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

108 5. Graphical Representation of Data Using Principal Componentsremain limitations on how many dimensions can be effectively shown simultaneously.The less sophisticated ideas of Tukey and Tukey (1981) stillhave a rôle to play in this respect.Even when dimensionality cannot be reduced to two or three, a reductionto as few dimensions as possible, without throwing away too muchinformation, is still often worthwhile before attempting to graph the data.Some techniques, such as Chernoff’s faces, impose a limit on the numberof variables that can be handled, although a modification due to Flury andRiedwyl (1981) increases the limit, and for most other methods a reductionin the number of variables leads to simpler and more easily interpretablediagrams. An obvious way of reducing the dimensionality is to replace theoriginal variables by the first few PCs, and the use of PCs in this contextwill be particularly successful if each PC has an obvious interpretation (seeChapter 4). Andrews (1972) recommends transforming to PCs in any case,because the PCs are uncorrelated, which means that tests of significancefor the plots may be more easily performed with PCs than with the originalvariables. Jackson (1991, Section 18.6) suggests that Andrews’ curvesof the residuals after ‘removing’ the first q PCs, that is, the sum of the last(r − q) terms in the SVD of X, may provide useful information about thebehaviour of residual variability.5.6.1 ExampleIn Jolliffe et al. (1986), 107 English local authorities are divided into groupsor clusters, using various methods of cluster analysis (see Section 9.2), onthe basis of measurements on 20 demographic variables.The 20 variables can be reduced to seven PCs, which account for over90% of the total variation in the 20 variables, and for each local authorityan Andrews’ curve is defined on the range −π ≤ t ≤ π by the functionf(t) = z 1√2+ z 2 sin t + z 3 cos t + z 4 sin 2t + z 5 cos 2t + z 6 sin 3t + z 7 cos 3t,where z 1 ,z 2 ,...,z 7 are the values of the first seven PCs for the local authority.Andrews’ curves may be plotted separately for each cluster. Thesecurves are useful in assessing the homogeneity of the clusters. For example,Figure 5.7 gives the Andrews’ curves for three of the clusters (Clusters 2,11 and 12) in a 13-cluster solution, and it can be seen immediately thatthe shape of the curves is different for different clusters.Compared to the variation between clusters, the curves fall into fairlynarrow bands, with a few exceptions, for each cluster. Narrower bands forthe curves imply greater homogeneity in the cluster.In Cluster 12 there are two curves that are somewhat different fromthe remainder. These curves have three complete oscillations in the range(−π, π), with maxima at 0 and ±2π/3. This implies that they are dominatedby cos 3t and hence z 7 . Examination of the seventh PC shows that