Jolliffe I. Principal Component Analysis (2ed., Springer, 2002)(518s)

56 3. Properties of Sample Principal ComponentsLanterman (2000) and Wang and Staib (2000) each use principal componentsin quantifying prior information in (different) image processingcontexts.Another possible use of PCA when a Bayesian approach to inference isadopted is as follows. Suppose that θ is a vector of parameters, and thatthe posterior distribution for θ has covariance matrix Σ. If we find PCsfor θ, then the last few PCs provide information on which linear functionsof the elements of θ can be estimated with high precision (low variance).Conversely, the first few PCs are linear functions of the elements of θ thatcan only be estimated with low precision. In this context, then, it wouldseem that the last few PCs may be more useful than the first few.3.8 Principal Components for PatternedCorrelation or Covariance MatricesAt the end of Chapter 2, and in Section 3.7.3, the structure of the PCsand their variances was discussed briefly in the case of a correlation matrixwith equal correlations between all variables. Other theoretical patterns incorrelation and covariance matrices can also be investigated; for example,Jolliffe (1970) considered correlation matrices with elements ρ ij for whichandρ 1j = ρ,ρ ij = ρ 2 ,j =2, 3....,p,2 ≤ i