Simultaneous construction of dual Borgen plots. I: The case of noise-free data

Abstract

In 1985, Borgen and Kowalski [DOI:10.1016/S0003-2670(00)84361-5] introduced a geometric construction algorithm for the regions of feasible nonnegative factorizations of spectral data matrices for three-component systems. The resulting Borgen plots represent the so-called area of feasible solutions (AFS). The AFS can be computed either for the spectral factor or for the factor of the concentration profiles. In the latter case, the construction algorithm is applied to the transposed spectral data matrix. The AFS is a low-dimensional representation of all possible nonnegative solutions, either of the possible spectra or of the possible concentration profiles. This work presents an improved algorithm for the simultaneous construction of the two dual Borgen plots for the spectra and for the concentration profiles. The new algorithm makes it possible to compute the two Borgen plots roughly at the costs of a single classical Borgen plot. The new algorithm comes without any loss of precision or spatial resolution. The new method is benchmarked against various program codes for the geometric-constructive and for the numerical optimization-based AFS computation.