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

262 10. Outlier Detection, Influential Observations and Robust EstimationComparisons are made with corresponding results derived from ‘random,’that is uncorrelated, data sets of the same size as the real data sets. This isdone with an objective similar to that of parallel analysis (see Section 6.1.3)in mind.A different approach to influence from that described in Section 10.2 wasproposed by Cook (1986) and called local influence. Shi (1997) developsthese ideas in the context of PCA. It is included in this section rather thanin 10.2 because, as with Tanaka and Tarumi (1985, 1987) and Benasseni(1986a, 1987a) weights are attached to each observation, and the weightsare varied. In Shi’s (1997) formulation the ith observation is expressed asx i (w) =w i (x i − ¯x),and w = (w 1 ,w 2 ,...,w n ) ′ is a vector of weights. The vector w 0 =(1, 1,...,1) ′ gives the unperturbed data, and Shi considers perturbations ofthe form w = w 0 +ɛh, where h is a fixed unit-length vector. The generalizedinfluence function for a functional θ is defined asθ(w 0 + ɛh) − θ(w 0 )GIF(θ, h) = lim.ε→0 ɛFor a scalar θ, such as an eigenvalue, local influence investigates the directionh in which GIF(θ, h) is maximized. For a vector θ, suchasaneigenvector, GIF(.) is converted into a scalar (a norm) via a quadraticform, and then h is found for which this norm is maximized.Yet another type of stability is the effect of changes in the PC coefficientsa k on their variances l k and vice versa. Bibby (1980) and Green (1977)both consider changes in variances, and other aspects such as the nonorthogonalityof the changed a k when the elements of a k are rounded. Thisis discussed further in Section 11.3.Krzanowski (1984b) considers what could be thought of as the oppositeproblem to that discussed by Bibby (1980). Instead of looking at the effectof small changes in a k on the value of l k , Krzanowski (1984b) examinesthe effect of small changes in the value of l k on a k , although he addressesthe problem in the population context and hence works with α k and λ k .He argues that this is an important problem because it gives informationon another aspect of the stability of PCs: the PCs can only be confidentlyinterpreted if their coefficients are stable with respect to small changes inthe values of the λ k .If λ k is decreased by an amount ε, then Krzanowski (1984b) looks fora vector α ε k that is maximally different from α k, subject to var(α ′ εkx)=λ k − ε. He finds that the angle θ between α k and α ε kis given byεcos θ =[1+(λ k − λ k+1 ) ]−1/2 , (10.3.1)and so depends mainly on the difference between λ k and λ k+1 .Ifλ k , λ k+1are close, then the kth PC, α ′ k x = z k, is more unstable than if λ k , λ k+1 are