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On the Longest Edge of the Minimal Spanning Tree

  • Evangelos Tabakis
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Summary

As part of the effort to (a) robustify the single link clustering method and (b) develop new inference approaches in clustering in the spirit of Bock (1985), we introduce upper and lower bounds for the length M n of the longest edge of the minimal spanning tree on n iid random variables drawn from a probability measure P, having Lebesgue density f. We prove that with probability converging to 1 as n → ∞:
$$ k_L \cdot \frac{1}{\Delta } \cdot \frac{{\log n}}{n} \leqslant M_n^d \leqslant k_U \cdot \frac{1}{\delta } \cdot \frac{{\log n}}{n} $$
where \( \delta = \inf \{ f(x),x\,\, \in \,\,\text{supp(P)\} ,}\,\Delta \,\, = \,\sup \{ f(x),x\,\, \in \,\text{supp(P)}\} \) and k L , k U are constants depending only on δsupp(P).

Keywords

Minimal Span Tree Neighbor Distance Empirical Measure Borel Probability Measure Breakdown Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. ALDOUS, D., and STEELE, J.M. (1992): Asymptotics for euclidean minimal spanning trees on random points. Probab. Theory Relat. Fields, 92, 247–258. CrossRefGoogle Scholar
  2. BOCK, H.H. (1985): On some significance tests in cluster analysis. Journal of Classification, 2, 77–108. CrossRefGoogle Scholar
  3. DEHEUVELS, P., EINMAHL, J., MASON, D., and RUYMGAART, F.H. (1988): The almost sure behavior of maximal and minimal multivariate k n spacings. Journal of Multivariate Analysis, 24, 155–176.CrossRefGoogle Scholar
  4. DETTE, H., and HENZE, N. (1989): The limit distribution of the largest nearest-neighbour link in the unit d-cube. Journal of Applied Probability, 26, 67–80. CrossRefGoogle Scholar
  5. HENZE, N. (1983): Ein asymptotischer Satz über den maximalen Minimalabstand von unabhängigen Zufallsvektoren mit Anwendung auf einen Anpassungstest im R d und auf der Kugel. Metrika, 30, 245–259. CrossRefGoogle Scholar
  6. HAMPEL, F.R., RONCHETTI, E.M., ROUSSEEUW, P.J., and STAHEL, W.A. (1986): Robust Statistics: The approach based on Influence Functions. Wiley, New York.Google Scholar
  7. HUBER, P.J. (1981): Robust Statistics. Wiley, New York.CrossRefGoogle Scholar
  8. LEBART, L., MORINEAU, A. and WARWICK, K.M. (1984): Multivariate Descriptive Statistical Analysis: Correspondence Analysis and Related Techniques for Large Matrices. Wiley, New York.Google Scholar
  9. LEVY, P. (1939): Sur la division d’un segment par des points choisis au hasard. C.R.Acad.Sci. Paris, 208, 147–149 Google Scholar
  10. SHOR, P.W., and YUKICH, J.E. (1991): Minimax grid matching and empirical measures. Annals of Probability, 19, 1338–1348. CrossRefGoogle Scholar
  11. STEELE, J.M. (1988): Growth rates of euclidean minimal spanning trees with power weighted edges. Annals of Probability, 16, 1767–1787. CrossRefGoogle Scholar
  12. STEELE, J.M., and TIERNEY, L. (1988): Boundary domination and the distribution of the largest nearest neighbor link in higher dimensions. Journal of Applied Probability, 23, 524–528. CrossRefGoogle Scholar
  13. TABAKIS, E. (1992): Asymptotic and Computational Problems in Single-Link Clustering. PhD thesis, Massachusetts Institute of Technology.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1996

Authors and Affiliations

  • Evangelos Tabakis
    • 1
  1. 1.Mathematisches Institut, LS Mathematik VIIUniversität BayreuthBayreuthFederal Republic of Germany

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