Smoothed functional principal components analysis by choice of norm

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.