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

oot of the corresponding eigenvalue of S, giving11.1. Rotation of Principal Components 273orã k =(lkn − 1) 12akà m =(n − 1) − 1 2 Am L m[Note that in this section we are using the notation of Section 5.3. Hencethe matrix L has elements l 1/2kthat are square roots of the eigenvalues ofX ′ X.AsS = X ′ X/(n−1), the square roots of its eigenvalues are ( l k 1/2].n−1)For this normalization, equation (11.1.4) becomes T ′ L 2 mT/(n − 1) and(11.1.5) is T ′ L 4 mT/(n − 1). Neither of these matrices is diagonal, so therotated PCs are correlated and have loadings that are not orthogonal.To obtain uncorrelated components a different normalization is needed inwhich each column a k of A m is divided by the square root of the correspondingeigenvalue of S. This normalization is used in the discussion ofoutlier detection in Section 10.1. Here it givesà m =(n − 1) 1 2 Am L −1mand equation (11.1.5) becomes (n − 1)T ′ T =(n − 1)I m , which is diagonal.Hence, the components are uncorrelated when this normalization is used.However, equation (11.1.4) is now (n − 1)T ′ L −2m T, which is not diagonal,so the loadings of the rotated components are not orthogonal.It would seem that the common normalization Ãm =(n − 1) − 1 2 A m L mshould be avoided, as it has neither orthogonal loadings nor uncorrelatedcomponents once rotation is done. However, it is often apparently usedin atmospheric science with a claim that the components are uncorrelated.The reason for this is that rotation is interpreted in a factor analysis frameworkrather than in terms of rotating PCA loadings. If the loadings B m areused to calculate rotated PC scores, then the properties derived above hold.However, in a factor analysis context an attempt is made to reconstruct Xfrom the underlying factors. With notation based on that of Chapter 7, wehave X ≈ FΛ ′ , and if PCs are used as factors this becomes X ≈ Z R mA ′ m.If a column of A m is multiplied by ( l k 1/2,n−1)then a corresponding columnof Z R m must be divided by the same quantity to preserve the approximationto X. This is similar to what happens when using the singular valuedecomposition (SVD) of X to construct different varieties of biplot (Section5.3). By contrast, in the PCA approach to rotation multiplicationof a column of A m by a constant implies that the corresponding columnof Z R m is multiplied by the same constant. The consequence is that whenthe factor analysis (or SVD) framework is used, the scores are found usingthe normalization à m =(n − 1) 1/2 A m L −1m when the normalizationà m =(n − 1) − 1 2 A m L m is adopted for the loadings, and vice versa. Jolliffe