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

330 12. PCA for Time Series and Other Non-Independent Datatheir Chapter 12 examines various combinations of real and complex-valuedharmonic analysis with PCA.Stoffer (1999) describes a different type of frequency domain PCA, whichhe calls the spectral envelope. Here a PCA is done on the spectral matrixF(λ) relative to the time domain covariance matrix Γ 0 .Thisisaformofgeneralized PCA for F(λ) with Γ 0 as a metric (see Section 14.2.2), andleads to solving the eigenequation [F(λ) − l(λ)Γ 0 ]a(λ) = 0 for varyingangular frequency λ. Stoffer (1999) advocates the method as a way of discoveringwhether the p series x 1 (t),x 2 (t),...x p (t) share common signalsand illustrates its use on two data sets involving pain perception and bloodpressure.The idea of cointegration is important in econometrics. It has a technicaldefinition, but can essentially be described as follows. Suppose that theelements of the p-variate time series x t are stationary after, but not before,differencing. If there are one or more vectors α such that α ′ x t is stationarywithout differencing, the p series are cointegrated. Tests for cointegrationbased on the variances of frequency domain PCs have been put forward bya number of authors. For example, Cubadda (1995) points out problemswith previously defined tests and suggests a new one.12.4.2 Growth Curves and Longitudinal DataA common type of data that take the form of curves, even if they are notnecessarily recorded as such, consists of measurements of growth for animalsor children. Some curves such as heights are monotonically increasing, butothers such as weights need not be. The idea of using principal componentsto summarize the major sources of variation in a set of growth curves datesback to Rao (1958), and several of the examples in Ramsay and Silverman(1997) are of this type. Analyses of growth curves are often concerned withpredicting future growth, and one way of doing this is to use principalcomponents as predictors. A form of generalized PC regression developedfor this purpose is described by Rao (1987).Caussinus and Ferré (1992) use PCA in a different type of analysis ofgrowth curves. They consider a 7-parameter model for a set of curves, andestimate the parameters of the model separately for each curve. These 7-parameter estimates are then taken as values of 7 variables to be analyzedby PCA. A two-dimensional plot in the space of the first two PCs givesa representation of the relative similarities between members of the set ofcurves. Because the parameters are not estimated with equal precision, aweighted version of PCA is used, based on the fixed effects model describedin Section 3.9.Growth curves constitute a special case of longitudinal data, also knownas ‘repeated measures,’ where measurements are taken on a number of individualsat several different points of time. Berkey et al. (1991) use PCAto model such data, calling their model a ‘longitudinal principal compo-