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

13.7. PCA in Statistical Process Control 367• One- or two-dimensional plots of PC scores. It was noted in Section10.1 that both the first few and the last few PCs may be usefulfor detecting (different types of) outliers, and plots of both are usedin process control. In the published discussion of Roes and Does(1995), Sullivan et al. (1995) argue that the last few PCs are perhapsmore useful in SPC than the first few, but in their reply tothe discussion Roes and Does disagree. If p is not too large, sucharguments can be overcome by using a scatterplot matrix to displayall two-dimensional plots of PC scores simultaneously. Plots can beenhanced by including equal-probability contours, assuming approximatemultivariate normality, corresponding to warning and actionlimits for those points that fall outside them (Jackson, 1991, Section1.7; Martin et al., 1999).• Hotelling’s T 2 . It was seen in Section 10.1 that this is a special casefor q = p of the statistic d 2 2i in equation (10.1.2). If multivariate normalityis assumed, the distribution of T 2 is known, and control limitscan be set based on that distribution (Jackson, 1991, Section 1.7).• The squared prediction error (SPE). This is none other than thestatistic d 2 1i in equation (10.1.1). It was proposed by Jackson andMudholkar (1979), who constructed control limits based on anapproximation to its distribution. They prefer d 2 1i to d2 2i for computationalreasons and because of its intuitive appeal as a sum ofsquared residuals from the (p − q)-dimensional space defined by thefirst (p − q) PCs. However, Jackson and Hearne (1979) indicate thatthe complement of d 2 2i , in which the sum of squares of the first fewrather than the last few renormalized PCs is calculated, may be usefulin process control when the objective is to look for groups of‘out-of-control’ or outlying observations, rather than single outliers.Their basic statistic is decomposed to give separate information aboutvariation within the sample (group) of potentially outlying observations,and about the difference between the sample mean and someknown standard value. In addition, they propose an alternative statisticbased on absolute, rather than squared, values of PCs. Jacksonand Mudholkar (1979) also extend their proposed control procedure,based on d 2 1i , to the multiple-outlier case, and Jackson (1991, Figure6.2) gives a sequence of significance tests for examining subgroups ofobservations in which each test is based on PCs in some way.Eggett and Pulsipher (1989) compare T 2 , SPE, and the complement ofd 2 2i suggested by Jackson and Hearne (1979), in a simulation study and findthe third of these statistics to be inferior to the other two. On the basis oftheir simulations, they recommend Hotelling’s T 2 for large samples, withSPE or univariate control charts preferred for small samples. They alsodiscussed the possibility of constructing CUSUM charts based on the threestatistics.