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Using combinatorial optimization in model-based trimmed clustering with cardinality constraints

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  • Gallegos, María Teresa
  • Ritter, Gunter
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    Abstract

    Statistical clustering criteria with free scale parameters and unknown cluster sizes are inclined to create small, spurious clusters. To mitigate this tendency a statistical model for cardinality-constrained clustering of data with gross outliers is established, its maximum likelihood and maximum a posteriori clustering criteria are derived, and their consistency and robustness are analyzed. The criteria lead to constrained optimization problems that can be solved by using iterative, alternating trimming algorithms of k-means type. Each step in the algorithms requires the solution of a [lambda]-assignment problem known from combinatorial optimization. The method allows one to estimate the numbers of clusters and outliers. It is illustrated with a synthetic data set and a real one.

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    File URL: http://www.sciencedirect.com/science/article/B6V8V-4X6MSNY-2/2/2b074aa9713f45300cb0ca60701884f8
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    Bibliographic Info

    Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

    Volume (Year): 54 (2010)
    Issue (Month): 3 (March)
    Pages: 637-654

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    Handle: RePEc:eee:csdana:v:54:y:2010:i:3:p:637-654

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    Web page: http://www.elsevier.com/locate/csda

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    References

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    1. Woodruff, David L. & Reiners, Torsten, 2004. "Experiments with, and on, algorithms for maximum likelihood clustering," Computational Statistics & Data Analysis, Elsevier, vol. 47(2), pages 237-253, September.
    2. Coleman, Dan & Dong, Xioapeng & Hardin, Johanna & Rocke, David M. & Woodruff, David L., 1999. "Some computational issues in cluster analysis with no a priori metric," Computational Statistics & Data Analysis, Elsevier, vol. 31(1), pages 1-11, July.
    3. H. Bock, 1985. "On some significance tests in cluster analysis," Journal of Classification, Springer, vol. 2(1), pages 77-108, December.
    4. Baibing Li, 2006. "A new approach to cluster analysis: the clustering-function-based method," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 457-476.
    5. Robert Tibshirani & Guenther Walther & Trevor Hastie, 2001. "Estimating the number of clusters in a data set via the gap statistic," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 411-423.
    6. Fraley C. & Raftery A.E., 2002. "Model-Based Clustering, Discriminant Analysis, and Density Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 611-631, June.
    7. Hanfeng Chen & Jiahua Chen & John D. Kalbfleisch, 2004. "Testing for a finite mixture model with two components," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 95-115.
    8. Glenn Milligan & Martha Cooper, 1985. "An examination of procedures for determining the number of clusters in a data set," Psychometrika, Springer, vol. 50(2), pages 159-179, June.
    9. María Gallegos & Gunter Ritter, 2009. "Trimming algorithms for clustering contaminated grouped data and their robustness," Advances in Data Analysis and Classification, Springer, vol. 3(2), pages 135-167, September.
    10. Neykov, N. & Filzmoser, P. & Dimova, R. & Neytchev, P., 2007. "Robust fitting of mixtures using the trimmed likelihood estimator," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 299-308, September.
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    Cited by:
    1. Gallegos, María Teresa & Ritter, Gunter, 2013. "Strong consistency of k-parameters clustering," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 14-31.
    2. Neykov, N.M. & Filzmoser, P. & Neytchev, P.N., 2012. "Robust joint modeling of mean and dispersion through trimming," Computational Statistics & Data Analysis, Elsevier, vol. 56(1), pages 34-48, January.

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