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Convexity Methods in Classification

  • Jean-Paul Rasson
Conference paper
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)

Summary

We investigate the solutions to the clustering and the discriminant analysis Problems when the points are supposed to be distributed according to Poisson Processes on convex supports. This leads to very intuitive criteria for homogeneous Poisson Processes based on the Lebesgue measures of convex hulls. For non homogeneous Poisson Processes, the Lebesgue measures have to be replaced by intensities integrated on convex hulls.

Similar geometrical tools, based on the Lebesgue measure, are used in the context of Pattern Recognition. First, a discriminant analysis algorithm is developed for estimating a convex domain when inside and outside points are available. Generalisation to non convex domains is explored.

Keywords

Discriminant Analysis Convex Hull Lebesgue Measure Convex Domain Cluster Problem 
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. Baufays, P., Ralson, J.-P. (1984): Une nouvelle règle de classement, utilisant l’enveloppe convexe et la mesure de Lebesgue, Statistique et Analyse. des Données, 2, pp. 31–47.Google Scholar
  2. Baufays, P., Rasson, J.P. (1984): Propriétés théoriques et pratiques et applications d’une nouvelle règle de classement, Statistique et Analyse des Données, vo1. 9 /3, pp. 1–10.Google Scholar
  3. Baufays, P., Rasson, J.P. (1985): A new geometric discriminant rule. Computational Statistics Quaterly, vol. 2, issue 1, 15–30.MATHGoogle Scholar
  4. Degytar, Y.U., Finkelsh Tein, M.Y. (1974): Classification Algorithms Based on Construction of Convex Hulls of Sets, Engineering Cybermetics, 12, pp. 150–154.Google Scholar
  5. Duda, R.O., Hart, P.E. (1973): Pattern Recognition and Scene Analysis, Wiley, Chichester. Efron, B. (1965): The Convex Hull of a Random Set of Points. Biometrika, 52, pp. 331–453.Google Scholar
  6. Fisher, L., Van Ness, J.W. (1971): Admissible Clustering Procedures. Biometrika, 58, pp. 91–104.MathSciNetMATHCrossRefGoogle Scholar
  7. Fukunaga, K. (1972): Introduction to Statistical Pattern Recognition, Academic Press, New York.Google Scholar
  8. Grenander, U. (1973): Statistical geometry: a tool for pattern analysis. Bulletin of the American Mathematical Society, vol. 79, 829–856.Google Scholar
  9. Hand, D.J. (1981): Discrimination and Classification,Wiley,Chichester.Google Scholar
  10. Hardy, A., Rasson, J.-P. (1982): Une Nouvelle Approche des Problèmes de Classification Automatique. Statistique et Analyse des Données, 7, pp. 41–56.MathSciNetMATHGoogle Scholar
  11. Hartigan, J.A. (1975): Clustering Algorithms, Wiley, Chichester.MATHGoogle Scholar
  12. Mac Lachlan, G.J. (1992): Discriminant Analysis and Statistical Pattern Recognition, Wiley, New York.CrossRefGoogle Scholar
  13. Moore, M., Lemay, Y. and Archambault, S. (1988): Algorithms to reconstruct a convex set from sample points. Computing Science and Statistics, In: Proceedings of the 20th Symposium on the Interface, Eds. E.J. Wegma.n, D.T. Gantz and J.J. Miller, ASA, Virginia, 553–558.Google Scholar
  14. Rasson, J.P. (1979): Estimation de domaines convexes du plan. Statistique et Analyse des Données, l, pp. 31–46.Google Scholar
  15. Rémon, M. (1994): The estimation of a convex domain when inside and outside observations are available. Supplemento ai Rendiconti del Circolo Matematico di Palermo, serie II, no 35, 227–235.Google Scholar
  16. Rémon, M. (1996): A Discriminant Analysis Algorithm for the Inside/Outside Problem. Computational Statistics and Data Analysis.Google Scholar
