Analysis of different modes of factor analysis as least squares fit problems

Paatero, P. and Tapper, U., 1993. Analysis of different modes of factor analysis as least squares fit problems. Chemometrics and intelligent Laboratory Systems, 18: 183–194. 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 σij are assumed for the measured data matrix Xij. Selecting the mode (e.g. Q-mode) implicitly selects a scaling transformation as a preparatory step. Each scaling corresponds optimally to a certain σ. It is shown that the customary modes (Q-mode and R-mode) corresponds to such error estimates which do not normally occur in chemistry or physics. The best posssible 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 X2 values produced by the new alternatives are smaller by a 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.