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

9.1. Discriminant Analysis 203Figure 9.2. Two data sets whose direction of separation is orthogonal to that ofthe first (within-group) PC.which there are large between-group differences. Such PCs therefore seemmore useful than those based on within-group covariance matrices, but thetechnique should be used with some caution, as it will work well only ifbetween-group variation dominates within-group variation.It is well known that, for two completely specified normal populations,differing only in mean, the probability of misclassification using the lineardiscriminant function is a monotonically decreasing function of the squaredMahalanobis distance ∆ 2 between the two populations (Rencher, 1998,Section 6.4). Here ∆ 2 is defined as∆ 2 =(µ 1 − µ 2 ) ′ Σ −1 (µ − µ 2 ). (9.1.1)Note that we meet a number of other varieties of Mahalanobis distance elsewherein the book. In equation (5.3.5) of Section 5.3, Mahalanobis distancebetween two observations in a sample is defined, and there is an obvious similaritybetween (5.3.5) and the definition given in (9.1.1) for Mahalanobisdistance between two populations. Further modifications define Mahalanobis