Skip to main content
SpringerLink
Log in

Extensions of Classical Multidimensional Scaling via Variable Reduction

  • Published:
Computational Statistics Aims and scope Submit manuscript

Summary

Classical multidimensional scaling constructs a configuration of points that minimizes a certain measure of discrepancy between the configuration’s interpoint distance matrix and a fixed dissimilarity matrix. Recent extensions have replaced the fixed dissimilarity matrix with a closed and convex set of dissimilarity matrices. These formulations replace fixed dissimilarities with optimization variables (disparities) that are permitted to vary subject to application-specific constraints. For example, simple bound constraints are suitable for distance matrix completion problems (Trosset, 2000) and for inferring molecular conformation from information about interatomic distances (Trosset, 1998b); whereas order constraints are suitable for nonmetric multidimensional scaling (Trosset, 1998a). This paper describes the computational theory that provides a common foundation for these formulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Byrd, R. H., Lu, P., Nocedal, J., and Zhu, C. (1995). A limited memory algorithm for bound constrained optimization. SIAM Journal on Scientific Computing, 16:1190–1208.

    Article  MathSciNet  Google Scholar 

  2. Critchley, F. (1988). On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra and Its Applications, 105:91–107.

    Article  MathSciNet  Google Scholar 

  3. de Leeuw, J. and Heiser, W. (1982). Theory of multidimensional scaling. In Krishnaiah, P. R. and Kanal, I. N., editors, Handbook of Statistics, volume 2, chapter 13, pages 285–316. North-Holland Publishing Company, Amsterdam.

    Article  MathSciNet  Google Scholar 

  4. Glunt, W., Hayden, T. L., and Raydan, M. (1993). Molecular conformations from distance matrices. Journal of Computational Chemistry, 14:114–120.

    Article  Google Scholar 

  5. Golub, G. H. and Pereyra, V. (1973). The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate. SIAM Journal on Numerical Analysis, 10:413–432.

    Article  MathSciNet  Google Scholar 

  6. Gower, J. C. (1966). Some distance properties of latent root and vector methods in multivariate analysis. Biometrika, 53:315–328.

    Article  MathSciNet  Google Scholar 

  7. Krippahl, L. and Barahona, P. (1999). Applying constraint programming to protein structure determination. In Principles and Practice of Constraint Programming, pages 289–302. Springer Verlag, New York.

    Google Scholar 

  8. Krippahl, L., Trosset, M., and Barahona, P. (2001). Combining constraint programming and multidimensional scaling to solve distance geometry problems. In Proceedings of the Third International Workshop on Integration of AI and OR Techniques. To appear.

  9. Kruskal, J. B. (1964a). Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29:1–27.

    Article  MathSciNet  Google Scholar 

  10. Kruskal, J. B. (1964b). Nonmetric multidimensional scaling: A numerical method. Psychometrika, 29:28–42.

    MathSciNet  MATH  Google Scholar 

  11. Lehoucq, R. B. (1995). Analysis and implementation of an implicitly restarted iteration. Technical Report 95-13, Department of Computational & Applied Mathematics, Rice University, 6100 Main Street, Houston, TX 77005–1892. Author’s Ph.D. thesis.

    Google Scholar 

  12. Lehoucq, R. B. and Sorensen, D. C. (1996). Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM Journal on Matrix Analysis and Applications, 17:789–821.

    Article  MathSciNet  Google Scholar 

  13. Lehoucq, R. B., Sorensen, D. C., and Yang, C. (1997). ARPACK Users’ Guide: Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. Department of Computational & Applied Mathematics, Rice University, 6100 Main Street, Houston, TX 77005–1892. Available at https://doi.org/www.caam.rice.edu/.

    Google Scholar 

  14. Lewis, A. S. (1996). Derivatives of spectral functions. Mathematics of Operations Research, 21:576–588.

    Article  MathSciNet  Google Scholar 

  15. Mardia, K. V. (1978). Some properties of classical multi-dimensional scaling. Communications in Statistics — Theory and Methods, A7:1233–1241.

