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On the Ultrametric Generated by Random Distribution of Points in Euclidean Spaces of Large Dimensions with Correlated Coordinates

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  • A. P. Zubarev

    (Samara State University of Railway Transport
    Samara University)

Abstract

Recently a general theorem stating that the matrix of normalized Euclidean distances on the set of specially distributed random points in the n-dimensional Euclidean space ℝ n with independent coordinates converges in probability as n→∞ to the ultrametric matrix had been proved. The main theorem of the present paper extends this result to the case of weakly correlated coordinates of random points. Prior to formulating and stating this result we give two illustrative examples describing particular algorithms of generation of such nearly ultrametric spaces.

Suggested Citation

  • A. P. Zubarev, 2017. "On the Ultrametric Generated by Random Distribution of Points in Euclidean Spaces of Large Dimensions with Correlated Coordinates," Journal of Classification, Springer;The Classification Society, vol. 34(3), pages 366-383, October.
    • Handle: RePEc:spr:jclass:v:34:y:2017:i:3:d:10.1007_s00357-017-9236-8
    DOI: 10.1007/s00357-017-9236-8
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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. Fionn Murtagh, 2004. "On Ultrametricity, Data Coding, and Computation," Journal of Classification, Springer;The Classification Society, vol. 21(2), pages 167-184, September.
    3. F. Murtagh, 2005. "Identifying the ultrametricity of time series," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 43(4), pages 573-579, February.
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