Random error bias in principal component analysis. Part II. Application of theoretical predictions to multivariate problems

In the first part of this paper expressions were derived for the prediction of random error bias in the eigenvalues of principal component analysis (PCA) and the singular values of singular value decomposition (SVD). The main issues of Part I were to investigate the question whether adequate prediction of this bias is possible and to discuss how the validation and evaluation of these predictions could proceed for a specific application in practice. The main issue of this second part is to investigate how random error bias should be taken into account. This question will be treated for a number of seemingly disparate multivariate problems. For example, the construction of confidence intervals for the bias-corrected quantities will be discussed with respect to the estimation of the number of significant principal components. The consequences of random error bias for calibration and prediction with ordinary least squares (OLS), principal component regression (PCR), partial least squares (PLS) and the generalized rank annihilation method (GRAM) will also be outlined. Finally, the derived bias expressions will be compared in detail with the corresponding results for OLS and GRAM.