largely on applications in mixture model-based learning, but the technique could be adapted for use with various other clustering/classification methods. Our approach is illustrated on both simulated and real data, highlighted by contrasting its performance with that of other comparable variable selection techniques on the real data sets.     

Metric inference for social networks

Using a natural metric on the space of networks, we define a probability measure for network-valued random variables. This measure is indexed by two parameters, which are interpretable as a location parameter and a dispersion parameter. From this structure, one can develop maximum likelihood estimates, hypothesis tests and confidence regions, all in the context of independent and identically distributed networks. The value of this perspective is illustrated through application to portions of the friedship cognitive social structure data gathered by Krackhardt (1987).We thank Ove Frank, David Krackhardt, the editor and the referees for their constructive comments and suggestions.     

Clustering criteria for discrete data and latent class models

We show that a well-known clustering criterion for discrete data, the information criterion, is closely related to the classification maximum likelihood criterion for the latent class model. This relation can be derived from the Bryant-Windham construction. Emphasis is placed on binary clustering criteria which are analyzed under the maximum likelihood approach for different multivariate Bernoulli mixtures. This alternative form of criterion reveals non-apparent aspects of clustering techniques. All the criteria discussed can be optimized with the alternating optimization algorithm. Some illustrative applications are included.

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Résumé Nous montrons que le critère de classification de l'information, souvent utilisé pour les données discrètes, est très lié au critère du maximum de vraisemblance classifiante appliqué au modèle des classes latentes. Ce lien peut être analysé sous l'approche de la paramétrisation de Bryant-Windham. L'accent est mis sur le cas des données binaires qui sont analysées sous l'approche du maximum de vraisemblance pour les mélanges de distributions multivariées de Bernoulli. Cette forme de critère permet de mettre en évidence des aspects cachés des méthodes de classification de données binaires. Tous les critères envisagés ici peuvent être optimisés avec l'algorithme d'optimisation alternée. Des exemples concluent cet article.

     

Percept Variance,Subadditivity and the Metric Classification of Similarity,and Dissimilarity Data

Percept variance is shown to change the additive property of city-block distances and make city-block distances more subadditive than Euclidean distances. Failure to account for percept variance will result in the misclassification of city-block data as Euclidean. A maximum likelihood estimation procedure is proposed for the multidimensional scaling of similarity data characterized by percept variance. Monte Carlo and empirical experiments are used to evaluate the proposed approach.     

Model-Based Clustering for Image Segmentation and Large Datasets via Sampling

The rapid increase in the size of data sets makes clustering all the more important to capture and summarize the information, at the same time making clustering more difficult to accomplish. If model-based clustering is applied directly to a large data set, it can be too slow for practical application. A simple and common approach is to first cluster a random sample of moderate size, and then use the clustering model found in this way to classify the remainder of the objects. We show that, in its simplest form, this method may lead to unstable results. Our experiments suggest that a stable method with better performance can be obtained with two straightforward modifications to the simple sampling method: several tentative models are identified from the sample instead of just one, and several EM steps are used rather than just one E step to classify the full data set. We find that there are significant gains from increasing the size of the sample up to about 2,000, but not from further increases. These conclusions are based on the application of several alternative strategies to the segmentation of three different multispectral images, and to several simulated data sets.     

A Copula-Based Algorithm for Discovering Patterns of Dependent Observations

The main aim of this work is the study of clustering dependent data by means of copula functions. Copulas are popular multivariate tools whose importance within clustering methods has not been investigated yet in detail. We propose a new algorithm (CoClust in brief) that allows to cluster dependent data according to the multivariate structure of the generating process without any assumption on the margins. Moreover, the approach does not require either to choose a starting classification or to set a priori the number of clusters; in fact, the CoClust selects them by using a criterion based on the log–likelihood of a copula fit. We test our proposal on simulated data for different dependence scenarios and compare it with a model–based clustering technique. Finally, we show applications of the CoClust to real microarray data of breast-cancer patients.     

