Abstract
One drawback of Procrustes Analysis is the lack of robustness. To overcome this limitation a procedure that applies the Generalised Procrustes methods, by way of a progressive sequence inspired to the “forward search”, was developed. Starting from an initial centroid, defined by the partial point configuration satisfying the LMS principle, this is extended by joining, at every step, a restricted subset of the remaining points. At every insertion, the updated centroid, redetermined by the new considered points, is compared with the previous by way of the common elements. If significant variations of the similarity transformation parameters occur, they reveal the presence of outliers or non stationary points among the new elements just inserted.
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References
ANDERSON, C.R. (1997): Object Recognition Using Statistical Shape Analysis. PhD thesis, University of Leeds (11, 13, 316).
ATKINSON, A.C. and RIANI M. (2000): Robust Diagnostic Regression Analysis. Springer, New York.
CROSILLA, F. and BEINAT, A. (2002): Use of Generalised Procrustes Analysis for Photogrammetric Block Adjustment by Independent Models. ISPRS Journal of Photogrammetry and Remote Sensing, Elsevier, 56(3), 195–209.
BORG, I. and GROENEN, P.J.F. (1997): Modern Multidimensional Scaling: Theory and Applications. Springer, New York.
CERIOLI, A. and RIANI, M. (2003): Robust Methods for the Analysis of Spatially Autocorrelated Data. Statistical Methods & Applications, 11, 335–358.
DRYDEN, I.L. and MARDIA, K.W. (1998): Statistical Shape Analysis. John Wiley & Sons, Chichester, England. 83–107.
GELFAND, M.S., MIRONOV, A.A. and PEVZNER, P.A. (1996): Gene Recognition Via Spliced Sequence Alignment. Proc. of the National Academy of Sciences. 93(19), 9061–9066.
GOODALL, C. (1991): Procrustes Methods in the Statistical Analysis of Shape. Journal Royal Statistical Society, Series B-Methodological, 53(2), 285–339.
GOWER, J.C. (1975): Generalized Procrustes Analysis. Psychometrika, 40(1), 33–51.
GROENEN, P.J.F., GIAQUINTO, P. and KIERS, H.A.L. (2005): An Improved Majorization Algorithm for Robust Procrustes Analysis. In: Vichi, M., Monari, P., Mignani, S. and Montanari, A. (Eds.): New developments in classification and data analysis. Springer, Heidelberg, 151–158.
KIERS, H.A.L. (2002): Setting up Alternating Least Squares and Iterative Majorization Algorithms for Solving Various Matrix Optimization Problems. Computational Statistics and, Data, Analysis. 41, 157–170.
KONG, Z. and CEGLAREK, D. (2003): Fixture Configuration Synthesis for Reconfigurable Assembly Using Procrustes-based Pairwise Optimization. Transactions of NAMRI, 31, 403–410.
KRISTOF, W. and WINGERSKY, B. (1971): Generalization of the Orthogonal Procrustes Rotation Procedure to More than Two Matrices. Proc. of the 79-th Annual Conv. of the American Psychological Ass., 6. 89–90.
LANGRON, S. P. and COLLINS, A. J. (1985): Perturbation Theory for Generalized Procrustes Analysis. Journal Royal Statistical Society, 47(2), 277–284.
ROHLF, F.J. and SLICE, D. (1992): Methods for Comparison of Sets of Landmarks. Syst. Zool., 39, 40–59.
ROUSSEEUW, P. J. (1984): Least Median of Squares Regression. Journal of the American Statistical Association. 79(388), 871–880.
SCHÖNEMANN, P. H. (1966): A Generalized Solution of the Orthogonal Procrustes Problem. Psychometrika, 31(1), 1–10.
SIBSON, R. (1979): Studies in the Robustness of Multidimensional Scaling: Perturbational Analysis of Classical Scaling. Journ. R. Statis. Soc., B, 41, 217–229.
SIEGEL, A.F. and BENSON, R.H. (1982): A Robust Comparison of Biological Shapes. Biometrics, 38, 341–350.
TEN BERGE, J. M. F. (1977): Orthogonal Procrustes Rotation for Two or More Matrices. Psychometrika, 42(2), 167–276.
XU, P. (2005): Sign-constrained Robust Least Squares, Subjective Breakdown Point and the Effect of Weights of Observations on Robustness. Journal of Geodesy, 79, 146–159.
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Crosilla, F., Beinat, A. (2006). A Forward Search Method for Robust Generalised Procrustes Analysis. In: Zani, S., Cerioli, A., Riani, M., Vichi, M. (eds) Data Analysis, Classification and the Forward Search. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-35978-8_23
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DOI: https://doi.org/10.1007/3-540-35978-8_23
Publisher Name: Springer, Berlin, Heidelberg
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