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

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

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

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206 9. <strong>Principal</strong> <strong>Component</strong>s Used with Other Multivariate Techniquesination, it may be that the opposite effects described by Mason and Gunst(1985) and by Takemura (1985) roughly balance in the case of using PCsin discriminant analysis.The fact that separation between populations may be in the directionsof the last few PCs does not mean that PCs should not be used at all indiscriminant analysis. They can still provide a reduction of dimensionalityand, as in regression, their uncorrelatedness implies that in linear discriminantanalysis each PC’s contribution can be assessed independently. Thisis an advantage compared to using the original variables x, where the contributionof one of the variables depends on which other variables are alsoincluded in the analysis, unless all elements of x are uncorrelated. The mainpoint to bear in mind when using PCs in discriminant analysis is that thebest subset of PCs does not necessarily simply consist of those with thelargest variances. It is easy to see, because of their uncorrelatedness, whichof the PCs are best at discriminating between the populations. However,as in regression, some caution is advisable in using PCs with very low variances,because at least some of the estimated coefficients in the discriminantfunction will have large variances if low variance PCs are included. Manyof the comments made in Section 8.2 regarding strategies for selecting PCsin regression are also relevant in linear discriminant analysis.Some of the approaches discussed so far have used PCs from the overallcovariance matrix, whilst others are based on the pooled within-groupcovariance matrix. This latter approach is valid for types of discriminantanalysis in which the covariance structure is assumed to be the same forall populations. However, it is not always realistic to make this assumption,in which case some form of non-linear discriminant analysis may benecessary. If the multivariate normality assumption is kept, the most usualapproach is quadratic discrimination (Rencher, 1998, Section 6.2.2). Withan assumption of multivariate normality and G groups with sample meansand covariance matrices ¯x g , S g ,g=1, 2,...,G, the usual discriminant ruleassigns a new vector of observations x to the group for which(x − ¯x g ) ′ S −1g (x − ¯x g )+ln(|S g |) (9.1.2)is minimized. When the equal covariance assumption is made, S g is replacedby the pooled covariance matrix S w in (9.1.2), and only the linear part ofthe expression is different for different groups. In the general case, (9.1.2) isa genuine quadratic function of x, leading to quadratic discriminant analysis.Flury (1995) suggests two other procedures based on his commonprincipal component (CPC) framework, whose assumptions are intermediatecompared to those of linear and quadratic discrimination. Furtherdetails will be given when the CPC framework is discussed in Section 13.5.Alternatively, the convenience of looking only at linear functions of xcan be kept by computing PCs separately for each population. In a numberof papers (see, for example, Wold, 1976; Wold et al., 1983), Wold andothers have described a method for discriminating between populations

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