Abstract
The aim of this paper is to analyze two scaling extensions of the Orthogonal Procrustes Problem (OPP) called the pre-scaling and the post-scaling approaches. We also discuss some problems related to these extensions and propose two new algorithms to find optimal solutions. These algorithms, which are based on the majorization principle, are shown to be monotonically convergent and their performance is examined.
Similar content being viewed by others
References
BATCHELOR, P.G., and FITZPATRICK, J.M. (2000), “A Study of the Anisotropically Weighted Procrustes Problem”, in Proceedings of the IEEE workshop on Mathematical Methods in Biomedical Image Analysis, WD, USA: IEEE Computer Society, pp. 212–218 .
BENNANI DOSSE,M. (2004), Extension de l’analyse de Procruste g´en´eralis´ee (Extension of Generalized Procrustes Analysis), Agrostat, Rennes, France, pp. 1–16.
BENNANI DOSSE, M., and TEN BERGE, J.M.F. (2008), “The Assumption of Proportional Components When Candecomp is Applied to Symmetric Matrices in the Context of Indscal”, Psychometrika, 73, 303–307.
BOLLA, M., MICHALETZKY, G., TUSNÁDY, G., and ZIERMANN, M. (1998), “Extrema of Sums of Heterogeneous Quadratic Forms”, Linear Algebra and Its Applications, 269, 331–365.
BROCK, J.E. (1968), “Optimal Matrices Describing Linear Systems”, AIAA Journal, 6, 1292–1296.
CHU, M.T., and TRENDAFILOV,N.T. (1998), “On a Differential Equation Approach to the Weighted Orthogonal Procrustes Problem”, Statistics and Computing, 8(2), 125–133.
CIARLET, P.G. (1989), Introduction to Numerical Linear Algebra and Optimization, Cambridge Texts in Applied Mathematics.
DE LEEUW, J. (1994), Block Relaxation Algorithms in Statistics, in Information Systems and Data Analysis, eds. H.-H. Bock, W. Lenski, and M.M. Richter, Berlin: Springer, pp. 308–324.
DRYDEN, I., and MARDIA, K. (1998), Statistical Shape Analysis, Chichester: John Wiley and Sons.
GIFI, A. (1990), Nonlinear Multivariate Analysis, Chichester: John Wiley and Sons.
GLASBEY, C.A., and MARDIA, K.V. (1998), “A Review of Image Warping Methods”, Journal of Applied Statistics, 25, 155–171.
GOODALL, C. (1991), “Procrustes Methods in the Statistical Analysis of Shape”, Journal of the Royal Statistical Society, Series B, 53, 285–339.
GOODALL, C., and GREEN, B. (1986), “Quantitative Analysis of Surface Growth”, Botanical Gazette, 147, 1–15.
GOWER, J.C. (1971), “Statistical Methods of Comparing Different Multivariate Analyses of the Same Data”, in Mathematics in the Archeological and Historical Sciences, eds. F.R. Hodson, D.G. Kendall, and P. Tautu, Edinburg: University Press, pp. 138–149.
GOWER, J.C. (1984), Multivariate Analysis : Ordination, Multidimensional Scaling and Allied Topics, in Handbook of Applicable Mathematics (Vol. VI: Statistics), ed. E.H. Lloyd, Chichester: John Wiley and Sons, pp. 727–781.
GOWER, J.C., and DIJKSTERHUIS,G.B. (2004), Procrustes Problems, Oxford University Press.
GREEN, B. (1952), “The Orthogonal Approximation of an Oblique Structure in Factor Analysis”, Psychometrika, 17(4), 429–440.
HENDERSON, H.V., and SEARLE, S.R. (1981), “The Vec-permutation Matrix, the Vec Operator and Kronecker Products: A Review, Linear and Multilinear Algebra, 9, 271–288.
HIGHAM, N.J. (1988), “The Symmetric Procrustes Problem”, BIT Numerical Mathematics, 28(1), 133–143.
HIGHAM, N.J. (1989), Matrix Nearness Problems and Applications, in Applications of Matrix Theory, eds. M.J.C. Gover and Barnett, Oxford University Press, pp. 1–27.
HIGHAM, N.J. (1995), Matrix Procrustes Problems, Department of Mathematics, University of Manchester.
HURLEY, J.R., and CATTELL, R.B. (1962), “The Procrustes Program : Producing Direct Rotation to Test a Hypothesised Factor Structure”, Behavioral Science, 7, 258–262.
JENSEN S.T., JOHANSEN S., and LAURITZEN, S.L. (1991), “Globally Convergent Algorithms for Maximizing a Likelihood Function”, Biometrika, 78, 867–877.
KIERS, H.A.L. (1995), “Maximization of Sums of Quotients of Quadratic Forms and Some Generalizations”, Psychometrika, 60, 221–245.
KIERS, H.A.L., and TEN BERGE, J.M.F. (1992), “Minimization of a Class ofMatrix Trace Functions by Means of Refined Majorization”, Psychometrika, 57, 371–382.
KOSCHAT, M.A., and SWAYNE, D.F. (1991), “A Weighted Procrustes Criterion”, Psychometrika, 56, 229–239.
MOOIJAART, A., and COMMANDEUR, J.J.F. (1990), “A General Solution of theWeighted Orthonormal Procrutes Problem”, Psychometrika, 55(4), 657–663.
OBERHOFERW., and KMENTA J. (1974), “A General Procedure for Obtaining Maximum Likelihood Estimates in Generalized Regression Models”, Econometrica, 42, 579–590.
ROHLF, F.J., and SLICE, D.E. (1990), “Extensions of the Procrustes Method for the Optimal Superimposition of Landmarks”, Systematic Zoology, 39, 40–59.
SCHAIBLE, S., and SHI, J. (2003), “Fractional Programming: The Sum-of-ratios Case”, Optimization Methods and Software, 18(2), 219–230.
SCHÖNEMANN, P.H. (1966), “A Generalized Solution to the Orthogonal Procrustes Problem”, Psychometrika 31, 1–10.
SCHO¨ NEMANN, 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.
TEN BERGE, J.M.F. (1977), “Orthogonal Procrustes Rotation for Two or More Matrices”, Psychometrika, 42, 267–276.
VIKLANDS, T. (2006), “Algorithms for Weighted Orthogonal Procrustes Problem and Other Least Squares Problems”, unpublished Ph. D. Thesis, Umea University.
WAHBA, G. (1965), “Problem 65-1 : A Least Squares Estimate of Satellite Attitude”, SIAM Review 7, 409; Solutions in SIAM Review 8, 384–386.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bennani Dosse, M., Ten Berge, J. Anisotropic Orthogonal Procrustes Analysis. J Classif 27, 111–128 (2010). https://doi.org/10.1007/s00357-010-9046-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00357-010-9046-8