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


The unfolding model is a model for preferential choice. It assumes that different individuals perceive various objects of choice in the same way but differ with respect to what they consider an ideal combination of the objects’ attributes. In unfolding, the data are usually preference scores (such as rank-orders of preference) of different individuals for a set of choice objects. These data can be conceived as proximities between the elements of two sets, individuals and choice objects. Technically, unfolding can be seen as a special case of MDS where the within-set proximities are missing. Individuals are represented as “ideal” points in the MDS space so that the distances from each ideal point to the object points correspond to the preference scores. We indicate how an unfolding solution can be computed by the majorization algorithm. Two variants for incorporating transformations are discussed: the conditional approach, which only considers the relations of the data values within rows (or columns), and the unconditional approach, which considers the relations among all data values as meaningful. It is found that if transformations are allowed on the data, then unfolding solutions are subject to many potential degeneracies. Stress forms that reduce the chances for degenerate solutions are discussed. Also, a mixed ordinal-linear method is suggested as a reasonable compromise.


Trivial Solution Ideal Point Object Point Attribute Point Proximity Matrix 
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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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