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Author: Dimitris Karlis
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
One of the estimating equations of the Maximum Likelihood Estimation method, for finite mixtures of the one parameter exponential family, is the first moment equation. This can help considerably in reducing the labor and the cost of calculating the Maximum Likelihood estimates. In this paper it is shown that the EM algorithm can be substantially improved by using this result when applied for mixture models. A short discussion about other methods proposed for the calculation of the Maximum Likelihood estimates are also reported showing that the above findings can help in this direction too.
Author: Dimitris Karlis
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
One of the estimating equations of the Maximum Likelihood Estimation method, for finite mixtures of the one parameter exponential family, is the first moment equation. This can help considerably in reducing the labor and the cost of calculating the Maximum Likelihood estimates. In this paper it is shown that the EM algorithm can be substantially improved by using this result when applied for mixture models. A short discussion about other methods proposed for the calculation of the Maximum Likelihood estimates are also reported showing that the above findings can help in this direction too.
Author: Geoffrey McLachlan
Publisher: John Wiley & Sons
ISBN: 047165406X
Category : Mathematics
Languages : en
Pages : 419
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Book Description
An up-to-date, comprehensive account of major issues in finitemixture modeling This volume provides an up-to-date account of the theory andapplications of modeling via finite mixture distributions. With anemphasis on the applications of mixture models in both mainstreamanalysis and other areas such as unsupervised pattern recognition,speech recognition, and medical imaging, the book describes theformulations of the finite mixture approach, details itsmethodology, discusses aspects of its implementation, andillustrates its application in many common statisticalcontexts. Major issues discussed in this book include identifiabilityproblems, actual fitting of finite mixtures through use of the EMalgorithm, properties of the maximum likelihood estimators soobtained, assessment of the number of components to be used in themixture, and the applicability of asymptotic theory in providing abasis for the solutions to some of these problems. The author alsoconsiders how the EM algorithm can be scaled to handle the fittingof mixture models to very large databases, as in data miningapplications. This comprehensive, practical guide: * Provides more than 800 references-40% published since 1995 * Includes an appendix listing available mixture software * Links statistical literature with machine learning and patternrecognition literature * Contains more than 100 helpful graphs, charts, and tables Finite Mixture Models is an important resource for both applied andtheoretical statisticians as well as for researchers in the manyareas in which finite mixture models can be used to analyze data.
Author: Dimitris Karlis
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Finite Poisson mixtures can be used in a variety of real applications to describe count data as they can describe situations where overdispersion relative to the simple Poisson model is present. They also admit a natural interpretation: the entire population is a mixture of k subpopulations each having a Poisson distribution giving rise to the k-finite Poisson distribution. Estimating the parameters of a k-finite Poisson mixture is not easy. However, the development of the EM algorithm for finite mixtures simplified the derivation of the maximum likelihood estimates. In this paper an improvement of the standard EM algorithm for finite Poisson mixtures is introduced. It is based on the result that one from the estimating equations for the Maximum Likelihood Estimates in the case of finite Poisson mixtures is the first moment equation. Hence, replacing one of the estimating equations by this simpler form help us considerably in reducing the labour and the cost of calculating the MLE. Tables verifying the results are also given.
Author: Maya R. Gupta
Publisher: Now Publishers Inc
ISBN: 1601984308
Category : Computers
Languages : en
Pages : 87
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Book Description
Introduces the expectation-maximization (EM) algorithm and provides an intuitive and mathematically rigorous understanding of this method. Theory and Use of the EM Algorithm is designed to be useful to both the EM novice and the experienced EM user looking to better understand the method and its use.
