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Global Optimization in Least-Squares Multidimensional Scaling by Distance Smoothing

Author

Listed:
  • P. J. F. Groenen
  • W. J. Heiser
  • J. J. Meulman

Abstract

Least-squares multidimensional scaling is known to have a serious problem of local minima, especially if one dimension is chosen, or if city-block distances are involved. One particular strategy, the smoothing strategy proposed by Pliner (1986, 1996), turns out to be quite successful in these cases. Here, we propose a slightly different approach, called distance smoothing. We extend distance smoothing for any Minkowski distance. In addition, we extend the majorization approach to multidimensional scaling to have a one-step update for Minkowski parameters larger than 2 and use the results for distance smoothing. We present simple ideas for finding quadratic majorizing functions. The performance of distance smoothing is investigated in several examples, including two simulation studies. Copyright Springer-Verlag New York Inc. 1999

Suggested Citation

  • P. J. F. Groenen & W. J. Heiser & J. J. Meulman, 1999. "Global Optimization in Least-Squares Multidimensional Scaling by Distance Smoothing," Journal of Classification, Springer;The Classification Society, vol. 16(2), pages 225-254, July.
    • Handle: RePEc:spr:jclass:v:16:y:1999:i:2:p:225-254
    DOI: 10.1007/s003579900055
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    Citations

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    Cited by:

    1. Michael Brusco & Stephanie Stahl, 2005. "Optimal Least-Squares Unidimensional Scaling: Improved Branch-and-Bound Procedures and Comparison to Dynamic Programming," Psychometrika, Springer;The Psychometric Society, vol. 70(2), pages 253-270, June.
    2. Julius Žilinskas, 2012. "Parallel branch and bound for multidimensional scaling with city-block distances," Journal of Global Optimization, Springer, vol. 54(2), pages 261-274, October.
    3. Groenen, P.J.F. & van de Velden, M., 2004. "Multidimensional scaling," Econometric Institute Research Papers EI 2004-15, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    4. Groenen, Patrick J. F. & Franses, Philip Hans, 2000. "Visualizing time-varying correlations across stock markets," Journal of Empirical Finance, Elsevier, vol. 7(2), pages 155-172, August.
    5. K. Deun & P. Groenen & W. Heiser & F. Busing & L. Delbeke, 2005. "Interpreting degenerate solutions in unfolding by use of the vector model and the compensatory distance model," Psychometrika, Springer;The Psychometric Society, vol. 70(1), pages 45-69, March.
    6. Michael Brusco & Hans-Friedrich Köhn & Stephanie Stahl, 2008. "Heuristic Implementation of Dynamic Programming for Matrix Permutation Problems in Combinatorial Data Analysis," Psychometrika, Springer;The Psychometric Society, vol. 73(3), pages 503-522, September.
    7. Dorit S. Hochbaum & Erick Moreno-Centeno & Phillip Yelland & Rodolfo A. Catena, 2011. "Rating Customers According to Their Promptness to Adopt New Products," Operations Research, INFORMS, vol. 59(5), pages 1171-1183, October.
    8. Groenen, P.J.F. & Kaymak, U. & van Rosmalen, J.M., 2006. "Fuzzy clustering with Minkowski distance," Econometric Institute Research Papers EI 2006-24, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    9. Antanas Žilinskas & Julius Žilinskas, 2008. "A hybrid method for multidimensional scaling using city-block distances," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(3), pages 429-443, December.
    10. K. Van Deun & P. J. F. Groenen, 2005. "Majorization Algorithms for Inspecting Circles, Ellipses, Squares, Rectangles, and Rhombi," Operations Research, INFORMS, vol. 53(6), pages 957-967, December.
    11. Groenen, P.J.F. & Winsberg, S. & Rodriguez, O. & Diday, E., 2006. "I-Scal: Multidimensional scaling of interval dissimilarities," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 360-378, November.

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