Skip to main content

Orthonormality-Constrained INDSCAL with Nonnegative Saliences

  • Conference paper
Computational Science and Its Applications – ICCSA 2004 (ICCSA 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3044))

Included in the following conference series:

  • 939 Accesses

Abstract

INDSCAL is a specific model for simultaneous metric multidimensional scaling (MDS) of several data matrices. In the present work the INDSCAL problem is reformulated and studied as a dynamical system on the product manifold of orthonormal and diagonal matrices. The problem for fitting of the INDSCAL model to the data is solved. The resulting algorithms are globally convergent. Numerical examples illustrate their application.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Boik, R.J.: Principal component models for correlation matrices. Biometrika 90, 679–701 (2003)

    Article  MathSciNet  Google Scholar 

  2. Carroll, J.D., Chang, J.J.: Analysis of individual differences in multidimensional scaling via an n−way generalization of “Eckart-Young” decomposition. Psychometrika 35, 283–319 (1970)

    Article  MATH  Google Scholar 

  3. Chu, M.T.: A continuous Jacobi-like approach to the simultaneous reduction of real matrices. Linear Algebra and its Applications 147, 75–96 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chu, M.T., Driessel, K.R.: The projected gradient method for least squares matrix approximations with spectral constraints. SIAM J. Numer. Anal. 27, 1050–1060 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chu, M.T., Trendafilov, N.T.: ORTHOMAX rotation problem. A differential equation approach. Behaviormetrika 25, 13–23 (1998)

    Article  Google Scholar 

  6. Chu, M.T., Trendafilov, N.T.: On a differential equation approach to the weighted orthogonal Procrustes problem. Statistics and Computing 8, 125–133 (1998a)

    Article  Google Scholar 

  7. Cox, T.F., Cox, M.A.A.: Multidimensional Scaling. Chapman & Hall, London (1995)

    Google Scholar 

  8. De Lathauwer, L.: Signal Processing Based on Multilinear Algebra, PhD Thesis, Katholieke Universiteit Leuven (1997), http://www.esat.kuleuven.ac.be/sista/members/delathau.html

  9. De Soete, G., Carroll, J.D., Chaturvedi, A.D.: A modified CANDECOMP method for fitting the extended INDSCAL model. Journal of Classification 10, 75–92 (1993)

    Article  MATH  Google Scholar 

  10. Diele, F., Lopez, L., Peluso, R.: The Cayley transform in the numerical solution of unitary differential systems. Advances in Computational Mathematics 8, 317–334 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Edelman, A., Arias, T., Smith, S.T.: The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. and Appl. 20, 303–353 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Engø, K., Marthinsen, A., Munthe-Kaas, H.: DiffMan–an object oriented MATLAB toolbox for solving differential equations on manifolds (User’s Guide) (1997), http://www.math.ntnu.no/num/synode/

  13. Flury, B.: Common Principal Components and Related Multivariate Models. John Wiley & Sons, New York (1988)

    MATH  Google Scholar 

  14. Helmke, U., Moore, J.B.: Optimization and Dynamical Systems. Springer, London (1994)

    Google Scholar 

  15. Hirsch, M.W., Smale, S.: Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, London (1974)

    MATH  Google Scholar 

  16. Iserles, A., Munte-Kaas, H., Norset, S.P., Zanna, A.: Lie group methods. Acta Numerika 9, 1–151 (2000)

    Article  Google Scholar 

  17. Kiers, H.A.L.: Simple structure in component analysis techniques for mixture of qualitative and quantitative variables. Psychometrika 56, 197–212 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kroonenberg, P.M.: Three Mode Principal Component Analysis: Theory and Applications. DSWO Press, Leiden (1983)

    Google Scholar 

  19. Lippert, R.A., Edelman, A.: Nonlinear eigenvalue problems (1998), http://www.mit.edu/people/ripper/Template/template.html

  20. Magnus, J.R., Neudecker, H.: Matrix Differential Calculus with Application in Statistics and Econometrics. Wiley, New York (1988)

    Google Scholar 

  21. Courcoux, P., Devaux, M.-F., Bouchet, B.: Simultaneous decomposition of multivariate images using three-way data analysis. Application to the comparison of cereal grains by confocal laser scanning microscopy. Chemometrics and Intelligent Laboratory Systems 62, 103–113 (2002)

    Article  Google Scholar 

  22. Owren, B., Welfert, B.: The Newton iteration on Lie groups. BIT 40, 121–145 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ramsay, J.O.: Maximum likelihood estimation in multidimensional scaling. Psychometrika 42, 241–266 (1977)

    Article  MATH  Google Scholar 

  24. Shampine, L.F., Reichelt, M.W.: The MATLAB ODE suite. SIAM Journal on Scientific Computing 18, 1–22 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  25. Stiefel, E.: Richtungsfelder und fernparallelismus in n-dimensionalel manning faltigkeiten. Commentarii Mathematici Helvetici 8, 305–353 (1935–1936)

    Google Scholar 

  26. Takane, Y., Young, F.W., De Leeuw, J.: Nonmetric individual differences multidimensional scaling: an alternating least squares method with optimal scaling features. Psychometrika 42, 7–67 (1977)

    Article  MATH  Google Scholar 

  27. ten Berge, J.M.F., Knol, D.L., Kiers, H.A.L.: A treatment of the ORTHOMAX rotation family in terms of diagonalization, and a re-examination of a singular value approach to VARIMAX rotation. Computational Statistics Quarterly 3, 207–217 (1988)

    Google Scholar 

  28. ten Berge, J.M.F., Kiers, H.A.L.: Some clarifications of the CANDECOMP algorithm applied to INDSCAL. Psychometrika 56, 317–326 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  29. ten Berge, J.M.F., Kiers, H.A.L., Krijnen, W.P.: Computational solutions for the problem of negative saliences and nonsymmetry in INDSCAL. Journal of Classification 10, 115–124 (1993)

    Article  MATH  Google Scholar 

  30. Turk, M., Pentland, A.: Face recognition using eigenfaces. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition, pp. 586–591 (1991)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Trendafilov, N.T. (2004). Orthonormality-Constrained INDSCAL with Nonnegative Saliences. In: Laganá, A., Gavrilova, M.L., Kumar, V., Mun, Y., Tan, C.J.K., Gervasi, O. (eds) Computational Science and Its Applications – ICCSA 2004. ICCSA 2004. Lecture Notes in Computer Science, vol 3044. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24709-8_100

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-24709-8_100

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22056-5

  • Online ISBN: 978-3-540-24709-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics