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

292 11. Rotation and Interpretation of Principal Components11.3 Simplified Approximations to PrincipalComponentsThe techniques of the previous section are alternatives to PCA that sacrificesome variance in order to enhance simplicity. A different approach toimproving interpretability is to find the PCs, as usual, but then to approximatethem. In Chapter 4, especially its first section, it was mentioned thatthere is usually no need to express the coefficients of a set of PCs to morethan one or two decimal places. Rounding the coefficients in this manner isone way of approximating the PCs. The vectors of rounded coefficients willno longer be exactly orthogonal, the rounded PCs will not be uncorrelatedand their variances will be changed, but typically these effects will not bevery great, as demonstrated by Green (1977) and Bibby (1980). The latterpaper presents bounds on the changes in the variances of the PCs (both inabsolute and relative terms) that are induced by rounding coefficients, andshows that in practice the changes are quite small, even with fairly severerounding.To illustrate the effect of severe rounding, consider again Table 3.2,in which PCs for eight blood chemistry variables have their coefficientsrounded to the nearest 0.2. Thus, the coefficients for the first PC, forexample, are given as0.2 0.4 0.4 0.4 − 0.4 − 0.4 − 0.2 − 0.2.Their values to three decimal place are, by comparison,0.195 0.400 0.459 0.430 − 0.494 − 0.320 − 0.177 − 0.171.The variance of the rounded PC is 2.536, compared to an exact varianceof 2.792, a change of 9%. The angle between the vectors of coefficientsdefining the exact and rounded PCs is about 8 ◦ . For the second, third andfourth PCs given in Table 3.2, the changes in variances are 7%, 11% and11%, respectively, and the angles between vectors of coefficients for exactand rounded PCs are 8 ◦ in each case. The angle between the vectors ofcoefficients for the first two rounded PCs is 99 ◦ and their correlation is−0.15. None of these changes or angles is unduly extreme considering theseverity of the rounding that is employed. However, in an example fromquality control given by Jackson (1991, Section 7.3) some of correlationsbetween PCs whose coefficients are approximated by integers are worringlylarge.Bibby (1980) and Jackson (1991, Section 7.3) also mention the possibilityof using conveniently chosen integer values for the coefficients.For example, the simplified first PC from Table 3.2 is proportional to2(x 2 + x 3 + x 4 )+x 1 − (x 7 + x 8 ) − 2(x 5 + x 6 ), which should be much simplerto interpret than the exact form of the PC. This is in the same spirit as thetechniques of Section 11.2.1, which restrict coefficients to be proportional