Principal components or principal axes

The expression principal components first appeared in the writings of the American statistician Harold Hotelling in 1933, but the technique was known earlier as principal axes and goes back to Karl Pearson (1901). In principle, principal components are appropriate for the analysis of variation within a sample coming from a single statistical population which follows the multivariate normal distribution (section 29.2). However, principal components are often used also in practice when the exact form of the probability distribution is uncertain or when data may come from several populations but the parent population of each individual observation is unknown. In principal component analysis, q linear combinations (sections 24.22 and 29.5) Y = [Y 1,..., Y q] of the original variates X = [X 1,..., X q]are determined in such a way that the new variates Y = [Y 1,...,Y q]have zero covariances, that their coordinate axes are mutually orthogonal, and that their variances range from a maximum value for Y 1, to a minimum value for Y q. The parametric (population) principal components correspond to the coordinates of observations on the principal axes of ellipses, ellipsoids or hyperellipsoids where the probability density is constant (sections 19.2, 23.1, 24.24 and 29.2): the first principal component corresponds to the major axis, the last principal component corresponds to the minor axis, and the other principal components correspond to successive intermediate axes. The simplest case is that of bivariate principal axes, which has already been discussed concerning the bivariate orthogonal estimation (regression) line (chapter 22) and the graphical representation of confidence, prediction or variation ellipses (section 30.8). The vectors of direction cosines (section 24.6) of principal axes are the latent vectors, and the variances of the new variates [Y 1,..., Y q]are the latent roots (sections 24.21 and 31.11) of the covariance matrix of original variates. The utilization of principal components has been discussed recently by Flury (1988) and Jackson (1991).