Robust principal component analysis for functional data

• Born, M. and E. Wolf (1980).Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Pergamon Press, New York.

  Google Scholar 

• Cootes, T. F., A. Hill, C.J. Taylor and J. Haslam (1993). The use of active shape models for locating structures in medical images.Information Processing in Medical Imaging (H.H. Barret and A.F. Gmitro, eds.) Lecture Notes in Computer Science, vol. 687. Springer Verlag, Berlin, 33–47.

  Google Scholar 

• Devijver, P. A. and J. Kittler (1982).Pattern Recognition: A Statistical Approach. Prentice Hall, London.

  MATH  Google Scholar 

• Dryden, I.L. and K.V. Mardia (1998).Statistical Shape Analysis. Wiley: New York.

  MATH  Google Scholar 

• Fan, J. and S.K. Lin (1998). Test of significance when the data are curves.Journal of the American Statistical Association,93, 1007–1021.

  Article  MATH  MathSciNet  Google Scholar 

• Gower, J.C. (1974). The mediancentre.Applied Statistics,23, 466–470.

  Article  Google Scholar 

• Haldane, J.B.S. (1948). Note on the median of a multivariate distribution.biometrika,35, 414–415.

  MATH  MathSciNet  Google Scholar 

• Hampel, F.R., E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel (1986).Robust Statistics: The Approach Based on Influence Functions Wiley, New York.

  MATH  Google Scholar 

• Huber, P.J. (1981).Robust Statistics. Wiley, New York.

  Book  MATH  Google Scholar 

• Inselberg, A. (1985). The plane with parallel coordinates.The Visual Computer,1, 69–91.

  Article  MATH  Google Scholar 

• Kelemen, A., G. Szekely and G. Gerig (1997). Three dimensional model-based segmentation. TR-178 Technical Report Image Science Lab, ETH Zurich.

• Milasevic, P. and G.R. Ducharme (1987). Uniqueness of the spatial median.Annals of Statistics,15, 1332–1333.

  MATH  MathSciNet  Google Scholar 

• Ramsay, J. O. and B.W. Silverman (1997).Functional Data Analysis. Springer Verlag, New York.

  MATH  Google Scholar 

• Rousseeuw, P.J. and A.M. Leroy (1987).Robust Regression and Outlier Detection. Wiley, New York.

  Book  MATH  Google Scholar 

• Small, G.C. (1990). A survey of multidimensional medians.International Statistical Review,58, 263–277.

  Article  Google Scholar 

• Staudte, R.G. and S.J. Sheather (1990).Robust Estimation and Testing, Wiley, New York.

  MATH  Google Scholar 

• Schwiegerling, J., J.E. Greivenkamp and J.M. Miller (1995). Representation of videokeratoscopic height data with Zernike polynomials.Journal of the Optical Society of America, A,12, 12105–2113.

  Article  Google Scholar 

• Wegman, E.J. (1990). Hyperdimensional data analysis using parallel coordinates.Journal of the American Statistical Association,85, 664–675.

  Article  Google Scholar 

References

• Besse, P. and J.O. Ramsay (1986). Principal component analysis of sampled functions.Psychometrika,51, 285–311.

  Article  MATH  MathSciNet  Google Scholar 

• Boente, G. and R. Fraiman (1998). Kernel-based functional principal components. Unpublished Manuscript.

• Dauxois, J., A. Pousse and Y. Romain (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference.Journal of Multivariate Analysis,12, 136–154.

  Article  MATH  MathSciNet  Google Scholar 

• Pezzulli, S.D. and B.W. Silverman (1993). Some properties of smoothed principal components analysis for functional data.Computational Statistics and Data Analysis,8, 1–16.

  MATH  MathSciNet  Google Scholar 

• Ramsay, J.O. and C.J. Dalzell (1991). Some tools for functional data analysis (with discussion).Journal of the Royal Statistics Society, B,53, 539–572.

  MATH  MathSciNet  Google Scholar 

• Ramsay, J.O. and B.W. Silverman (1997).Functional Data Analysis. Springer Series in Statistics, Springer-Verlag, New York.