    Article  MathSciNet  Google Scholar 

  16. Mardia, K. V., Kent, J. T., and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, Orlando.

    MATH  Google Scholar 

  17. McCormick, G. P. and Tapia, R. A. (1972). The gradient projection method under mild differentiability conditions. SIAM Journal on Control, 10:93–98.

    Article  MathSciNet  Google Scholar 

  18. Parks, T. A. (1985). Reducible nonlinear programming problems. Technical Report 85-8, Department of Mathematical Sciences, Rice University, Houston, TX. Author’s Ph.D. dissertation.

    Google Scholar 

  19. Saito, T. (1978). The problem of the additive constant and eigenvalues in metric multidimensional scaling. Psychometrika, 43:193–201.

    Article  Google Scholar 

  20. Schoenberg, I. J. (1935). Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espaces distanciés vectoriellement applicable sur l’espace de Hilbert”. Annals of Mathematics, 38:724–732.

    Article  Google Scholar 

  21. Sibson, R. (1979). Studies in the robustness of multidimensional scaling: Perturbational analysis of classical scaling. Journal of the Royal Statistical Society, Series B, 41:217–229.

    MathSciNet  MATH  Google Scholar 

  22. Sorensen, D. C. (1992). Implicit application of polynomial filters in a k-step Arnoldi method. SIAM Journal on Matrix Analysis and Applications, 13:357–385.

    Article  MathSciNet  Google Scholar 

  23. Torgerson, W. S. (1952). Multidimensional scaling: I. Theory and method. Psychometrika, 17:401–419.

    Article  MathSciNet  Google Scholar 

  24. Trefethen, L. N. and Bau, D. (1997). Numerical Linear Algebra. SIAM, Philadelphia, PA.

    Book  Google Scholar 

  25. Trosset, M. W. (1997). Numerical algorithms for multidimensional scaling. In Klar, R. and Opitz, P., editors, Classification and Knowledge Organization, pages 80–92, Berlin. Springer. Proceedings of the 20th annual conference of the Gesellschaft für Klassifikation e.V., held March 6–8, 1996, in Freiburg, Germany.

    Chapter  Google Scholar 

  26. Trosset, M. W. (1998a). Applications of multidimensional scaling to molecular conformation. Computing Science and Statistics, 29: 148–152.

    Google Scholar 

  27. Trosset, M. W. (1998b). A new formulation of the nonmetric STRAIN problem in multidimensional scaling. Journal of Classification, 15:15–35.

    Article  Google Scholar 

  28. Trosset, M. W. (2000). Distance matrix completion by numerical optimization. Computational Optimization and Applications, 17: 11–22.

    Article  MathSciNet  Google Scholar 

  29. Trosset, M. W., Baggerly, K. A., and Pearl, K. (1996). Another look at the additive constant problem in multidimensional scaling. Technical Report 96-7, Department of Statistics—MS 138, Rice University, Houston, TX 77005–1892.

    Google Scholar 

  30. Young, G. and Householder, A. S. (1938). Discussion of a set of points in terms of their mutual distances. Psychometrika, 3:19–22.

    Article  Google Scholar 

  31. Zhu, C., Byrd, R. H., Lu, P., and Nocedal, J. (1994). L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization. Technical Report NAM-11, Department of Electrical Engineering & Computer Science, Northwestern University, Evanston, IL 60208. Revised October 8, 1996.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of William & Mary, P.O. Box 8795, Williamsburg, VA, 23187-8795, USA

    Michael W. Trosset

Authors
  1. Michael W. Trosset
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported by grant DMS-9622749 from the National Science Foundation and by the W. M. Keck Center for Computational Biology under Medical Informatics Training grant 1T1507093 from the National Library of Medicine. Many people contributed to this research. I am especially grateful to Pedro Barahona, Jan de Leeuw, Timothy Havel, Bruce Hendrickson, Ludwig Krippahl, Chi-Kwong Li, Roy Mathias, Thomas Milligan, George Phillips, Richard Tapia, and Pablo Tarazaga.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Trosset, M.W. Extensions of Classical Multidimensional Scaling via Variable Reduction. Computational Statistics 17, 147–163 (2002). https://doi.org/10.1007/s001800200099

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s001800200099

Keywords

Search

Navigation