Functional PCA and Base-Line Logit Models

In many statistical applications data are curves measured as functions of a continuous parameter as time. Despite of their functional nature and due to discrete-time observation, these type of data are usually analyzed with multivariate statistical methods that do not take into account the high correlation between observations of a single curve at nearby time points. Functional data analysis methodologies have been developed to solve these type of problems. In order to predict the class membership (multi-category response variable) associated to an observed curve (functional data), a functional generalized logit model is proposed. Base-line category logit formulations will be considered and their estimation based on basis expansions of the sample curves of the functional predictor and parameters. Functional principal component analysis will be used to get an accurate estimation of the functional parameters and to classify sample curves in the categories of the response variable. The good performance of the proposed methodology will be studied by developing an experimental study with simulated and real data.     

Wavelet-based Fuzzy Clustering of Time Series

Traditional procedures for clustering time series are based mostly on crisp hierarchical or partitioning methods. Given that the dynamics of a time series may change over time, a time series might display patterns that may enable it to belong to one cluster over one period while over another period, its pattern may be more consistent with those in another cluster. The traditional clustering procedures are unable to identify the changing patterns over time. However, clustering based on fuzzy logic will be able to detect the switching patterns from one time period to another thus enabling some time series to simultaneously belong to more than one cluster. In particular, this paper proposes a fuzzy approach to the clustering of time series based on their variances through wavelet decomposition. We will show that this approach will distinguish between time series with different patterns in variability as well identifying time series with switching patterns in variability.     

Parameter modification for clustering criteria

The more ways there are of understanding a clustering technique, the more effectively the results can be analyzed and used. I will give a general procedure, calledparameter modification, to obtain from a clustering criterion a variety of equivalent forms of the criterion. These alternative forms reveal aspects of the technique that are not necessarily apparent in the original formulation. This procedure is successful in improving the understanding of a significant number of clustering techniques.The insight obtained will be illustrated by applying parameter modification to partitioning, mixture and fuzzy clustering methods, resulting in a unified approach to the study of these methods and a general algorithm for optimizing them.The author wishes to thank Professor Doctor Hans-Hermann Bock for many stimulating discussions.     

A Run Length Transformation for Discriminating Between Auto Regressive Time Series

We describe a simple time series transformation to detect differences in series that can be accurately modelled as stationary autoregressive (AR) processes. The transformation involves forming the histogram of above and below the mean run lengths. The run length (RL) transformation has the benefits of being very fast, compact and updatable for new data in constant time. Furthermore, it can be generated directly from data that has already been highly compressed. We first establish the theoretical asymptotic relationship between run length distributions and AR models through consideration of the zero crossing probability and the distribution of runs. We benchmark our transformation against two alternatives: the truncated Autocorrelation function (ACF) transform and the AR transformation, which involves the standard method of fitting the partial autocorrelation coefficients with the Durbin-Levinson recursions and using the Akaike Information Criterion stopping procedure. Whilst optimal in the idealized scenario, representing the data in these ways is time consuming and the representation cannot be updated online for new data. We show that for classification problems the accuracy obtained through using the run length distribution tends towards that obtained from using the full fitted models. We then propose three alternative distance measures for run length distributions based on Gower’s general similarity coefficient, the likelihood ratio and dynamic time warping (DTW). Through simulated classification experiments we show that a nearest neighbour distance based on DTW converges to the optimal faster than classifiers based on Euclidean distance, Gower’s coefficient and the likelihood ratio. We experiment with a variety of classifiers and demonstrate that although the RL transform requires more data than the best performing classifier to achieve the same accuracy as AR or ACF, this factor is at worst non-increasing with the series length, m, whereas the relative time taken to fit AR and ACF increases with m. We conclude that if the data is stationary and can be suitably modelled by an AR series, and if time is an important factor in reaching a discriminatory decision, then the run length distribution transform is a simple and effective transformation to use.