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

226 9. Principal Components Used with Other Multivariate Techniqueswhere µ 1 , µ 2 are vectors of means, Λ 1 , Λ 2 , Γ 1 , Γ 2 are matrices of coefficients,z is a vector of latent variables common to both x p1 and x p2 , y 1 , y 2are vectors of latent variables specific to x p1 , x p2 ,ande 1 , e 2 are vectorsof errors. Tucker (1958) fits the model using the singular value decompositionof the (p 1 × p 2 ) matrix of correlations between two batteries oftests x p1 , x p2 , and notes that his procedure is equivalent to finding linearcombinations of the two batteries that have maximum covariance. Browne(1979) demonstrates some algebraic connections between the results of thistechnique and those of CCA.The method was popularised in atmospheric science by Bretherton etal. (1992) and Wallace et al. (1992) under the name singular value decomposition(SVD) analysis. This name arose because, as Tucker (1958)showed, the analysis can be conducted via an SVD of the (p 1 × p 2 ) matrixof covariances between x p1 and x p2 , but the use of this general term for aspecific technique is potentially very confusing. The alternative canonicalcovariance analysis, which Cherry (1997) notes was suggested in unpublishedwork by Muller, is a better descriptor of what the technique does,namely that it successively finds pairs of linear functions of x p1 and x p2that have maximum covariance and whose vectors of loadings are orthogonal.Even better is maximum covariance analysis, which is used by vonStorch and Zwiers (1999, Section 14.1.7) and others (Frankignoul, personalcommunication), and we will adopt this terminology. Maximum covarianceanalysis differs from CCA in two ways: covariance rather than correlationis maximized, and vectors of loadings are orthogonal instead of derivedvariates being uncorrelated. The rationale behind maximum covarianceanalysis is that it may be important to explain a large proportion ofthe variance in one set of variables using the other set, and a pair ofvariates from CCA with large correlation need not necessarily have largevariance.Bretherton et al. (1992) and Wallace et al. (1992) discuss maximumcovariance analysis (SVD analysis) in some detail, make comparisons withcompeting techniques and give examples. Cherry (1997) and Hu (1997)point out some disadvantages of the technique, and Cherry (1997) alsodemonstrates a relationship with PCA. Suppose that separate PCAs aredone on the two sets of variables and that the values (scores) of the nobservations on the first q PCs are given by the (n × q) matrices Z 1 , Z 2for the two sets of variables. If B 1 , B 2 are orthogonal matrices chosen tominimize ‖Z 1 B 1 − Z 2 B 2 ‖, the resulting matrices Z 1 B 1 , Z 2 B 2 contain thevalues for the n observations of the first q pairs of variates from a maximumcovariance analysis. Thus, maximum covariance analysis can be viewed astwo PCAs, followed by rotation to match up the results of the two analysesas closely as possible.Like maximum covariance analysis, redundancy analysis attempts to incorporatevariance as well as correlation in its search for relationshipsbetween two sets of variables. The redundancy coefficient was introduced