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

320 12. PCA for Time Series and Other Non-Independent DataThe third computational method described by Ramsay and Silverman(1997) involves applying numerical quadrature schemes to the integral onthe left-hand side of (12.3.1). Castro et al. (1986) used this approach in anearly example of functional PCA. Quadrature methods can be adapted tocope with irregularly spaced data (Ratcliffe and Solo, 1998). Aguilera et al.(1995) compare a method using a trigonometric basis with one based ona trapezoidal scheme, and find that the behaviour of the two algorithmsis similar except at the extremes of the time interval studied, where thetrapezoidal method is superior.Preisendorfer and Mobley (1988, Section 2d) have two interesting approachesto finding functional PCs in the case where t represents spatialposition and different observations correspond to different discrete times. Inthe first the eigenequation (12.3.1) is replaced by a dual eigenequation, obtainedby using a relationship similar to that between X ′ X and XX ′ ,whichwas noted in the proof of Property G4 in Section 3.2. This dual problem isdiscrete, rather than continuous, and so is easier to solve. Its eigenvectorsare the PC scores for the continuous problem and an equation exists forcalculating the eigenvectors of the original continuous eigenequation fromthese scores.The second approach is similar to Ramsay and Silverman’s (1997) useof basis functions, but Preisendorfer and Mobley (1988) also compare thebasis functions and the derived eigenvectors (EOFs) in order to explorethe physical meaning of the latter. Bouhaddou et al. (1987) independentlyproposed the use of interpolating basis functions in the implementation ofa continuous version of PCA, given what is necessarily a discrete-valueddata set.12.3.3 Example - 100 km Running DataHere we revisit the data that were first introduced in Section 5.3, but ina slightly different format. Recall that they consist of times taken for ten10 km sections by 80 competitors in a 100 km race. Here we convert thedata into speeds over each section for each runner. Ignoring ‘pitstops,’ itseems reasonable to model the speed of each competitor through the race asa continuous curve. In this example the horizontal axis represents positionin (one-dimensional) space rather than time. Figure 12.9 shows the speedfor each competitor, with the ten discrete points joined by straight lines.Despite the congestion of lines on the figure, the general pattern of slowingdown is apparent. Figure 12.10 shows the coefficients for the first threeordinary PCs of these speed data. In Figure 12.11 the piecewise linear plotsof Figure 12.10 are smoothed using a spline basis. Finally, Figure 12.12 givesthe eigenfunctions from a FPCA of the data, using spline basis functions,implemented in S-Plus.There are strong similarities between Figures 12.10–12.12, though somedifferences exist in the details. The first PC, as with the ‘time taken’ version