J. Banfield and A. Raftery. Model-based gaussian and non-gaussian clustering. Biometrics, 49:803--821, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....percentage assignment problems. Finally, we do a postprocessing step of ML clustering in situations where strict balancing is not required. There are several motivations behind our approach. First, probabilistic model based clustering provides a principled and general approach to clustering [5]. For example, the number of clusters may be estimated using Bayesian model selection, though this is not done in this paper. Second, the two step view of partitional clustering is natural and has been discussed by Kalton, k l 2 o 3 o n o W G Figure 1: A bipartite graph view of model based ....

J. D. Banfield and A. E. Raftery. Model-based Gaussian and non-Gaussian clustering. Biometrics, 49:803-- 821, 1993.

....2000] proposed OPOSSUM, a similarity based clustering approach based on constrained,weighted graph partitioning. OPOSSUM is based on Jaccard Similarity and is particularly attuned to real life market baskets, characterized by high dimensional sparse customer product matrices. Ban eld and Raftery[Ban eld and Raftery, 1993] described the approach based on mixture model for clustering. As pointed out in[Fasulo, 1999] there are several problems with this approach. First the approach does not focus on eciency. Second, a large degree of manual intervention is required. Last but not the least, mixture models rely on the ....

J. Ban eld and A. Raftery, \Model-based Gaussian and nonGaussian clustering," Biometrics, 49:803-821, 1993.

....under sample dense regions and over sample sparse regions of the data. A memory ecient single pass algorithm is proposed that approximates density biased sampling. An excellent detailed exposition of algorithms in [51] 16] and [44] can be found in [84] An excellent survey of the algorithms in [48, 2, 11, 13] is given in [41] 7 Mining the web The challenge in mining the web for useful information is the huge size and unstructured placement of data. Search engines aim to search the web for a speci c topic and give the most important web pages on this topic to the user. A considerable amount of ....

J. Ban eld and A. Raftery. Model-based gaussian and non-gaussian clustering. Biometrics, 49:803-821, 1993.

....based on empirical arguments. The last decade, however, there is an increased interest in model based methodologies, which allow for clustering procedures based on statistical arguments and methodologies. The majority of such procedures are based on the multivariate normal distribution, see [3, 16] and others. The central idea of such models is the use of finite mixtures of multivariate normal distributions. In general, in model based clustering, the observed data are assumed to arise from a number of apriori unknown segments that are mixed in unknown proportions. The objective is then to ....

Banfield, J.D., and Raftery, A.E., Model-based Gaussian and non-Gaussian clustering, in: Biometrics, Vol. 49, pp. 803-821.

....model for which the merge leads to the smallest decrease in log likelihood. As we are not maximizing the mixture likelihood (1) use of the BIC for estimating the number of components is not justified; instead we estimate the number of groups by maximizing the Approximate Weight of Evidence [3]: G = argmax G (2 2r (3 2 log(n) 3) The hierarchical approach can be expected to work well if the groups are clearly separated. Unfortunately, straightforward implementation of hierarchical model based clustering leads to an O(n ) algorithm. In contrast, the algorithms presented in ....

J. D. Banfield and A. Raftery. Model-based Gaussian and non-Gaussian clustering. Biometrics, 49:803--821, 1993.

....that integrate segment visual similarity and duration in a joint model, rely on strong temporal adjacency, and account for the fact that clusters are composed of few segments. One such method is described in the next section. V. Our Approach HAC algorithms can be based on probability models [2]. We propose to build models of visual similarity, duration, and temporal adjacency de ned on pairs of segments. A HAC algorithm can be thought of as a sequential binary classi er, which at each step decides whether a pair of segments should be merged. The formulation as a two class classi ....

J. D. Ban eld and A. E. Raftery, \Model-based Gaussian and Non-Gaussian Clustering," Biometrics, Vol. 49, No. 3, pp. 803-821, Sept. 1993.

....common covariance is assumed, then N(k) k 1) kd d(d 1) 2. Several EM based approaches also use approximate versions of the Bayes factor (the correct Bayesian model selection criterion [9] such as the evidencebased Bayesian (EBB) criterion [25] the approximate weight of evidence (AWE) [1], and Schwarz s Bayesian inference criterion (BIC) 5] Although derived in a different framework, BIC formally coincides with MDL and is also given by Eq. 11) The minimum message length (MML) criterion [20] Akaike s information criterion (AIC) 35] and Bezdek s partition coefficient (PC) 3] ....

