An improved algorithm for estimating pure component spectra in exploratory chemometric studies based on entropy minimization

Abstract

It is assumed that multiple spectroscopic measurements are made on a sequence of solutions containing unknown components. Further, it is assumed that a singular value decomposition of the spectral data set is performed whereby the right singular vectors associated with the pure component spectra (absorptivities) are obtained. Sasaki et al. [ Applied Optics, 1983, 1984] have shown that a set of estimated absorptivities can be obtained by solving a minimization problem involving a set of spectroscopic basic vectors – without relying on any further information. The resulting estimated absorptivities often closely resemble the true absorptivities. The objective function then proposed involves second order derivatives, i.e., a classical entropy functional. In the present contribution, the use of higher order derivatives is investigated, viz., fourth order derivatives, in the objective function. Synthetic infrared spectroscopic data are constructed for two-component solutions, and sets of right singular vectors are obtained by singular value decomposition. It is shown that for spectra composed of strongly overlapping spectroscopic features, the fourth derivative objective function provides considerable improvement over the second derivative objective function for the estimation of absorptivities. It is also shown that for spectra composed of mildly overlapping spectroscopic features, the fourth derivative algorithm and the second derivative algorithm produce estimated absorptivities which are, for most practical purposes, identical to the real absorptivities. Strongly overlapping spectroscopic features are common to most sets of NIR, UV and XPS spectra, and IR and NMR data collected over a very limited spectral range.