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

214 9. Principal Components Used with Other Multivariate Techniques(2001). Suppose that the p variables are divided into G groups or clusters,and that x g denotes the vector of variables in the gth group, g =1, 2,...,G.Vigneau and Qannari (2001) seek vectors a 11 , a 21 ,...,a G1 that maximize∑ Gg=1 var(a′ g1x g ), where var(a ′ g1x g ) is the sample variance of the linearfunction a ′ g1x g . This sample variance is clearly maximized by the first PCfor the variables in the gth group, but simultaneously we wish to find thepartition of the variables into G groups for which the sum of these variancesis maximized. An iterative procedure is presented by Vigneau and Qannari(2001) for solving this problem.The formulation of the problem assumes that variables with large squaredcorrelations with the first PC in a cluster should be assigned to that cluster.Vigneau and Qannari consider two variations of their technique. In thefirst, the signs of the correlations between variables and PCs are important;only those variables with large positive correlations with a PC should bein its cluster. In the second, relationships with external variables are takeninto account.9.2.1 ExamplesOnly one example will be described in detail here, although a number ofother examples that have appeared elsewhere will be discussed briefly. Inmany of the published examples where PCs have been used in conjunctionwith cluster analysis, there is no clear-cut cluster structure, and clusteranalysis has been used as a dissection technique. An exception is the wellknownexample given by Jeffers (1967), which was discussed in the contextof variable selection in Section 6.4.1. The data consist of 19 variables measuredon 40 aphids and, when the 40 observations are plotted with respectto the first two PCs, there is a strong suggestion of four distinct groups; referto Figure 9.3, on which convex hulls (see Section 5.1) have been drawnaround the four suspected groups. It is likely that the four groups indicatedon Figure 9.3 correspond to four different species of aphids; thesefour species cannot be readily distinguished using only one variable at atime, but the plot with respect to the first two PCs clearly distinguishesthe four populations.The example introduced in Section 1.1 and discussed further in Section5.1.1, which has seven physical measurements on 28 students, also shows (inFigures 1.3, 5.1) how a plot with respect to the first two PCs can distinguishtwo groups, in this case men and women. There is, unlike the aphid data,a small amount of overlap between groups and if the PC plot is used toidentify, rather than verify, a cluster structure, then it is likely that somemisclassification between sexes will occur. A simple but specialized use ofPC scores, one PC at a time, to classify seabird communities is describedby Huettmann and Diamond (2001).In the situation where cluster analysis is used for dissection, the aim of atwo-dimensional plot with respect to the first two PCs will almost always be