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

260 10. Outlier Detection, Influential Observations and Robust Estimationone the weights of one or more observations are reduced, without removingthem entirely. This type of ‘sensitivity’ is discussed in general for multivariatetechniques involving eigenanalyses by Tanaka and Tarumi (1985, 1987),with PCA as a special case. Benasseni (1986a) also examines the effect ofdiffering weights for observations on the eigenvalues in a PCA. He givesbounds on the perturbed eigenvalues for any pattern of perturbations ofthe weights for both covariance and correlation-based analyses. The workis extended in Benasseni (1987a) to include bounds for eigenvectors as wellas eigenvalues. A less structured perturbation is investigated empiricallyby Tanaka and Tarumi (1986). Here each element of a (4 × 4) ‘data’ matrixhas an independent random perturbation added to it.In Tanaka and Mori (1997), where the objective is to select a subsetof variables reproducing all the p variables as well as possible and hencehas connections with Section 6.3, the influence of variables is discussed.Fujikoshi et al. (1985) examine changes in the eigenvalues of a covariancematrix when additional variables are introduced. Krzanowski (1987a) indicateshow to compare data configurations given by sets of retained PCs,including all the variables and with each variable omitted in turn. Thecalculations are done using an algorithm for computing the singular valuedecomposition (SVD) with a variable missing, due to Eastment and Krzanowski(1982), and the configurations are compared by means of Procrustesrotation (see Krzanowski and Marriott 1994, Chapter 5). Holmes-Junca(1985) gives an extensive discussion of the effect of omitting observationsor variables from a PCA. As in Krzanowski (1987a), the SVD plays a prominentrôle, but the framework in Holmes-Junca (1985) is a more general onein which unequal weights may be associated with the observations, and ageneral metric may be associated with the variables (see Section 14.2.2).A different type of stability is investigated by Benasseni (1986b). He considersreplacing each of the np-dimensional observations in a data set bya p-dimensional random variable whose probability distribution is centredon the observed value. He relates the covariance matrix in the perturbedcase to the original covariance matrix and to the covariance matrices ofthe n random variables. From this relationship, he deduces bounds on theeigenvalues of the perturbed matrix. In a later paper, Benasseni (1987b)looks at fixed, rather than random, perturbations to one or more of theobservations. Expressions are given for consequent changes to eigenvaluesand eigenvectors of the covariance matrix, together with approximationsto those changes. A number of special forms for the perturbation, for examplewhere it affects only one of the p variables, are examined in detail.Corresponding results for the correlation matrix are discussed briefly.Dudziński et al. (1975) discuss what they call ‘repeatability’ of principalcomponents in samples, which is another way of looking at the stabilityof the components’ coefficients. For each component of interest the angleis calculated between the vector of coefficients in the population and thecorresponding vector in a sample. Dudziński et al. (1975) define a repeata-