Open Access
February 1996 Smoothed functional principal components analysis by choice of norm
Bernard W. Silverman
Ann. Statist. 24(1): 1-24 (February 1996). DOI: 10.1214/aos/1033066196


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.


Download Citation

Bernard W. Silverman. "Smoothed functional principal components analysis by choice of norm." Ann. Statist. 24 (1) 1 - 24, February 1996.


Published: February 1996
First available in Project Euclid: 26 September 2002

zbMATH: 0853.62044
MathSciNet: MR1389877
Digital Object Identifier: 10.1214/aos/1033066196

Primary: 62G07 , 62H25
Secondary: 65D10 , 73P20

Keywords: Biomechanics , consistency , cross-validation , Functional data analysis , mean integrated square error , PCA , roughness penalty , smoothing

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 1 • February 1996
Back to Top