Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

12.3. Functional PCA 327series than to those closer to the present time from where the future is tobe predicted. Besse et al. (2000) also use functional PCA to predict timeseries, but in the context of a smoothed first order autoregessive model.The different replications of the function correspond to different years,while the function ranges over the months within years. A ‘local’ versionof the technique is developed in which the assumption of stationarity isrelaxed.Hall et al. (2001) advocate the use of functional PCA as a dimensionreducingstep in the context of discriminating between different types ofradar signal. Although this differs from the usual set-up for PCA in discriminantanalysis (see Section 9.1) because it notionally has an infinitenumber of variables in a continuum, there is still the possibility that someof the later discarded components may contain non-trivial discriminatorypower.RotationAs with ordinary PCA, the interpretation of FPCA may be improved byrotation. In addition to the conventional rotation of coefficients in a subsetof PCs (see Section 11.1), Ramsay and Silverman (1997, Section 6.3.3)suggest that the coefficients b 1 , b 2 ,...,b m of the first m eigenfunctionswith respect to a chosen set of basis functions, as defined in Section 12.3.2,could also be rotated to help interpretation. Arbuckle and Friendly (1977)propose a variation on the usual rotation criteria of Section 11.1 for variablesthat are measurements at discrete points on a continuous curve. Theysuggest rotating the results of an ordinary PCA towards smoothness ratherthan towards simplicity as usually defined (see Section 11.1).Density EstimationKneip and Utikal (2001) discuss functional PCA as a means of examiningcommon structure and differences in a set of probability density functions.The densities are first estimated from a number of data sets using kerneldensity estimators, and these estimates are then subjected to functionalPCA. As well as a specific application, which examines how densities evolvein data sets collected over a number of years, Kneip and Utikal (2001)introduce some new methodology for estimating functional PCs and fordeciding how many components should be retained to represent the densityestimates adequately. Their paper is followed by published discussion fromfour sets of discussants.RobustnessLocantore et al. (1999) consider robust estimation of PCs for functionaldata (see Section 10.4).