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Mathematical Methods of Operations Research

, Volume 68, Issue 3, pp 429–443 | Cite as

A hybrid method for multidimensional scaling using city-block distances

  • Antanas Žilinskas
  • Julius ŽilinskasEmail author
Original Article

Abstract

The problem of multidimensional scaling with city-block distances in the embedding space is reduced to a two level optimization problem consisting of a combinatorial problem at the upper level and a quadratic programming problem at the lower level. A hybrid method is proposed combining randomized search for the upper level problem with a standard quadratic programming algorithm for the lower level problem. Several algorithms for the combinatorial problem have been tested and an evolutionary global search algorithm has been proved most suitable. An experimental code of the proposed hybrid multidimensional scaling algorithm is developed and tested using several test problems of two- and three-dimensional scaling.

Keywords

Multidimensional scaling Global optimization City-block distances 

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References

  1. Arabie P (1991) Was Euclid an unnecessarily sophisticated psychologist?. Psychometrika 56: 567–587zbMATHCrossRefGoogle Scholar
  2. Borg I, Groenen P (2005) Modern multidimensional scaling, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  3. Bortz J (1974) Kritische Bemerkungen über den Einsatz nicht-euklidischer Metriken im Rahmen der multidimensionalen Skalierung. Archiv für Psychologie 126: 196–212Google Scholar
  4. Brusco MJ (2001) A simulated annealing heuristics for unidimensional and multidimensional (city block) scaling of symmetric proximity matrices. J Classif 18: 3–33zbMATHMathSciNetGoogle Scholar
  5. Cox T, Cox M (2001) Multidimensional scaling. Chapman & Hall/CRC, London/Boca RatonzbMATHGoogle Scholar
  6. de Leeuw J (1984) Differentiability of Kruskal’s stress at a local minimum. Psychometrika 49: 111–113CrossRefMathSciNetGoogle Scholar
  7. Everett JE (2001) Algorithms for multidimensional scaling. In: Chambers LD(eds) The practical handbook of genetic algorithms, 2nd edn. Chapman & Hall/CRC, London/Boca Raton, pp 203–233Google Scholar
  8. Green P, Carmone F, Smith S (1989) Multidimensional scaling: concepts and applications. Allyn and Bacon, BostonGoogle Scholar
  9. Groenen PJF, Heiser WJ, Meulman JJ (1998) City-block scaling: smoothing strategies for avoiding local minima. In: Balderjahn I, Mathar R, Schader M(eds) Classification, data analysis, and data highways. Springer, Heidelberg, pp 46–53Google Scholar
  10. Groenen PJF, Heiser WJ, Meulman JJ (1999) Global optimization in least-squares multidimensional scaling by distance smoothing. J Classif 16: 225–254zbMATHCrossRefMathSciNetGoogle Scholar
  11. Groenen PJF, Mathar R, Heiser WJ (1995) The majorization approach to multidimensional scaling for Minkowski distances. J Classif 12: 3–19zbMATHCrossRefMathSciNetGoogle Scholar
  12. Groenen P, Mathar R, Trejos J (2000) Global optimization methods for multidimensional scaling applied to mobile communication. In: Gaul W, Opitz O, Schander M(eds) Data analysis: scientific modeling and practical applications. Springer, Heidelberg, pp 459–475Google Scholar
  13. Hooker JN (1995) Testing heuristics: we have it all wrong. J Heuristics 1: 33–42zbMATHCrossRefGoogle Scholar
  14. Hubert L, Arabie P, Hesson-Mcinnis M (1992) Multidimensional scaling in the city-block metric: a combinatorial approach. J Classif 9: 211–236CrossRefGoogle Scholar
  15. Hubert L, Arabie P, Meulman J (2006) The structural representation of proximity matrices with Matlab. SIAM, PhiladelphiazbMATHGoogle Scholar
  16. Leung PL, Lau K (2004) Estimating the city-block two-dimensional scaling model with simulated annealing. Eur J Oper Res 158: 518–524zbMATHCrossRefGoogle Scholar
  17. Mathar R, Žilinskas A (1993) On global optimization in two-dimensional scaling. Acta Appl Math 33: 109–118zbMATHCrossRefMathSciNetGoogle Scholar
  18. Michalewicz Z (1996) Genetic algorithms + data structures = evolution programs. Springer, BerlinzbMATHGoogle Scholar
  19. Murillo A, Vera JF, Heiser WJ (2005) A permutation-translation simulated annealing algorithm for L1 and L2 unidimensional scaling. J Classif 22: 119–138zbMATHCrossRefMathSciNetGoogle Scholar
  20. TTörn A, Žilinskas A (1989) Global optimization. Lect Notes Comput Sci 350: 1–250Google Scholar
  21. Žilinskas A, Žilinskas J (2006) Parallel hybrid algorithm for global optimization of problems occurring in MDS-based visualization. Comput Math Appl 52: 211–224zbMATHCrossRefMathSciNetGoogle Scholar
  22. Žilinskas A, Žilinskas J (2007) Two level minimization in multidimensional scaling. J Global Optim 38: 581–596zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

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