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

9.3. Canonical Correlation Analysis and Related Techniques 225Table 9.3. Coefficients for the first two canonical variates in a canonical correlationanalysis of species and environmental variables.First canonical variates Second canonical variates⎧x 1 0.03 0.17x 2 0.51 0.52x 3 0.56 0.49Environment⎪⎨x 4 0.37 0.67variables x 5 0.01 −0.08x 6 0.03 0.07⎪⎩ x 7 −0.00 0.04x 8 0.53 −0.02⎧x 9 0.97 −0.19x 10 −0.06 −0.25x 11 0.01 −0.28Species⎪⎨x 12 0.14 0.58variablesx 13 0.19 0.00⎪⎩ x 14 0.06 0.46x 15 0.01 0.53Canonicalcorrelation 0.559 0.334variables and the first canonical variate for the species variables are eachdominated by a single PC, and the second canonical variate for the speciesvariables has two non-trivial coefficients. Thus, the canonical variates forPCs look, at first sight, easier to interpret than those based on the originalvariables. However, it must be remembered that, even if only one PCoccurs in a canonical variate, the PC itself is not necessarily an easily interpretedentity. For example, the environmental PC that dominates thefirst canonical variate for the environmental variables has six large coefficients.Furthermore, the between-group relationships found by CCA of theretained PCs are different in this example from those found from CCA onthe original variables.9.3.3 Maximum Covariance Analysis (SVD Analysis),Redundancy Analysis and Principal PredictorsThe first technique described in this section has been used in psychology formany years, dating back to Tucker (1958), where it is known as inter-batteryfactor analysis. This method postulates a model in whichx p1 = µ 1 + Λ 1 z + Γ 1 y 1 + e 1 (9.3.1)x p2 = µ 2 + Λ 2 z + Γ 2 y 2 + e 2 ,