Least squares algorithms under unimodality and non-negativity constraints

Abstract

In this paper a least squares method is developed for minimizing jjY À XB T jj 2 F over the matrix B subject to the constraint that the columns of B are unimodal, i.e. each has only one peak, and jjMjj 2 F being 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. Typical problems arise in certain types of time series analysis such as chromatography or flow injection analysis. A fundamental and surprising result of this work is that unimodal least squares regression (including optimization of mode location) is not any more difficult than two simple Kruskal monotone regressions. This had not been realized earlier, leading to the use of either undesirable ad hoc methods or very time-consuming exhaustive search algorithms. The new method is useful in and exemplified with two-and multi-way methods based on alternating least squares regression solving problems from fluorescence spectroscopy and flow injection analysis.

Hyperspectral vibrational photoacoustic imaging of lipids and collagen

“…This algorithm iteratively optimizes the spectral matrix S T and the distribution matrix C, with the aid of various constraints (e.g. non-negativity, unimodality, closure and equality) according to the chemical properties and origin of mathematics to reduce the ambiguity [21,22].…”

In situ transmission Fourier transform infrared (FT-IR) spectroscopy investigations of the surface chemistry of gettering common gases in vacuum systems with a common commercial flashed getter alloy as a function of temperature

“…Our non-negativity algorithm is an improvement on Bro's approach [11] that greatly reduces the computational time required [10].…”

“…In this paper we present the use of in situ Fourier transform infrared (FT-IR) spectroscopy coupled with multivariate curve resolution (MCR) to understand the chemistry that occurs www.elsevier.com/locate/vibspec Vibrational Spectroscopy 44 (2007) [9][10][11][12][13][14][15][16][17][18] between gases commonly produced in vacuum electronic environments (CO, O 2 , CO 2 , and CH 4 ) and a getter film as a function of temperature. Use of FT-IR allows extraction of information from the entire mid IR region of the spectrum (4000-700 cm À1 ) and allows monitoring the formation of several different chemical species simultaneously.…”

Modelling of spectroscopic batch process data using grey models to incorporate external information

“…If non-exact external information is available, it may still be possible to incorporate this by implementing soft constraints [48] on the model parameters, such as non-negativity. For example, if pure compound spectra are not known, the knowledge that they are unimodal and non-negative can still be used to restrict the solution parameter estimates.…”