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A fast and recursive algorithm for clustering large datasets with k-medians

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  • Cardot, Hervé
  • Cénac, Peggy
  • Monnez, Jean-Marie

Abstract

Clustering with fast algorithms large samples of high dimensional data is an important challenge in computational statistics. A new class of recursive stochastic gradient algorithms designed for the k-medians loss criterion is proposed. By their recursive nature, these algorithms are very fast and are well adapted to deal with large samples of data that are allowed to arrive sequentially. It is proved that the stochastic gradient algorithm converges almost surely to the set of stationary points of the underlying loss criterion. A particular attention is paid to the averaged versions which are known to have better performances. A data-driven procedure that permits a fully automatic selection of the value of the descent step is also proposed. The performance of the averaged sequential estimator is compared on a simulation study, both in terms of computation speed and accuracy of the estimations, with more classical partitioning techniques such as k-means, trimmed k-means and PAM (partitioning around medoids). Finally, this new online clustering technique is illustrated on determining television audience profiles with a sample of more than 5000 individual television audiences measured every minute over a period of 24 hours.

Suggested Citation

  • Cardot, Hervé & Cénac, Peggy & Monnez, Jean-Marie, 2012. "A fast and recursive algorithm for clustering large datasets with k-medians," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1434-1449.
    • Handle: RePEc:eee:csdana:v:56:y:2012:i:6:p:1434-1449
    DOI: 10.1016/j.csda.2011.11.019
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    1. Luis García-Escudero & Alfonso Gordaliza & Carlos Matrán & Agustín Mayo-Iscar, 2010. "A review of robust clustering methods," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 4(2), pages 89-109, September.
    2. Monnez, Jean-Marie, 2006. "Almost sure convergence of stochastic gradient processes with matrix step sizes," Statistics & Probability Letters, Elsevier, vol. 76(5), pages 531-536, March.
    3. Croux, Christophe & Gallopoulos, Efstratios & Van Aelst, Stefan & Zha, Hongyuan, 2007. "Machine Learning and Robust Data Mining," Computational Statistics & Data Analysis, Elsevier, vol. 52(1), pages 151-154, September.
    4. García-Treviño, E.S. & Barria, J.A., 2012. "Online wavelet-based density estimation for non-stationary streaming data," Computational Statistics & Data Analysis, Elsevier, vol. 56(2), pages 327-344.
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    1. Gaunand, A. & Hocdé, A. & Lemarié, S. & Matt, M. & Turckheim, E.de, 2015. "How does public agricultural research impact society? A characterization of various patterns," Research Policy, Elsevier, vol. 44(4), pages 849-861.
    2. Monnez, Jean-Marie & Skiredj, Abderrahman, 2021. "Widening the scope of an eigenvector stochastic approximation process and application to streaming PCA and related methods," Journal of Multivariate Analysis, Elsevier, vol. 182(C).
    3. Godichon-Baggioni, Antoine & Lu, Wei, 2024. "Online stochastic Newton methods for estimating the geometric median and applications," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    4. Hervé Cardot & Antoine Godichon-Baggioni, 2017. "Fast estimation of the median covariation matrix with application to online robust principal components analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 461-480, September.
    5. Godichon-Baggioni, Antoine, 2016. "Estimating the geometric median in Hilbert spaces with stochastic gradient algorithms: Lp and almost sure rates of convergence," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 209-222.

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