Celeux G., Govaert, G. : A Classification EM algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14(3):315--332. (1992)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....between modeling p(x) and p(c x) We present two alternative forms for the model of p(x) The first defines piece wise for the Voronoi regions as (unnormalized) Gaussians: For x V j , 1 e #VQ #x m , 6) where #VQ 0. The model is also interpretable as a classification mixture [4]. Despite the piecewise definition, the density is everywhere continuous with respect to x, for the borders of Voronoi regions are always half way between the cluster prototypes. If the normalization factor Z( m j is interpreted as a prior, the model for p(x) appears in the total likelihood as ....

G. Celeux and G. Govaert, "A Classification EM algorithm for clustering and two stochastic versions," Computational Statistics & Data Analysis, vol. 14, pp. 315--332, 1992.

.... and Raftery 1998, Fraley and Raftery 1998) The classical iterative k means clustering algorithm, first proposed as a heuristic clustering algorithm, has been shown to be closely related to model based clustering using the equal volume spherical model (EI) as computed by the EM algorithm (Celeux and Govaert 1992). K means has 3 been successfully used for a wide variety of clustering tasks, including clustering of gene expression data. This is not surprising, from the model based perspective, given k means interpretation as an approximate estimation method for a parsimonious model of simple independent ....

Celeux, G. and G. Govaert (1992). A classification em algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis 14, 315--332.

.... and Raftery 1998, Fraley and Raftery 1998) The classical iterative k means clustering algorithm, first proposed as a heuristic clustering algorithm, has been shown to be closely related to model based clustering using the equal volume spherical model (EI) as computed by the EM algorithm (Celeux and Govaert 1992). K means has 3 been successfully used for a wide variety of clustering tasks, including clustering of gene expression data. This is not surprising, from the model based perspective, given k means interpretation as an approximate estimation method for a parsimonious model of simple independent ....

Celeux, G. and G. Govaert (1992). A classification em algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis 14, 315--332.

.... Raftery, 1998) Fraley and Raftery, 1998) The classical iterative k means clustering algorithm, first proposed as a heuristic clustering algorithm, has been shown to be very closely related to model based clustering using the equal volume spherical model (EI) as computed by the EM algorithm (Celeux and Govaert, 1992). K means has been successfully used for a wide variety of clustering tasks, including clustering of gene expression data. This is not surprising, given k means interpretation as a parsimoniousmodel of simple independent Gaussians, which is adequate to describe data arising in many contexts. ....

Celeux, G. and Govaert, G. (1992) A classification em algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14, 315--332.

....either a density estimation [13] or a clustering [6] or a discriminant [15] analysis point of view. Estimation of the mixture parameters is performed either through maximum likelihood via the EM [14] or the SEM [9] algorithms or through classification maximum likelihood via the CEM algorithm [10]. These three algorithms can be chained (e.g. CEM then EM with results of CEM) to use advantages of each of them in the estimation process. It is possible to constrain some parameters. For example, a partition may be totally (discriminant analysis situation) or partially fixed, centers may be ....

....z) Pr(x; zj ) n X i=1 K X k=1 z ik ln(p k OE(x i ja k ) Maximization is now performed on the couple ( z) So, we estimate both and z in one exercise. But it can be shown that the CML estimator of z corresponds to the MAP of the conditional probabilities of the CML estimator of [10]. This maximization can be performed with the CEM algorithm. 6 The CEM algorithm The CEM algorithm [10] is also a variant of EM. After the E step, a classification step (C step) is added. It gives a partition z m from the MAP of the probabilities t m . Then the M step uses the partition z ....

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G. Celeux and G. Govaert. A Classification EM Algorithm for Clustering and Two Stochastic Versions. Computational Statistics & Data Analysis, 14:315--332, 1992.

