Fitting Graphs and Trees with Multidimensional Scaling Methods

  • Willem J. Heiser
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)


The symmetric difference between sets of qualitative elements (called features) forms the basis of a distance model that can be used as a general framework for fitting a particular class of graphs, which includes additive trees, hierarchical trees and circumplex structures. It is shown how to parametrize this fitting problem in terms of a lattice of subsets, and how inclusion relations between feature sets lead to additivity of distance along paths in a graph. An algorithm based on alternating least squares and on the recent method of cluster differences scaling is described, and illustrated for the general case.


Loss Function Edge Length Latent Node Feature Distance Feature Graph 
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  1. Abdi, H. (1990): Additive tree representations, In: Trees and Hierarchical Structures, Dress, A. et al. (Eds.), 43–59, Springer Verlag, Berlin.CrossRefGoogle Scholar
  2. Arabie, P., and Carroll, J.D. (1980): MAPCLUS: A mathematical programming approach to fitting the ADCLUS model, Psychometrika, 45, 211–235.MATHCrossRefGoogle Scholar
  3. Arabie, P., and Hubert, L. (1992): Combinatorial data analysis, Annual Review of Psychology, 43, 169–203.CrossRefGoogle Scholar
  4. Barthélemy, J.-P. and Guénoche, A. (1991): Trees and Proximity Representations, Wiley, New York.MATHGoogle Scholar
  5. Boorman, S.A. and Arabie, P. (1972): Structural measures and the method of sorting, In: Multidimensional Scaling: Theory and Applications in the Behavioral Sciences, Shepard, R.N. et al. (Eds.), 225–249, Seminar Press, New York.Google Scholar
  6. Buneman, P. (1971): The recovery of trees from measures of dissimilarity, In: Mathematics in the Archaeological and Historical Sciences, Hodson, F.R. et al. (Eds.), 387–395, Edinburgh University Press, Edinburgh.Google Scholar
  7. Carroll, J.D. (1976): Spatial, non-spatial and hybrid models for scaling, Psychometrika, 41, 439–463.MATHCrossRefGoogle Scholar
  8. Chandon, J.L., Lemaire, J., and Pouget, J. (1980): Construction de l’ultramétrique la plus proche d’une dissimilarité au sens des moindres carrés, R.A.I.R.O. Recherche Opérationelle, 14, 157–170.MathSciNetMATHGoogle Scholar
  9. Corter, J.E., and Tversky, A. (1986): Extended similarity trees, Psychometrika, 51, 429–451.Google Scholar
  10. Cunningham, J.P. (1978): Free trees and bidirectional trees as representations of psychological distance, Journal of Mathematical Psychology, 17, 165–188.MathSciNetMATHCrossRefGoogle Scholar
  11. De Soete, G. (1983): A least squares algorithm for fitting additive trees to proximity data, Psychometrika, 48, 621–626.CrossRefGoogle Scholar
  12. Felsenstein, J. (Ed.)(1983): Numerical Taxonomy,Springer Verlag, Heidelberg.Google Scholar
  13. Flament, C. (1963): Applications of Graph Theory to Group Structure, Prentice-Hall, Englewood Cliffs, New Jersey.Google Scholar
  14. Goodman, N. (1951): The Structure ofAppearence, Bobbs-Merrill, Indianapolis, Indiana. Goodman, N. (1977): The Structure ofAppearence ( 3rd ed. ), Reidel, Dordrecht, Holland.Google Scholar
  15. Groenen, P.J.F., Mathar, R., and Heiser, W.J. (1995): The majorization approach to multidimensional scaling for Minkowski distances, Journal of Classification, 12, 3–19.MathSciNetMATHCrossRefGoogle Scholar
  16. Guttman, L. (1954): A new approach to factor analysis: The radex, In: Mathematical thinking in the social sciences, Lazarsfeld, P.F. (Ed.), 258–348, The Free Press, Glencoe, Illinois.Google Scholar
  17. Hakimi, S.L., and Yau, S.S. (1965): Distance matrix of a graph and its realizability, Quarterly of Applied Mathematics, 22, 305–317.MathSciNetMATHGoogle Scholar
  18. Hartigan, J.A. (1967): Representation of similarity matrices by trees, Journal of the American Statistical Association, 62, 1140–1158.MathSciNetCrossRefGoogle Scholar
  19. Heiser, W.J. (1981): Unfolding analysis of proximity data, Unpublished doctoral dissertation, University of Leiden, The Netherlands.Google Scholar
  20. Heiser, W.J., and Groenen, P.J.F. (1996): Cluster differences scaling with a within-clusters loss component and a fuzzy successive approximation strategy to avoid local minima, Psychometrika,61 in press.Google Scholar
  21. Henley, N.M. (1969): A psychological study of the semantics of animal terms, Journal of Verbal Learning and Verbal Behavior, 8, 176–184.CrossRefGoogle Scholar
  22. Holman, E.W. (1995): Axioms for Guttman scales with unknown polarity, Journal of Mathematical Psychology, 39, 400–402.MATHCrossRefGoogle Scholar
  23. Hutchinson, J.W. (1989): NETSCAL: A network scaling algorithm for nonsymmetric proximity data, Psychometrika, 54, 25–52.CrossRefGoogle Scholar
  24. Klauer, K.C. (1994): Representing proximities by network models, In: New Approaches in Classification and Data Analysis, Diday, E. et al. (eds.), 493–501, Springer Verlag, Heidelberg.CrossRefGoogle Scholar
  25. Klauer, K.C., and Carroll, J.D. (1989): A mathematical programming approach to fitting general graphs, Journal of Classification, 6, 247–270.CrossRefGoogle Scholar
  26. Lawson, C.L., and Hanson, R.J. (1974): Solving least squares problems, Prentice Hall, Englewood Cliffs, NJ.Google Scholar
  27. Mirkin, B.G. (1987): Additive clustering and qualitative factor analysis methods for similarity matrices, Journal of Classification, 4, 7–31.MathSciNetMATHCrossRefGoogle Scholar
  28. Restle, F. (1959): A metric and an ordering on sets, Psychometrika, 24, 207–220. Restle, F. (1961): Psychology of Judgment and Choice, Wiley, New York.Google Scholar
  29. Roberts, F.S. (1976): Discrete Mathematical Models, with Applications to Social, Biological, and Environmental Problems, Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
  30. Sattath, S., and Tversky, A. (1977): Additive similarity trees, Psychometrika, 42, 319–345.Google Scholar
  31. Shepard, R.N., and Arabic, P. (1979): Additive clustering: Representation of similarities as combinations of discrete overlapping properties, Psychological Review, 86, 87–123.CrossRefGoogle Scholar
  32. Takane, Y., Young, F.W., and De Leeuw, J. (1977): Nonmetric individual differences in multidimensional scaling: An alternating Ieast quares method with optimal scaling features, Psychometrika, 42, 7–67.MATHCrossRefGoogle Scholar
  33. Tversky, A. (1977): Features of similarity, Psychological Review, 84, 327–352.CrossRefGoogle Scholar

Copyright information

© Springer Japan 1998

Authors and Affiliations

  • Willem J. Heiser
    • 1
  1. 1.Department of Data TheoryLeiden UniversityLeidenThe Netherlands

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