Skip to main content
SpringerLink
Log in

Generalized Classifications

  • Chapter
Towards a General Theory of Classifications

Part of the book series: Studies in Universal Logic ((SUL))

  • 1360 Accesses

Abstract

One of the main problems for a general theory of classifications is that classifications do not amount to hierarchical classifications. There are in fact many other kinds of classification (Sect. 5.2), and the most recent ones in the domain of library science, for example, may have the same subjects inserted in many classes, which obviously means that classes intersect.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This inequality defines in fact what we call now a “linear dissimilarity”.

  2. 2.

    It is obviously a particular choice. Another solution would have been to start from the characteristics of the objects without using a dissimilarity.

  3. 3.

    This empirical way of introducing an order in a system of classes must not hide the algebraic viewpoint on the question. The French mathematician Claude Frasnay, a disciple of Roland Fraïssé, already proved, in the 1960s, some conjectures of Fraïssé concerning the interpretation of special m-ary relations by a total order (see [181]).

  4. 4.

    We shall see that our general theory of classifications tries to solve this question before any else.

  5. 5.

    For a bibliographic comment on the question, see [302], 109–128.

References

  1. Bandelt, H.J., Dress, A.W.: Weak hierarchies associated with similarity measures—an additive clustering technique. Bull. Math. Biol. 51, 133–166 (1989).

    MathSciNet  MATH  Google Scholar 

  2. Bandelt, H.J.: Four point characterization of the dissimilarity functions obtained from indexed closed weak hierarchies. Mathematisches Seminar, Universität Hamburg, Germany, 1992

    Google Scholar 

  3. Batbedat, A.: Les isomorphismes HTS et HTE (d’après la bijection de Benzécri-Johnson). Metron 46(1–4), 47–59 (1988)

    MathSciNet  MATH  Google Scholar 

  4. Batbedat, A.: Les dissimilarités médas ou arbas. Stat. Anal. Don. 14(3), 1–18 (1989)

    MathSciNet  Google Scholar 

  5. Belmandt, Z.: Manuel de prétopologie et ses applications. Hermès, Paris (1993)

    MATH  Google Scholar 

  6. Benzécri, J.-P., et al.: L’Analyse des données, tome 1, taxinomie. Dunod, Paris (1973)

    Google Scholar 

  7. Benzécri, J.-P., et al.: L’Analyse des données, tome 2, correspondances. Dunod, Paris (1973)

    Google Scholar 

  8. Berge, C.: Graphes et hypergraphes. Dunod, Paris (1970)

    MATH  Google Scholar 

  9. Bertrand, P., Janowitz, M.F.: Pyramids and weak hierarchies in the ordinal model for clustering. Discrete Appl. Math. 122(1–3), 55–81 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brucker, F.: Modèles de classification en classes empiétantes. Thèse pour le doctorat en mathématiques et informatique, EHESS, 6 juillet 2001

    Google Scholar 

  11. Brucker, F., Barthélemy, J.-P.: Eléments de classifications. Hermès-Lavoisier, Paris (2007)

    Google Scholar 

  12. Buneman, P.: The recovery of trees for measure of dissimilarity. In: Hodson, F.R., Kendal, D.G., Tauty, P. (eds.) Mathematics in Archeological and Historical Sciences, pp. 387–395. Edinburgh University Press, Edinburgh (1971)

    Google Scholar 

  13. Chandon, J.-L., Pinson, S.: Analyse typologique. Masson, Paris (1980)

    Google Scholar 

  14. Diatta, J.: Une extension de la classification hiérarchique: les quasi-hiérarchies. Thèse de doctorat, Mathématiques appliquées, Université de Provence, Aix-Marseille I, France, 1996

    Google Scholar 

  15. Diatta, J., Fichet, B.: From Asprejan hierarchies and Bandelt-Dress weak hierarchies to quasi-hierarchies. In: Diday, E., Lechevallier, Y., Schader, M., Burtschy, B. (eds.) New Approaches in Classification and Data Analysis, pp. 111–118. Springer, Berlin (1994)

    Chapter  Google Scholar 

  16. Diday, E.: Une représentation visuelle des classes empiétantes: les Pyramides. Rapport de recherche 291, Institut National de Recherche en Informatique et en Automatique (INRIA), centre de Rocquencourt, Domaine de Voluceau, B P. 105, 78153, Le Chesnay, Cedex, France, 1984

    Google Scholar 

  17. Diday, E.: Orders and overlapping clusters in pyramids. In: De Leeuw, J., Heiser, W., Meulman, J., Critchley, F. (eds.) Multidimensional Data Analysis Proceedings, pp. 201–234. DSWO Press, Leiden (1986)

