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An Ordinal Model for Cluster Analysis — 15 Years in Retrospect

  • M. F. Janowitz
Conference paper
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Summary

A model for cluster analysis based on order theoretic techniques was introduced some 15 years ago. The paper seeks to provide a coherent introduction to this model, and to give an indication of some of the applications which it generated.

Keywords

Cluster Algorithm Cluster Method Principal Ideal Residual Mapping Order Ideal 
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. BANDELT, H.-J. and DRESS, A. (1994): An order theoretic framework for overlapping clustering, preprint.Google Scholar
  2. BAULIEU, F. (1985): Order theoretic classification of monotone equivariant cluster methods, Algebra Universalis, 20, 351–367. CrossRefGoogle Scholar
  3. BAULIEU, F. (1991): Classification of normalized cluster methods in an order theoretic model, Discrete Applied Mathematics, 32, (1991), 1–29. CrossRefGoogle Scholar
  4. CRITCHLEY, F. and VAN CUTSEM, B. (1993): An order-theoretic unification and generalisation of certain fundanetal bijections in mathematical classification - I, preprint.Google Scholar
  5. CRITCHLEY, F. and VAN CUTSEM, B. (1993a): An order-theoretic unification and generalisation of certain fundanetal bijections in mathematical classification - II, preprint.Google Scholar
  6. DIDAY, E. (1987): Orders and overlapping clusters by pyramids, Rapports de Recherche INRIA 730, INRIA-Rocqencourt.Google Scholar
  7. HERDEN, G. (1984): Some aspects of clustering functions, SIAM Journal on Algebraic and Discrete Methods, 5, 101–116.CrossRefGoogle Scholar
  8. HUBERT, L. (1977): A set-theoretical approach to the problem of hierarchical clustering, Journal of Math. Psychology, 15, 70–88. CrossRefGoogle Scholar
  9. JANOWITZ, M. F. (1978): An order theoretic model for cluster analysis, SIAM Journal on Applied Mathematics, 34, 55–72. CrossRefGoogle Scholar
  10. JANOWITZ, M. F. (1978a): Semiflat X-cluster methods,Discrete Mathematics, 21, 47–60. CrossRefGoogle Scholar
  11. JANOWITZ, M. F. (1978b): Monotone equivariant cluster methods, SIAM Journal on Applied Mathematics, 34, 148–165. CrossRefGoogle Scholar
  12. JANOWITZ, M. F. (1981): Continuous X-cluster methods, Discrete Applied Mathematics, 3, 107–112. CrossRefGoogle Scholar
  13. JANOWITZ, M. F. and STINEBRICKNER, R. (1993): Preservation of weak order equivalence, Mathematical Social Sciences, 25, 181–197. CrossRefGoogle Scholar
  14. JANOWITZ, M. F. and STINEBRICKNER, R. (1993a): Compatibility in a graph-theoretic setting, Mathematical Social Sciences, 25, 251–279. CrossRefGoogle Scholar
  15. JANOWITZ, M. F. and SCHWEIZER, B. (1989): Ordinal and percentile clustering, Mathematical Social Sciences, 18, 135–186. CrossRefGoogle Scholar
  16. JANOWITZ, M. F. and WILLE, R. (1993): Compatibility with respect to isotone mappings, to appear.Google Scholar
  17. JARDINE, N. and SIBSON, R. (1968): The construction of hierarchic and non-hierarchic classifications, Computer Journal, 11, 177–184 Google Scholar
  18. JARDINE, N. and SIBSON, R. (1968a): A model for taxonomy, Mathematical Biosciences, 2, 465–482. CrossRefGoogle Scholar
  19. JARDINE, N. and SIBSON, R. (1971): Choice of methods for automatic classification, Computer Journal, 14, 404–406. CrossRefGoogle Scholar
  20. JARDINE, N. and SIBSON, R. (1971a): Mathematical Taxonomy, Wiley, New York.Google Scholar
  21. MATULA, D. W. (1971): Graph theoretic techniques for cluster algorithms. In: J. van Ryzin, (ed.): Classification and clustering, Academic Press, New York, 95–129.Google Scholar
  22. MICHALSKI, R. S., and STEPP, R. E., III (1983): Automated construction of classifications: conceptual clustering versus numerical taxonomy.IEEE Transactions on Pattern Analysis and Machine Intelligence, 5, 396–410. CrossRefGoogle Scholar
  23. POWERS, R. C. (1988): Order theoretic classification of percentile cluster methods, Ph.D. dissertation, University of Massachusetts, Amherst, MA, USA.Google Scholar
  24. SCHWEIZER, B. and SKLAR, A. (1983): Probabilistic Metric Spaces, North- Holland, New York.Google Scholar
  25. SIBSON, R. (1970): A model for taxonomy II, Mathematical Biosciences, 6, 405- 430. Google Scholar
  26. WILLE, R. (1982): Restructuring lattice theory: an approach based on hierarchies of concepts. In: I. Rival (ed.): Ordered Sets, NATO ASI Series 83, Reidel, Dordrecht.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1996

Authors and Affiliations

  • M. F. Janowitz
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA

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