Comparison of Curve Fitting Algorithms Using for Separation of Overlapping Peaks. Case Study: Spectral Doublets

Joseph Dubrovkin, Computer Department, Western Galilee College, 2421 Acre, Israel

Mathematical separation of overlapping signals (peaks) in analytical chemistry is widely used in research labs and in the process analytical technology. Usually parameters of overlapping peaks are estimated by a nonlinear optimization method (a curve-fitting procedure). However, the curve-fitting algorithms were tested using a small number of measured or simulated spectra.

Therefore, the findings are unlikely to be common. Our error analysis in determining parameters of overlapping Gaussian doublets, triplets and quartets was based on more than one million of simulated spectra [J. Dubrovkin ,International Journal of Emerging Technologies in Computational and Applied Sciences, 2015, 1-12, 1; 2-12, 174; 1-13, 36]. Although it is not possible to mimic all real-life situations, these models allowed studying most practical cases of overlapping peaks encountered in spectra.

In this report we present the results of our research on the problem of the separation of 667,106 Gaussian doublets disturbed by constant additive, proportional and correlated (AR(1)) noise. The following algorithms were used: Gauss-Newton with the Levenberg–Marquard modification (GNLM), Genetic (GN), Interior-reflective Newton, Broyden–Fletcher–Goldfarb–Shanno (BFGS), Davidon–Fletcher–Powell (DFP) and the Steepest Descent (SD). We evaluated the probability (Pr) that the relative error in estimating each peak parameter will be less than some predefined value (val) for a given fitting error.

Statistical parameters of the distributions of the relative errors were estimated. It was found that the relative number of the good fitting models is low than 6% for all algorithms in the case of proportional noise; for DFP and SD for all types of noise. However, for most models, Pr >0.8 for val=1%. We found that SD is not applicable to the decomposition problem solving. The results showed that the "best" algorithms for decomposition of overlapping peaks are GNLM and BFGS. GN are the most time-consuming method.