Dimensionally Reduced Model-Based Clustering Through Mixtures of Factor Mixture Analyzers

Dimensionally reduced model-based clustering methods are recently receiving a wide interest in statistics as a tool for performing simultaneously clustering and dimension reduction through one or more latent variables. Among these, Mixtures of Factor Analyzers assume that, within each component, the data are generated according to a factor model, thus reducing the number of parameters on which the covariance matrices depend. In Factor Mixture Analysis clustering is performed through the factors of an ordinary factor analysis which are jointly modelled by a Gaussian mixture. The two approaches differ in genesis, parameterization and consequently clustering performance. In this work we propose a model which extends and combines them. The proposed Mixtures of Factor Mixture Analyzers provide a unified class of dimensionally reduced mixture models which includes the previous ones as special cases and could offer a powerful tool for modelling non-Gaussian latent variables.     

Constrained clustering and Kohonen Self-Organizing Maps

The Self-Organizing Feature Maps (SOFM; Kohonen 1984) algorithm is a well-known example of unsupervised learning in connectionism and is a clustering method closely related to the k-means. Generally the data set is available before running the algorithm and the clustering problem can be approached by an inertia criterion optimization. In this paper we consider the probabilistic approach to this problem. We propose a new algorithm based on the Expectation Maximization principle (EM; Dempster, Laird, and Rubin 1977). The new method can be viewed as a Kohonen type of EM and gives a better insight into the SOFM according to constrained clustering. We perform numerical experiments and compare our results with the standard Kohonen approach.     

Unclassed matrix shading and optimal ordering in hierarchical cluster analysis

A method is presented for the graphic display of proximity matrices as a complement to the common data analysis techniques of hierarchical clustering. The procedure involves the use of computer generated shaded matrices based on unclassed choropleth mapping in conjunction with a strategy for matrix reorganization. The latter incorporates a combination of techniques for seriation and the ordering of binary trees.Partial support for this research was provided by NIJ Grant #82-IJ-CX-0019 and NSF Grant #SES82-06067. The authors wish to acknowledge the assistance of Professors L.J. Hubert, R.G. Golledge, and W.R. Tobler.     

Minkowski Generalizations of Ward’s Method in Hierarchical Clustering

In this paper, we consider several generalizations of the popular Ward’s method for agglomerative hierarchical clustering. Our work was motivated by clustering software, such as the R function hclust, which accepts a distance matrix as input and applies Ward’s definition of inter-cluster distance to produce a clustering. The standard version of Ward’s method uses squared Euclidean distance to form the distance matrix. We explore the effect on the clustering of using other definitions of distance, such as the Minkowski distance.     

Classification of Asymmetric Proximity Data

When clustering asymmetric proximity data, only the average amounts are often considered by assuming that the asymmetry is due to noise. But when the asymmetry is structural, as typically may happen for exchange flows, migration data or confusion data, this may strongly affect the search for the groups because the directions of the exchanges are ignored and not integrated in the clustering process. The clustering model proposed here relies on the decomposition of the asymmetric dissimilarity matrix into symmetric and skew-symmetric effects both decomposed in within and between cluster effects. The classification structures used here are generally based on two different partitions of the objects fitted to the symmetric and the skew-symmetric part of the data, respectively; the restricted case is also presented where the partition fits jointly both of them allowing for clusters of objects similar with respect to the average amounts and directions of the data. Parsimonious models are presented which allow for effective and simple graphical representations of the results.     

Optimization Strategies for Two-Mode Partitioning

Two-mode partitioning is a relatively new form of clustering that clusters both rows and columns of a data matrix. In this paper, we consider deterministic two-mode partitioning methods in which a criterion similar to k-means is optimized. A variety of optimization methods have been proposed for this type of problem. However, it is still unclear which method should be used, as various methods may lead to non-global optima. This paper reviews and compares several optimization methods for two-mode partitioning. Several known methods are discussed, and a new fuzzy steps method is introduced. The fuzzy steps method is based on the fuzzy c-means algorithm of Bezdek (1981) and the fuzzy steps approach of Heiser and Groenen (1997) and Groenen and Jajuga (2001). The performances of all methods are compared in a large simulation study. In our simulations, a two-mode k-means optimization method most often gives the best results. Finally, an empirical data set is used to give a practical example of two-mode partitioning. We would like to thank two anonymous referees whose comments have improved the quality of this paper. We are also grateful to Peter Verhoef for providing the data set used in this paper.     

