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Clustering Empirical Bootstrap Distribution Functions Parametrized by Galton–Watson Branching Processes

Author

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  • Lauri Varmann

    (Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal)

  • Helena Mouriño

    (Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal
    Centro de Estatística e Aplicações, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal)

Abstract

The nonparametric bootstrap has been used in cluster analysis for various purposes. One of those purposes is to account for sampling variability. This can be achieved by obtaining a bootstrap approximation of the sampling distribution function of the estimator of interest and then clustering those distribution functions. Although the consistency of the nonparametric bootstrap in estimating transformations of the sample mean has been known for decades, little is known about how it carries over to clustering. Here, we investigated this problem with a simulation study. We considered single-linkage agglomerative hierarchical clustering and a three-type branching process for parametrized transformations of random vectors of relative frequencies of possible types of the index case of each process. In total, there were nine factors and 216 simulation scenarios in a fully-factorial design. The ability of the bootstrap-based clustering to recover the ground truth clusterings was quantified by the adjusted transfer distance between partitions. The results showed that in the best 18 scenarios, the average value of the distance was less than 20 percent of the maximum possible distance value. We noticed that the results most notably depended on the number of retained clusters, the distribution for sampling the prevalence of types, and the sample size appearing in the denominators of relative frequency types. The comparison of the bootstrap-based clustering results with so-called uninformed random partitioning results showed that in the vast majority of scenarios considered, the bootstrap-based approach led, on average, to remarkably lower classification errors than the random partitioning.

Suggested Citation

  • Lauri Varmann & Helena Mouriño, 2024. "Clustering Empirical Bootstrap Distribution Functions Parametrized by Galton–Watson Branching Processes," Mathematics, MDPI, vol. 12(15), pages 1-25, August.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:15:p:2409-:d:1448698
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    References listed on IDEAS

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    1. Joeri Hofmans & Eva Ceulemans & Douglas Steinley & Iven Mechelen, 2015. "On the Added Value of Bootstrap Analysis for K-Means Clustering," Journal of Classification, Springer;The Classification Society, vol. 32(2), pages 268-284, July.
    2. Irene Charon & Lucile Denoeud & Alain Guenoche & Olivier Hudry, 2006. "Maximum Transfer Distance Between Partitions," Journal of Classification, Springer;The Classification Society, vol. 23(1), pages 103-121, June.
    3. Qidi Peng & Nan Rao & Ran Zhao, 2019. "Some Developments in Clustering Analysis on Stochastic Processes," Biostatistics and Biometrics Open Access Journal, Juniper Publishers Inc., vol. 9(3), pages 72-77, April.
    4. Mahmoudi, Mohammad Reza & Baleanu, Dumitru & Mansor, Zulkefli & Tuan, Bui Anh & Pho, Kim-Hung, 2020. "Fuzzy clustering method to compare the spread rate of Covid-19 in the high risks countries," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
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