SMOOTHED FUNCTIONAL PRINCIPAL, COMPONENTS-ANALYSIS BY CHOICE OF NORM

The principal components analysis of functional data is often enhanced by the use of smoothing. It is shown that an attractive method of inc orporating smoothing is to replace the usual L(2)-orthonormality const raint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions. T he method is easily implemented in practice by making use of appropria te function transforms (Fourier transforms for periodic data) and stan dard principal components analysis programs. Several alternative possi ble interpretations of the smoothed principal components as obtained b y the method are presented. Some theoretical properties of the method are discussed: the estimates are shown to be consistent under appropri ate conditions, and asymptotic expansion techniques are used to invest igate their bias and variance properties. These indicate that the form of smoothing proposed is advantageous under mild conditions, indeed m ilder than those for existing methods of smoothed functional principal components analysis. The choice of smoothing parameter by cross-valid ation is discussed. The methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the b ehaviour of the human thumb-forefinger system.