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An application of the minimal spanning tree approach to the cluster stability problem

Author

Listed:
  • Z. Volkovich
  • Z. Barzily
  • G.-W. Weber
  • D. Toledano-Kitai
  • R. Avros

Abstract

Among the areas of data and text mining which are employed today in OR, science, economy and technology, clustering theory serves as a preprocessing step in the data analyzing. An important component of clustering theory is determination of the true number of clusters. This problem has not been satisfactorily solved. In our paper, this problem is addressed by the cluster stability approach. For several possible numbers of clusters, we estimate the stability of the partitions obtained from clustering of samples. Partitions are considered consistent if their clusters are stable. Clusters validity is measured by the total number of edges, in the clusters’ minimal spanning trees, connecting points from different samples. Actually, we use the Friedman and Rafsky two sample test statistic. The homogeneity hypothesis of well mingled samples, within the clusters, leads to an asymptotic normal distribution of the considered statistic. Resting upon this fact, the standard score of the mentioned edges quantity is set, and the partition quality is represented by the worst cluster, corresponding to the minimal standard score value. It is natural to expect that the true number of clusters can be characterized by the empirical distribution having the shortest left tail. The proposed methodology sequentially creates the described distribution and estimates its left-asymmetry. Several presented numerical experiments demonstrate the ability of the approach to detect the true number of clusters. Copyright Springer-Verlag 2012

Suggested Citation

  • Z. Volkovich & Z. Barzily & G.-W. Weber & D. Toledano-Kitai & R. Avros, 2012. "An application of the minimal spanning tree approach to the cluster stability problem," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 20(1), pages 119-139, March.
    • Handle: RePEc:spr:cejnor:v:20:y:2012:i:1:p:119-139
    DOI: 10.1007/s10100-010-0157-4
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    References listed on IDEAS

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    1. 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.
    2. Sugar, Catherine A. & James, Gareth M., 2003. "Finding the Number of Clusters in a Dataset: An Information-Theoretic Approach," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 750-763, January.
    3. Glenn Milligan & Martha Cooper, 1985. "An examination of procedures for determining the number of clusters in a data set," Psychometrika, Springer;The Psychometric Society, vol. 50(2), pages 159-179, June.
    4. J. Hartigan, 1985. "Statistical theory in clustering," Journal of Classification, Springer;The Classification Society, vol. 2(1), pages 63-76, December.
    5. Glenn Milligan & Richard Cheng, 1996. "Measuring the influence of individual data points in a cluster analysis," Journal of Classification, Springer;The Classification Society, vol. 13(2), pages 315-335, September.
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    Cited by:

    1. Josefa Mula & Marija Bogataj, 2021. "OR in the industrial engineering of Industry 4.0: experiences from the Iberian Peninsula mirrored in CJOR," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 29(4), pages 1163-1184, December.

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