Quantitative Risk Management: A Practical Guide to Financial Risk, + Website

Appendix C: Principal Components Overview

The factors obtained by principal components analysis are new random variables, which are linear combinations of the original variables:

(A10.9)

We want the variance-covariance matrix of the factors to be diagonal (so factors are uncorrelated):

(A10.10)

Principal components analysis sets the columns of the matrix A to the eigenvectors (characteristic vectors) of the variance-covariance matrix, with columns ordered by size of the eigenvalues. The eigenvectors are a convenient choice. They work because by the definition of the eigenvectors of the matrix Σ_Y:²²

(A10.11)

where Diag(λ_·) is the matrix with zeros off-diagonal and λ_i in the diagonal element (i,i). This diagonalization gives a diagonal matrix for the variance-covariance of the variables F, E[F·F′]:

(A10.12)

The reverse transformation from factors to original variables is:

(A10.13)

The ...