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

11.2. Alternatives to Rotation 289As t decreases from √ p, the SCoTLASS components move progressivelyaway from PCA and some of the loadings become zero. Eventually, for t =1, a solution is reached in which only one variable has a non-zero loading oneach component, as with ψ = 1 in SCoT (Section 11.1.2). Examples follow.Mediterranean SSTWe return to the Mediterranean SST example and Figures 11.2–11.5 oncemore. As with SCoT in Section 11.1.2, results are given for one value of thetuning parameter, in this case t, chosen to give a compromise between varianceretention and simplicity. In the autumn, SCoTLASS behaves ratherlike rotated PCA, except that its patterns are more clearcut, with severalgrid boxes having zero rather than small loadings. Its first componentconcentrates on the Eastern Mediterranean, as does rotated PCA, but itssecond component is centred a little further west than the second rotatedPC. The patterns found in winter are fairly similar to those in autumn,and again similar to those of rotated PCA except for a reversal in order.In autumn and winter, respectively, the first two SCoTLASS componentsaccount for 70.5%, 65.9% of the total variation. This compares with 78.2%,71.0% for PCA and 65.8%, 57.9% for rotated PCA. The comparison withrotated PCA is somewhat unfair to the latter as we have chosen to displayonly two out of three rotated PCs. If just two PCs had been rotated,the rotated PCs would account for the same total variation as the first twoPCs. However, the fact that the first two SCoTLASS components are muchsimplified versions of the two displayed rotated PCs, and at the same timehave substantially larger variances, suggests that SCoTLASS is superior torotated PCA in this example.PitpropsTables 11.3 and 11.4 give coefficients and cumulative variances for the firstand fourth SCoTLASS components for Jeffers’ (1967) pitprop data. Thefirst component sacrifices about the same amount of variance comparedto PCA as the first simple component. Both achieve a greatly simplifiedpattern of coefficients compared to PCA, but, as in the SST example, ofquite different types. For the fourth component, the cumulative varianceis again similar to that of the simple components, but the SCoTLASScomponent is clearly simpler in this case. Further examples and discussionof the properties of SCoTLASS can be found in Jolliffe et al. (2002a) andUddin (1999).11.2.3 Empirical Orthogonal TeleconnectionsVan den Dool et al. (2000) propose a technique, the results of which theyrefer to as empirical orthogonal teleconnections (EOTs). The data they considerhave the standard atmospheric science form in which the p variables