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Advances in Operations Research
Volume 2011, Article ID 263762, 18 pages
http://dx.doi.org/10.1155/2011/263762
Research Article

Outlier-Resistant 𝐿 𝟏 Orthogonal Regression via the Reformulation-Linearization Technique

Department of Statistical Sciences and Operations Research, Virginia Commonwealth University, P.O. Box 843083, 1015 Floyd Avenue, Richmond, VA 23284, USA

Received 9 September 2010; Revised 7 January 2011; Accepted 14 January 2011

Academic Editor: I. L. Averbakh

Copyright © 2011 J. Paul Brooks and Edward L. Boone. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. L. Brown, “Robust line estimation with errors in both variables,” Journal of the American Statistical Association, vol. 77, pp. 71–79, 1982. View at Google Scholar
  2. R. J. Carroll and P. P. Gallo, “Aspects of robustness in the functional errors-in-variables regression model,” Communications in Statistics, vol. 11, pp. 2573–2585, 1982. View at Google Scholar
  3. R. H. Zamar, “Robust estimation in the errors-in-variables model,” Biometrika, vol. 76, pp. 149–160, 1989. View at Google Scholar
  4. H. Späth and G. A. Watson, “On orthogonal linear 1 approximation,” Numerische Mathematik, vol. 51, no. 5, pp. 531–543, 1987. View at Publisher · View at Google Scholar · View at MathSciNet
  5. N. A. Campbell, “Robust procedures in multivariate analysis—I: robust covariance estimation,” Applied Statistics, vol. 29, pp. 231–237, 1980. View at Google Scholar
  6. S. J. Devlin, R. Gnandesikan, and J. R. Kettenring, “Robust estimation of dispersion matrices and principal components,” Journal of the American Statistical Association, vol. 76, pp. 354–362, 1981. View at Google Scholar
  7. J. S. Galpin and D. M. Hawkins, “Methods of L1 estimation of a covariance matrix,” Computational Statistics and Data Analysis, vol. 5, no. 4, pp. 305–319, 1987. View at Google Scholar
  8. R. A. Naga and G. Antille, “Stability of robust and non-robust principal components analysis,” Computational Statistics and Data Analysis, vol. 10, no. 2, pp. 169–174, 1990. View at Google Scholar
  9. J. I. Marden, “Some robust estimates of principal components,” Statistics and Probability Letters, vol. 43, no. 4, pp. 349–359, 1999. View at Google Scholar
  10. C. Croux and G. Haesbroeck, “Principal component analysis based on robust estimators of the covariance or correlation matrix: influence functions and efficiencies,” Biometrika, vol. 87, no. 3, pp. 603–618, 2000. View at Google Scholar
  11. H. Kamiya and S. Eguchi, “A class of robust principal component vectors,” Journal of Multivariate Analysis, vol. 77, no. 2, pp. 239–269, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  12. G. Li and Z. Chen, “Projection-pursuit approach to robust dispersion matrices and principal components: primary theory and Monte Carlo,” Journal of the American Statistical Association, vol. 80, pp. 759–766, 1985. View at Google Scholar
  13. Y. Xie, J. Wang, Y. Liang, L. Sun, X. Song, and R. Yu, “Robust principal components analysis by projection pursuit,” Journal of Chemometrics, vol. 7, pp. 527–541, 1993. View at Google Scholar
  14. R. Maronna, “Principal components and orthogonal regression based on robust scales,” Technometrics, vol. 47, no. 3, pp. 264–273, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  15. C. Croux and A. Ruiz-Gazen, “High breakdown estimators for principal components: the projection-pursuit approach revisited,” Journal of Multivariate Analysis, vol. 95, no. 1, pp. 206–226, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  16. N. Kwak, “Principal component analysis based on L1-norm maximization,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 30, no. 9, pp. 1672–1680, 2008. View at Publisher · View at Google Scholar · View at PubMed
  17. S. B. Norton, Using Biological Monitoring Data to Distinguish Among Types of Stress in Streams of the Eastern Corn Belt Plains Ecoregion, Ph.D. thesis, George Mason University, Fairfax, Va, USA, 1999.
  18. I. Lipkovich, E. P. Smith, and K. Ye, “Evaluating the impact environmental stressors on Benthic macroinvertebrate communities via Bayesian model averaging,” in Case Studies in Bayesian Statistics, pp. 267–283, 2002. View at Google Scholar
  19. R. Noble, E. P. Smith, and K. Ye, “Model selection in canonical correlation analysis (CCA) using Bayesian model averaging,” Environmetrics, vol. 15, no. 4, pp. 291–311, 2004. View at Publisher · View at Google Scholar
  20. E. L. Boone, K. Ye, and E. P. Smith, “Evaluating the relationship between ecological and habitat conditions using hierarchical models,” Journal of Agricultural, Biological, and Environmental Statistics, vol. 10, no. 2, pp. 131–147, 2005. View at Publisher · View at Google Scholar
  21. Ohio Environmental Protection Agency, The Qualitative Habitat Evaluation Index (QHEI): Rationale, Methods and Application, State of Ohio Environmental Protection Agency, 1989.
  22. Ohio Environmental Protection Agency, Biological Criteria for the Protection of Aquatic Life: Volume II: Users Manual for Biological Assessment of Ohio Surface Waters, State of Ohio Environmental Protection Agency, 1988, WQMA-SWS-6.
  23. A. Baccini, P. Besse, and A. de Faguerolles, “A L1-norm PCA and heuristic approach,” in Proceedings of the International Conference on Ordinal and Symbolic Data Analysis, pp. 359–368, 1987.
  24. S. Agarwal, M. K. Chandraker, F. Kahl, D. Kriegman, and S. Belongie, “Practical global optimization for multiview geometry,” Lecture Notes in Computer Science, vol. 3951, pp. 592–605, 2006. View at Google Scholar
  25. S. Zwanzig, “On L1-norm estimators in nonlinear regression and nonlinear errors-in-variables models,” IMS Lecture Notes—Monograph Series, vol. 35, pp. 101–118, 1997. View at Google Scholar
  26. P. J. Rousseeuw and A. Struyf, “Computing location depth and regression depth in higher dimensions,” Statistics and Computing, vol. 8, no. 3, pp. 193–203, 1998. View at Google Scholar
  27. H. D. Sherali and C. H. Tuncbilek, “A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique,” Journal of Global Optimization, vol. 2, no. 1, pp. 101–112, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  28. I. T. Jolliffe, Principal Component Analysis, Springer, New York, NY, USA, 2nd edition, 2002.
  29. R. A. Maronna and R. H. Zamar, “Robust estimates of location and dispersion for high-dimensional datasets,” Technometrics, vol. 44, no. 4, pp. 307–317, 2002. View at Google Scholar
  30. R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2009.
  31. P. Filzmozer, H. Fritz, and K. Kalcher, pcaPP: Robust PCA by Projection Pursuit, 2009.
  32. J. Lindsey, rmutil: Utilities for Nonlinear Regression and Repeated Measurements, 2009.