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

268 10. Outlier Detection, Influential Observations and Robust EstimationFinally, we mention that Ruymgaart (1981) discusses a class of robust PCestimators. However, his discussion is restricted to bivariate distributions(that is p = 2) and is entirely theoretical in nature. Ibazizen (1986) suggeststhat it would be difficult to generalize Ruymgaart’s (1981) ideas to morethan two dimensions.10.5 Concluding RemarksThe topics discussed in this chapter pose difficult problems in data analysis.Much research has been done and is continuing on all of them. It is usefulto identify potentially outlying observations, and PCA provides a numberof ways of doing so. Similarly, it is important to know which observationshave the greatest influence on the results of a PCA.Identifying potential outliers and influential observations is, however,only part of the problem; the next, perhaps more difficult, task is to decidewhether the most extreme or influential observations are sufficientlyextreme or influential to warrant further action and, if so, what that actionshould be. Tests of significance for outliers were discussed only briefly inSection 10.1 because they are usually only approximate, and tests of significancefor influential observations in PCA have not yet been widely used.Perhaps the best advice is that observations that are much more extreme orinfluential than most of the remaining observations in a data set should bethoroughly investigated, and explanations sought for their behaviour. Theanalysis could also be repeated with such observations omitted, althoughit may be dangerous to act as if the deleted observations never existed.Robust estimation provides an automatic way of dealing with extreme (orinfluential) observations but, if at all possible, it should be accompaniedby a careful examination of any observations that have been omitted orsubstantially downweighted by the analysis.