....required) To compare MMDL versus MDL BIC, wehave performed experiments on real and synthetic data. All the experiments confirm that MMDL allows a better fit to the observed data. Finally,we mention the parameterization of the covariance matrices (based on eigen decomposition) introduced in [1] (see also [4] That parameterization allows taking selected characteristics of the components to be common (for example, same shape, arbitrary orientation) MMDL can also be used to perform model selection among the options provided by that approach.Thegoalistosimultaneously choose the number ....

J. Banfield and A. Raftery. Model-based Gaussian and non-Gaussian clustering. Biometrics, 49:803--821, 1993.

....(iii) account for the fact that clusters are likely to contain only a few elements. These observations motivated the methodology described in the next section. 5 Our Approach: Probabilistic Hierarchical Clustering of Home Video Segments HAC algorithms can be based on probability models [28] [3], 4] To capture the inherent characteristics of home video clusters, we propose to build statistical models of visual similarity and temporal duration and adjacency de ned on pairs of segments in a HAC framework. In particular, a HAC algorithm can be thought of as a sequential binary classi er, ....

J. D. Baneld and A. E. Raftery, Model-based Gaussian and Non-Gaussian Clustering, Biometrics, Vol. 49, No. 3, pp. 803-821, Sept. 1993.

.... [Sch78] Ris89] Minimum Message Length (MML) criterion [WF87] WD94] Bayesian Information Criterion (BIC) Sch78] FR98] tkaike s Information Criterion (AIC) Boz83] Non coding Information Theoretic Criterion (ICOMP) Boz94] Approximate FFeight of Evidence (AWE) criterion [BF93]. Bayes Factors [KR95] and others [Bock96] All these criteria are expressed through combinations of loglikelihood L, number of clusters k, number of parameters per cluster, total number of estimated parameters p, and different flavors of Fisher information matrix. For example, 28 MDL(k) ....

Banfield, J. and Raftery, A. Model-based Gaussian and non-Gaussian clustering. Biometrics, 49, 803-821, 1993.

....few exceptions [18] connected operators have been formulated in 2 DANIEL GATICA PEREZ et al. deterministic terms. In this paper, we argue that the introduction of probability models in the design of connected operators is useful both to generalize existing formulations and to define new designs [1], 3] 6] In particular, we introduce an operator based on the formulation of hierarchical clustering as a sequential binary classification process, and on the development of statistical models of visual similarity among image regions. Its performance is illustrated on image collections and ....

....An alternative to encode the a priori knowledge of the problem consists on the use of probability models [6] 3. A probabilistic view of attribute based connected operators Hierarchical agglomerative clustering algorithms based on probability models are increasingly used in pattern recognition [1], 3] 6] The simplest algorithm Fig. 1. a) Giraffe. b c) Filtered images with area connected operator. All components with area less than (b) 100 and (c) 1000 pixels, are removed. The merging order depends on both area (minimum size) and intensity (color difference) attributes. consists of a ....

J.D. Banfield and A. Raftery, "Model-based Gaussian and Non-Gaussian Clustering," Biometrics, Vol. 49, No. 3, pp. 803-821, Sept. 1993.

....covariance is assumed, then N(k) k Gamma 1) kd d(d 1) 2. Several EM based approaches also use approximate versions of the Bayes factor (the correct Bayesian model selection criterion [9] such as the evidencebased Bayesian (EBB) criterion [25] the approximate weight of evidence (AWE) [1], and Schwarz s Bayesian inference criterion (BIC) 5] Although derived in a different framework, BIC formally coincides with MDL and is also given by Eq. 11) The minimum message length (MML) criterion [20] Akaike s information criterion (AIC) 35] and Bezdek s partition coefficient (PC) 3] ....