  17. Ripley, B.D., Rasson, J.P. (1977): Finding the edge of a Poisson forest. Journal of Applied Pro bab ility, 14, 483–491.MathSciNetMATHCrossRefGoogle Scholar
  18. Toussaint, G.T. (1980): Pattern Recognition and Geometrical Complexity, In: Proc. Fith Int. Conf Pattern Recognition, pp. 1324–1347, IEEE.Google Scholar
  19. Baufays, P., Ralson, J.-P. (1984): Une nouvelle règle de classement, utilisant l’enveloppe convexe et la mesure de Lebesgue, Statistique et Analyse. des Données, 2, pp. 31–47.Google Scholar
  20. Baufays, P., Ralson, J.-P. (1984): Une nouvelle règle de classement, utilisant l’enveloppe convexe et la mesure de Lebesgue, Statistique et Analyse. des Données, 2, pp. 31–47.Google Scholar
  21. Baufays, P., Rasson, J.P. (1984): Propriétés théoriques et pratiques et applications d’une nouvelle règle de classement, Statistique et Analyse des Données, vo1. 9 /3, pp. 1–10.Google Scholar
  22. Baufays, P., Rasson, J.P. (1985): A new geometric discriminant rule. Computational Statistics Quaterly, vol. 2, issue 1, 15–30.MATHGoogle Scholar
  23. Degytar, Y.U., Finkelsh Tein, M.Y. (1974): Classification Algorithms Based on Construction of Convex Hulls of Sets, Engineering Cybermetics, 12, pp. 150–154.Google Scholar
  24. Duda, R.O., Hart, P.E. (1973): Pattern Recognition and Scene Analysis, Wiley, Chichester. Efron, B. (1965): The Convex Hull of a Random Set of Points. Biometrika, 52, pp. 331–453.Google Scholar
  25. Fisher, L., Van Ness, J.W. (1971): Admissible Clustering Procedures. Biometrika, 58, pp. 91–104.MathSciNetMATHCrossRefGoogle Scholar
  26. Fukunaga, K. (1972): Introduction to Statistical Pattern Recognition, Academic Press, New York.Google Scholar
  27. Grenander, U. (1973): Statistical geometry: a tool for pattern analysis. Bulletin of the American Mathematical Society, vol. 79, 829–856.Google Scholar
  28. Hand, D.J. (1981): Discrimination and Classification,Wiley,Chichester.Google Scholar
  29. Hardy, A., Rasson, J.-P. (1982): Une Nouvelle Approche des Problèmes de Classification Automatique. Statistique et Analyse des Données, 7, pp. 41–56.MathSciNetMATHGoogle Scholar
  30. Hartigan, J.A. (1975): Clustering Algorithms, Wiley, Chichester.MATHGoogle Scholar
  31. Mac Lachlan, G.J. (1992): Discriminant Analysis and Statistical Pattern Recognition, Wiley, New York.CrossRefGoogle Scholar
  32. Moore, M., Lemay, Y. and Archambault, S. (1988): Algorithms to reconstruct a convex set from sample points. Computing Science and Statistics, In: Proceedings of the 20th Symposium on the Interface, Eds. E.J. Wegma.n, D.T. Gantz and J.J. Miller, ASA, Virginia, 553–558.Google Scholar
  33. Rasson, J.P. (1979): Estimation de domaines convexes du plan. Statistique et Analyse des Données, l, pp. 31–46.Google Scholar
  34. Rémon, M. (1994): The estimation of a convex domain when inside and outside observations are available. Supplemento ai Rendiconti del Circolo Matematico di Palermo, serie II, no 35, 227–235.Google Scholar
  35. Rémon, M. (1996): A Discriminant Analysis Algorithm for the Inside/Outside Problem. Computational Statistics and Data Analysis.Google Scholar
  36. Ripley, B.D., Rasson, J.P. (1977): Finding the edge of a Poisson forest. Journal of Applied Pro bab ility, 14, 483–491.MathSciNetMATHCrossRefGoogle Scholar
  37. Toussaint, G.T. (1980): Pattern Recognition and Geometrical Complexity, In: Proc. Fith Int. Conf Pattern Recognition, pp. 1324–1347, IEEE.Google Scholar

Copyright information

© Springer Japan 1998

Authors and Affiliations

  • Jean-Paul Rasson
    • 1
  1. 1.Département de MathématiqueF.U.N.D.P.NamurBelgium

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