RANDOM ERROR BIAS IN PRINCIPAL COMPONENT ANALYSIS .1. DERIVATION OF THEORETICAL PREDICTIONS

Principal component analysis (PCA) or singular value decomposition (SV D) are multivariate techniques that are often used to compress large d ata matrices to a relevant size. Subsequent data analysis then proceed s with the model representation of the data. In this first paper expre ssions are derived for the prediction of the bias in the eigenvalues o f PCA and singular values of SVD that results from random measurement errors in the data. Theoretical expressions for the prediction of this ''random error bias'' have been given in the statistics literature. T hese results are, however, restricted to the case that only one princi pal component (PC) is significant. The first objective of this paper i s to extend these results to an arbitrary number of significant PCs. F or the generalization Malinowski's error functions are used. A signal- to-noise ratio is defined that describes the error situation for each individual PC. This definition enhances the interpretability of the de rived expressions. The adequacy of the derived expressions is tested b y a limited Monte Carlo study. This finally leads to the second object ive of this paper. Simulation results are always restricted to the cla ss of data that is well represented in the study. Thus rather than giv ing extensive simulation results it is outlined how the validation and evaluation of theoretical predictions can proceed for a specific appl ication in practice.