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

278 11. Rotation and Interpretation of Principal Componentsand associates exactly one component with each cluster. The criterion tobe optimized is the sum over variables of squared loadings for each variablefor the single component associated with that variable’s cluster. The finalmethod is similar to the one that maximizes the quartimax criterion, butit relaxes the requirement of orthogonality of components.Gains in simplicity achieved by any of the methods are paid for by aloss of variance explained compared to rotated PCA, but none of the fourmethods explicitly takes into account the desirability of minimizing thevariance lost. Kiers (1993) investigates the trade-off between simplicity gainand variance loss for the four methods in a simulation study. He finds thatneither of the first two methods offers any advantage over rotated PCA.The third and fourth methods show some improvement over rotated PCAin recovering the simple structures that are built into the simulated data,and the fourth is better than the third in terms of retaining variance.Two of the disadvantages of standard rotation noted earlier are the loss ofthe successive optimization property and the possible sensitivity of the resultsto the choice of m. All four techniques compared by Kiers (1993) sharethese disadvantages. We now describe a method that avoids these drawbacksand explicitly takes into account both variance and simplicity. LikeKiers’ (1993) methods it replaces the two stages of rotated PCA by a singlestep. Linear combinations of the p variables are successively found thatmaximize a criterion in which variance is combined with a penalty functionthat pushes the linear combination towards simplicity. The method isknown as the Simplified Component Technique (SCoT) (Jolliffe and Uddin,2000).Let c ′ k x i be the value of the kth simplified component (SC) for the ithobservation. Suppose that Sim(c k ) is a measure of simplicity for the vectorc k , for example the varimax criterion, and Var(c ′ kx) denotes the samplevariance c ′ k Sc k of c ′ kx. Then SCoT successively maximizes(1 − ψ)Var(c ′ kx)+ψSim(c k ) (11.1.6)subject to c ′ k c k = 1 and (for k ≥ 2, h