Bayesian Estimation of Minimum Uncertainties in Determining Parameters of Analytical Signal Consisting of Overlapping Peaks

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

Mathematical separation of overlapping peaks is one of the most common preprocessing techniques for instrumental analytical chemistry. It necessarily requires error analysis of parameter estimates. Uncertainty in determining peak maximum position (x0) using the second order derivatives and deconvolution (performed by gradient and genetic algorithms) was estimated by [J. Dubrovkin, International Journal of Emerging Technologies in Computational and Applied Sciences, 2015, 2-11 , 102; 2014, 2-10 , 192; 2014, 3-9 , 248]. The upper bounds of the total error in determining all peak parameters [V. A. Lorenz-Fonfria and E.Pardos, Analyst, 2004, 129, 1243] and of the errors in determining x0: UB(x0) ∝s√w/Am, peak width: UB(w) ∝s√w/Am and peak maximum intensity: UB(Am)= ∝s/√w (s is the standard noise deviation) were obtained using Least Squares Estimation (LSE) [J. Dubrovkin, Euroanalysis-2015, France].

In this report we present the results of the error analysis of Bayesian estimates (BE) of peak parameters. The posterior distribution Pr(P|F) (where P and F are vectors of unknown parameters and of the measured analytical signal, respectively) was obtained for a normal nonlinear model of F by numerical integration. Pr(P|F) was evaluated in the best fitting point (P=Pbf) and in the interval abs(P-Pbf)<dP (where dP is an increment of the P). The minimum errors in determining each parameter in the wide range of varied parameters of Gaussian and Lorentzian peaks and for different s-values were calculated for the probabilities 0.5, 0.75, 0.90 and 0.95.

Algebraic expressions of the dependences of the minimum uncertainties in determining peak parameters on these parameters and the noise level obtained by the Bayesian estimation are identical to the formulae given above.

Unlike LSE BE approach does not require cumbersome algebraic calculations. It is applicable to any peak shape, and it is not a very time-consuming process.

Also we discuss Markov chain Monte-Carlo estimation of Pr(P|F).