An Expectation–Maximization algorithm for the Wishart mixture model: Application to movement clustering

This article presents an Expectation–Maximization algorithm for the Wishart mixture model in which realizations are matrices. Given a set of matrices, an iterative algorithm for estimating the parameters of such a mixture model is proposed. The obtained estimates can be interpreted in terms of mean...

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Vydáno v:	Pattern recognition letters Ročník 31; číslo 14; s. 2318 - 2324
Hlavní autoři:	Hidot, Sullivan, Saint-Jean, Christophe
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 15.10.2010 Elsevier
Témata:	Applied sciences Clustering Computer Science Detection, estimation, filtering, equalization, prediction EM algorithm Engineering Sciences Exact sciences and technology Information, signal and communications theory Movement recognition Second-order cross moments Signal and communications theory Signal and Image Processing Signal representation. Spectral analysis Signal, noise Telecommunications and information theory Wishart mixture model Second-order cross moments Wishart mixture model Movement recognition Clustering EM algorithm Second order Automatic classification Parameter estimation Mixture theory Iterative method Scale factor Signal classification A posteriori estimation Geometrical method
ISSN:	0167-8655, 1872-7344
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Abstract	This article presents an Expectation–Maximization algorithm for the Wishart mixture model in which realizations are matrices. Given a set of matrices, an iterative algorithm for estimating the parameters of such a mixture model is proposed. The obtained estimates can be interpreted in terms of mean matrices and scale factors. By applying the maximum a posteriori rule, we get an algorithm for the clustering of a set of matrices. This mixture model is then modified in order to deal with a set of samples. Unfortunately, the samples may be of different sizes. We propose to tackle this problem by considering the cross-product matrix as a signature for each sample. This set of cross-product matrices may be fitted with the proposed Wishart pseudo-mixture model in which the scale parameters of the distribution are not estimated but fixed. Again, we easily get a clustering algorithm from final parameter estimates. The different estimators are studied empirically through an analysis of their bias and variance and are validated onto an artificial dataset. Finally, we apply the Wishart pseudo-mixture model for analyzing motion-captured movements. Given the successive 3D positions of markers over the time, a cross-product matrix is constructed for each movement and put into the proposed classifier. We observe that the recognition rates are higher with our proposed approach than those with other geometric methods. Limits and constraints of the provided models are finally discussed.
AbstractList	This article presents an Expectation-Maximization algorithm for the Wishart mixture model in which realizations are matrices. Given a set of matrices, an iterative algorithm for estimating the parameters of such a mixture model is proposed. The obtained estimates can be interpreted in terms of mean matrices and scale factors. By applying the maximum a posteriori rule, we get an algorithm for the clustering of a set of matrices. This mixture model is then modified in order to deal with a set of samples. Unfortunately, the samples may be of different sizes. We propose to tackle this problem by considering the cross-product matrix as a signature for each sample. This set of cross-product matrices may be fitted with the proposed Wishart pseudo-mixture model in which the scale parameters of the distribution are not estimated but fixed. Again, we easily get a clustering algorithm from final parameter estimates. The different estimators are studied empirically through an analysis of their bias and variance and are validated onto an artificial dataset. Finally, we apply the Wishart pseudo-mixture model for analyzing motion-captured movements. Given the successive 3D positions of markers over the time, a cross-product matrix is constructed for each movement and put into the proposed classifier. We observe that the recognition rates are higher with our proposed approach than those with other geometric methods. Limits and constraints of the provided models are finally discussed. This article presents an Expectation–Maximization algorithm for the Wishart mixture model in which realizations are matrices. Given a set of matrices, an iterative algorithm for estimating the parameters of such a mixture model is proposed. The obtained estimates can be interpreted in terms of mean matrices and scale factors. By applying the maximum a posteriori rule, we get an algorithm for the clustering of a set of matrices. This mixture model is then modified in order to deal with a set of samples. Unfortunately, the samples may be of different sizes. We propose to tackle this problem by considering the cross-product matrix as a signature for each sample. This set of cross-product matrices may be fitted with the proposed Wishart pseudo-mixture model in which the scale parameters of the distribution are not estimated but fixed. Again, we easily get a clustering algorithm from final parameter estimates. The different estimators are studied empirically through an analysis of their bias and variance and are validated onto an artificial dataset. Finally, we apply the Wishart pseudo-mixture model for analyzing motion-captured movements. Given the successive 3D positions of markers over the time, a cross-product matrix is constructed for each movement and put into the proposed classifier. We observe that the recognition rates are higher with our proposed approach than those with other geometric methods. Limits and constraints of the provided models are finally discussed.
Author	Hidot, Sullivan Saint-Jean, Christophe
Author_xml	– sequence: 1 givenname: Sullivan surname: Hidot fullname: Hidot, Sullivan organization: Department of Computer Science, Faculty of Engineering, University of Mons, 7000 Mons, Belgium – sequence: 2 givenname: Christophe surname: Saint-Jean fullname: Saint-Jean, Christophe email: christophe.saint-jean@univ-lr.fr organization: Laboratory of Mathematics, Image and Applications, University of La Rochelle, 17000 La Rochelle, France
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Keywords	Second-order cross moments Wishart mixture model Movement recognition Clustering EM algorithm Second order Automatic classification Parameter estimation Mixture theory Iterative method Scale factor Signal classification A posteriori estimation Geometrical method
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Snippet	This article presents an Expectation–Maximization algorithm for the Wishart mixture model in which realizations are matrices. Given a set of matrices, an... This article presents an Expectation-Maximization algorithm for the Wishart mixture model in which realizations are matrices. Given a set of matrices, an...
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SubjectTerms	Applied sciences Clustering Computer Science Detection, estimation, filtering, equalization, prediction EM algorithm Engineering Sciences Exact sciences and technology Information, signal and communications theory Movement recognition Second-order cross moments Signal and communications theory Signal and Image Processing Signal representation. Spectral analysis Signal, noise Telecommunications and information theory Wishart mixture model
Title	An Expectation–Maximization algorithm for the Wishart mixture model: Application to movement clustering
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