Algorithms in Multidimensional Scaling

  • Rudolf Mathar
Conference paper


The ideas described in this paper are motivated by a basic problem in Multidimensional Scaling which is solved not quite satisfactory up to now. Suppose that there are given pairwise dissimilarities δij, 1 ≤ i, j ≤ n, as nonnegative real numbers between n objects. These originate from the special type of application and may be, for example in cartography, measurements of pairwise distances disturbed by additive random errors or, in psychology, individual assessments of the deviation in behaviour of n persons measured on a certain real scale. There are many other applications in a variety of fields, a structured overview may be obtained from de Leeuw & Heiser (1980, 1982).


Unit Ball Multidimensional Scaling Homogeneous Function Nonnegative Real Number Euclidean Case 
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. de Leeuw J. (1977) Applications of convex analysis to Multidimensional Scaling. In: J.R. Barra et al. (eds.), Recent Developments in Statistics, North-Holland Publ. Comp., 133–145.Google Scholar
  2. de Leeuw J., Heiser W. (1980) Multidimensional Scaling with restrictions on the configuration. In: P.R. Krishnaiah (ed.), Multivariate Analysis-V, North-Holland, Amsterdam, 501–522.Google Scholar
  3. de Leeuw J., Heiser W. (1982) Theory of Multidimensional Scaling. In: P.R. Krishnaiah, L.N. Kamal (eds.), Handbook of Statistics, Vol 2, North-Holland, Amsterdam, 285–316.Google Scholar
  4. Gaffke N., Mathar R. (1989) A cyclic projection algorithm via duality. Metrika, 36, 29–54.MathSciNetMATHCrossRefGoogle Scholar
  5. Ihm P. (1986) A problem on bacterial taxonomy. Working paper, EURATOM, Ispra.Google Scholar
  6. Luenberger D.G. (1969) Optimization by Vector Space Methods. J. Wiley, New York.MATHGoogle Scholar
  7. Mathar R. (1985) The best Euclidian fit to a given distance matrix in prescribed dimensions. Linear Algebra Appl., 67, 1–6.MathSciNetMATHCrossRefGoogle Scholar
  8. Mathar R (1988) Dimensionality in constrained scaling. In: H.H. Bock (ed.), Classification and Related Methods of Data Analysis, North-Holland, Amsterdam, 479–488.Google Scholar
  9. Mathar R. (1989) Cyclic projections in data analysis. To appear: Proceedings of the DGOR Conference, Berlin 1988.Google Scholar
  10. Rockafellar R.T. (1970) Convex Analysis. Princeton University Press, Princeton.MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1989

Authors and Affiliations

  • Rudolf Mathar
    • 1
  1. 1.Institute of MathematicsUniversity of AugsburgGermany

Personalised recommendations