MDS Models and Measures of Fit

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


MDS models are defined by specifying how given similarity or dissimilarity data, the proximities p ij , are mapped into distances of an m-dimensional MDS configuration X. The mapping is specified by a representation function, f : P ij d ij (X), which specifies how the proximities should be related to the distances. In practice, one usually does not attempt to strictly satisfy f. Rather, what is sought is a configuration (in a given dimensionality) whose distances satisfy f as closely as possible. The condition “as closely as” is quantified by a badness-of-fit measure or loss function. The loss function is a mathematical expression that aggregates the representation errors, e ij = f (p ij )−d ij (X), over all pairs (i, j). A normed sum-of-squares of these errors defines Stress, the most common loss function in MDS. How Stress should be evaluated is a major issue in MDS. It is discussed at length in this chapter, and various criteria are presented.


True Dimensionality True Distance Dissimilarity Data Monotone Regression Shepard Diagram 
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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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