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Asymptotic properties of univariate sample k-means clusters

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Abstract

A random sample of sizeN is divided intok clusters that minimize the within clusters sum of squares locally. Some large sample properties of this k-means clustering method (ask approaches ∞ withN) are obtained. In one dimension, it is established that the sample k-means clusters are such that the within-cluster sums of squares are asymptotically equal, and that the sizes of the cluster intervals are inversely proportional to the one-third power of the underlying density at the midpoints of the intervals. The difficulty involved in generalizing the results to the multivariate case is mentioned.

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References

  • BILLINGSLEY, P. (1968),Convergence of Probability Measures, New York: John Wiley and Sons.

    Google Scholar 

  • BLASHFIELD, R.K., and Aldenderfer, M.S. (1978), “The Literature on Cluster Analysis,”Multivariate Behavioral Research, 13, 271–95.

    Google Scholar 

  • ELIAS, P. (1970), “Bounds on Performance of Optimum Quantizers,”IEEE Transactions on Information Theory, 16, 172–184.

    Google Scholar 

  • FELLER, W. (1971),An Introduction to Probability Theory and Its Applications, Volume II, New York: John Wiley and Sons, 549–553.

    Google Scholar 

  • HARTIGAN, J.A. (1975),Clustering Algorithms, New York: John Wiley and Sons.

    Google Scholar 

  • HARTIGAN, J.A. (1978), “Asymptotic Distributions for Clustering Criteria,”Annals of Statistics, 6, 117–131.

    Google Scholar 

  • HARTIGAN, J.A., and WONG, M.A. (1979), “Algorithm AS136: A K-means Clustering Algorithm,”Applied Statistics, 28, 100–108.

    Google Scholar 

  • MACQUEEN, J.B. (1967), “Some Methods for Classification and Analysis of Multivariate Observations”,Proceedings of the Fifth Berkeley Symposium on Probability and Statistics, 281–297.

  • POLLARD, D. (1981), “Strong Consistency of K-means Clustering,”Annals of Statistics, 9, 135–140.

    Google Scholar 

  • WONG, M.A. (1982a), “Asymptotic Properties of Univariate Population K-means Cluster,”The Classification Society Bulletin, 5, 44–50.

    Google Scholar 

  • WONG, M.A. (1982b), “Asymptotic Properties of Bivariate K-means Clusters,”Communications in Statistics, Volume A-11, 1155–1172.

    Google Scholar 

  • WONG, M.A. (1985), “Using the K-means Clustering Method as a Density Estimation Procedure,”Journal of Organizational Behavior and Statistics, (to appear).

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Authors and Affiliations

  1. Sloan School of Management, Massachusetts Institute of Technology, 02139, Cambridge, MA

    M. Anthony Wong

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  1. M. Anthony Wong
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Additional information

This research was supported in part by the National Science Foundation under Grant MCS75-08374. The author would like to thank John Hartigan and David Pollard for helpful discussions and comments.

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Wong, M.A. Asymptotic properties of univariate sample k-means clusters. Journal of Classification 1, 255–270 (1984). https://doi.org/10.1007/BF01890126

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  • DOI: https://doi.org/10.1007/BF01890126

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