LEAST-SQUARES ALGORITHMS UNDER UNIMODALITY AND NON-NEGATIVITY CONSTRAINTS

In this paper a least squares method is developed for minimizing \\Y - XBT \\(2)(F) over the matrix B subject to the constraint that the col umns of B are unimodal, i.e. each has only one peak, and \\M\\(2)(F) b eing the sum of squares of all elements of M. This method is directly applicable in many curve resolution problems, but also for stabilizing other problems where unimodality is known to be a valid assumption. T ypical problems arise in certain types of time series analysis such as chromatography or flow injection analysis. A fundamental and surprisi ng result of this work is that unimodal least squares regression (incl uding optimization of mode location) is not any more difficult than tw o simple Kruskal monotone regressions. This had not been realized earl ier, leading to the use of either undesirable ad hoc methods or very t ime-consuming exhaustive search algorithms. The new method is useful i n and exemplified with two- and multi-way methods based on alternating least squares regression solving problems from fluorescence spectrosc opy and flow injection analysis. (C) 1998 John Wiley & Sons, Ltd.