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

12.4. PCA and Non-Independent Data—Some Additional Topics 331nent model.’ As they are in Rao’s (1987) generalized principal componentregression, the PCs are used for prediction.Growth curves are also the subject of James et al. (2000). Their objectiveis to model growth curves when the points at which the curves aremeasured are possibly irregular, different for different individuals (observations),and sparse. Various models are proposed for such data and thedifferences and connections between models are discussed. One of theirmodels is the reduced rank modelq∑x i (t) =µ(t)+ a k (t)z ik + ɛ i (t), i=1, 2,...,n, (12.4.1)k=1where x i (t) represents the growth curve for the ith individual, µ(t) isamean curve, ɛ i (t) isanerrortermfortheith individual and a k (t),z ik arecurves defining the principal components and PC scores, respectively, as inSection 12.3.1. James et al. (2000) consider a restricted form of this modelin which µ(t) anda k (t) are expressed in terms of a spline basis, leading toa modelx i = Φ i b 0 + Φ i Bz + ɛ i ,i=1, 2,...,n. (12.4.2)Here x i , ɛ i are vectors of values x i (t),ɛ i (t) at the times for which measurementsare made on the ith individual; b 0 , B contain coefficients in theexpansions of µ(t),a k (t), respectively in terms of the spline basis; and Φ iconsists of values of that spline basis at the times measured for the ith individual.When all individuals are measured at the same time, the subscripti disappears from Φ i in (12.4.2) and the error term has covariance matrixσ 2 I p , where p is the (common) number of times that measurements aremade. James et al. (2000) note that the approach is then equivalent to aPCA of the spline coefficients in B. More generally, when the times of measurementare different for different individuals, the analysis is equivalent toa PCA with respect to the metric Φ ′ iΦ i . This extends the idea of metricbasedPCA described in Section 14.2.2 in allowing different (non-diagonal)metrics for different observations. James et al. (2000) discuss how to choosethe number of knots in the spline basis, the choice of q in equation (12.4.1),and how to construct bootstrap-based confidence intervals for the meanfunction, the curves defining the principal components, and the individualcurves.As noted in Section 9.3.4, redundancy analysis can be formulated as PCAon the predicted responses in a multivariate regression. Van den Brink andter Braak (1999) extend this idea so that some of the predictor variablesin the regression are not included among the predictors on which the PCAis done. The context in which they implement their technique is wherespecies abundances depend on time and on which ‘treatment’ is applied.The results of this analysis are called principal response curves.