Estimating the boundaries of the feasible profiles in the bilinear decomposition of multi-component data matrices

Abstract

The boundaries of the range of feasible profiles in the bilinear decomposition of a data matrix are estimated by suitably modifying the already published N-BANDS algorithm. Instead of maximizing and minimizing the profile norms, the new sensor-wise N-BANDS approach calculates the profiles showing extreme values of the elements at each sensor in both data modes. This produces two sets of profiles for each sensor, whose envelops define the boundaries of the range of feasible solutions: the upper envelop for maximum element values represents the upper boundary, and the lower envelop the lower boundary. A three-component data set was studied in this manner under non-negativity constraints in both data modes. The sensor-wise N-BANDS results agree with those provided by the FACPACK program, which computes the area of feasible solutions and the associated set of feasible profiles. Computer times required for truly multi-component systems appear to be feasible using sensor-wise N-BANDS (up to eight sample components were checked). As an example, a six-component system was analyzed with the presently described approach, with results which are consistent with the expectations based on the consideration of the degree of profile overlapping among the participating species.