Journal of Statistical Theory and Practice

, Volume 8, Issue 2, pp 400–413 | Cite as

A Modification of Silverman’s Method for Smoothed Functional Principal Components Analysis

  • S. Mohammad E. Hosseini-NasabEmail author


Statistical procedures for analyzing data that are in the form of curves and of infinite dimension are provided by functional data analysis. Functional principal component analysis is widely used in the study of functional data, since it allows finite-dimensional analysis of a problem that is intrinsically infinite dimensional. In this article, when considering smoothed functional principal component analysis (SFPCA), we first briefly review Silverman’s method for SFPCA. Then we give a modification of the Silverman’s method for SFPCA and investigate the performance of the modification through stochastic expansions. The modification is based on considering another parameter with the smoothing parameter proposed by Silverman (1996). We study the consistency under suitable conditions theoretically and show that adding the new parameter partly improves the performance of the eigenfunctions estimators toward having smaller error. We also show this improvement through a simulation study, numerically.


Eigenfunction Eigenvalue Functional data analysis Smoothed functional principal component analysis Stochastic expansion 

AMS Subject Classification

62H25 62M99 60H25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adams, R. A. 1975. Sobolev spaces. New York, NY: Academic Press.zbMATHGoogle Scholar
  2. Besse, P. 1992. PCA stability and choice of dimensionality. Stat. Prob. Lett., 13, 405–410.MathSciNetCrossRefGoogle Scholar
  3. Besse, P., and J. O., Ramsay. 1986. Principal components-analysis of sampled functions. Psychometrika, 51, 285–311.MathSciNetCrossRefGoogle Scholar
  4. Boente, G., and R. Fraiman. 2000. Kernel-based functional principal components. Statist. Probab. Lett., 48, 335–345.MathSciNetCrossRefGoogle Scholar
  5. Bosq, D. 2000. Linear processes in function spaces. Lect. Notes Stat., 149.Google Scholar
  6. Brumback, B. A., and J. A., Rice. 1998. Smoothing spline models for the analysis of nested and crossed samples of curves. J. Am. Stat. Assoc. 93, 961–976.MathSciNetCrossRefGoogle Scholar
  7. Cai, T., and P., Hall. 2007. Prediction in functional linear regression. Ann. Stat., 34, 2159–2179.MathSciNetCrossRefGoogle Scholar
  8. Cardot, H. 2000. Nonparametric estimation of smoothed principal components analysis of sampled noisy functions. J. Nonparam. Stat., 12, 503–538.MathSciNetCrossRefGoogle Scholar
  9. Cardot, H., F. Ferraty, and P. Sarda. 2000. Etude asymptotique d’un estimateur spline hybride pourle modμele lineaire fonctionnel. C. R. Acad. Sci. Paris Ser. I, 330, 501–504.CrossRefGoogle Scholar
  10. Cardot, H., F. Ferraty, and P. Sarda. 2003. Spline estimators for the functional linear model. Stat. Sin., 13, 571–591.MathSciNetzbMATHGoogle Scholar
  11. 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. J. Multivariate Anal., 12, 136–154.MathSciNetCrossRefGoogle Scholar
  12. Hall, P., and M. Hosseini-Nasab. 2006. On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68, 109–126.MathSciNetCrossRefGoogle Scholar
  13. Hall, P., and M. Hosseini-Nasab. 2009. Theory for high-order bounds in functional principal components analysis. Math. Proc. Camb. Philos. Soc., 149, 225–56.MathSciNetCrossRefGoogle Scholar
  14. He, G. Z., H.-G. Muller, and J.-L. Wang. 2003. Functional canonical analysis for square integrable stochastic processes. J. Multivar. Anal., 85, 54–77.MathSciNetCrossRefGoogle Scholar
  15. Hoerl, A. E., and R. W. Kennard. 1970a. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12, 55–67.CrossRefGoogle Scholar
  16. Hoerl, A. E., and R. W. Kennard. 1970b. Ridge regression: Application for nonorthogonal problems. Technometrics, 12, 69–82.CrossRefGoogle Scholar
  17. Hosseini-Nasab, M., and K. Z. Mirzaei. 2014. Functional analysis of glaucoma data. Stat. Med., doi: 10.1002/sim.6061.CrossRefMathSciNetGoogle Scholar
  18. James, G. M., T. J. Hastie, and C. A. Sugar. 2000. Principal component models for sparse functional data. Biometrika, 87, 587–602.MathSciNetCrossRefGoogle Scholar
  19. Mas, A. 2002. Weak convergence for the covariance operators of a Hilbertian linear process. Stochastic Process. Appl., 99, 117–135.MathSciNetCrossRefGoogle Scholar
  20. Ocana, F. A., A. M. Aguilera, and M. J. Valderrama. 1999. Functional principal components analysis by choice of norm. J. Multivariate Anal., 71, 262–276.MathSciNetCrossRefGoogle Scholar
  21. Pezzulli, S., and B. W. Silverman. 1993. Some properties of smoothed principal components analysis for functional data. Comput. Stat., 8, 1–16.MathSciNetzbMATHGoogle Scholar
  22. Qi, X., and H. Zhao. 2011. Some theoretical properties of Silverman’s method for smoothed functional principal component analysis. J. Multivaraite Anal., 102, 741–767.MathSciNetCrossRefGoogle Scholar
  23. Ramsay, J. O., and C. J. Dalzell. 1991. Some tools for functional data analysis. (With discussion.) J. R. Stat. Soc. Ser. B, 53, 539–572.zbMATHGoogle Scholar
  24. Ramsay, J. O. and B. W. Silverman. 2002. Applied functional data analysis: Methods and case studies. New York, NY: Springer.CrossRefGoogle Scholar
  25. Ramsay, J. O. and B. W. Silverman. 2005. Functional data analysis, 2nd ed., New York, NY: Springer.zbMATHGoogle Scholar
  26. Rice, J. A., and B. W. Silverman. 1991. Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. Ser. B, 53, 233–243.MathSciNetzbMATHGoogle Scholar
  27. Silverman, B. W. 1995. Incorporating parametric effects into functional principal components analysis. J. R. Stat. Soc. Ser. B, 57, 673–689.MathSciNetzbMATHGoogle Scholar
  28. Silverman, B. W. 1996. Smoothed functional principal components analysis by choice of norm. Ann. Stat., 24, 1–24.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2014

Authors and Affiliations

  1. 1.Department of Statistics, Faculty of Mathematical SciencesShahid Beheshti UniversityTehranIran

Personalised recommendations