Page 60 - Jolliffe I. Principal Component Analysis

3

Mathematical and Statistical
Properties of Sample Principal

Components

The first part of this chapter is similar in structure to Chapter 2, except
that it deals with properties of PCs obtained from a sample covariance
(or correlation) matrix, rather than from a population covariance (or cor-
relation) matrix. The first two sections of the chapter, as in Chapter 2,
describe, respectively, many of the algebraic and geometric properties of
PCs. Most of the properties discussed in Chapter 2 are almost the same for
samples as for populations. They will be mentioned again, but only briefly.
There are, in addition, some properties that are relevant only to sample
PCs, and these will be discussed more fully.
The third and fourth sections of the chapter again mirror those of Chap-
ter 2. The third section discusses, with an example, the choice between
correlation and covariance matrices, while the fourth section looks at the
implications of equal and/or zero variances among the PCs, and illustrates
the potential usefulness of the last few PCs in detecting near-constant
relationships between the variables.
The last five sections of the chapter cover material having no counterpart
in Chapter 2. Section 3.5 discusses the singular value decomposition, which
could have been included in Section 3.1 as an additional algebraic property.
However, the topic is sufficiently important to warrant its own section, as
it provides a useful alternative approach to some of the theory surrounding
PCs, and also gives an efficient practical method for actually computing
PCs.
The sixth section looks at the probability distributions of the coefficients
and variances of a set of sample PCs, in other words, the probability distri-
butions of the eigenvectors and eigenvalues of a sample covariance matrix.