The use of matrix methods in the modeling of spectroscopic data sets

We describe a general approach to the model-based analysis of sets of spectroscopic data that is built upon the techniques of matrix analysis. A model hypothesis may often be expressed by writing a matrix of measured spectra as the product of a matrix of spectra of individual molecular species and a matrix of corresponding species populations as a function of experimental conditions. The modeling procedure then requires the simultaneous determination of a set of species spectra and a set of model parameters (from which the populations are derived), such that this product yields an optimal description of the measured spectra. This procedure may be implemented as an optimization problem in the space of the (possibly nonlinear) model parameters alone, coupled with the efficient solution of a corollary linear optimization problem using matrix decomposition methods to obtain a set of species spectra corresponding to any set of model parameters. Known species spectra, as well as other information and assumptions about spectral shapes, may be incorporated into this general framework, using parametrized analytical functional forms and basis-set techniques. The method by which assumed relationships between global features (e.g., peak positions) of different species spectra may be enforced in the modeling without otherwise specifying the shapes of the spectra will be shown. We also consider the effect of measurement errors on this approach and suggest extensions of the matrix-based least-squares procedures applicable to situations in which measurement errors may not be assumed to be normally distributed. A generalized analysis procedure is introduced for cases in which the species spectra vary with experimental conditions.