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Angle-Constrained Alternating Least Squares

When resolving mixture data sets using self-modeling mixture analysis techniques, there are generally a range of possible solutions. There are cases, however, in which a unique solution is possible. For example, variables may be present (e.g., m/z values in mass spectrometry) that are characteristic for each of the components (pure variables), in which case the pure variables are proportional to the actual concentrations of the components. Similarly, the presence of pure spectra in a data set leads to a unique solution. This paper will show that these solutions can be obtained by applying angle constraints in combination with non-negativity to the solution vectors (resolved spectra and resolved concentrations). As will be shown, the technique goes beyond resolving data sets with pure variables and pure spectra by enabling the analyst to selectively enhance contrast in either the spectral or concentration domain. Examples will be given of Fourier transform infrared (FT-IR) microscopy of a polymer laminate, secondary ion mass spectrometry (SIMS) images of a two-component mixture, and energy dispersive spectrometry (EDS) of alloys.