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

5.3. Biplots 95The vector gi ∗ consists of the first two elements of g i, which are simply thevalues of the first two PCs for the ith observation.The properties of the h j that were demonstrated above for α = 0 willno longer be valid exactly for α = 1, although similar interpretations canstill be made, at least in qualitative terms. In fact, the coordinates of h ∗ jare simply the coefficients of the jth variable for the first two PCs. Theadvantage of superimposing the plots of the gi ∗ and h ∗ j is preserved forα =1,asx ij still represents the projection of g i onto h j . In many ways,the biplot with α = 1 is nothing new, since the gi∗ give PC scores andthe h ∗ j give PC coefficients, both of which are widely used on their own.The biplot, however, superimposes both the gi∗ and h∗ j to give additionalinformation.Other values of α could also be used; for example, Gabriel (1971) mentionsα = 1 2, in which the sum of squares of the projections of plotted pointsonto either one of the axes is the same for observations as for variables (Osmond,1985), but most applications seem to have used α = 0, or sometimesα = 1. For other values of α the general qualitative interpretation of therelative positions of the g i and the h j remains the same, but the exactproperties that hold for α =0andα = 1 are no longer valid.Another possibility is to superimpose the gi∗ and the h ∗ j correspondingto different values of α. Choosing a single standard value of α for boththe gi∗ and h∗ j may mean that the scales of observations and variables areso different that only one type of entity is visible on the plot. Digby andKempton (1987, Section 3.2) choose scales for observations and variablesso that both can easily be seen when plotted together. This is done ratherarbitrarily, but is equivalent to using different values of α for the two typesof entity. Mixing values of α in this way will, of course, lose the propertythat x ij is the projection of g i onto h j , but the relative positions of thegi ∗ and h∗ j still give qualitative information about the size of each variablefor each observation. Another way of mixing values of α is to use gi∗ correspondingto α =1andh ∗ j corresponding to α =0,sothattheg∗ i givea PC plot, and the h ∗ j have a direct interpretation in terms of variancesand covariances. This is referred to by Gabriel (2001) as a ‘correspondenceanalysis’ (see Section 5.4) plot. Gower and Hand (1996) and Gabriel (2001),among others, have noted that different plotting positions can be chosento give optimal approximations to two, but not all three, of the following:(a) the elements of X, as given by the scalar products g ∗′i h∗ j ;(b) Euclidean distances between the rows of X;(c) the covariance structure in the columns of X.We noted earlier that for α = 0, (b) is fitted less well than (c). For α =1,(c) rather than (b) is sacrificed, while the correspondence analysis plot loses(a). Choosing α = 1 2approximates (a) optimally, but is suboptimal for (b)and (c). For each of these four choices Gabriel (2001) investigates how