....required) To compare MMDL versus MDL BIC, we have performed experiments on real and synthetic data. All the experiments confirm that MMDL allows a better fit to the observed data. Finally, we mention the parameterization of the covariance matrices (based on eigen decomposition) introduced in [1] (see also [4] That parameterization allows taking selected characteristics of the components to be common (for example, same shape, arbitrary orientation) MMDL can also be used to perform model selection among the options provided by that approach. The goal is to simultaneously choose the ....

J. Banfield and A. Raftery. Model-based Gaussian and non-Gaussian clustering. Biometrics, 49:803--821, 1993.

....data clustering, Bayesian networks, Bayesian multinets, constructive induction, EM algorithm, BC EM method. 1 Introduction One of the main problems that arises in a great variety of elds, including pattern recognition, machine learning and statistics, is the so called data clustering problem [1, 3, 7, 14, 15, 22, 25]. Data clustering can be viewed as a data partitioning problem, where we partition data into di erent clusters based on a quality or similarity criterion (e.g. as in K Means [30] Alternatively, data clustering is one way of representing the joint probability distribution of a database. We ....

....to a given database as incomplete when the classi cation is not given. Parameter estimation and model comparison in classical and Bayesian statistics provide a solution to the data clustering problem. The most frequently used approaches include mixture density models (e.g. Gaussian mixture models [3]) and Bayesian networks (e.g. AutoClass [8] 1 We aim to automatically recover the joint probability distribution from a given incomplete database by learning recursive Bayesian multinets (RBMNs) Roughly, a recursive Bayesian multinet is a decision tree [4, 44] where each decision path (i.e. ....

Baneld, J., & Raftery, A. (1993). Model-based Gaussian and non-Gaussian Clustering. Biometrics, 49, 803-821.

....database. Keywords: Data clustering, conditional Gaussian networks, feature selection, edge exclusion tests. 1 Introduction One of the basic problems that arises in a great variety of elds, including pattern recognition, machine learning and statistics, is the so called data clustering problem [1, 2, 10, 11, 18, 22]. Despite the di erent interpretations and expectations it gives rise to, the generic data clustering problem involves the assumption that, in addition to the observed variables (also referred to as predictive attributes or, simply, features) there is a hidden variable. This last unobserved ....

J.D. Baneld and A.E. Raftery, \Model-Based Gaussian and Non-Gaussian Clustering, " Biometrics, vol. 49, pp. 803-821, 1993.

....clustering, conditional Gaussian networks, naive Bayes models, tree augmented naive Bayes models, extended naive Bayes models. 1 Introduction A basic problem that arises in a variety of elds, such as pattern recognition, machine learning, and statistics, is the so called data clustering problem [1, 2, 7, 8, 16, 18]. From the point of view adopted in this paper, the data clustering problem may be de ned as the inference of a generalized joint probability distribution from a database. We assume that, in addition to the observed random variables or predictive attributes, there is a hidden random variable. This ....

Baneld, J. and Raftery, A., Model-based Gaussian and non-Gaussian Clustering, Biometrics 49, 803-821, 1993.

....as x q = arg max fxj f 0 (x) 0; f 00 (x) 0g f 00 (x) where f 0 and f 00 are the first and second derivatives of f . That is, we can take the point at which the smooth density estimate most rapidly flattens out into a tail. We use a model based clustering method (see [4]) to cluster the intervals containing perturbations. This clustering method is well suited for overlapping ellipsoidal clusters of varying sizes and orientations. We use all five variables in clustering the intervals into four clusters, which are represented by red, magenta, green, and cyan in ....

Jeffrey D. Banfield and Adrian E. Raftery. Model-based gaussian and non-gaussian clustering. Biometrics, 49:803--821, 1993.

....with Noise (MBGCN) as implemented in the 8 SIMULATIONS 28 p = 0 p = 1 p = 2 p = 3 n = 25 30 30 too large n = 50 10 30 30 too large n = 100 6 10 15 30 n = 200 6 6 10 30 n = 300 10 Table 8. 2: Smallest c from simulations with n r 1:5 software package mclust based on the work of Banfield and Raftery (1993). A current version is treated in DasGupta and Raftery (1998) They assume the points z i = x i ; y i ) i = 1; n; as i.i.d. distributed according to L(z i ) ffl 0 U C s X j=1 ffl j N a j ; Sigma j ; where U C denotes the uniform distribution on some convex set C, a j 2 IR p 1 ....

