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Strong consistency of factorial $$K$$ K -means clustering

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  • Yoshikazu Terada

Abstract

Factorial $$k$$ k -means (FKM) clustering is a method for clustering objects in a low-dimensional subspace. The advantage of this method is that the partition of objects and the low-dimensional subspace reflecting the cluster structure are obtained, simultaneously. In some cases that reduced $$k$$ k -means (RKM) clustering does not work well, FKM clustering can discover the cluster structure underlying a lower dimensional subspace. Conditions that ensure the almost sure convergence of the estimator of FKM clustering as the sample size increases unboundedly are derived. The result is proved for a more general model including FKM clustering. Moreover, it is also shown that there exist some cases in which RKM clustering becomes equivalent to FKM clustering as the sample size goes to infinity. Copyright The Institute of Statistical Mathematics, Tokyo 2015

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  • Yoshikazu Terada, 2015. "Strong consistency of factorial $$K$$ K -means clustering," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 335-357, April.
    • Handle: RePEc:spr:aistmt:v:67:y:2015:i:2:p:335-357
    DOI: 10.1007/s10463-014-0454-0
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    References listed on IDEAS

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    1. Timmerman, Marieke E. & Ceulemans, Eva & Kiers, Henk A.L. & Vichi, Maurizio, 2010. "Factorial and reduced K-means reconsidered," Computational Statistics & Data Analysis, Elsevier, vol. 54(7), pages 1858-1871, July.
    2. Yoshikazu Terada, 2014. "Strong Consistency of Reduced K-means Clustering," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 913-931, December.
    3. Wei‐Chien Chang, 1983. "On Using Principal Components before Separating a Mixture of Two Multivariate Normal Distributions," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 32(3), pages 267-275, November.
    4. Vichi, Maurizio & Kiers, Henk A. L., 2001. "Factorial k-means analysis for two-way data," Computational Statistics & Data Analysis, Elsevier, vol. 37(1), pages 49-64, July.
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    2. Chakraborty, Saptarshi & Das, Swagatam, 2021. "On uniform concentration bounds for Bi-clustering by using the Vapnik–Chervonenkis theory," Statistics & Probability Letters, Elsevier, vol. 175(C).

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    Keywords

    Subspace clustering; $$K$$ K -means;

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