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

274 11. Rotation and Interpretation of Principal Components(1995) and Mestas-Nuñez (2000) discuss the effects of different normalizations.Mestas-Nuñez distinguishes between the PCA and factor analysisapproaches as corresponding to the ‘analysis’ and ‘synthesis’ formulae, respectively.This terminology comes from Preisendorfer and Mobley (1988,Section 2b). Other terminology used by Mestas-Nuñez is less helpful. Herefers to unit-length eigenvectors as defining the ‘EOF model,’ and eigenvectorswith squared lengths equal to their eigenvalues as the ‘PCA model.’This distinction, which was used by Richman (1986), is confusing—it ismore usual to refer to the eigenvectors as EOFs, and the derived variablesas PCs, whatever their normalization.Von Storch and Zwiers (1999, Section 13.1.11) note a further implicationof the two approaches. In PCA using a covariance matrix with all variablesmeasured in the same units, it is clear that that the loadings in the PCs aredimensionless, and the PCs themselves have the same units as the originalvariables. For example, if each of the measured variables is a temperature,so are the PCs. In the factor analysis approach, dividing each PC by itsstandard deviation makes the resulting components dimensionless, and theloadings are now measured in the same units as the original variables.Things are different when PCA is done on a correlation matrix, as thestandardized variables forming the input to the analysis are themselvesdimensionless.Arbuckle and Friendly (1977) describe a different way of rotating PCs.Their data consist of p measurements over time of the same quantity, forn individuals. There is an assumption that the measurements representdiscrete points on an underlying smooth curve, so that the coefficients inany PC should also vary smoothly over time. Such data can be analysedusing functional PCA (see Section 12.3), but Arbuckle and Friendly (1997)treat smoothness as a form of simplicity and attempt to rotate a chosensubset of PCs towards smoothness rather than towards the more usualform of simple structure. The criterion which they minimize over possiblerotations is the sum of squared first differences of the coefficients when theyare arranged in time order.11.1.1 ExamplesMediterranean Sea Surface TemperaturesThe data presented here were originally analysed by Bartzokas et al. (1994).They consist of values of sea surface temperatures (SST) for sixteen 5 ◦ ×5 ◦grid boxes covering most of the Mediterranean, averaged over 3-month seasons,for each of the 43 years 1946–1988. Here we consider data for autumn.Figure 11.1 shows the varimax rotated loadings of the ‘first’ rotated PCwhen three PCs are rotated. The three plots within Figure 11.1 provideresults from analyses in which three different normalization constraints areused, corresponding to A m , à m and à m . Of course, the term ‘first’ ro-