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

326 12. PCA for Time Series and Other Non-Independent Dataon the smoothed data. Kneip (1994) independently suggested smoothingthe data, followed by PCA on the smoothed data, in the context of fittinga model in which a small number of functions is assumed to underlie aset of regression curves. Theoretical properties of Kneip’s (1994) procedureare examined in detail in his paper. Champely and Doledec (1997) uselo(w)ess (locally weighted scatterplot smoothing—Cleveland, 1979, 1981)to fit smooth trend and periodic curves to water quality data, and thenapply FPCA separately to the trend and periodic curves.Principal Differential AnalysisPrincipal differential analysis (PDA), a term coined by Ramsay (1996) anddiscussed in Ramsay and Silverman (1997, Chapter 14) is another methodof approximating a set of curves by a smaller number of functions. AlthoughPDA has some similarities to FPCA, which we note below, it concentrateson finding functions that have a certain type of smoothness, rather thanmaximizing variance. Define a linear differential operatorL = w 0 I + w 1 D + ...+ w m−1 D m−1 + D m ,where D i , as before, denotes the ith derivative operator and I is the identityoperator. PDA finds weights w 0 ,w 1 ,...,w m−1 for which [Lx i (t)] 2 is smallfor each observed curve x i (t). Formally, we minimize ∑ n∫i=1 [Lxi (t)] 2 dtwith respect to w 0 ,w 1 ,...,w m−1 .Once w 0 ,w 1 ,...,w m−1 and hence L are found, any curve satisfyingLx(t) = 0 can be expressed as a linear combination of m linearly independentfunctions spanning the null space of the operator L. Anyobservedcurve x i (t) can be approximated by expanding it in terms of these m functions.This is similar to PCA, where the original data can be approximatedby expanding them in terms of the first few (say m) PCs. The difference isthat PCA finds an m-dimensional space with a least squares fit to the originaldata, whereas PDA finds a space which penalizes roughness. This lastinterpretation follows because Lx i (t) is typically rougher than x i (t), andPDA aims to make Lx i (t), or rather [Lx i (t)] 2 , as small as possible whenaveraged over i and t. An application of PDA to the study of variations inhandwriting is presented by Ramsay (2000).Prediction and DiscriminationAguilera et al. (1997, 1999a) discuss the idea of predicting a continuoustime series by regressing functional PCs for the future on functional PCsfrom the past. To implement this methodology, Aguilera et al. (1999b)propose cutting the series into a number of segments n of equal length,which are then treated as n realizations of the same underlying process.Each segment is in turn divided into two parts, with the second, shorter,part to be predicted from the first. In calculating means and covariancefunctions, less weight is given to the segments from the early part of the