Section 4.4 has shown that, in a multinormal distribution, the first principal axis is the line that goes through the greatest dimension of the concentration ellipsoid describing the distribution. In the same way, the following principal axes (orthogonal to one another, i.e. at right angles to one another, and successively shorter) go through the following greatest dimensions of the p-dimensional ellipsoid. A maximum of p principal axes may be derived from a data table containing p variables (Fig. 4.9). The principal axes of a dispersion matrix S are found by solving (eq. 4.23):
whose characteristic equation
is used to compute the eigenvalues Xk. The eigenvectors u^ associated with the Xk are found by putting the different Xk values in turn into eq. 9.1. These eigenvectors are the principal axes of dispersion matrix S (Section 4.4). The eigenvectors are normalized (i.e. scaled to unit length, Section 2.4) before computing the principal components, which give the coordinates of the objects on the successive principal axes. Principal component analysis (PCA) is due to Hotelling (1933). The method and several of its
Principal components implications for data analysis are clearly presented in the seminal paper of Rao (1964). PCA possesses the following properties, which make it a powerful instrument for the analysis of ecological data:
1) Since any dispersion matrix S is symmetric, its principal axes are orthogonal to one another. In other words, they correspond to linearly independent directions in the concentration ellipsoid of the distribution of objects (Section 2.9).
2) The eigenvalues Xk of a dispersion matrix S give the amount of variance corresponding to the successive principal axes (Section 4.4).
3) Because of the first two properties, principal component analysis can often summarize, in a few dimensions, most of the variability of a dispersion matrix of a large number of descriptors. It also provides a measure of the amount of variance explained by these few independent principal axes.
The present Section shows how to compute the relationships among objects and among descriptors, as well as the relationships between the principal axes and the original descriptors. A simple numerical example is developed, involving five objects and two quantitative descriptors:
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