Resolving factor analysis

Bilinear data matrices may be resolved into abstract factors by factor analysis. The underlying chemical processes that generated the data may be deduced from the abstract factors by hard (model fitting) or soft (model-free) analyses. We propose a novel approach that combines the advantages of both hard and soft methods, in that only a few parameters have to be fitted, but the assumptions made about the system are very general and common to a range of possible models: The true chemical factors span the same space as the abstract factors and may be mapped onto the abstract factors by a transformation matrix T, since they are a linear combination of the abstract factors. The difference between the true factors and any other linear combination of the abstract factors is that the true factors conform to known chemical constraints (for instance, nonnegativity of absorbance spectra or monomodality of chromatographic peaks). Thus, by excluding linear combinations of the abstract factors that are not physically possible (assuming a unique solution), we can find the true chemical factors. This is achieved by passing the elements of a transformation matrix to a nonlinear optimization routine, to find the best estimate of T that fits the criteria. The optimization routine usually converges to the correct minimum with random starting parameters, but more difficult problems require starting parameters based on some other method, for instance EFA. We call the new method resolving factor analysis (RFA). The use of RFA is demonstrated with computer-generated kinetic and chromatographic data and with real chromatographic (HPLC) data. RFA produces correct solutions with data sets that are refractory to other methods, for instance, data with an embedded nonconstant baseline.