....structure for estimating the parameters. CEM uses classification instead of the probability matrix. Similarly to k means, CEM forms the classification C (s 1) using the parameters (s) Theta (s) and estimates the new parameters (s 1) Theta (s 1) using the classification C (s) [3]. 2.3 Similarity of Classifications Suppose that a set X t is classified several times with different initial values of the EM algorithm. This yields in general a set of different local optima. We are interested in the differences between these results at the classification level, namely the ....

Celeux, G. and Govaert, G.: 1992, "A Classification EM algorithm for clustering and two stochastic versions", Computational Statistics and Data Analysis, 14, 315-332.

....Expectation part which calculates the likelihood of the model, and the Maximization part which maximizes the likelihood of the data with respect to the centers and radii. A recent experimental investigation into several clustering algorithms [18] has shown that the EM approach to clustering [17] [19] outperforms naive center initialization by using random perturbations or using the output of a hierarchical agglomerative clustering [20] A refinement of the random initialization improved the performance of K Means clustering [21] They used several small subsamples of the Training Set, and ....

G. Celeux and G. Govart, "A classification em algorithm for clustering and two stochasitc versions," Computational statistics and data analysis, vol. 14, pp. 315--332, 1992.

....for clustering presented above have been studied. These include the stochastic EM or SEM algorithm (Broniatowski, Celeux and Diebolt [18] Celeux and Diebolt [22] in which the z ik are simulated rather than estimated in the E step, and the classification EM or CEM algorithm (Celeux and Govaert [23]) which converts the z ik from the E step to a discrete classification before performing the Mstep. The standard k means algorithm can be shown to be a version of the CEM algorithm corresponding to the uniform spherical Gaussian model # k = #I [23] 5 initialize z ik (this can be from a ....

....EM or CEM algorithm (Celeux and Govaert [23] which converts the z ik from the E step to a discrete classification before performing the Mstep. The standard k means algorithm can be shown to be a version of the CEM algorithm corresponding to the uniform spherical Gaussian model # k = #I [23]. 5 initialize z ik (this can be from a discrete classification (5) repeat M step: maximize (6) given z ik (f k as in (3) n k # P n i=1 z ik # k n k n k P n i=1 z ik x i n k # k : depends on the model see Celeux and Govaert [24] E step: compute z ik given the parameter ....

G. Celeux and G. Govaert. A classification EM algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14:315--332, 1992.

....through the Danish Computational Neural Network Center (connect) and the THOR Center for Neuroinformatics. APPENDIX A: INFORMATION CRITERIA FOR K MEANS From a statistical point of view, the K means algorithm is a particular instance of the Classification Expectation Maximisation algorithm (see Celeux and Govaert, 1992, for definition and theoretical results) for a Gaussian mixture model with equal mixture weights and equal isotropic variances. The underlying density, for a data point u j 2 IR Q , is expressed as: P (u j jM ; oe 2 ) 1 K K X k=1 1 p 2 oe 2 exp Gamma 1 2 (u j Gamma k ) 2 ....

Celeux, G. and Govaert, G. (1992). A Classification EM algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14:315--332.

....The decrease occurs because the Bayesian criterion has a built in penalty for complexity. We examine several algorithms for learning the parameters of a given model structure: the Expectation Maximization (EM) algorithm (Dempster, Laird, Rubin, 1977) the Classification EM (CEM) algorithm (Celeux Govaert, 1992), and model based hierarchical agglomerative clustering (HAC) e.g. Banfield Raftery, 1993) Sometimes, we shall refer to these parameter learning algorithms simply as clustering algorithms. The EM algorithm is iterative, consisting of two alternating steps: the Expectation (E) step and the ....

Celeux, G. and Govart, G. (1992). A classification EM algorithm for clustering and two stochastic versions. Computational statistics and data anlaysis, 14:315--332.