.... of eigenvectors, A j = diag(1; ff 2j ; ff (p 1)j ) The software mclust computes Maximum Likelihood estimators using the EM algorithm for the parameters ffl 0 ; ffl 1 ; a 1 ; Sigma 1 ) ffl s ; Sigma s ) from starting values given by some hierarchical model based method from Banfield and Raftery (1993). The component memberships of the points can be estimated by analogy to the MLCLR procedure. The Bayesian Information Criterion BIC (Schwarz 1978) was used for the estimation of the number of components s. The form of the covariance matrices may be restricted. DasGupta and Raftery propose to ....

Banfield, J. D. and Raftery, A. E. (1993): Model-Based Gaussian and Non-Gaussian Clustering, Biometrics 49, p. 803-821.

.... matrices are restricted to being equal, or even diagonal as in the AutoClass program of Cheeseman and Stutz (1996) Less restrictive constraints can be imposed by a reparameterization of the component covariance matrices in terms of their eigenvalue decompo2 sitions as, for example, in Banfield and Raftery (1993). In the latest version of AutoClass (http: ic.arc.nasa.gov ic projects bayes group autoclass autoclass c program.html) the covariance matrices are unrestricted In other software for the fitting of mixture models, there are MCLUST and EMCLUST which are a suite of S PLUS functions for ....

.... the covariance matrices are unrestricted In other software for the fitting of mixture models, there are MCLUST and EMCLUST which are a suite of S PLUS functions for hierarchical clustering EM, and BIC, respectively based on parameterized Gaussian mixture models; see Banfield and Raftery (1993), Byers and Raftery (1998) Campbell et al. 1998) DasGupta and Raftery (1998) and Fraley and Raftery (1998) MCLUST (http: stat.washington.edu fraley software.shtml) and EMCLUST (http: stat.washington.edu fraley software.shtml) are written in FORTRAN with an interface to the S PLUS ....

Banfield, J.D., and Raftery, A. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics 49, 803--821.

....log p i Yj b (k) j P (k) where P (k) in an increasing function penalizing higher values of k. Cost functions of this type have been proposed under a Bayesian model selection framework. Examples are the evidence based Bayesian (EBB) criterion [17] the approximate weight of evidence (AWE) [18], and Schwarz s Bayesian inference criterion (BIC) 19] 20] 8 Other criteria include Rissanen s minimum description length (MDL) 21] which formally coincides with BIC) the minimum message length (MML) 22] 23] 24] Akaike s information criterion (AIC) 25] and Bezdek s partition ....

J. Banfield and A. Raftery, "Model-based Gaussian and non-Gaussian clustering," Biometrics, vol. 49, pp. 803--821, 1993.

....It can be useful to impose constraints on the model, for instance by assuming that all variance matrices are identical, Sigma 1 = Sigma K . Constraints on the mixture model can be derived from parameterizing the matrix Sigma k of a component in terms of its eigenvalue decomposition [1] [11] 2] Sigma k = k D k A k D 0 k where k = j Sigma k j 1=d , D k is the matrix of eigenvectors of Sigma k and A k is a diagonal matrix, such that jA k j = 1, with the normalized eigenvalues of Sigma k on the diagonal in a decreasing order. The parameter k determines the volume of ....

....00 j ) m X i=1 K X k=1 z ik ln(p k OE(x i ja k ) n X i=m 1 K X k=1 z ik ln(p k OE(x i ja k ) Application with XEMgaus . Situation where the three first elements of the partition z are fixed (we assumed here that n = 5) bestpmS,besttz] XEMgaus(x, K ,2, z ,z, zfix ,[1 1 1 0 0] ) 9 5.2 Adaptation to choose models Criteria ICL, BIC or ICL can be used in exactly the same way that described in the clustering context, except that now likelihood and classification likelihood are replaced by their new respective expression. But note that new information is now available ....

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J.D. Banfield and A.E. Raftery. Model-based Gaussian and non-Gaussian Clustering. Biometrics, 49:803--821, 1993.

