Psychometrika

, Volume 48, Issue 4, pp 575–595 | Cite as

Monotone spline transformations for dimension reduction

  • S. Winsberg
  • J. O. Ramsay
Article

Abstract

Consider a set of data consisting of measurements ofn objects with respect top variables displayed in ann ×p matrix. A monotone transformation of the values in each column, represented as a linear combination of integrated basis splines, is assumed determined by a linear combination of a new set of values characterizing each row object. Two different models are used: one, an Eckart-Young decomposition model, and the other, a multivariate normal model. Examples for artificial and real data are presented. The results indicate that both methods are helpful in choosing dimensionality and that the Eckart-Young model is also helpful in displaying the relationships among the objects and the variables. Also, results suggest that the resulting transformations are themselves illuminating.

Key words

I-splines Eckart-Young decomposition transformation to normality multivariate normal covariance structure maximum likelihood estimation 

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References

  1. Andrews, D. F., Gnanadesikan, R. & Warner, J. L. Transformations of multivariate data.Biometrika, 1971,27, 825–840.Google Scholar
  2. Box, G. E. P. & Cox, D. R. Analysis of transformations (with discussion).Journal of the Royal Statistical Society, Series B, 1964,26, 211–252.Google Scholar
  3. Curry, H. B. & Schoenberg, I. J. On spline distribution functions and their limits: the Pólya distribution functions. Abstract 380t.Bulletin of the American Mathematical Society, 1947,53, 1114.Google Scholar
  4. Curry, H. B. & Schoenberg, I. J. On Pólya frequency functions. IV: The fundamental spline functions and their limits.Journal d'Analyse Mathématique, 1966,17, 71–107.Google Scholar
  5. de Boor, C.A Practical Guide to Splines. New York: Springer-Verlag, 1978.Google Scholar
  6. Eckart, C. & Young, G. The approximation of one matrix by another of lower rank.Psychometrika, 1936,1, 211–218.Google Scholar
  7. Gifi, A.Nonlinear Multivariate Analysis. Leiden: Department of Datatheorie. 1981.Google Scholar
  8. Harman, H. H.Modern Factor Analysis. Chicago: University of Chicago Press, 1967 (revised edition).Google Scholar
  9. Kruskal, J. B. & Shepard, R. N. A nonmetric variety of linear factor analysis.Psychometrika, 1974,38, 123–157.Google Scholar
  10. Ramsay, J. O. Monotonic weighted power transformations to additivity.Psychometrika, 1977,42, 83–109(a).Google Scholar
  11. Ramsay, J. O. Maximum likelihood estimation in multidimensional scaling.Psychometrika, 1977,42, 241–266(b).Google Scholar
  12. Ramsay, J. O. Some statistical approaches to multidimensional scaling data. (with discussion).Journal of the Royal Statistical Society, Series A, 1982,145, 285–312.Google Scholar
  13. van Rijckevorsel, J. Canonical analysis with B-splines, in H. Caussinus, P. Ettinger, & R. Tomassone (Eds.),COMPST AT 1982, Part I, Wien, Physica Verlag, 1982.Google Scholar
  14. SAS Institute Inc.SAS User's Guide: Statistics, 1982 Edition. Cary, NC: SAS Institute Inc., 1982.Google Scholar
  15. Thurstone, L. L.Multiple Factor Analysis. Chicago: University of Chicago, 1947.Google Scholar
  16. Winsberg, S. & Ramsay, J. O. Monotonic transformations to additivity using splines.Biometrika, 1980,67, 669–674.Google Scholar
  17. Winsberg, S. & Ramsay, J. O. Analysis of pairwise preferences data using integrated B-splines.Psychometrika, 1981,46, 171–186.Google Scholar
  18. Young, F. W., Takane, Y. & de Leeuw, J. The principal components of mixed measurement level multivariate data: An alternating least squares method with optimal scaling features.Psychometrika, 1978,43, 279–281.Google Scholar

Copyright information

© The Psychometric Society 1983

Authors and Affiliations

  • S. Winsberg
    • 1
  • J. O. Ramsay
    • 2
  1. 1.Université de MontréalCanada
  2. 2.Mcgill UniversityUSA

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