## The Annals of Statistics

### Smoothed functional principal components analysis by choice of norm

Bernard W. Silverman

#### Abstract

The principal components analysis of functional data is often enhanced by the use of smoothing. It is shown that an attractive method of incorporating smoothing is to replace the usual $L^2$-orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions. The method is easily implemented in practice by making use of appropriate function transforms (Fourier transforms for periodic data) and standard principal components analysis programs. Several alternative possible interpretations of the smoothed principal components as obtained by the method are presented. Some theoretical properties of the method are discussed: the estimates are shown to be consistent under appropriate conditions, and asymptotic expansion techniques are used to investigate their bias and variance properties. These indicate that the form of smoothing proposed is advantageous under mild conditions, indeed milder than those for existing methods of smoothed functional principal components analysis. The choice of smoothing parameter by cross-validation is discussed. The methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb-forefinger system.

#### Article information

Source
Ann. Statist., Volume 24, Number 1 (1996), 1-24.

Dates
First available in Project Euclid: 26 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1033066196

Digital Object Identifier
doi:10.1214/aos/1033066196

Mathematical Reviews number (MathSciNet)
MR1389877

Zentralblatt MATH identifier
0853.62044

#### Citation

Silverman, Bernard W. Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 (1996), no. 1, 1--24. doi:10.1214/aos/1033066196. https://projecteuclid.org/euclid.aos/1033066196

#### References

• ADAMS, R. A. 1975. Sobolev Spaces. Academic Press, New York. Z.
• DALZELL, C. J. and RAMSAY, J. O. 1993. Computing reproducing kernels with arbitrary boundary constraints. SIAM J. Sci. Comput. 14 511 518. Z.
• DAUXOIS, J., POUSSE, A. and ROMAIN, Y. 1982. Asy mptotic theory for the principal component analysis of a vector random function: some applications to statistical inference. J. Multivariate Anal. 12 136 154. Z.
• GREEN, P. J. and SILVERMAN, B. W. 1994. Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman and Hall, London. Z.
• LEURGANS, S. E., MOy EED, R. A. and SILVERMAN, B. W. 1993. Canonical correlation analysis when the data are curves. J. Roy. Statist. Soc. Ser. B 55 725 740. Z.
• PEZZULLI, S. D. and SILVERMAN, B. W. 1993. Some properties of smoothed principal components analysis for functional data. Comput. Statist. Data Anal. 8 1 16. Z.
• RAMSAY, J. O. 1995. Some tools for the multivariate analysis of functional data. In Recent Z. Advances in Descriptive Multivariate Analy sis W. Krzanowski, ed. 269 282. Clarendon, Oxford. Z. Z
• RAMSAY, J. O. and DALZELL, C. J. 1991. Some tools for functional data analysis with discus. sion. J. Roy. Statist. Soc. Ser. B 53 539 572. Z.
• RAMSAY, J. O., FLANAGAN, R. and WANG, X. 1995. The functional data analysis of the pinch force of human fingers. J. Roy. Statist. Soc. Ser. C 44 17 30. Z.
• 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. Z.
• SPIVAK, M. 1967. Calculus. Addison-Wesley, Reading, MA. Z.
• TAy LOR, A. E. and LAY, D. C. 1980. Introduction to Functional Analy sis. Wiley, New York. Z.
• TUKEY, J. W. 1977. Exploratory Data Analy sis. Addison-Wesley, Reading, MA.