Advertisement

Journal of Classification

, Volume 9, Issue 2, pp 237–256 | Cite as

Resistant orthogonal procrustes analysis

  • Peter Verboon
  • Willem J. Heiser
Article

Abstract

In this paper two alternative loss criteria for the least squares Procrustes problem are studied. These alternative criteria are based on the Huber function and on the more radical biweight function, which are designed to be resistant to outliers. Using iterative majorization it is shown how a convergent reweighted least squares algorithm can be developed. In asimulation study it turns out that the proposed methods perform well over a specific range of contamination. When a uniform dilation factor is included, mixed results are obtained. The methods also yield a set of weights that can be used for diagnostic purposes.

Keywords

Resistance Procrustes analysis Outliers Iterative majorization Reweighted least squares 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BORG, I., and LINGOES, J. (1987),Multidimensional Similarity Structure Analysis, New York: Springer-Verlag.Google Scholar
  2. CLIFF, N. (1966), “Orthogonal Rotation to Congruence,”Psychometrika, 31, 33–42.CrossRefMathSciNetGoogle Scholar
  3. DE LEEUW, J. (1988), “Convergence of the Majorization Method for Multidimensional Scaling,”Journal of Classification, 5, 163–180.zbMATHCrossRefMathSciNetGoogle Scholar
  4. GOODALL, C. (1983), “M-estimators of Location: an Introduction to the Theory,” inUnderstanding Robust and Exploratory Data Analysis, Eds. D. C. Hoaglin, F. Mosteller and J. W. Tukey, New York: Wiley, 211–243.Google Scholar
  5. GOWER, J. C. (1975), “Generalized Procrustes Analysis,”Psychometrika, 40, 33–51.zbMATHCrossRefMathSciNetGoogle Scholar
  6. GREEN, B. F. (1952), “The Orthogonal Approximation of an Oblique Structure in Factor Analysis,”Psychometrika, 17, 429–440.zbMATHCrossRefMathSciNetGoogle Scholar
  7. HAMPEL, F. R., RONCHETTI, E. M., ROUSSEEUW, P. J., and STAHEL, W. A. (1986),Roust Statistics—The Approach Based on Influence Functions, New York: Wiley.Google Scholar
  8. HEISER, W. J. (1987), “Correspondence Analysis with Least Absolute Residuals,”Computational Statistics and Data Analysis, 5, 337–356.zbMATHCrossRefGoogle Scholar
  9. HUBER, P. J. (1981),Robust Statistics, New York: Wiley.zbMATHCrossRefGoogle Scholar
  10. MOSTELLER, F., and TUKEY, J. W. (1977),Data Analysis and Regression, Massachusetts: Addison-Wesley.Google Scholar
  11. ROUSSEEUW, P. J., and LEROY, A. M. (1987),Robust Regression and Outlier Detection, New York: Wiley.zbMATHCrossRefGoogle Scholar
  12. SCHÖNEMANN, P. H. (1966), “A Generalized Solution of the Orthogonal Procrustes Problem,”Psychometrika, 31, 1–10.zbMATHCrossRefMathSciNetGoogle Scholar
  13. SCHÖNEMANN, P. H., and CARROLL, R. M. (1970), “Fitting One Matrix to Another under Choice of a Central Dilation and a Rigid Motion,”Psychometrika, 35, 245–255.CrossRefGoogle Scholar
  14. SPENCE, I., and LEWANDOWSKY, S. (1989), “Robust Multidimensional Scaling,”Psychometrika, 54, 501–513.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Peter Verboon
    • 1
  • Willem J. Heiser
    • 1
  1. 1.Department of Data Theory, FSWLeiden UniversityLeidenThe Netherlands

Personalised recommendations