Principal components

Abstract

In this chapter we assume that x ∈ ℝ^p with E x = µ and var x = Σ = (σij). When the dimension p is large, the principal components method seeks to replace x by y ∈ ℝ^k, where k < p (and hopefully much smaller), without losing too much “information.” This is sometimes particularly useful for a graphical description of the data since it is much easier to view vectors of low dimension. Section 10.2 defines principal components and gives their interpretation as normalized linear combinations with maximum variance. In Section 10.3, we explain an optimal property of principal components as best approximating subspace of dimension k in terms of squared prediction error. Section 10.4 introduces the sample principal components; they give the coordinates of the projected data which is closest, in terms of euclidian distance, to the original data. Section 10.5 treats the sample principal components calculated from the correlation matrix. Finally, Section 10.6 presents a simple test for multivariate normality which generalizes the univariate Shapiro and Wilk’s statistic. A book entirely devoted to principal component analysis is that of Jolliffe (1986).