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

362 13. Principal Component Analysis for Special Types of Datacomponents which diagonalize the covariance or correlation matrices reflectthe most important sources of variation in the data.’Muller (1982) suggests that canonical correlation analysis of PCs (seeSection 9.3) provides a way of comparing the PCs based on two sets ofvariables, and cites some earlier references. When the two sets of variablesare, in fact, the same variables measured for two groups of observations,Muller’s analysis is equivalent to that of Krzanowski (1979b); the latterpaper notes the links between canonical correlation analysis and its owntechnique.In a series of five technical reports, Preisendorfer and Mobley (1982) examinevarious ways of comparing data sets measured on the same variablesat different times, and part of their work involves comparison of PCs fromdifferent sets (see, in particular, their third report, which concentrates oncomparing the singular value decompositions (SVDs, Section 3.5) of twodata matrices X 1 , X 2 ). Suppose that the SVDs are writtenX 1 = U 1 L 1 A ′ 1X 2 = U 2 L 2 A ′ 2.Then Preisendorfer and Mobley (1982) define a number of statistics thatcompare U 1 with U 2 , A 1 with A 2 , L 1 with L 2 or compare any two of thethree factors in the SVD for X 1 with the corresponding factors in the SVDfor X 2 . All of these comparisons are relevant to comparing PCs, since Acontains the coefficients of the PCs, L provides the standard deviations ofthe PCs, and the elements of U are proportional to the PC scores (see Section3.5). The ‘significance’ of an observed value of any one of Preisendorferand Mobley’s statistics is assessed by comparing the value to a ‘referencedistribution’, which is obtained by simulation. Preisendorfer and Mobley’s(1982) research is in the context of atmospheric science. A more recent applicationin this area is that of Sengupta and Boyle (1998), who illustratethe use of Flury’s (1988) common principal component model to comparedifferent members of an ensemble of forecasts from a general circulationmodel (GCM) and to compare outputs from different GCMs. Applicationsin other fields of the common PC model and its variants can be found inFlury (1988, 1997).When the same variables are measured on the same n individuals in thedifferent data sets, it may be of interest to compare the configurations ofthe points defined by the n individuals in the subspaces of the first few PCsin each data set. In this case, Procrustes analysis (or generalized Procrustesanalysis) provides one possible way of doing this for two (more than two)data sets (see Krzanowski and Marriott (1994, Chapter 5)). The techniquein general involves the SVD of the product of one data matrix and thetranspose of the other, and because of this Davison (1983, Chapter 8) linksit to PCA.