ANALYSIS OF DIFFERENT MODES OF FACTOR-ANALYSIS AS LEAST-SQUARES FIT PROBLEMS

It is shown that each mode of principal component analysis or 'factor analysis' is equivalent to solving a certain least squares problem where certain error estimators sigma(ij) are assumed for the measured data matrix X(ij). Selecting the mode (e.g. Q-mode) implicitly selects a scaling transformation as a preparatory step. Each scaling corresponds optimally to a certain sigma. It is shown that the customary modes (Q-mode and R-mode) correspond to such error estimates which do not normally occur in chemistry or physics. The best possible scaling ('optimal scaling') and a near-optimal scaling are introduced. The Quail Roost II air pollution simulation data sets are studied as examples: it is shown that the chi2 Values produced by the new alternatives are smaller by a factor of 10. Thus one would also expect that the factors are more precise. However, the values of the factors are not monitored in the present work.