  MATH  Google Scholar 

• Rice, J. and B.W. Silverman (1991). Estimating the mean and covariance structure nonparametrically when the data are curves.Journal of the Royal Statistics Society, B,53, 233–243.

  MATH  MathSciNet  Google Scholar 

• Silverman, B.W. (1996). Smoothed functional principal components analysis by choice of norm.Anals of Statistics,24, 1–24.

  Article  MATH  Google Scholar 

• Tyler, D. (1991). Some issues in the robust estimation of multivariate location and scatter. InDirections in Robust Statistics and Diagnostics (W. Stahel and S. Weisberg eds.) Springer Verlag, New York, 327–336.

  Google Scholar 

References

• Brumback, B.A. and J.A. Rice (1998). Smoothing spline models for the analysis of nested and crossed samples of curves (with discussion).Journal of the American Statistical Association,93, 961–994.

  Article  MATH  MathSciNet  Google Scholar 

• Rice, J.A. and B.W. Silverman (1991). Estimating the mean and covariance structure nonparametrically when the data are curves.Journal of the Royal Statistical Society, B,53, 233–243.

  MATH  MathSciNet  Google Scholar 

• Silverman, B.W. (1996). Smoothed functional principal components analysis by choice of norm.Annals of Statistics,24, 1–24.

  Article  MATH  MathSciNet  Google Scholar 

References

• Bedall, F.K. and H. Zimmermann (1979). Algorithm AS 143. The mediancentre.Applied Statistics,28, 325–328.

  Article  MATH  Google Scholar 

• Critchley, F. (1985). Influence in principal component analysis.Biometrika,72, 627–636.

  Article  MATH  MathSciNet  Google Scholar 

• Croux, C. and G. Haesbroeck (1999). Principal component analysis based on robust estimators of the covariance or correlation matrix: influence functions and efficiencies. Preprint, University of Brussels (ULB), http://www.sig.eggs.ulg.ac.be/Haesbroeck/

• Devlin, S.J., R. Guanadesikan and J.R. Kettenring (1981). Robust estimation of dispersion matrices and principal components.Journal of the American Statistical Association,76, 354–362.

  Article  MATH  Google Scholar 

• Hössjer, O. and C. Croux (1995). Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter.Nonparametric Statistics,4, 293–308.

  MathSciNet  MATH  Google Scholar 

• Li, G. and Z. Chen (1985). Projection-pursuit approach to robust dispersion matrices and principal components: primary theory and Monte Carlo.Journal of the American Statistical Association,80, 759–766.

  Article  MATH  Google Scholar 

References

• Fan, J. and S.K. Lin (1998). Test of significance when the data are curves.Journal of the American Statistical Association,93, 1007–1021.

  Article  MATH  MathSciNet  Google Scholar 

• Liu, R.Y. and K. Singh (1992). Ordering directional data: concepts of data depth on circles and spheres.Annals of Statistics,20, 1468–1484.

  MATH  MathSciNet  Google Scholar 

• Rice, J.A. and B.W. Silverman (1991). Estimating the mean and covariance structure nonparametrically when the data are curves.Journal of the Royal Statistical Society, B,53, 233–243.

  MATH  MathSciNet  Google Scholar 

References

• Bosq, D. (1991). Modelization, non-parametric estimation and prediction for continuous time processes.NATO, ASI Series, Springer Verlag, New York.

  Google Scholar 

• Kneip, A. (1994). Nonparametric estimation of common regressors for similar curve data.Annals of Statistics,22, 1386–1428.

  MATH  MathSciNet  Google Scholar 

• Kneip, A. and T. Gasser (1992). Statistical tools to analyze data representing a sample of curves.Annals of Statistics,20, 1266–1305.

  MATH  MathSciNet  Google Scholar 

• Kneip, A., X. Li, B. MacGibbon and J.O. Ramsay (1998). Curve registration by local regression.Canadian Journal of Statistics, to appear.

• Kneip, A. and K. Utikal (1999). Inference for density families using functional principal component analysis. Manuscript.

• Ramsay, J.O. and X. Li (1996). Curve registration.Journal of the Royal Statistical Society, to appear.

