Inclusion of chemical constraints in factor analysis to extract a unique set of solutions from spectroscopic and environmental data

The paper reviews methods to introduce chemical information in the decomposition of a multivariate data matrix D from an analytical instrument. The correct estimation of the number of factors and the unique extraction of a chemically informative score matrix R and a loading matrix C are required. Several constraints that may be applied during decomposition to extract the unique set have been proposed, but many of them are based only on mathematical concepts. Here we report the incorporation of chemical constraints in factor analysis with the development of Factor Analysis with Equilibrium Constraints (FAEC) for analysis of spectroscopic data and Oblique Rotational Factor Analysis with Partially Non-negative Constraint (ORFA-PNNC) for analysis of environmental data. When FAEC is applied to a system consisting of optically active species in chemical equilibria, it is sufficiently powerful to give equilibrium constants, pure component spectra, and concentrations of species. ORFA-PNNC has been applied by our group to data from the analysis of acid rain. It is based on the constraint of non-negative concentrations, but the hydrogen ion is excluded from the constraint to allow for the neutralization reaction between acidic and alkaline pollutants where alkaline conditions are seen as negative values of hydorgen ion concentration. Consequently, the evaluation of acidity (or basicity) of the pollutant source is possible. In this paper, the concept, procedure and applications of these two methods are reported.