Psychometrika

, Volume 40, Issue 1, pp 33–51 | Cite as

Generalized procrustes analysis

  • J. C. Gower
Article

Abstract

SupposePi(i) (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σi=1m Σi=1n Δ2(Pj(i)Gi), whereGi is the centroid of them pointsPi(i) (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.

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References

  1. Banfield, C. F. & Harries, J. M. A technique for comparing judges' performance in sensory tests.J. Food Technology (In press).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, 1970,35, 283–320.Google Scholar
  3. Eckart, V. & Young, G. The approximation of one matrix by another of lower rank.Psychometrika, 1935,1, 211–218.Google Scholar
  4. Gower, J. C. Statistical methods of comparing different multivariate analyses of the same data. InMathematics in the Archaeological & Historical Sciences. Hodson, F. R., Kendall, D. G., & Tautu, P. Edinburgh: Edinburgh Univ. Press, 1971, 138–149.Google Scholar
  5. Gower, J. C. The determinant of an orthogonal matrix. Algorithm AS 82.Applied Statistics, (In press)24.Google Scholar
  6. Gruvaeus, G. T. A general approach to Procrustes pattern rotation.Psychometrika, 1970,35, 493–505.Google Scholar
  7. Kristof, W. & Wingersky, B. Generalization of the orthogonal Procrustes rotation procedure to more than two matrices, 1971. Proceedings, 79th Annual Convention, A. P. A., 81–90.Google Scholar
  8. Krzanowski, W. J. A comparison of some distance measures applicable to multinomial data, using a rotational fit technique.Biometrics, 1971,27, 1062–1068.Google Scholar
  9. Nelder, J. A. and Members of the Rothamsted Statistics Department.Genstat Reference Manual, Inter-University/Research Councils Series, Report No. 3, Second Edition. Edinburgh: Program Library Unit, Edinburgh Regional Computing Centre, 1973.Google Scholar
  10. Pomeroy, R. W., Williams, D. R., Harries, J. M. & Ryan, P. O. Material, measurements, jointing and tissue separation.Journal of Agricultural Science, (In press).Google Scholar
  11. Pomeroy, R. W., Williams, D. R., Harries, J. M. & Ryan, P. O. The use of regression equations to estimate total tissue components from observations on intact and quartered sides and partial dissection data.Journal of Agricultural Science, (In press).Google Scholar
  12. Schönemann, P. H. On two-sided Procrustes problems.Psychometrika, 1968,33, 19–34.Google Scholar
  13. Schönemann, P. H. & Carroll, R. M. Fitting one matrix to another under choice of a central dilation and a rigid motion.Psychometrika, 1970,35, 245–256.Google Scholar

Copyright information

© Psychometric Society 1975

Authors and Affiliations

  • J. C. Gower
    • 1
  1. 1.Rothamsted Experimental StationHarpenden

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