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

3Mathematical and StatisticalProperties of Sample PrincipalComponentsThe first part of this chapter is similar in structure to Chapter 2, exceptthat it deals with properties of PCs obtained from a sample covariance(or correlation) matrix, rather than from a population covariance (or correlation)matrix. The first two sections of the chapter, as in Chapter 2,describe, respectively, many of the algebraic and geometric properties ofPCs. Most of the properties discussed in Chapter 2 are almost the same forsamples as for populations. They will be mentioned again, but only briefly.There are, in addition, some properties that are relevant only to samplePCs, and these will be discussed more fully.The third and fourth sections of the chapter again mirror those of Chapter2. The third section discusses, with an example, the choice betweencorrelation and covariance matrices, while the fourth section looks at theimplications of equal and/or zero variances among the PCs, and illustratesthe potential usefulness of the last few PCs in detecting near-constantrelationships between the variables.The last five sections of the chapter cover material having no counterpartin Chapter 2. Section 3.5 discusses the singular value decomposition, whichcould have been included in Section 3.1 as an additional algebraic property.However, the topic is sufficiently important to warrant its own section, asit provides a useful alternative approach to some of the theory surroundingPCs, and also gives an efficient practical method for actually computingPCs.The sixth section looks at the probability distributions of the coefficientsand variances of a set of sample PCs, in other words, the probability distributionsof the eigenvectors and eigenvalues of a sample covariance matrix.