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

11.2. Alternatives to Rotation 279which are described in Section 11.2. The normalization of loadings and theshading in Figures 11.2–11.5 follow the same conventions as in Figure 11.1,except that any loadings which are exactly zero are unshaded in the figures.For the autumn data, the first two SCoT components may be viewedas slightly simpler versions of PC1 and PC2. The largest loadings becomelarger and the smallest become smaller, as is often observed during rotation.The strong similarity between SCoT and PCA components is reflected bythe fact that the first two SCoT components account for 78.0% of thetotal variation compared to 78.2% for the first two PCs. Turning to thewinter data, the first two PCs and RPCs in Figures 11.4, 11.5 are not toodifferent from those in autumn (Figures 11.2, 11.3), but there are biggerdifferences for SCoT. In particular, the second SCoT component is verymuch dominated by a single grid box in the Eastern Mediterranean. Thisextreme simplicity leads to a reduction in the total variation accounted forby the first two SCoT components to 55.8%, compared to 71.0% for thefirst two PCs.Results are presented for only one value of the tuning parameter ψ, andthe choice of this value is not an easy one. As ψ increases, there is oftena rapid jump from components that are very close to the correspondingPC, like those in autumn, to components that are dominated by a singlevariable, as for SC2 in winter. This jump is usually accompanied by a largedrop in the proportion of variation accounted for, compared to PCA. Thisbehaviour as ψ varies can also be seen in a later example (Tables 11.3,11.4), and is an unsatisfactory feature of SCoT. Choosing different valuesψ 1 ,ψ 2 ,... of ψ for SC1, SC2, ..., only partially alleviates this problem.The cause seems to be the presence of a number of local maxima for thecriterion defined by (11.1.6). As ψ changes, a different local maximum maytake over as the global maximum, leading to a sudden switch betweentwo quite different solutions. Further discussion of the properties of SCoT,together with additional examples, is given by Jolliffe and Uddin (2000)and Uddin (1999).Filzmoser (2000) argues that sometimes there is simple structure in aplane but not in any single direction within that plane. He derives a wayof finding such simply structured planes—which he calls principal planes.Clearly, if a data set has such structure, the methods discussed in thissection and the next are unlikely to find simple single components withouta large sacrifice in variance.11.2 Alternatives to RotationThis section describes two ideas for constructing linear functions of the poriginal variables having large variances; these techniques differ from PCAin imposing additional constraints on the loadings or coefficients of the