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

3.7. Inference Based on Sample Principal Components 49If a distribution other than the multivariate normal is assumed, distributionalresults for PCs will typically become less tractable. Jackson (1991,Section 4.8) gives a number of references that examine the non-normal case.In addition, for non-normal distributions a number of alternatives to PCscan reasonably be suggested (see Sections 13.1, 13.3 and 14.4).Another deviation from the assumptions underlying most of the distributionalresults arises when the n observations are not independent. Theclassic examples of this are when the observations correspond to adjacentpoints in time (a time series) or in space. Another situation where nonindependenceoccurs is found in sample surveys, where survey designs areoften more complex than simple random sampling, and induce dependencebetween observations (see Skinner et al. (1986)). PCA for non-independentdata, especially time series, is discussed in detail in Chapter 12.As a complete contrast to the strict assumptions made in most workon the distributions of PCs, Efron and Tibshirani (1993, Section 7.2) lookat the use of the ‘bootstrap’ in this context. The idea is, for a particularsample of n observations x 1 , x 2 ,..., x n , to take repeated random samplesof size n from the distribution that has P [x = x i ]= 1 n,i=1, 2,...,n,calculate the PCs for each sample, and build up empirical distributions forPC coefficients and variances. These distributions rely only on the structureof the sample, and not on any predetermined assumptions. Care needs tobe taken in comparing PCs from different bootstrap samples because ofpossible reordering and/or sign switching in the PCs from different samples.Failure to account for these phenomena is likely to give misleadingly widedistributions for PC coefficients, and distributions for PC variances thatmay be too narrow.3.7 Inference Based on Sample PrincipalComponentsThe distributional results outlined in the previous section may be usedto make inferences about population PCs, given the sample PCs, providedthat the necessary assumptions are valid. The major assumption that x hasa multivariate normal distribution is often not satisfied and the practicalvalue of the results is therefore limited. It can be argued that PCA shouldonly ever be done for data that are, at least approximately, multivariatenormal, for it is only then that ‘proper’ inferences can be made regardingthe underlying population PCs. As already noted in Section 2.2, this isa rather narrow view of what PCA can do, as it is a much more widelyapplicable tool whose main use is descriptive rather than inferential. Itcan provide valuable descriptive information for a wide variety of data,whether the variables are continuous and normally distributed or not. Themajority of applications of PCA successfully treat the technique as a purely