    Google Scholar 

  18. Duchet, P.: Représentations, noyaux, en théorie des graphes et des hypergraphes. Thèse, Université de Paris VI, 1979

    Google Scholar 

  19. Durand, C.: Sur la représentation pyramidale en Analyse des Données. Mémoire de DEA en Mathématiques Appliquées, Université de Provence, Marseille, France, 1986

    Google Scholar 

  20. Durand, C.: Ordres et graphes pseudo-hiérarchiques: théorie et optimisation algorithmique. Thèse de doctorat, Mathématiques appliquées. Université de Provence, Aix-Marseille I, France, 1989

    Google Scholar 

  21. Durand, C., Fichet, B.B.: One-to-one correspondences in pyramidal representation: a unified approach. In: Bock, H.H. (ed.) Classification and Related Methods of Data Analysis, pp. 85–90. North-Holland, Amsterdam (1988)

    Google Scholar 

  22. Fichet, B.: Sur une extension de la notion de hiérarchie et son équivalence avec quelques matrices de Robinson. In: Actes de la Journée de statistique de la Grande Motte, 12-12-1984

    Google Scholar 

  23. Flament, D.: Hypergraphes arborés. Discrete Math. 21, 223–227 (1978)

    Article  MathSciNet  Google Scholar 

  24. Frasnay, C.: Quelques problèmes combinatoires concernant les ordres totaux et les relations monomorphes. Ann. Inst. Fourier 15(2), 415–524 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gondran, M.: La structure algébrique des classifications hiérarchiques. Ann. INSEE 22–23, 181–190 (1976)

    MathSciNet  Google Scholar 

  26. Janowitz, M.F.: An order theoretic model for cluster analysis. SIAM J. Appl. Math. 34, 55–72 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jardine, N., Sibson, R.: A model for taxonomy. Math. Biosci. 2, 465–482 (1968)

    Article  Google Scholar 

  28. Jardine, N., Sibson, R.: The construction of hierarchic and non-hierarchic classifications. Comput. J. 11, 177–184 (1968)

    Article  MATH  Google Scholar 

  29. Jardine, N., Sibson, R.: Numerical Taxonomy. Wiley, New York (1971)

    Google Scholar 

  30. Kleinberg, J.: An impossibility theorem for clustering. In: Advances in Neural Information Processing Systems (NIPS), Proceedings of the 2002 Conference, British Columbia, Canada, pp. 529–536 (2002)

    Google Scholar 

  31. Lance, G.C., Williams, W.T.: A generalised sorting strategy for computer classification. Nature 212, 218 (1966)

    Article  Google Scholar 

  32. Lance, G.C., Williams, W.T.: A general theory of classification sorting. Comput. J. 9, 373–380 (1967)

    Article  Google Scholar 

  33. Leclerc, B., Cucumel, G.: Consensus en classification: une revue bibliographique. Math. Sci. Hum. 100, 109–128 (1987)

    MathSciNet  MATH  Google Scholar 

  34. Lehel, J., Mc Morris, F.R., Powers, R.C.: Consensus methods for pyramids and other hypergraphs. In: Hayashi, C., Yajima, K., Tanaka, Y., Bock, H.-H., Baba, Y. (eds.) Data Science, Classification and Related Methods, pp. 187–190 (1998)

    Google Scholar 

  35. Puzicha, J., Hofmann, T., Buhmann, J.: A theory of proximity based clustering: Structure detection by optimization. Pattern Recognit. 33(4), 617–634 (2000)

    Article  Google Scholar 

  36. Robinson, W.S.: A method for chronologically ordering archaeological deposits. Am. Antiquity 16, 293–301 (1951)

    Article  Google Scholar 

  37. Sokal, R.R., Sneath, P.H.: Principles of Numerical Taxonomy. Freeman, San Francisco (1973)

    MATH  Google Scholar 

  38. Sørensen, T.: A method of establishing groups of equal amplitude in plant sociology based on similarity of species content. Konge Dan. Vidensk. Selsk., Biol. Skr. 5, 1–34 (1948)

    Google Scholar 

  39. Van Cutsem, B. (ed.): Classification and Dissimilarity Analysis. Springer, New York (1994)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Philosophy, Université Jean Moulin – Lyon III, Lyon, France

    Daniel Parrochia

  2. Ecole Nationale Supérieure des Sciences de l’Information et de la Bibliothèque, Villeurbanne, France

    Pierre Neuville

Authors
  1. Daniel Parrochia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre Neuville
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this chapter

Cite this chapter

Parrochia, D., Neuville, P. (2013). Generalized Classifications. In: Towards a General Theory of Classifications. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0609-1_5

Download citation

Publish with us

Policies and ethics

Search

Navigation