Robust functional principal components: a projection-pursuit approach. (English) Zbl 1246.62145

Summary: In many situations, data are recorded over a period of time and may be regarded as realizations of a stochastic process. In this paper, robust estimators for the principal components are considered by adapting the projection pursuit approach to the functional data setting. Our approach combines robust projection-pursuit with different smoothing methods. Consistency of the estimators are shown under mild assumptions. The performance of the classical and robust procedures are compared in a simulation study under different contamination schemes.


62H25 Factor analysis and principal components; correspondence analysis
62G35 Nonparametric robustness
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)


fda (R); robustbase
Full Text: DOI arXiv Euclid


[1] Bali, J. L. and Boente, G. (2009). Principal points and elliptical distributions from the multivariate setting to the functional case. Statist. Probab. Lett. 79 1858-1865. · Zbl 1169.62326
[2] Bali, J. L., Boente, G., Tyler, D. E. and Wang, J. L. (2010). Robust functional principal components: A projection-pursuit approach. Available at . · Zbl 1246.62145
[3] Bali, J. L., Boente, G., Tyler, D. E. and Wang, J. L. (2011a). Supplement A to “Robust functional principal components: A projection-pursuit approach.” DOI: . · Zbl 1246.62145
[4] Bali, J. L., Boente, G., Tyler, D. E. and Wang, J. L. (2011b). Supplement B to “Robust functional principal components: A projection-pursuit approach.” DOI: . · Zbl 1246.62145
[5] Beaton, A. E. and Tukey, J. W. (1974). The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data. Technometrics 16 147-185. · Zbl 0282.62057
[6] Boente, G. and Fraiman, R. (1999). Discussion of “Robust principal components for functional data,” by Locantore et al. Test 8 28-35.
[7] Boente, G. and Fraiman, R. (2000). Kernel-based functional principal components. Statist. Probab. Lett. 48 335-345. · Zbl 0997.62024
[8] Cantoni, E. and Ronchetti, E. (2001). Resistant selection of the smoothing parameter for smoothing splines. Stat. Comput. 11 141-146.
[9] Croux, C. (1994). Efficient high-breakdown M -estimators of scale. Statist. Probab. Lett. 19 371-379. · Zbl 0791.62034
[10] Croux, C. (1999). Discussion of “Robust principal components for functional data,” by Locantore et al. Test 8 41-46.
[11] Croux, C., Filzmoser, P. and Oliveira, M. R. (2007). Algorithms for projection-pursuit robust principal component analysis. Chemometrics and Intelligent Laboratory Systems 87 218-225.
[12] Croux, C. and Ruiz-Gazen, A. (1996). A fast algorithm for robust principal components based on projection pursuit. In Compstat : Proceedings in Computational Statistics (A. Prat, ed.) 211-217. Physica-Verlag, Heidelberg. · Zbl 0900.62300
[13] Croux, C. and Ruiz-Gazen, A. (2005). High breakdown estimators for principal components: The projection-pursuit approach revisited. J. Multivariate Anal. 95 206-226. · Zbl 1065.62040
[14] Cui, H., He, X. and Ng, K. W. (2003). Asymptotic distributions of principal components based on robust dispersions. Biometrika 90 953-966. · Zbl 1436.62222
[15] Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136-154. · Zbl 0539.62064
[16] Gervini, D. (2006). Free-knot spline smoothing for functional data. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 671-687. · Zbl 1110.62044
[17] Gervini, D. (2008). Robust functional estimation using the median and spherical principal components. Biometrika 95 587-600. · Zbl 1437.62469
[18] Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 109-126. · Zbl 1141.62048
[19] Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493-1517. · Zbl 1113.62073
[20] Hampel, F. R. (1971). A general qualitative definition of robustness. Ann. Math. Statist. 42 1887-1896. · Zbl 0229.62041
[21] Huber, P. J. (1981). Robust Statistics . Wiley, New York. · Zbl 0536.62025
[22] Hyndman, R. J. and Shahid Ullah, M. (2007). Robust forecasting of mortality and fertility rates: A functional data approach. Comput. Statist. Data Anal. 51 4942-4956. · Zbl 1162.62434
[23] Li, G. and Chen, Z. (1985). Projection-pursuit approach to robust dispersion matrices and principal components: Primary theory and Monte Carlo. J. Amer. Statist. Assoc. 80 759-766. · Zbl 0595.62060
[24] Locantore, N., Marron, J. S., Simpson, D. G., Tripoli, N., Zhang, J. T. and Cohen, K. L. (1999). Robust principal component analysis for functional data. Test 8 1-73. · Zbl 0980.62049
[25] Marden, J. I. (1999). Some robust estimates of principal components. Statist. Probab. Lett. 43 349-359. · Zbl 0939.62055
[26] Maronna, R. A., Martin, R. D. and Yohai, V. J. (2006). Robust Statistics : Theory and Methods . Wiley, Chichester. · Zbl 1094.62040
[27] Martin, R. D. and Zamar, R. H. (1993). Bias robust estimation of scale. Ann. Statist. 21 991-1017. · Zbl 0787.62038
[28] Pezzulli, S. and Silverman, B. W. (1993). Some properties of smoothed principal components analysis for functional data. Comput. Statist. 8 1-16. · Zbl 0775.62146
[29] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis , 2nd ed. Springer, New York. · Zbl 1079.62006
[30] Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233-243. · Zbl 0800.62214
[31] Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1-24. · Zbl 0853.62044
[32] Tyler, D. E. (2010). A note on multivariate location and scatter statistics for sparse data sets. Statist. Probab. Lett. 80 1409-1413. · Zbl 1201.62065
[33] Varadarajan, V. S. (1958). On the convergence of sample probability distributions. Sankhyā 19 23-26. · Zbl 0082.34201
[34] Visuri, S., Koivunen, V. and Oja, H. (2000). Sign and rank covariance matrices. J. Statist. Plann. Inference 91 557-575. · Zbl 0965.62049
[35] Wang, F. T. and Scott, D. W. (1994). The L 1 method for robust nonparametric regression. J. Amer. Statist. Assoc. 89 65-76. · Zbl 0791.62044
[36] Yao, F. and Lee, T. C. M. (2006). Penalized spline models for functional principal component analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 3-25. · Zbl 1141.62050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.