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

266 10. Outlier Detection, Influential Observations and Robust Estimationdure for both the first and suitably defined second, third, . . . robust PCs.Xu and Yuille (1995) present a robust PCA using a ‘statistical physics’ approachwithin a neural network framework (see Sections 14.1.3, 14.6.1). Thefunction optimized in their algorithm can be regarded as a generalizationof a robust redescending M-estimator.Locantore et al. (1999) discuss robust PCA for functional data (see Section12.3). In fact, their data are images rather than functions, but apre-processing step turns them into functions. Both means and covariancematrices are robustly estimated, the latter by shrinking extreme observationsto the nearest point on the surface of a hypersphere or hyperellipsoidcentred at the robustly estimated mean, a sort of multivariate Winsorization.Locantore et al.’s (1999) paper is mainly a description of an interestingcase study in opthalmology, but it is followed by contributions from 9 setsof discussants, and a rejoinder, ranging over many aspects of robust PCAin a functional context and more generally.A different type of approach to the robust estimation of PCs is discussedby Gabriel and Odoroff (1983). The approach relies on the fact that PCsmay be computed from the singular value decomposition (SVD) of the(n × p) data matrix X (see Section 3.5 and Appendix Al). To find theSVD, and hence the PCs, a set of equations involving weighted meansof functions of the elements of X must be solved iteratively. Replacingthe weighted means by medians, weighted trimmed means, or some othermeasure of location which is more robust than the mean leads to estimatesof PCs that are less sensitive than the usual estimates to the presence of‘extreme’ observations.Yet another approach, based on ‘projection pursuit,’ is proposed by Liand Chen (1985). As with Gabriel and Odoroff (1983), and unlike Campbell(1980) and Devlin et al. (1981), the PCs are estimated directly withoutfirst finding a robustly estimated covariance matrix. Indeed, Li and Chensuggest that it may be better to estimate the covariance matrix Σ fromthe robust PCs via the spectral decomposition (see Property A3 of Sections2.1 and 3.1), rather than estimating Σ directly. Their idea is to findlinear functions of x that maximize a robust estimate of scale, rather thanfunctions that maximize variance. Properties of their estimates are investigated,and the estimates’ empirical behaviour is compared with that ofDevlin et al. (1981)’s estimates using simulation studies. Similar levels ofperformance are found for both types of estimate, although, as noted byCroux and Haesbroeck (2000), methods based on robust estimation of covariancematrices have poor properties for large values of p and projectionpursuit-based techniques may be preferred in such cases. One disadvantageof Li and Chen’s (1985) approach is that it is complicated to implement.Both Xie et al. (1993) and Croux and Ruiz-Gazen (1995) give improvedalgorithms for doing so.Another way of ‘robustifying’ PCA directly, rather than doing so viaa robust covariance or correlation matrix, is described by Baccini et al.