Simultaneous Visualization and Clustering Methods as an Alternative to Kohonen Maps

  • H. H. Bock
Part of the International Centre for Mechanical Sciences book series (CISM, volume 382)


Kohonen maps are often used for visualizing high-dimensional feature vectors in lowdimensional space. This approach is often recommended for supporting the clustering of data. In this paper an alternative approach is proposed which is more in the lines of multivariate statistics and provides a simultaneous visualization and clustering of data. This approach combines projection and embedding methods (such as principal components or multidimensional scaling) with clustering criteria and corresponding optimization algorithms. Four distinct methods are proposed: projection pursuit clustering for quantitative data vectors, two MDS clustering methods for dissimilarity data (either with or without a representation of classes) and a group difference scaling method (known from. literature).


Multidimensional Scaling Class Representative Cluster Criterion Average Dissimilarity Simultaneous Visualization 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ambroise, Ch., G. Govaert (1996a): Constrained clustering and Kohonen self-organizing maps. J. of Classification 13, 299–313.CrossRefMATHMathSciNetGoogle Scholar
  2. Ambroise, Ch., G. Govaert (1996b): Analysing dissimilarity matrices via Kohonen maps. Lecture given at the 5th Conference of the International Federation of Classification Societies, Kobe/Japan, March 1996, Abstract volume II, p. 96–99.Google Scholar
  3. Arabie, Ph., Hubert, L. and G. De Soete (eds.) (1996): Clustering and classification. World Science Publishers, River Edge/NJ.MATHGoogle Scholar
  4. Bezdek, J.C. (1981): Pattern recognition with fuzzy objective function algorithms. Plenum Press, New York.CrossRefMATHGoogle Scholar
  5. Bock, H.H. (1974): Automatische Klassifikation (Cluster-Analyse). Vandenhoeck Ruprecht, Göttingen.MATHGoogle Scholar
  6. Bock, H.H. (1979a): Clusteranalyse mit unscharfen Partitionen. In: Bock, H.H. (ed.): Klassifikation und Erkenntnis III: Numerische Klassifikation. Indeks-Verlag, Frankfurt, 137–163.Google Scholar
  7. Bock, H.H. (1979b): Fuzzy clustering procedures. In: R. Tomassone (ed.): Analyse des donées et informatique. Institut de Recherche en Informatique et en Automatique (INRIA), Le Chesnay, France, 205–218.Google Scholar
  8. Bock, H.H. (1985): On some significance tests in cluster analysis. J. of Classification 2, 77–108.CrossRefMATHMathSciNetGoogle Scholar
  9. Bock, H.H. (1986): Multidimensional scaling in the framework of cluster analysis. In: P.O. Degens, 11.-J. Hermes, O. Opitz (Eds.): Classification and its environment. Studien zur Klassifikation no. 17. Indeks-Verlag, Frankfurt, 1986, 247–258.Google Scholar
  10. Bock, H.H. (1987a): On the interface between cluster analysis, principal component analysis, and multidimensional scaling. In: H. Bozdogan, A.K. Gupta (Eds.): Multivariate statistical modeling and data analysis. D. Reidel, Dordrecht, 1987, 17–34.CrossRefGoogle Scholar
  11. Bock, H.H. (1987b): Metrische Modelle bei der Klassifikation mit Unähnlichkeitsmatrizen. In: H. Iserman et al. (Eds.): Operations Research Proceedings 1986. Springer-Verlag, Berlin, 1987, 440–446.Google Scholar
  12. Bock, H.H. (1996a): Probabilistic models in partitional cluster analysis. In: A. Ferligoj and A. Kramberger (eds.): Developments in data analysis. FDV, Metodoloski zvezki, 12, Ljubljana, Slovenia, 1996, 3–25.Google Scholar
  13. Bock, H.H. (1996b): Probabilistic methods in cluster analysis. Computational Statistics and Data Analysis 3, 5–28.CrossRefGoogle Scholar
  14. Bock, H.H. (1996c): Probability models and hypotheses testing in partitioning cluster analysis. In: P. Arabic, L. Hubert and G. De Soete (eds.): Clustering and classification. World Science Publishers, River Edge/NJ, 1996, 377–453.CrossRefGoogle Scholar
  15. Bock, H.H. (1997): Simultaneous clustering and visualization methods with a view towards Kohonen’s neural networks. In: G. Della Riccia (ed.): Learning, networks and statistics. Proc. ISSEK Workshop, University of Udine, September 1996. CISM Courses and Lectures, Springer-Verlag, Wien, 1997 (in press).Google Scholar