Structural Classification Analysis of Three-Way Dissimilarity Data

The paper presents a methodology for classifying three-way dissimilarity data, which are reconstructed by a small number of consensus classifications of the objects each defined by a sum of two order constrained distance matrices, so as to identify both a partition and an indexed hierarchy. Specifically, the dissimilarity matrices are partitioned in homogeneous classes and, within each class, a partition and an indexed hierarchy are simultaneously fitted. The model proposed is mathematically formalized as a constrained mixed-integer quadratic problem to be fitted in the least-squares sense and an alternating least-squares algorithm is proposed which is computationally efficient. Two applications of the methodology are also described together with an extensive simulation to investigate the performance of the algorithm.     

A validation study of a variable weighting algorithm for cluster analysis

De Soete (1986, 1988) proposed a variable weighting procedure when Euclidean distance is used as the dissimilarity measure with an ultrametric hierarchical clustering method. The algorithm produces weighted distances which approximate ultrametric distances as closely as possible in a least squares sense. The present simulation study examined the effectiveness of the De Soete procedure for an applications problem for which it was not originally intended. That is, to determine whether or not the algorithm can be used to reduce the influence of variables which are irrelevant to the clustering present in the data. The simulation study examined the ability of the procedure to recover a variety of known underlying cluster structures. The results indicate that the algorithm is effective in identifying extraneous variables which do not contribute information about the true cluster structure. Weights near 0.0 were typically assigned to such extraneous variables. Furthermore, the variable weighting procedure was not adversely effected by the presence of other forms of error in the data. In general, it is recommended that the variable weighting procedure be used for applied analyses when Euclidean distance is employed with ultrametric hierarchical clustering methods.     

A modification of the SINDCLUS algorithm for fitting the ADCLUS and INDCLUS models

The SINDCLUS algorithm for fitting the ADCLUS and INDCLUS models deals with a parameter matrix that occurs twice in the model by considering the two occurrences as independent parameter matrices. This procedure has been justified empirically by the observation that upon convergence of the algorithm to the global optimum, the two independently treated parameter matrices turn out to be equal. In the present paper, results are presented that contradict this finding, and a modification of SINDCLUS is presented which obviates the need for independently treating two occurrences of the same parameter matrix.     

A New Representation of Interval Symbolic Data and Its Application in Dynamic Clustering

In this study, we consider the type of interval data summarizing the original samples (individuals) with classical point data. This type of interval data are termed interval symbolic data in a new research domain called, symbolic data analysis. Most of the existing research, such as the (centre, radius) and [lower boundary, upper boundary] representations, represent an interval using only the boundaries of the interval. However, these representations hold true only under the assumption that the individuals contained in the interval follow a uniform distribution. In practice, such representations may result in not only inconsistency with the facts, since the individuals are usually not uniformly distributed in many application aspects, but also information loss for not considering the point data within the intervals during the calculation. In this study, we propose a new representation of the interval symbolic data considering the point data contained in the intervals. Then we apply the city-block distance metric to the new representation and propose a dynamic clustering approach for interval symbolic data. A simulation experiment is conducted to evaluate the performance of our method. The results show that, when the individuals contained in the interval do not follow a uniform distribution, the proposed method significantly outperforms the Hausdorff and city-block distance based on traditional representation in the context of dynamic clustering. Finally, we give an application example on the automobile data set.     

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.     

Design of Blurring Mean-Shift Algorithms for Data Classification

The mean-shift algorithm is an iterative method of mode seeking and data clustering based on the kernel density estimator. The blurring mean-shift is an accelerated version which uses the original data only in the first step, then re-smoothes previous estimates. It converges to local centroids, but may suffer from problems of asymptotic bias, which fundamentally depend on the design of its smoothing components. This paper develops nearest-neighbor implementations and data-driven techniques of bandwidth selection, which enhance the clustering performance of the blurring method. These solutions can be applied to the whole class of mean-shift algorithms, including the iterative local mean method. Extended simulation experiments and applications to well known data-sets show the goodness of the blurring estimator with respect to other algorithms.     

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.     

