# Estimation of the uncertainty in determining the bell-shaped peak position and area in the presence of the random errors in the x-axis

Joseph Dubrovkin, Multidisciplinary Department, Western Galilee College, Zikhron Yaakov, Israel (dubrovkin@013.net.il)

Bell-shaped peak parameters contain essential analytical information. Among them the peak maxima are most frequently used in spectroscopy; the amounts of each component of the chromatographically separated mixture correlate with the area of asymmetrical peaks.

However, the uncertainty in determining the peak maximum position (MaxPos) was only estimated when measurements were performed in the points distributed with a constant step (h) along the x-axis (signal abscissa) [1]. These x-points in practice can be burdened with errors ("errors-in-variables model [2]").

We considered the errors in determining MaxPos when the abscissas are randomly changed near their actual values using for the measurements of the peak intensity. All peak intensities were reconstructed by fitting the peak shape models (Gaussian and Lorentzian symmetrical and Gaussian asymmetrical functions (GS, LS, GA, respectively)) to the noisy data numerically using the nonlinear regression [2]. The mean values of 100,000 estimations of the relative errors in determining MaxPos were calculated. Theoretical analysis showed that the mean of estimated positions of GS had the precise value. The dependences of the confidence interval radius of the errors on the h demonstrated that the errors may be equal to the tens percent of the h. For small h, the errors obtained for LS were more significant than those of GS. The impact of the asymmetry parameter on the errors was small.

Similarly, we estimated the dependences of the relative bias and relative error of the GA's area on the h. It turned out that the errors may be some percent. For large h, the area estimates were biased depending on peak intensity, width and asymmetry.

References

1. Dubrovkin J. (2018). *Mathematical Processing of Spectral Data in Analytical Chemistry: A Guide to Error Analysis*. Cambridge Scholars Publishing.

2. Seber, G. A. F. & Wild, C. J. (1989). *Nonlinear Regression*, Wiley.

Short Biography of Presenting Author

Joseph Dubrovkin gained a degree in Automatics from the Aviation Institute, Russia, in 1968, and Doctoral degrees in Technical Sciences and Physics and Mathematics from Leningrad State University, Russia, in 1979 and 1989 respectively. He was a Lecturer at the Aviation Institute and the Pharmaceutical Institute, Russia, and Western Galilee College (department of Bar-Ilan University), Israel, before retiring.