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

324 12. PCA for Time Series and Other Non-Independent DataKeeping one or more parameters for each curve, defining how registrationwas done, leads to what Ramsay and Silverman (1997) called ‘mixed data.’Each observation for such data consists of a curve, together with p other‘ordinary’ variables. Ramsay and Silverman (1997, Chapter 8) discuss theanalysis of such data.Bivariate FPCAIn other cases, the data are not ‘mixed’ but there is more than one curveassociated with each individual. An example involving changes of angles inboth hip and knee during a gait cycle is described by Ramsay and Silverman(1997, Section 6.5). They discuss the analysis of bivariate curves from thisexample using bivariate FPCA. Suppose that the two sets of curves arex 1 (t),x 2 (t),...,x n (t); y 1 (t),y 2 (t),...,y n (t). Define a bivariate covariancefunction S(s, t) as[ ]SXX (s, t) S XY (s, t),S YX (s, t) S YY (s, t)where S XX (s, t), S YY (s, t) are covariance functions defined, as earlier, for(x(s),x(t)) and (y(s),y(t)), respectively, and S XY (s, t) has elements thatare covariances between x(s) andy(t). Suppose that∫∫z Xi = a X (t)x i (t) dt, z Yi = a Y (t)y i (t)dt.∑1 nFinding a X (t), a Y (t) to maximizen−1 i=1 (z2 Xi + z2 Yi ) leads to theeigenequations∫∫S XX (s, t)a X (t) dt + S XY (s, t)a Y (t) dt = la X (t)∫∫S YX (s, t)a X (t) dt + S YY (s, t)a Y (t) dt = la Y (t).This analysis can be extended to the case of more than two curves perindividual.SmoothingIf the data are not smooth, the weighting functions a(t) in FPCA may notbe smooth either. With most curves, an underlying smoothness is expected,with the superimposed roughness being due to noise that should ideallybe removed. Ramsay and Silverman (1997, Chapter 7) tackle this problem.Their main approach incorporates a roughness penalty into FPCA’svariance-maximizing problem. The second derivative of a curve is oftentaken as a measure of its roughness and, if D 2 a(t) represents the secondderivative of a(t), a smooth curve requires a small value of D 2 a(t). Ramsayand Silverman’s approach is to maximize∑1 nn−1 i=1 [∫ a(t)x i (t)dt] 2∫a2 (t)dt + λ ∫ (12.3.4)(D 2 a(t)) 2 dt