Mathematical and Statistical Properties of Population Principal Components
In this chapter many of the mathematical and statistical properties of PCs are described, based on a known population covariance (or correlation) matrix Σ. Further properties are included in Chapter 3 but in the context of sample, rather than population, PCs. As well as being derived from a statistical viewpoint, PCs can be found using purely mathematical arguments; they are given by an orthogonal linear transformation of a set of variables which optimizes a certain algebraic criterion. In fact, the PCs optimize several different algebraic criteria and these optimization properties, together with their statistical implications, are described in the first section of the chapter.
KeywordsCovariance Matrix Covariance Matrice Correlation Matrice Generalize Varianee Constant Probability
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