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Multidimensional Scaling Using Majorization: SMACOF in R

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  • Jan de Leeuw
  • Patrick Mair
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    Abstract

    In this paper we present the methodology of multidimensional scaling problems (MDS) solved by means of the majorization algorithm. The objective function to be minimized is known as stress and functions which majorize stress are elaborated. This strategy to solve MDS problems is called SMACOF and it is implemented in an R package of the same name which is presented in this article. We extend the basic SMACOF theory in terms of configuration constraints, three-way data, unfolding models, and projection of the resulting configurations onto spheres and other quadratic surfaces. Various examples are presented to show the possibilities of the SMACOF approach offered by the corresponding package.

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    Bibliographic Info

    Article provided by American Statistical Association in its journal Journal of Statistical Software.

    Volume (Year): 31 ()
    Issue (Month): i03 ()
    Pages:

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    Handle: RePEc:jss:jstsof:31:i03

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    Web page: http://www.jstatsoft.org/

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    1. Roger Shepard, 1974. "Representation of structure in similarity data: Problems and prospects," Psychometrika, Springer, vol. 39(4), pages 373-421, December.
    2. J. Carroll & Jih-Jie Chang, 1970. "Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition," Psychometrika, Springer, vol. 35(3), pages 283-319, September.
    3. Yoshio Takane & Forrest Young & Jan Leeuw, 1977. "Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features," Psychometrika, Springer, vol. 42(1), pages 7-67, March.
    4. Ingwer Borg & James Lingoes, 1980. "A model and algorithm for multidimensional scaling with external constraints on the distances," Psychometrika, Springer, vol. 45(1), pages 25-38, March.
    5. Jacqueline Meulman, 1992. "The integration of multidimensional scaling and multivariate analysis with optimal transformations," Psychometrika, Springer, vol. 57(4), pages 539-565, December.
    6. Patrick Groenen & Willem Heiser, 1996. "The tunneling method for global optimization in multidimensional scaling," Psychometrika, Springer, vol. 61(3), pages 529-550, September.
    7. Peter Schönemann, 1972. "An algebraic solution for a class of subjective metrics models," Psychometrika, Springer, vol. 37(4), pages 441-451, December.
    8. Jan Leeuw, 1988. "Convergence of the majorization method for multidimensional scaling," Journal of Classification, Springer, vol. 5(2), pages 163-180, September.
    9. Jan Leeuw & Jacqueline Meulman, 1986. "A special Jackknife for Multidimensional Scaling," Journal of Classification, Springer, vol. 3(1), pages 97-112, March.
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    Cited by:
    1. Köhn, Hans-Friedrich, 2010. "Representation of individual differences in rectangular proximity data through anti-Q matrix decomposition," Computational Statistics & Data Analysis, Elsevier, vol. 54(10), pages 2343-2357, October.
    2. Michael Greenacre & Patrick J. F. Groenen, 2013. "Weighted Euclidean biplots," Economics Working Papers 1380, Department of Economics and Business, Universitat Pompeu Fabra.
    3. Theuβl, Stefan & Reutterer, Thomas & Hornik, Kurt, 2014. "How to derive consensus among various marketing journal rankings?," Journal of Business Research, Elsevier, vol. 67(5), pages 998-1006.

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