• Ramsay, J. O. and B.W. Silverman (1997).Functional Data Analysis, Springer Verlag, New York.

  MATH  Google Scholar 

• Silverman, B.W. (1995). Incorporating parametric effects into functional principal component analysis.Journal of the Royal Statistical Society B,57, 673–689.

  MATH  Google Scholar 

• Wang, K. and T. Gasser (1995). Alignment of curves by dynamic time warping.Biometrics,14, 1–17.

  MATH  Google Scholar 

References

• Brown, B.M. (1983). Statistical use of the spatial median.Journal of the Royal Statistical Society, B,45, 25–30.

  MATH  Google Scholar 

• Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data.Journal of the American Statistical Association,91, 862–872.

  Article  MATH  MathSciNet  Google Scholar 

• Marden, J.I. (1999). Some robust estimates of principal components.Statistics and Probability Letters,43 (to appear).

• Visuri, S., V. Koivunen, H. and Oja (1999). Sign and rank covariance matrices.

References

• Bennett, R.J. (1979).Spatial Time Series. Chapman and Hall.

• Donoho, D.L. (1982). Breakdown Properties of Multivariate Location Estimators. Ph.D. qualifying paper, Harvard University, Department of Statistics.

• Droesbeke, F. (ed.) (1987).Spatial Processes and Spatial Time Series Analysis. Facultes Universitaires Saint-Louis.

• Fan, J. and S.K. Lin (1998). Test of significance when the data are curves.Journal of the American Statistical Association,93, 1007–1021.

  Article  MATH  MathSciNet  Google Scholar 

• Huber, P.J. (1985). Projection pursuit.Annals of Statistics,13, 435–475.

  MATH  MathSciNet  Google Scholar 

• Jones, M.C. and R. Sibson (1987). What is projection pursuit?.Journal of the Royal Statistical Society, A,150, 1–36.

  Article  MATH  MathSciNet  Google Scholar 

• Peña, D. and F.J. Prieto (1997). Robust covariance matrix estimation and multivariate outlier detection. Working Paper WP97-08, Universidad Carlos III de Madrid.

• Piccolo, D. (1990). A distance measure for classifying ARIMA models.Journal of Time Series Analysis,11, 153–164.

  MATH  Google Scholar 

• Stahel, W.A. (1981). Robuste Schätzungen: Infinitesimale Optimalität und Schätzungen von Kovarianzmatrizen. Ph.D. Thesis, ETH Zurich.

References

• Brumback, B. and J. Rice (1998). Smoothing spline models for the analysis of nested and crossed samples of curves (with discussion).Journal of the American Statistical Association,93, 961–994.

  Article  MATH  MathSciNet  Google Scholar 

References

• Aguilera, A. M., R. Gutiérrez and M.J. Valderrama (1996). Approximation of estimators in the PCA of a stochastic process using B-splines.Communications in Statistics: Simulation and Computation,25, 671–690.

  MATH  MathSciNet  Google Scholar 

• Aguilera, A.M., F.A. Ocaña and M.J. Valderrama (1999). Principal component analysis of Hilbertian random variables on finite-dimensional spaces. Technical Report 02/99, Department of Statistics and Operations Research, University of Granada.

• Croux, C. and A. Ruiz-Gazen (1996). A fast algorithm for robust principal components based on projection pursuit.Proceedings in Computational Statistics 1996, (A. Prat, ed.) Barcelona, Spain, 211–216.

• Ruiz, M.D. and M.J. Valderrama (1997). Orthogonal representations of random fields and an application to geophysies data.Journal of Applied Probability,34, 458–476.

  Article  MATH  MathSciNet  Google Scholar 

References

• Carroll, R.J., D. Ruppert and L.A. Stefanski (1995).Measurement, Error in Nonlinear Models, Chapman and Hall, London.

  MATH  Google Scholar 

• Good, I.J. (1969). Some applications of the singular value decomposition of a matrix.Technometrics,11, 823–831.

  Article  MATH  Google Scholar 

• He, X. and D.G. Simpson (1992). Robust direction estimation.Annals of Statistics,20, 351–369.

  MATH  MathSciNet  Google Scholar 

Download references