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

11.1. Rotation of Principal Components 277Table 11.1. Unrotated and rotated loadings for components 3 and 4: artisticqualities data.PC3 PC4 RPC3 RPC4Composition −0.59 −0.41 −0.27 0.66Drawing 0.60 −0.50 0.78 −0.09Colour 0.49 −0.22 0.53 0.09Expression 0.23 0.73 −0.21 0.74Percentage of 10.0 7.3 9.3 8.0total variationtogether they account for 83% of the total variation. Rotating them maylose information on individual dominant sources of variation. On the otherhand, the last two PCs have similar eigenvalues and are candidates for rotation.Table 11.1 gives unrotated (PC3, PC4) and rotated (RPC3, RPC4)loadings for these two components, using varimax rotation and the normalizationconstraints a ′ k a k = 1. Jolliffe (1989) shows that using alternativerotation criteria to varimax makes almost no difference to the rotated loadings.Different normalization constraints also affect the results very little.Using ã ′ kãk = l k gives vectors of rotated loadings whose angles with thevectors for the constraint a ′ k a k =1areonly6 ◦ and 2 ◦ .It can be seen from Table 11.1 that rotation of components 3 and 4 considerablysimplifies their structure. The rotated version of component 4 isalmost a pure contrast between expression and composition, and component3 is also simpler than either of the unrotated components. As well asillustrating rotation of components with similar eigenvalues, this examplealso serves as a reminder that the last few, as well as the first few, componentsare sometimes of interest (see Sections 3.1, 3.7, 6.3, 8.4–8.6, 9.1 and10.1).In such cases, interpretation of the last few components may have asmuch relevance as interpretation of the first few, and for the last fewcomponents close eigenvalues are more likely.11.1.2 One-step Procedures Using Simplicity CriteriaKiers (1993) notes that even when a PCA solution is rotated to simplestructure, the result may still not be as simple as required. It may thereforebe desirable to put more emphasis on simplicity than on variancemaximization, and Kiers (1993) discusses and compares four techniquesthat do this. Two of these explicitly attempt to maximize one of the standardsimplicity criteria, namely varimax and quartimax respectively, overall possible sets of orthogonal components, for a chosen number of componentsm. A third method divides the variables into non-overlapping clusters