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Asymptotics of hierarchical clustering for growing dimension

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  • Borysov, Petro
  • Hannig, Jan
  • Marron, J.S.

Abstract

Modern day science presents many challenges to data analysts. Advances in data collection provide very large (number of observations and number of dimensions) data sets. In many areas of data analysis an informative task is to find natural separations of data into homogeneous groups, i.e. clusters. In this paper we study the asymptotic behavior of hierarchical clustering in situations where both sample size and dimension grow to infinity. We derive explicit signal vs noise boundaries between different types of clustering behaviors. We also show that the clustering behavior within the boundaries is the same across a wide spectrum of asymptotic settings.

Suggested Citation

  • Borysov, Petro & Hannig, Jan & Marron, J.S., 2014. "Asymptotics of hierarchical clustering for growing dimension," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 465-479.
    • Handle: RePEc:eee:jmvana:v:124:y:2014:i:c:p:465-479
    DOI: 10.1016/j.jmva.2013.11.010
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    References listed on IDEAS

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    1. Peter Hall & J. S. Marron & Amnon Neeman, 2005. "Geometric representation of high dimension, low sample size data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(3), pages 427-444, June.
    2. Glenn Milligan, 1980. "An examination of the effect of six types of error perturbation on fifteen clustering algorithms," Psychometrika, Springer;The Psychometric Society, vol. 45(3), pages 325-342, September.
    3. Jianqing Fan & Jinchi Lv, 2008. "Sure independence screening for ultrahigh dimensional feature space," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(5), pages 849-911, November.
    4. Yata, Kazuyoshi & Aoshima, Makoto, 2012. "Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations," Journal of Multivariate Analysis, Elsevier, vol. 105(1), pages 193-215.
    5. Qiao, Xingye & Zhang, Hao Helen & Liu, Yufeng & Todd, Michael J. & Marron, J. S., 2010. "Weighted Distance Weighted Discrimination and Its Asymptotic Properties," Journal of the American Statistical Association, American Statistical Association, vol. 105(489), pages 401-414.
    6. Jeongyoun Ahn & J. S. Marron & Keith M. Muller & Yueh-Yun Chi, 2007. "The high-dimension, low-sample-size geometric representation holds under mild conditions," Biometrika, Biometrika Trust, vol. 94(3), pages 760-766.
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    Citations

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    Cited by:

    1. Gautier Marti & S'ebastien Andler & Frank Nielsen & Philippe Donnat, 2016. "Clustering Financial Time Series: How Long is Enough?," Papers 1603.04017, arXiv.org, revised Apr 2016.
    2. Gautier Marti & Frank Nielsen & Miko{l}aj Bi'nkowski & Philippe Donnat, 2017. "A review of two decades of correlations, hierarchies, networks and clustering in financial markets," Papers 1703.00485, arXiv.org, revised Nov 2020.
    3. Gautier Marti & Frank Nielsen & Philippe Donnat & S'ebastien Andler, 2016. "On clustering financial time series: a need for distances between dependent random variables," Papers 1603.07822, arXiv.org.
    4. Richard Audoly & Rory McGee & Sergio Ocampo & Gonzalo Paz-Pardo, 2024. "The Life-Cycle Dynamics of Wealth Mobility," Staff Reports 1097, Federal Reserve Bank of New York.
    5. Egashira, Kento & Yata, Kazuyoshi & Aoshima, Makoto, 2024. "Asymptotic properties of hierarchical clustering in high-dimensional settings," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    6. Gautier Marti & Sébastien Andler & Frank Nielsen & Philippe Donnat, 2016. "Clustering Financial Time Series: How Long is Enough?," Post-Print hal-01400395, HAL.
    7. Modarres, Reza, 2022. "A high dimensional dissimilarity measure," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    8. Nakayama, Yugo & Yata, Kazuyoshi & Aoshima, Makoto, 2021. "Clustering by principal component analysis with Gaussian kernel in high-dimension, low-sample-size settings," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    9. Gautier Marti & Philippe Very & Philippe Donnat & Frank Nielsen, 2015. "A proposal of a methodological framework with experimental guidelines to investigate clustering stability on financial time series," Papers 1509.05475, arXiv.org.
    10. Luis Lorenzo & Javier Arroyo, 2023. "Online risk-based portfolio allocation on subsets of crypto assets applying a prototype-based clustering algorithm," Financial Innovation, Springer;Southwestern University of Finance and Economics, vol. 9(1), pages 1-40, December.
    11. Kazuyoshi Yata & Makoto Aoshima, 2020. "Geometric consistency of principal component scores for high‐dimensional mixture models and its application," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 899-921, September.
    12. Patrick K. Kimes & Yufeng Liu & David Neil Hayes & James Stephen Marron, 2017. "Statistical significance for hierarchical clustering," Biometrics, The International Biometric Society, vol. 73(3), pages 811-821, September.

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