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

11Rotation and Interpretation ofPrincipal ComponentsIt was noted earlier, especially in Chapter 4, that PCs are particularlyuseful if they can be simply interpreted. The word reification is sometimesused for this process of interpretation, and a number of examples have beenseen in previous chapters for which this has been successful. However, theconstruction of PCs as linear combinations of all the measured variablesmeans that interpretation is not always easy. Various suggestions have beenmade to simplify the process of interpretation. These are the subject of thischapter.One way to aid interpretation is to rotate the components, as is donewith the factor loadings in factor analysis (see Chapter 7). Rotation of PCsis discussed in Section 11.1. The approach can provide useful simplificationin some cases, but it has a number of drawbacks, and some alternativeapproaches are described. In one the two steps of PCA followed by rotationare replaced by a single optimization problem which takes into account bothvariance maximization and simplicity. In others simplification criteria areemphasized at the expense of variance maximization.Section 11.2 describes some alternatives to PCA which aim to providesimpler ‘components.’ Some techniques restrict the coefficients of the variablesin each component to integer values, whilst another drives some of thecoefficients to zero. The purpose of these techniques is to provide replacementsfor PCs that are simpler to interpret, but which do not sacrificemuch variance. In other circumstances, the objective may be to approximatethe PCs in a way which makes them simpler to interpret, againwithout much loss of variance. The most common way of doing this is toignore (effectively set to zero) coefficients whose absolute values fall below