.... for multivariate data analysis and statistical pattern recognition (see for instance McLachlan 1992 and Ripley 1996) Recently several authors have exploited the eigenvalue decomposition of the group variance matrices in Gaussian mixtures to propose numerous and powerful models for clustering (Banfield and Raftery 1993, Celeux and Govaert 1995, Bensmail, Celeux, Raftery and Robert 1997) and discriminant analysis Flury, Schmid and Narayanan 1993, Bensmail and Celeux 1996) This parametrization of variance matrices of the mixture components provides a general and flexible framework to give raise to efficient, ....

....the classification matrix t from the maximum likelihood estimates of the mixture parameter (p k ; a k )k = 1; K and CLM(M;K) L(M;K) Gamma LP (M;K) where LP (M;K) is derived from the classification maximum likelihood estimates of the mixture parameters. Finally, we want to mention that Banfield and Raftery (1993) have suggested Bayesian solution to the choice of the number of clusters based on an approximation of the integrated classification likelihood, in the same spirit of BIC. Their approximation leads to the so called approximate weight of evidence of the form AWE(M;K) Gamma2C (M;K) 2(M; K) 3 ....

Banfield, J. D. and Raftery, A. E. (1993). Model-based Gaussian and non Gaussian clustering. Biometrics, 49, 803-821.

....defined in terms of the number of observations. This work related to problem spaces of dimensionality two, with generalization possible to three dimensional spaces [4] It may be helpful to distinguish this work from clustering understood as mixture distribution modelling. Banfield and Raftery [5], for example, discuss algorithms for optimal cluster modelling and fitting. Murtagh and Starck s work [2] on O(1) clustering algorithms is based on noise modelling. It can accurately be defined as data background modelling. In this article we describe an effective approach for clustering ....

Banfield, J. D. and Raftery, A. E. (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics, 49, 803-- 821.

.... case of multivariate data, particularly for the covariance matrices which might benefit from an eigen decomposition, enabling the specification of priors which favour components which are a similar size, shape, orientation, or some combination of these three (such a decomposition is considered by Banfield and Raftery, 1993). 37 2.4.2 Full conditional posterior distributions The full conditional posterior distributions of the parameters and hyperparameters, for use in the Gibbs sampler, are as follows: p(z j = i j Delta Delta Delta ) i N r (x j ; i ; Sigma i ) 2.45) fi j Delta Delta Delta W r i 2g ....

Banfield, J. D. and Raftery, A. E. (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics, 49, 803--821.

....method. Keywords: K Means algorithm, K Means initialization, partitional clustering, Genetic Algorithms. 1 Introduction One of the basic problems that arises in a great variety of fields, including pattern recognition, machine learning and statistics, is the so called clustering problem [1, 2, 5, 8, 14, 18]. The fundamental data clustering problem may be defined as discovering groups in data or grouping similar objects together. Each of these groups is called a cluster, a region in which the density of objects is locally higher than in other regions. In this paper, data clustering is viewed as a ....

Banfield, J. and Raftery, A. (1993). Model-based Gaussian and non-Gaussian Clustering. Biometrics, 49, 803-821.

....clustering, Bayesian networks, EM algorithm, Bayesian Structural EM algorithm, Bound and Collapse method. 1 Introduction One of the basic problems that arises in a great variety of fields, including pattern recognition, machine learning and statistics, is the so called data clustering problem [1,5,6,10,12]. From the point of view adopted in this paper, the data clustering problem may be defined as the inference of a probability distribution for a database. We assume that, in addition to the observed variables, there is a hidden variable. This last unobserved variable would reflect the cluster ....

J. Banfield and A. Raftery, Model-based Gaussian and non-Gaussian Clustering, Biometrics 49 (1993) 803-821. 11

....performance in terms of identifying the main features of interest in the image. We chose to study three cases, K = 3; 4 and 10. Model based statistical methods for clustering multivariate observations are very exible and have been applied successfully in many domains of practical interest ( e.g. [1, 10, 11, 9]) However for complex data such as those associated with tissue segmentation in medical imaging, these methods sometimes produce rather fragmented results that do not correspond directly to a meaningful classi cation, because they do not take into account spatial location and dependence. For ....

