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

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

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

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166 7. <strong>Principal</strong> <strong>Component</strong> <strong>Analysis</strong> and Factor <strong>Analysis</strong>be ‘borrowed’ for PCA without any implication that a factor model is beingassumed. Once PCA has been used to find an m-dimensional subspacethat contains most of the variation in the original p variables, it is possibleto redefine, by rotation, the axes (or derived variables) that form a basisfor this subspace. The rotated variables will together account for the sameamount of variation as the first few PCs, but will no longer successivelyaccount for the maximum possible variation. This behaviour is illustratedby Tables 7.1 and 7.2; the four rotated PCs in Table 7.2 together accountfor 75.7% of the total variation, as did the unrotated PCs in Table 7.1.However, the percentages of total variation accounted for by individualfactors (rotated PCs) are 27.4, 21.9, 14.2 and 12.1, compared with 47.7,11.3, 9.6 and 7.1 for the unrotated PCs. The rotated PCs, when expressedin terms of the original variables, may be easier to interpret than the PCsthemselves because their coefficients will typically have a simpler structure.This is discussed in more detail in Chapter 11. In addition, rotated PCs offeradvantages compared to unrotated PCs in some types of analysis basedon PCs (see Sections 8.5 and 10.1).

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