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MDS Models and Measures of Fit

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

Abstract

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.

Keywords

True Dimensionality True Distance Dissimilarity Data Monotone Regression Shepard Diagram 
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.

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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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