Classical Scaling

  • Ingwer Borg
  • Patrick Groenen
Part of the Springer Series in Statistics book series (SSS)


Since the first practical method available for MDS was a technique due to Torgerson (1952, 1958) and Gower (1966), classical scaling is also known under the names Torgerson scaling and Torgerson-Gower scaling. It is based on theorems by Eckart and Young (1936) and by Young and Householder (1938). The basic idea of classical scaling is to assume that the dissimilarities are distances and then find coordinates that explain them. In (7.5) a simple matrix expression is given between the matrix of squared distances D (2)(X) — we also write D (2) for short — and the coordinate matrix X which shows how to get squared Euclidean distances from a given matrix of coordinates and then scalar products from these distances. In Section 7.9, the reverse was discussed, i.e., how to find the coordinate matrix given a matrix of scalar products B = XX′. Classical scaling uses the same procedure but operates on squared dissimilarities Δ (2) instead of D (2), since the latter is unknown. This method is popular because it gives an analytical solution, requiring no iterations.


Scalar Product Loss Function Negative Eigenvalue External Variable Classical Scaling 
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Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Ingwer Borg
    • 1
  • Patrick Groenen
    • 2
  1. 1.Zentrum für Umfragen, Methoden und AnalysenMannheimGermany
  2. 2.Department of Data TheoryLeiden UniversityLeidenThe Netherlands

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