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High breakdown estimators for principal components: the projection-pursuit approach revisited

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Author Info

  • Croux, Christophe
  • Ruiz-Gazen, Anne

Abstract

Li and Chen (J. Amer. Statist. Assoc. 80 (1985) 759) proposed a method for principal components using projection-pursuit techniques. In classical principal components one searches for directions with maximal variance, and their approach consists of replacing this variance by a robust scale measure. Li and Chen showed that this estimator is consistent, qualitative robust and inherits the breakdown point of the robust scale estimator. We complete their study by deriving the influence function of the estimators for the eigenvectors, eigenvalues and the associated dispersion matrix. Corresponding Gaussian efficiencies are presented as well. Asymptotic normality of the estimators has been treated in a paper of Cui et al. (Biometrika 90 (2003) 953), complementing the results of this paper. Furthermore, a simple explicit version of the projection-pursuit based estimator is proposed and shown to be fast to compute, orthogonally equivariant, and having the maximal finite-sample breakdown point property. We will illustrate the method with a real data example.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Multivariate Analysis.

Volume (Year): 95 (2005)
Issue (Month): 1 (July)
Pages: 206-226

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Handle: RePEc:eee:jmvana:v:95:y:2005:i:1:p:206-226

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Related research

Keywords: Breakdown point Dispersion matrix Influence function Principal components analysis Projection-pursuit Robustness;

References

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  1. Li, Baibing & Martin, Elaine B. & Morris, A. Julian, 2002. "On principal component analysis in L1," Computational Statistics & Data Analysis, Elsevier, vol. 40(3), pages 471-474, September.
  2. Croux, Christophe, 1994. "Efficient high-breakdown M-estimators of scale," Statistics & Probability Letters, Elsevier, vol. 19(5), pages 371-379, April.
  3. N. Locantore & J. Marron & D. Simpson & N. Tripoli & J. Zhang & K. Cohen & Graciela Boente & Ricardo Fraiman & Babette Brumback & Christophe Croux & Jianqing Fan & Alois Kneip & John Marden & Daniel P, 1999. "Robust principal component analysis for functional data," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 8(1), pages 1-73, June.
  4. Hengjian Cui, 2003. "Asymptotic distributions of principal components based on robust dispersions," Biometrika, Biometrika Trust, vol. 90(4), pages 953-966, December.
  5. Abul Naga, Ramses & Antille, Gerard, 1990. "Stability of robust and non-robust principal components analysis," Computational Statistics & Data Analysis, Elsevier, vol. 10(2), pages 169-174, October.
  6. Hossjer, O. & Croux, C. & Rousseeuw, P. J., 1994. "Asymptotics of Generalized S-Estimators," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 148-177, October.
  7. Graciela Boente, 2002. "Influence functions and outlier detection under the common principal components model: A robust approach," Biometrika, Biometrika Trust, vol. 89(4), pages 861-875, December.
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Citations

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Cited by:
  1. Choulakian, V. & Allard, J. & Almhana, J., 2006. "Robust centroid method," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 737-746, November.
  2. Heinrich Fritz & Peter Filzmoser & Christophe Croux, 2012. "A comparison of algorithms for the multivariate L 1 -median," Computational Statistics, Springer, vol. 27(3), pages 393-410, September.
  3. Lee, Seokho & Shin, Hyejin & Billor, Nedret, 2013. "M-type smoothing spline estimators for principal functions," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 89-100.
  4. Lanius, Vivian & Gather, Ursula, 2010. "Robust online signal extraction from multivariate time series," Computational Statistics & Data Analysis, Elsevier, vol. 54(4), pages 966-975, April.
  5. Serneels, Sven & Verdonck, Tim, 2009. "Principal component regression for data containing outliers and missing elements," Computational Statistics & Data Analysis, Elsevier, vol. 53(11), pages 3855-3863, September.
  6. Hyndman, Rob J. & Shahid Ullah, Md., 2007. "Robust forecasting of mortality and fertility rates: A functional data approach," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4942-4956, June.
  7. Hubert, Mia & Rousseeuw, Peter & Verdonck, Tim, 2009. "Robust PCA for skewed data and its outlier map," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2264-2274, April.
  8. Serneels, Sven & Verdonck, Tim, 2008. "Principal component analysis for data containing outliers and missing elements," Computational Statistics & Data Analysis, Elsevier, vol. 52(3), pages 1712-1727, January.
  9. Pires, Ana M. & Branco, João A., 2010. "Projection-pursuit approach to robust linear discriminant analysis," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2464-2485, November.
  10. Kondylis, Athanassios & Hadi, Ali S., 2006. "Derived components regression using the BACON algorithm," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 556-569, November.
  11. Valentin Todorov & Peter Filzmoser, . "An Object-Oriented Framework for Robust Multivariate Analysis," Journal of Statistical Software, American Statistical Association, vol. 32(i03).
  12. Lanius, Vivian & Gather, Ursula, 2007. "Robust online signal extraction from multivariate time series," Technical Reports 2007,38, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
  13. Boente, Graciela & Molina, Julieta & Sued, Mariela, 2010. "On the asymptotic behavior of general projection-pursuit estimators under the common principal components model," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 228-235, February.
  14. Debruyne, Michiel & Hubert, Mia & Van Horebeek, Johan, 2010. "Detecting influential observations in Kernel PCA," Computational Statistics & Data Analysis, Elsevier, vol. 54(12), pages 3007-3019, December.
  15. Rob J. Hyndman & Han Lin Shang, 2008. "Rainbow plots, Bagplots and Boxplots for Functional Data," Monash Econometrics and Business Statistics Working Papers 9/08, Monash University, Department of Econometrics and Business Statistics.

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