....having a certain number of components is applied to a mixture in which there are actually fewer groups, then it may fail due to ill conditioning. A number of variants of the EM algorithm for clustering presented above have been studied. The classification EM or CEM algorithm (Celeux and Govaert [22]) converts the z ik from the E step to a discrete classification before performing the M step. The standard k means algorithm can be shown to be a version of the CEM algorithm corresponding to the uniform spherical Gaussian model Sigma k = I [22] 2.4 Bayesian Model Selection in Clustering One ....

....EM or CEM algorithm (Celeux and Govaert [22] converts the z ik from the E step to a discrete classification before performing the M step. The standard k means algorithm can be shown to be a version of the CEM algorithm corresponding to the uniform spherical Gaussian model Sigma k = I [22]. 2.4 Bayesian Model Selection in Clustering One advantage of the mixture model approach to clustering is that it allows the use of approximate Bayes factors to compare models. This gives a means of selecting not only the parameterization of the model (and hence the clustering method) but also ....

Celeux, G. and Govaert, G. (1992). A Classification EM Algorithm for Clustering and two stochastic versions. Computational Statistics and Data Analysis, 14, 315--332.

.... ) log f i (y i j z i ; and fi (q) arg max fi X i2S X z i P z (q) z i j y; Psi (q Gamma1) log P z (q) z i j fi) 21) 5 Related algorithms In place of the EM algorithm, other algorithms can be considered in step (2) as the Classification EM (CEM) algorithm (Celeux and Govaert 1992) or the Stochastic EM (SEM) algorithm (Celeux and Diebolt 1985) They both consist of generating a configuration z (q) after the E step and use it as an image restoration in the following M step. In the CEM algorithm, z (q) is generated according to a maximum a posteriori (MAP) rule (C step) ....

Celeux, G. and G. Govaert (1992). A classification EM algorithm for clustering and two stochastic versions.

....algorithm may be easily adapted for seeking a hard partition (c ik 2 f0; 1g) ffl The easiest way is to consider the resulting fuzzy partition at the convergence and to assigned each individual to the most probable class according to the a posteriori probabilities. ffl As in the CEM algorithm (Celeux and Govaert 1992), it is possible to add an intermediate classification step between the E and the M step. 3 A Bayesian Interpretation Notice that it is possible to have a Bayesian interpretation of the NEM algorithm. Maximizing the criterion U(c; Phi) is equivalent to maximizing expfU(c; Phi)g = expfL(c; ....

Celeux, G. and G. Govaert (1992). `A classification em algorithm for clustering and two stochastic versions'. Computational statistics and data analysis 14, 315--332.

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Celeux G., Govaert, G. : A Classification EM algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14(3):315--332. (1992)

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G. Celeux, G. Govaert, A classification EM algorithm for clustering and two stochastic versions, Computational Statistics & Data Analysis 14 (1992) 315-- 332. 38

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Gilles Celeux and G'erard Govaert. A Classification EM algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14:315-- 332, 1992.

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Celeux G., Govaert, G. : A Classification EM algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14(3):315--332. (1992)

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G. Celeux and G. Govaert. A classification em algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14:315--332, 1992.

No context found.

G. Celeux and G. Govaert. A classification em algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14:315--332, 1992.

No context found.

G. Celeux and G. Govaert. A Classification EM algorithm for clustering and two stochastic versions. Computational Statistics and Data Analysis, 14(3):315--332, 1992.

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Celeux, G. and Govaert, G. (1992). A Classification EM Algorithm for Clustering and Two Stochastic Versions. Computational Statistics and Data Analysis 14(3), 315-332.

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G. Celeux and G. Govaert, "A classification EM algorithm for clustering and two stochastic versions," Computational Statistics and Data Analysis, vol. 14, pp. 315--332, 1992.

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Celeux, G. and Govaert, G. (1992), "A Classification EM Algorithm for Clustering and Two Stochastic Versions," Computational Statistics and Data Analysis, 14, 315 -- 332.

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Celeux, G. and Govaert, G. (1992) "A classification EM algorithm for clustering and two stochastic versions", Computational Statistics and Data Analysis, 14, 315 -- 332.

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