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

368 13. Principal Component Analysis for Special Types of DataThe control limits described so far are all based on the assumption ofapproximate multivariate normality. Martin and Morris (1996) introducea non-parametric procedure that provides warning and action contourson plots of PCs. These contours can be very different from the normalbasedellipses. The idea of the procedure is to generate bootstrap samplesfrom the data set and from each of these calculate the value of a (possiblyvector-valued) statistic of interest. A smooth approximation to theprobability density of this statistic is then constructed using kernel densityestimation, and the required contours are derived from this distribution.Coleman (1985) suggests that when using PCs in quality control, the PCsshould be estimated robustly (see Section 10.4). Sullivan et al. (1995) dothis by omitting some probable outliers, identified from an initial scan ofthe data, before carrying out a PCA.When a variable is used to monitor a process over time, its successivevalues are likely to be correlated unless the spacing between observations islarge. One possibility for taking into account this autocorrelation is to plotan exponentially weighted moving average of the observed values. Wold(1994) suggests that similar ideas should be used when the monitoringvariables are PC scores, and he describes an algorithm for implementing‘exponentially weighted moving principal components analysis.’Data often arise in SPC for which, as well as different variables and differenttimes of measurement, there is a third ‘mode,’ namely different batches.So-called multiway, or three-mode, PCA can then be used (see Section 14.5and Nomikos and MacGregor (1995)). Grimshaw et al. (1998) note thepossible use of multiway PCA simultaneously on both the variables monitoringthe process and the variables measuring inputs or initial conditions,though they prefer a regression-based approach involving modifications ofHotelling’s T 2 and the SPE statistic.Boyles (1996) addresses the situation in which the number of variablesexceeds the number of observations. The sample covariance matrix S isthen singular and Hotelling’s T 2 cannot be calculated. One possibility isto replace S −1 by ∑ rk=1 l−1 k a ka ′ kfor r < n, based on the first r termsin the spectral decomposition of S (the sample version of Property A3 inSection 2.1). However, the data of interest to Boyles (1996) have variablesmeasured at points of a regular lattice on the manufactured product. Thisstructure implies that a simple pattern exists in the population covariancematrix Σ. Using knowledge of this pattern, a positive definite estimate ofΣ can be calculated and used in T 2 in place of S. Boyles finds appropriateestimates for three different regular lattices.Lane et al. (2001) consider the case where a several products or processesare monitored simultaneously. They apply Flury’s common PC subspacemodel (Section 13.5) to this situation. McCabe (1986) suggests the useof principal variables (see Section 6.3) to replace principal components inquality control.