Using of Undetectable Perturbations for Determining Confidence Intervals for Nonlinear Curve Fitting of Overlapping Analytical Signals

Joseph Dubrovkin, Western Galilee College, Acre, Israel

Decomposition of overlapping analytical signals (peaks) is usually performed using nonlinear fitting a priory given model to the measured data. Different methods for determining confidence intervals of evaluated model parameters have been developed [1]. In these methods the squared norm of residuals as a measure of the goodness of fit has minimum value. However, the minimum does not guarantee the unique solution of the fitting. Therefore, modified method of undetectable perturbations (UPs) [2] in determining overlapping peak parameters was suggested. Correlations between relative errors of evaluated parameters have been taken into account [3]. 

In the present communication the confidence intervals of asymmetrical peak parameters were evaluated using linear approximation, bootstrap [1] and the UPs methods.  The fluorescence spectra of solutions of fluorescein, trans-stilbene and benzene were decomposed. The polynomial modified Gaussians (PMG), weighted sum of Lorentzian and PMG; double asymmetrical sigmoid and asymmetrical log-normal functions were used as the models.  

A multiple band-decomposition procedure based on the variation of the initial peak parameters was chosen to guarantee a unique solution with a minimal uncertainty [2, 4].   The estimated model parameters were constrained by the  values which have physical meaning. It was found that the linear approximation and bootstrap methods often overestimate radii of the confidence intervals.  Use Ups allowed us to control the radii according to a priory given level of the squared norm of residuals of curve fitting as opposed to the former methods.

Mathematical and physical justifications of model parameters evaluated by decomposition were discussed.

[1] Pomerantsev, A.L. (1999)  Chemom. and Intel. Lab. Syst. 49, 41.

[2] Dubrovkin, J.(2016) Chemom. Intell. Lab. Syst. 153, 9.

[3] Dubrovkin, J. Estimation of the feasible solutions in determining parameters of  symmetrical overlapping peaks, Isranalyitica - 2017.

[4] Dubrovkin, J., Tomin, V., Ushakou, D. (2016)  , J. Appl. Spectr. 83, 534.