.... algorithm to estimate the model parameters (e.g. 20] 4] For multidimensional images, the relevant distributions are multivariate, and there has been much recent progress on estimation methods that combine agglomerative hierarchical clustering methods based on maximum classi cation likelihood ([1]; 8] with the EM algorithm ( 5] 10] When the number of pixels is very large, the method cannot be used without modi cation because of its computational requirements. One possible approach is to take a sample of the pixels rst, as was done for an MRI brain scan in [1] Initialization via ....

[Article contains additional citation context not shown here]

J. D. Baneld and A. E. Raftery. Model-based Gaussian and Non-Gaussian Clustering. Biometrics, 49:803821, 1993.

....tissue classification in biomedical images, identification of objects in astronomy, analysis of images from molecular spectroscopy, and recognition and classification of surface defects in manufactured products. Agglomerative hierarchical clustering (Murtagh and Raftery [8] Banfield and Raftery [1]) the EM algorithm and related iterative techniques (Celeux and Govaert [3] or some combination of these (Dasgupta and Raftery [4] are effective computational techniques for obtaining partitions from these models. The subject of efficient computation in this context has however received little ....

....to maximize the likelihood L(x; fl) i=1 f fl i (x; 1) Our focus is on the case where f k (x; is multivariate normal (Gaussian) with mean vector k and variance matrix Sigma k . The overall approach is much more general and is not restricted to multivariate normal distributions [1]. However, experience to date suggests that clustering based on the multivariate normal distribution is useful in a great many situations of interest ( 8] 1] 9] 3] 4] When f k (x; is multivariate normal, the likelihood (1) has the form L(x; 1 ; G ; Sigma 1 ; Sigma ....

[Article contains additional citation context not shown here]

J. D. Banfield and A. E. Raftery. Model-based Gaussian and non-Gaussian clustering. Biometrics, 49:803--821, 1993.

....algorithms. In particular, the model based approach assumes that the data is generated by a finite mixture of underlying probability distributions such as multivariate normal distributions. The Gaussian mixture model has been shown to be a powerful tool for many applications (for example Banfield and Raftery 1993, Celeux and Govaert 1993, McLachlan and Basford 1988) With the underlying probability model, the problems of determining the number of clusters and of choosing an appropriate clustering method become statistical model choice problems (Dasgupta and Raftery 1998, Fraley and Raftery 1998, Fraley ....

....normal distribution with parameters k (mean vector) and k (covariance matrix) f k (y i j k ; k ) expf 1 2 (y i k ) T 1 k (y i k )g q det(2 k ) 2) Geometric features (shape, volume, orientation) of each component k are determined by the covariance matrix k . Banfield and Raftery (1993) proposed a general framework for representing the covariance matrix in terms of its eigenvalue decomposition k = k D k A k D T k ; 3) 2 where D k is the orthogonal matrix of eigenvectors, A k is a diagonal matrix whose elements are proportional to the eigenvalues of k , and k is a ....

Banfield, J. D. and A. E. Raftery (1993). Model-based gaussian and non-gaussian clustering. Biometrics 49, 803--821.

....algorithms. In particular, the model based approach assumes that the data is generated by a finite mixture of underlying probability distributions such as multivariate normal distributions. The Gaussian mixture model has been shown to be a powerful tool for many applications (for example Banfield and Raftery 1993, Celeux and Govaert 1993, McLachlan and Basford 1988) With the underlying probability model, the problems of determining the number of clusters and of choosing an appropriate clustering method become statistical model choice problems (Dasgupta and Raftery 1998, Fraley and Raftery 1998, Fraley ....

....distribution with parameters 7 (mean vector) and 8 ) covariance matrix) 7 . 8 ) 9 ; A CB D , B 7 . E 8GF ) B 7 . H I J : K L M 8 ) 2) Geometric features (shape, volume, orientation) of each component are determined by the covariance matrix 8 ) Banfield and Raftery (1993) proposed a general framework for representing the covariance matrix in terms of its eigenvalue decomposition 8 ) ON) PQ) RS) E ) 3) 2 where PQ) is the orthogonal matrix of eigenvectors, R ) is a diagonal matrix whose elements are proportional to the eigenvalues of 8 ) and ) is a ....

Banfield, J. D. and A. E. Raftery (1993). Model-based gaussian and non-gaussian clustering.

