Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Resistant GPA algorithms based on the M and LMS estimation

  • Hyun, Geehong (Department of Epidemiology and Cancer Control, St. Jude Children's Research Hospital) ;
  • Lee, Bo-Hui (Department of Statistics, Pusan National University) ;
  • Choi, Yong-Seok (Department of Statistics, Pusan National University)
  • Received : 2018.10.11
  • Accepted : 2018.11.13
  • Published : 2018.11.30
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Abstract

Procrustes analysis is a useful technique useful to measure, compare shape differences and estimate a mean shape for objects; however it is based on a least squares criterion and is affected by some outliers. Therefore, we propose two generalized Procrustes analysis methods based on M-estimation and least median of squares estimation that are resistant to object outliers. In addition, two algorithms are given for practical implementation. A simulation study and some examples are used to examine and compared the performances of the algorithms with the least square method. Moreover since these resistant GPA methods are available for higher dimensions, we need some methods to visualize the objects and mean shape effectively. Also since we have concentrated on resistant fitting methods without considering shape distributions, we wish to shape analysis not be sensitive to particular model.

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References

  1. Dryden IL and Mardia KV (2016). Statistical Shape Analysis with Applications in R (2nd ed), John Wiley & Sons, New York.
  2. Dryden IL and Walker G (1999). Highly resistant regression and object matching, Biometrics, 55, 820-825. https://doi.org/10.1111/j.0006-341X.1999.00820.x
  3. Er F (1998). Robust methods in statistical shape analysis (Ph.D. thesis), University of Leeds, Leeds, UK.
  4. Goodall C (1991). Procrustes methods in the statistical analysis of shape (with discussion), Journal of Royal Statistical Society, Series B (Methodological), 53, 285-339. https://doi.org/10.1111/j.2517-6161.1991.tb01825.x
  5. Gower JC (1975). Generalized Procrustes analysis, Psychometrika, 40, 33-51. https://doi.org/10.1007/BF02291478
  6. Kendall DG (1984). Shape manifolds, Procrustean metrics, and complex projective spaces, Bulletin of the London Mathematical Society, 16, 81-121. https://doi.org/10.1112/blms/16.2.81
  7. Kent JT (1994). The complex Bingham distribution and shape analysis, Journal of Royal Statistical Society, Series B (Methodological) , 56, 285-299. https://doi.org/10.1111/j.2517-6161.1994.tb01978.x
  8. Rohlf FJ and Slice D (1990). Extensions of the Procrustes method for the optimal superimposition of landmarks, Systematic Zoology, 39, 40-59. https://doi.org/10.2307/2992207
  9. Rousseeuw PJ (1984). Least median of squares regression, Journal of the American Statistical Association, 79, 871-880. https://doi.org/10.1080/01621459.1984.10477105
  10. Rousseeuw PJ and Leroy AM (1987). Robust Regression and Outlier Detection, Wiley, New York.
  11. Siegel AF and Benson RH (1982). A robust comparison of biological shapes, Biometrics, 38, 341-350. https://doi.org/10.2307/2530448
  12. Ten Berge JMF (1977). Orthogonal Procrustes rotation for two or more matrices, Psychometrika, 42, 267-276. https://doi.org/10.1007/BF02294053
  13. Verboon P and Heiser WJ (1992). Resistant orthogonal Procrustes analysis, Journal of Classification, 9, 237-256. https://doi.org/10.1007/BF02621408