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

358 13. Principal Component Analysis for Special Types of Datafound pairs of vectors, one each in the two subspaces spanned by A 1q , A 2q ,that have minimum angles between them, subject to being orthogonalto previous pairs of vectors. Krzanowski (1979b) notes that the sum ofeigenvalues tr(A ′ 1qA 2q A ′ 2qA 1q ) can be used as an overall measure of thesimilarity between the two subspaces. Crone and Crosby (1995) define atransformed version of this trace as an appropriate measure of subspacesimilarity, examine some of its properties, and apply it to an example fromsatellite meteorology.Another extension is to G>2 groups of individuals with covariancematrices S 1 , S 2 ,...,S G and matrices A 1q , A 2q ,...,A Gq containing the firstq eigenvectors (PC coefficients) for each group. We can then look for avector that minimizesG∑∆= cos 2 δ g ,g=1where δ g is the angle that the vector makes with the subspace defined bythe columns of A gq . This objective is achieved by finding eigenvalues andeigenvectors ofG∑A gq A ′ gq,g=1and Krzanowski (1979b) shows that for g = 2 the analysis reduces to thatgiven above.In Krzanowski (1979b) the technique is suggested as a descriptive tool—if δ (for G = 2) or ∆ (for G>2) is ‘small enough’ then the subsets of qPCs for the G groups are similar, but there is no formal definition of ‘smallenough.’ In a later paper, Krzanowski (1982) investigates the behaviour ofδ using simulation, both when all the individuals come from populationswith the same covariance matrix, and when the covariance matrices aredifferent for the two groups of individuals. The simulation encompassesseveral different values for p, q and for the sample sizes, and it also includesseveral different structures for the covariance matrices. Krzanowski (1982)is therefore able to offer some limited guidance on what constitutes a ‘smallenough’ value of δ, based on the results from his simulations.As an example, consider anatomical data similar to those discussed inSections 1.1, 4.1, 5.1, 10.1 and 10.2 that were collected for different groups ofstudents in different years. Comparing the first three PCs found for the 1982and 1983 groups of students gives a value of 2.02 ◦ for δ; the correspondingvalue for 1982 and 1984 is 1.25 ◦ , and that for 1983 and 1984 is 0.52 ◦ .Krzanowski (1982) does not have a table of simulated critical angles for thesample sizes and number of variables relevant to this example. In addition,his tables are for covariance matrices whereas the student data PCAs arefor correlation matrices. However, for illustration we note that the threevalues quoted above are well below the critical angles corresponding to the