Computational modeling of spectral data fitting with nonlinear distortions

Substances such as chemical compounds and biological agents are invisible to human eyes, they are usually captured by sensing equipments with their spectral fingerprints. Although spectra of pure substances can be identified by visual inspection, the spectra of their mixtures take a variety of complicated forms. Given the knowledge of spectral references of the constituent substances, the task of data fitting is to solve their weights, which usually can be solved by a least squares. Complications occur if the basis functions (reference spectra) may not be used directly to best fit the data. In fact, random spectral distortions such as shifting, compression, and expansion have been observed in some spectra. In this paper, we formulate mathematical model for such distortions and build them into data fitting algorithms. If minimal knowledge of the distortions is available, a deterministic approach termed augmented least squares is developed by fitting the spectral references together with their derivatives to the mixtures. If the distribution of the distortions is known a prior, we propose to solve the problem with maximum likelihood estimators which incorporate the distortions into the variance matrix. The proposed methods are substantiated with numerical examples including data from Raman spectroscopy (RS), nuclear magnetic resonance (NMR), and differential optical absorption spectroscopy (DOAS) and show satisfactory results. Yuanchang Sun · Wensong Wu Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA. E-mail: [email protected]; [email protected] Jack Xin Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA. E-mail: [email protected]