....algorithms. In particular, the model based approach assumes that the data is generated by a finite mixture of underlying probabilitydistributions such as multivariate normal distributions. The Gaussian mixture model has been shown to be a powerful tool for many applications (for example, (Banfield and Raftery, 1993), Celeux and Govaert, 1993) McLachlan and Basford, 1988) With the underlying probability model, the problems of determining the number of clusters and of choosing an appropriate clustering method become statistical model choice problems ( Dasgupta and Raftery, 1998) Fraley and Raftery, ....

....normal distribution with parameters #k (mean vector) and # k (covariance matrix) f k #y i j# k ; # k #= expf, 1 2 #y i , # k # T # ,1 k #y i , # k #g p det#2## k # : 2) Geometric features (shape, volume, orientation) of each component k are determined by the covariance matrix # k . (Banfield and Raftery, 1993) proposed a general framework for exploiting the representation of the covariance matrix in terms of its eigenvalue decomposition # k = # k D k A k D T k ; 3) where D k is the orthogonal matrix of eigenvectors, A k is a diagonal matrix whose elements are proportional to the eigenvalues of # k ....

Banfield, J. D. and Raftery, A. E. (1993) Model-basedGaussianand non-Gaussian clustering. Biometrics, 49, 803--821.

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Baneld, J. D. and Raftery, A. E. (1993). Model-Based Gaussian and Non-Gaussian Clustering. Biometrics, 49, 803-821.

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Baneld, J. D. and A. E. Raftery (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics 49 (3), 803-822.

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J. Banfield and A. Raftery. Model-based gaussian and non-gaussian clustering. Biometrics, 49:803--821, 1993.

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Jeffrey D. Banfield, Adrian E. Raftery. Modelbased Gaussian and Non-Gaussian Clustering. Biometrics 49, September 1993.

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Banfield, J. and Raftery, A. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics, 49:803--821.

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J. D. Banfield and A. E. Raftery. Model-based Gaussian and non-Gaussian clustering. Biometrics, 49:803--821, 1993.

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J. Banfield & A. Raftery. Model-based Gaussian and non-Gaussian clustering. Biomentrics, vol. 49, pages 803--821, 1993.

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Banfield, J. D. and Raftery, A. E. (1993). Model-based Gaussian and non-gaussian clustering.

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Banfield, J.D., Raftery, A.E.: Model-based gaussian and non-gaussian clustering. Biometrics 49 (1993) 803--821

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J. D. Banfield and A. E. Raftery. Model-based gaussian and non-gaussian clustering. Biometrics, 49(3):803--821, 1993.

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J. D. Banfield and A. E. Raftery, "Model-based Gaussian and nonGaussian clustering," Biometrics, vol. 49, pp. 803--821, 1993.

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Banfield, J. D. and Rafferty, A. E. (1993). Model-Based Gaussian and NonGaussian Clustering, Biometrics 49, 803-821.

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J. Banfield and A. Raftery. Model-based gaussian and non-gaussian clustering. Biometrics, 49:803--821, 1993.

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J. D. Banfield and A. E. Raftery, "Model-based Gaussian and non-Gaussian clustering," Biometrics, vol. 49, no. 3, pp. 803--821, Sept. 1993.

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J. D. Banfield and A. E. Raftery. Model-based Gaussian and non-Gaussian clustering. Biometrics, 49:803--821, September 1993.

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Banfield, J. D. e Raftery, A. E. (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics, 49, 803-822.

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Ban eld, J. D. and Raftery, A. E. (1993). Model-based Gaussian and nonGaussian clustering. Biometrics, 49, 803-821.

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Banfield, J. D. and Raftery, A. E. (1993). Model-Based Gaussian and Non-Gaussian Clustering. Biometrics, 49, 803-821.

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Banfield J. and Raftery A., "Model-based Gaussian and Non-Gaussian Clustering", Biometrics, Vol. 49:803-821, pp. 15-34.

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J.D. Banfield and A.E. Raftery. Model-Based Gaussian and Non-Gaussian Clustering. Biometrics, 49:803--821, 1993.

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Baneld, J.D. and Raftery, A. E. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics, 49, 803-821.

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