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Ultrametricity indices for the Euclidean and Boolean hypercubes

Ultrametricity indices for the Euclidean and Boolean hypercubes Motivated by Murtagh’s experimental observation that sparse random samples of the hypercube become more and more ultrametric as the dimension increases, we consider a strict version of his ultrametricity coefficient, an index derived from Rammal’s degree of ultrametricity, and a topological ultrametricity index. First, we prove that the three ultrametricity indices converge in probability to one as dimension increases, if the sample size remains fixed. This is done for uniformly and normally distributed samples in the Euclidean hypercube, and for uniformly distributed samples in F2 N with Hamming distance, as well as for very general probability distributions. Further, this holds true for random categorial data in complete disjunctive form. A second result is that the ultrametricity indices vanish in the limit for the full hypercube F2 N as dimensionN increases,whereby Murtagh’s ultrametricity index is largest, and the topological ultrametricity index smallest, if N is large. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png P-Adic Numbers, Ultrametric Analysis, and Applications Springer Journals

Ultrametricity indices for the Euclidean and Boolean hypercubes

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References (15)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Pleiades Publishing, Ltd.
Subject
Mathematics; Algebra
ISSN
2070-0466
eISSN
2070-0474
DOI
10.1134/S2070046616040038
Publisher site
See Article on Publisher Site

Abstract

Motivated by Murtagh’s experimental observation that sparse random samples of the hypercube become more and more ultrametric as the dimension increases, we consider a strict version of his ultrametricity coefficient, an index derived from Rammal’s degree of ultrametricity, and a topological ultrametricity index. First, we prove that the three ultrametricity indices converge in probability to one as dimension increases, if the sample size remains fixed. This is done for uniformly and normally distributed samples in the Euclidean hypercube, and for uniformly distributed samples in F2 N with Hamming distance, as well as for very general probability distributions. Further, this holds true for random categorial data in complete disjunctive form. A second result is that the ultrametricity indices vanish in the limit for the full hypercube F2 N as dimensionN increases,whereby Murtagh’s ultrametricity index is largest, and the topological ultrametricity index smallest, if N is large.

Journal

P-Adic Numbers, Ultrametric Analysis, and ApplicationsSpringer Journals

Published: Nov 6, 2016

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