  16. Borg, I., Groenen, P. (1997): Modern multidimensional scaling: Theory and applications. Springer-Verlag, New York.CrossRefMATHGoogle Scholar
  17. Braverman, E.M. (1966): The method of potential functions in the problem of training machines to recognize patterns witout a teacher. Automation Remote Control 27, 1748–1771.Google Scholar
  18. Dorofeyuk, A.A. (1966): Teaching algorithms for a pattern recognition machine witout a teacher based on the method of potential functions. Automation Remote Control 27, 1728–1737.Google Scholar
  19. Fort, J.-C., Pagès, G. (1996): About the Kohonen algorithm: Strong or weak sel-organization? Neural networks 9, 773–785.CrossRefGoogle Scholar
  20. Groenen, P.J.F., Heiser, W.J. (1996): The tunneling method for global optimization in multidimensional scaling. Psychometrika 61, 529–550.CrossRefMATHGoogle Scholar
  21. Heiser, W. (1993): Clustering in low-dimensional spaces. In: O. Opitz, B. Lausen, R. Klar (Eds.): Information and classification–Concepts, methods and applications. Springer-Verlag, Heidelberg, 1993, 162–173.CrossRefGoogle Scholar
  22. Heiser, W., Groenen, P. (1997): Cluster differences scaling with a within-clusters loss component and a fuzzy successive approximation strategy to avoid local minima. Psychometrika 62, 63–83.CrossRefMATHMathSciNetGoogle Scholar
  23. Höppner, F., Klawonn, F. and R. Kruse (1997): Fuzzy-Clusteranalyse. Verfahren für die Bilderkennung, Klassifizierung und Datenanalyse. Verlag Vieweg, Wiesbaden.Google Scholar
  24. Jokusch, S., Ritter, H. (1994): Self-organizing maps: Local competition and evolutionary optimization. Neural Networks 7, 1229–1239.Google Scholar
  25. Keller, J.B. (1962): Factorization of matrices by least squares. Biometrika 49, 239–242.MATHMathSciNetGoogle Scholar
  26. Kohonen, T. (1990): The self-organizing map. Proceedings of the IEEE 78, 1464–1480.CrossRefGoogle Scholar
  27. Kohonen, T. (1991): Artificial neural networks 1,2. North Holland, Amsterdam.Google Scholar
  28. Krzanowski, W.J. (1994): Ordination in the presence of group structure for general multivariate data. J. of Classification 11, 195–207.CrossRefMATHGoogle Scholar
  29. MacQueen, J. (1967): Some methods for classification and analysis of multivariate observations. In: L. LeCam, J. Neyman (eds.): Proc. 5th Berkeley Symp. Math, Statist. Prob. 1965/66. Univ. California Press, Berkeley, 1967, Vol. 1, 281–297.Google Scholar
  30. Mathar, R. (1985): The best Euclidean fit to a given distance matrix in prescribed dimensions. Linear Algebra and its Applications 67, 1–6.CrossRefMATHMathSciNetGoogle Scholar
  31. Mathar, R. (1994): Multidimensional scaling with /p-distances, a unifying approach. In: H.H. Bock, W. Lenski, M.M. Richter (eds.): Information sytsems and data analysis. Springer-Verlag, Heidelberg, 325–331.CrossRefGoogle Scholar
  32. Murtagh, F., Hernandez-Pajares, M. (1995): The Kohonen self-organizing map method: an assessment. J. of Classification 12, 165–190.CrossRefMATHGoogle Scholar
  33. Ritter, H. (1997): Neural networks for rapid learning in computer vision and robotics. This volume.Google Scholar
  34. Ritter, H., Schulten, K. (1986): On the stationary state of Kohonen’s self-organizing sensory mapping. Biological Cybernetics 54, 99–106.CrossRefMATHGoogle Scholar
  35. Ritter, H., T. Martinetz and K. Schulten (1991): Neuronale Netze. Eine Einführung in die Neuroinformatik selbstorganisierender Netzwerke. Addison-Wesley, Bonn.Google Scholar
  36. Tolat, V.V. (1990): An analysis of Kohorte ‘s self-organizing maps using a system of energy functions. Biological Cybernetics 64, 155–164.CrossRefMATHGoogle Scholar
  37. Ultsch, A. (1993): Self-organizing neural networks for visualisation and classification. In: O. Opitz, B. Lausen, R. Klar (Eds.): Information and classification–Concepts, methods and applications. Springer-Verlag, Heidelberg, 1993, 301–306.CrossRefGoogle Scholar
  38. Varfis, A., Versino, C. (1992): Clustering of socio-economic data with Kohonen maps. International Journal on Neural and Mass-Parallel Computing and Information Systems “Neural Network World”, 2, 813–833.Google Scholar

Copyright information

© Springer-Verlag Wien 1997

Authors and Affiliations

  • H. H. Bock
    • 1
  1. 1.RWTH AachenAachenGermany

Personalised recommendations