Chapter 14 - A Smoothness Constraint in Multivariate Curve Resolution-Alternating Least Squares of Spectroscopy Data

Abstract

Obtaining a correct resolution of a chemical problem with multivariate curve resolution-alternating least squares (MCR-ALS) highly depends on the use of the appropriate constraints. With an increasing complexity of the data handled in analytical spectroscopy, as, e.g., hyperspectral imaging (HSI) data, new constraints might be necessary to encode spatial and spectral information on the mixture components. We therefore present in this chapter a constraint that exploits signal characteristics in order to impose smoothness of the MCR-ALS components. The proposed approach relies on the use of P-splines and can be applied to one-dimensional signals, such as concentration profiles and spectra, or two-dimensional signals, such as concentration distribution maps. Within this chapter, the concept of smoothness is explained, as well as a background on the method used for smoothing signals. Additionally, if the smooth character of a signal can be used as a constraint in MCR-ALS, we can also consider the rough character of a signal to be constrained. It is introduced how the constraint is implemented within the MCR-ALS framework, supported by simple simulated examples to illustrate the flexibility and viability of the constraint. The last section presents two case studies. One example deals with ultrafast time-resolved absorption spectroscopy data and the other with HSI data. In both applications, applying the smoothness constraint resulted in some improvement of the solutions obtained.