A Survey on Feature Weighting Based K-Means Algorithms

In a real-world data set, there is always the possibility, rather high in our opinion, that different features may have different degrees of relevance. Most machine learning algorithms deal with this fact by either selecting or deselecting features in the data preprocessing phase. However, we maintain that even among relevant features there may be different degrees of relevance, and this should be taken into account during the clustering process.With over 50 years of history, K-Means is arguably the most popular partitional clustering algorithm there is. The first K-Means based clustering algorithm to compute feature weights was designed just over 30 years ago. Various such algorithms have been designed since but there has not been, to our knowledge, a survey integrating empirical evidence of cluster recovery ability, common flaws, and possible directions for future research. This paper elaborates on the concept of feature weighting and addresses these issues by critically analyzing some of the most popular, or innovative, feature weighting mechanisms based in K-Means.     

An error variance approach to two-mode hierarchical clustering

A new agglomerative method is proposed for the simultaneous hierarchical clustering of row and column elements of a two-mode data matrix. The procedure yields a nested sequence of partitions of the union of two sets of entities (modes). A two-mode cluster is defined as the union of subsets of the respective modes. At each step of the agglomerative process, the algorithm merges those clusters whose fusion results in the smallest possible increase in an internal heterogeneity measure. This measure takes into account both the variance within the respective cluster and its centroid effect defined as the squared deviation of its mean from the maximum entry in the input matrix. The procedure optionally yields an overlapping cluster solution by assigning further row and/or column elements to clusters existing at a preselected hierarchical level. Applications to real data sets drawn from consumer research concerning brand-switching behavior and from personality research concerning the interaction of behaviors and situations demonstrate the efficacy of the method at revealing the underlying two-mode similarity structure.     

Between-group analysis with heterogeneous covariance matrices: The common principal component model

Analysis of between-group differences using canonical variates assumes equality of population covariance matrices. Sometimes these matrices are sufficiently different for the null hypothesis of equality to be rejected, but there exist some common features which should be exploited in any analysis. The common principal component model is often suitable in such circumstances, and this model is shown to be appropriate in a practical example. Two methods for between-group analysis are proposed when this model replaces the equal dispersion matrix assumption. One method is by extension of the two-stage approach to canonical variate analysis using sequential principal component analyses as described by Campbell and Atchley (1981). The second method is by definition of a distance function between populations satisfying the common principal component model, followed by metric scaling of the resulting between-populations distance matrix. The two methods are compared with each other and with ordinary canonical variate analysis on the previously introduced data set.     

Model-based Clustering with Noise: Bayesian Inference and Estimation

This paper proposes a new way of overcoming the existing limitations. It generalizes the model used in the previous approaches by introducing a more comprehensive portfolio of covariance matrix structures. Further, this paper proposes a Bayesian solution in the presence of the noise in clustering problems. The performace of the proposed method is first studied by simulation; the procedure is also applied to the analysis of data concerning species of butterflies and diabetes patients.     

Joint Orthomax Rotation of the Core and Component Matrices Resulting from Three-mode Principal Components Analysis

The analysis of a three-way data set using three-mode principal components analysis yields component matrices for all three modes of the data, and a three-way array called the core, which relates the components for the different modes to each other. To exploit rotational freedom in the model, one may rotate the core array (over all three modes) to an optimally simple form, for instance by three-mode orthomax rotation. However, such a rotation of the core may inadvertently detract from the simplicity of the component matrices. One remedy is to rotate the core only over those modes in which no simple solution for the component matrices is desired or available, but this approach may in turn reduce the simplicity of the core to an unacceptable extent. In the present paper, a general approach is developed, in which a criterion is optimized that not only takes into account the simplicity of the core, but also, to any desired degree, the simplicity of the component matrices. This method (in contrast to methods for either core or component matrix rotation) can be used to find solutions in which the core and the component matrices are all reasonably simple.     

Block-Relaxation Approaches for Fitting the INDCLUS Model

A well-known clustering model to represent I?×?I?×?J data blocks, the J frontal slices of which consist of I?×?I object by object similarity matrices, is the INDCLUS model. This model implies a grouping of the I objects into a prespecified number of overlapping clusters, with each cluster having a slice-specific positive weight. An INDCLUS model is fitted to a given data set by means of minimizing a least squares loss function. The minimization of this loss function has appeared to be a difficult problem for which several algorithmic strategies have been proposed. At present, the best available option seems to be the SYMPRES algorithm, which minimizes the loss function by means of a block-relaxation algorithm. Yet, SYMPRES is conjectured to suffer from a severe local optima problem. As a way out, based on theoretical results with respect to optimally designing block-relaxation algorithms, five alternative block-relaxation algorithms are proposed. In a simulation study it appears that the alternative algorithms with overlapping parameter subsets perform best and clearly outperform SYMPRES in terms of optimization performance and cluster recovery.