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Multidimensional Scaling by Majorization: A Review

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  • Groenen, Patrick J. F.
  • van de Velden, Michel

Abstract

A major breakthrough in the visualization of dissimilarities between pairs of objects was the formulation of the least-squares multidimensional scaling (MDS) model as defined by the Stress function. This function is quite flexible in that it allows possibly nonlinear transformations of the dissimilarities to be represented by distances between points in a low dimensional space. To obtain the visualization, the Stress function should be minimized over the coordinates of the points and the over the transformation. In a series of papers, Jan de Leeuw has made a significant contribution to majorization methods for the minimization of Stress in least-squares MDS. In this paper, we present a review of the majorization algorithm for MDS as implemented in the smacof package and related approaches. We present several illustrative examples and special cases.

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  • Groenen, Patrick J. F. & van de Velden, Michel, 2016. "Multidimensional Scaling by Majorization: A Review," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 73(i08).
    • Handle: RePEc:jss:jstsof:v:073:i08
    DOI: http://hdl.handle.net/10.18637/jss.v073.i08
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    1. de Leeuw, Jan, 2006. "Principal component analysis of binary data by iterated singular value decomposition," Computational Statistics & Data Analysis, Elsevier, vol. 50(1), pages 21-39, January.
    2. Butts, Carter T., 2008. "network: A Package for Managing Relational Data in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 24(i02).
    3. C. Horan, 1969. "Multidimensional scaling: Combining observations when individuals have different perceptual structures," Psychometrika, Springer;The Psychometric Society, vol. 34(2), pages 139-165, June.
    4. de Leeuw, Jan & Mair, Patrick, 2009. "Multidimensional Scaling Using Majorization: SMACOF in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 31(i03).
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