Author: Geoffrey J. McLachlan
Publisher: John Wiley & Sons
ISBN: 0470191600
Category : Mathematics
Languages : en
Pages : 399
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Book Description
The only single-source——now completely updated and revised——to offer a unified treatment of the theory, methodology, and applications of the EM algorithm Complete with updates that capture developments from the past decade, The EM Algorithm and Extensions, Second Edition successfully provides a basic understanding of the EM algorithm by describing its inception, implementation, and applicability in numerous statistical contexts. In conjunction with the fundamentals of the topic, the authors discuss convergence issues and computation of standard errors, and, in addition, unveil many parallels and connections between the EM algorithm and Markov chain Monte Carlo algorithms. Thorough discussions on the complexities and drawbacks that arise from the basic EM algorithm, such as slow convergence and lack of an in-built procedure to compute the covariance matrix of parameter estimates, are also presented. While the general philosophy of the First Edition has been maintained, this timely new edition has been updated, revised, and expanded to include: New chapters on Monte Carlo versions of the EM algorithm and generalizations of the EM algorithm New results on convergence, including convergence of the EM algorithm in constrained parameter spaces Expanded discussion of standard error computation methods, such as methods for categorical data and methods based on numerical differentiation Coverage of the interval EM, which locates all stationary points in a designated region of the parameter space Exploration of the EM algorithm's relationship with the Gibbs sampler and other Markov chain Monte Carlo methods Plentiful pedagogical elements—chapter introductions, lists of examples, author and subject indices, computer-drawn graphics, and a related Web site The EM Algorithm and Extensions, Second Edition serves as an excellent text for graduate-level statistics students and is also a comprehensive resource for theoreticians, practitioners, and researchers in the social and physical sciences who would like to extend their knowledge of the EM algorithm.
Author: M.I. Jordan
Publisher: Springer Science & Business Media
ISBN: 9401150141
Category : Computers
Languages : en
Pages : 658
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Book Description
In the past decade, a number of different research communities within the computational sciences have studied learning in networks, starting from a number of different points of view. There has been substantial progress in these different communities and surprising convergence has developed between the formalisms. The awareness of this convergence and the growing interest of researchers in understanding the essential unity of the subject underlies the current volume. Two research communities which have used graphical or network formalisms to particular advantage are the belief network community and the neural network community. Belief networks arose within computer science and statistics and were developed with an emphasis on prior knowledge and exact probabilistic calculations. Neural networks arose within electrical engineering, physics and neuroscience and have emphasised pattern recognition and systems modelling problems. This volume draws together researchers from these two communities and presents both kinds of networks as instances of a general unified graphical formalism. The book focuses on probabilistic methods for learning and inference in graphical models, algorithm analysis and design, theory and applications. Exact methods, sampling methods and variational methods are discussed in detail. Audience: A wide cross-section of computationally oriented researchers, including computer scientists, statisticians, electrical engineers, physicists and neuroscientists.
Author: Lei Xu
Publisher:
ISBN:
Category :
Languages : en
Pages : 9
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Book Description
Author: Otmar Scherzer
Publisher: Springer Science & Business Media
ISBN: 0387929193
Category : Mathematics
Languages : en
Pages : 1626
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Book Description
The Handbook of Mathematical Methods in Imaging provides a comprehensive treatment of the mathematical techniques used in imaging science. The material is grouped into two central themes, namely, Inverse Problems (Algorithmic Reconstruction) and Signal and Image Processing. Each section within the themes covers applications (modeling), mathematics, numerical methods (using a case example) and open questions. Written by experts in the area, the presentation is mathematically rigorous. The entries are cross-referenced for easy navigation through connected topics. Available in both print and electronic forms, the handbook is enhanced by more than 150 illustrations and an extended bibliography. It will benefit students, scientists and researchers in applied mathematics. Engineers and computer scientists working in imaging will also find this handbook useful.
Author: Dominik Liebl
Publisher:
ISBN:
Category :
Languages : en
Pages : 80
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Book Description
Author: Masahiro Kuroda
Publisher:
ISBN:
Category : Electronic books
Languages : en
Pages : 0
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Book Description
Mixture models become increasingly popular due to their modeling flexibility and are applied to the clustering and classification of heterogeneous data. The EM algorithm is largely used for the maximum likelihood estimation of mixture models because the algorithm is stable in convergence and simple in implementation. Despite such advantages, it is pointed out that the EM algorithm is local and has slow convergence as the main drawback. To avoid the local convergence of the EM algorithm, multiple runs from several different initial values are usually used. Then the algorithm may take a large number of iterations and long computation time to find the maximum likelihood estimates. The speedup of computation of the EM algorithm is available for these problems. We give the algorithms to accelerate the convergence of the EM algorithm and apply them to mixture model estimation. Numerical experiments examine the performance of the acceleration algorithms in terms of the number of iterations and computation time.
