Error Analysis in Determining Parameters of Overlapping Peaks Using Separation of Complex Spectra into Individual Components. Case Study: Doublets (original) (raw)
Based on the concept of Big Data Modeling, the errors in determining peak parameters of noisy Gaussian quartets using nonlinear least squares curve fitting have been evaluated. The probability that the relative error in estimating each model parameter is not greater than a priory given limit for a given fitting error has been found. Obtained results showed that small fitting error does not guarantee that the fitting algorithm does converge to the correct peak parameters. It was found that the mean probability is a useful measure of the effectiveness of the curve fitting procedure.
Based on the concept of Big Data Modeling, the errors in determining peak parameters of noisy Gaussian quartets using nonlinear least squares curve fitting have been evaluated. The probability that the relative error in estimating each model parameter is not greater than a priory given limit for a given fitting error has been found. Obtained results showed that small fitting error does not guarantee that the fitting algorithm does converge to the correct peak parameters. It was found that the mean probability is a useful measure of the effectiveness of the curve fitting procedure.
Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms, 1984
In this paper, we report several fast separation methods using very simple algorithms and time-saving computational processes with reasonable precision, considering the statistical quality of the data. A comparison is made between the different procedures using synthetic peak doublets. The influence of the ratio and statistics of the two peaks is tested. The quality of the spectrum energy calibration, which is critical in some methods, is studied in detail. Two applications dealing with atmospheric pollution and biomedical trace element studies are given.
Reliability of the parameters in the resolution of overlapped Gaussian peaks
Analytica Chimica Acta, 1993
The reliability of the parameters obtained in the decomposition of two overlapped Gaussian peaks has been studied both as a function of the degree of overlapping between them and of the height and halfband width ratios. To do this a method based on the examination of the elements of dispersion matrix, obtained in a Gauss Newton non linear least squares procedure has been adopted. This is a potent tool for pointing out the functional dependence of the parameter errors on the set of numerical values defining the model and on the choice of experimental conditions. It has been shown that the ratio between haifband widths and the degree of overlapping of the bands are the leading factors affecting the reliability of results in this kind of curve decomposition.
Journal of Magnetic Resonance, 2000
We have derived analytical expressions of the Cramer-Rao lower bounds on spectral parameters for singlet, doublet, and triplet peaks in noise. We considered exponential damping (Lorentzian lineshape) and white Gaussian noise. The expressions, valid if a sufficiently large number of samples is used, were derived in the time domain for algebraic convenience. They enable one to judge the precision of any unbiased estimator as a function of the spectral and experimental parameters, which is useful for quantitation objectives and experimental design. The influence of constraints (chemical prior knowledge) on parameters of the peaks of doublets and triplets is demonstrated both analytically and numerically and the inherent benefits for quantitation are shown. Our expressions also enable analysis of spectra comprising many peaks. Copyright 2000 Academic Press.
Talanta, 1986
The resolving power of multicomponent spectra analysis and the computation reliability of the stability constants and molar absorptivities determined for nine variously protonated anions of three sulphonephthaleins and an impurity, by analysis of data for a mixture by programs SQUAD(84) and PSEQUAD(83), has been examined by use of synthetic and experimentally measured spectra containing severely overlapping spectral bands. The model mixture of Bromocresol Green, Phenol Red, Thymol Blue and azoxine as impurity, with five yellow, three blue and one red species in the pH range from 2 to 10, was used to examine the influence of precision of spectral data and of use of the spectra of the individual components, on the precision and accuracy of the estimated parameters when the chemical model is known. An efficient computation strategy has been found and both programs were shown to lead to the same values and reliability of the parametric estimates. Of the various diagnostics considered, the goodness-of-fit achieved is used as the criterion of whether the parameters found adequately represent the data.
Mathematical Processing of Spectral Data in Analytical Chemistry
Cambridge Scholars Publishing , 2018
Mathematical Processing of Spectral Data in Analytical Chemistry: A Guide to Error Analysis By Joseph Dubrovkin Cambridge Scholars Publishing Release Date: Aug. 1, 2018 This book will appeal to both practitioners and researchers in industrial and university analytical laboratories, as well as students specializing in analytical spectroscopy and chemometrics. The subjects covered include the advanced principles of calibration (univariate and multivariate) and the estimation of the peak parameters in spectra with overlapping components. This book differs from existing studies on the subject in that it provides easily reproducible computer calculations illustrating its significant theoretical statements. As such, it can also serve as a practical guide to lecturers in analytical spectrometry and chemometrics. ABOUT THE 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. 20% discount on online orders using the discount code ERROR20 – purchase through www.cambridgescholars.com or email orders@cambridgescholars.com
A statistical study of the analysis of congested spectra by total spectrum fitting
Journal of Molecular Spectroscopy, 2004
Heavily overlapped, or congested spectra often display much structure but few individual ''lines.'' Methods have been devised for analyzing such spectra through nonlinear least-squares fitting of the intensity as a function of wavelength or wavenumber. Such total spectrum fitting (TSF) methods are examined statistically for a simple diatomic model and compared with the standard ''measure-assign-fit'' (MAF) approach in use since the dawn of spectroscopy. Monte Carlo computations on typically 1000 synthetic spectra at a time verify that the predictions of the variance-covariance matrix are reliable under many circumstances. However in regions where the P and R branches double up, the predicted standard errors in the key spectroscopic constants rise sharply and greatly exceed estimates from the Monte Carlo ensemble statistics. In the same regions, the MAF method actually gives better precision. However, for imperfectly overlapped R and P branches, the MAF standard errors are typically three times larger than the TSF values; moreover, the MAF statistical errors are dwarfed by bias. The TSF approach, while clearly superior in these tests, has a practical drawback: it, too, can give significant bias if the spectra are analyzed with an incorrect model, as illustrated here through calculations employing the wrong function to describe the spectral lineshape.
Surface and Interface Analysis, 2000
Standard test data for x-ray photoelectron spectroscopy (XPS-STD) have been developed for determining bias and random error in peak parameters derived from curve fitting in XPS. The XPS-STD are simulated C 1s spectra from spline polynomial models of measured C 1s polymer spectra. Some have a single peak, but most are doublet spectra. The doublets were created from a factorial design with three factors: peak separation, relative intensity of the component peaks, and fractional Poisson noise. These doublet spectra simulated XPS measurements made on different two-component polymer specimens. This, the second of a three-part study, focuses on bias and random errors in determining peak intensities. We report the errors in results from 20 analysts who used a variety of programs and curve-fitting approaches. Peak intensities were analyzed as a ratio of the intensity of the larger peak in a doublet to the total intensity, or as a ratio of intensities for singlet peaks in separate but related spectra. For spectra that were correctly identified as doublets, bias and random errors in peak intensities depended on the amount of separation between the component peaks and on their relative intensities. Median biases for doublets calculated on a relative, unitless scale from −1 to 1 ranged from −0.33 to 0.17, whereas random errors for doublets calculated on the same scale ranged from 0.016 to 0.18. In most cases the magnitude of the median bias exceeded the median random error. On this scale, errors of −0.33 and 0.18 corresponded to errors of factors of 4 and 2, respectively, in determinations of the relative intensities as a ratio of the larger peak in a doublet to the smaller peak. Analysts may evaluate uncertainties in their own analyses of the XPS-STD by visiting the web site http://www.acg.nist.gov/std/.
Evaluation of the Uncertainty in Spectral Peak Location Case Study: Symmetrical Lines J
The dependences of the relative peak shifts of Gaussian and Lorentzian symmetrical doublets on the peak separation were evaluated theoretically. It has been shown that these dependences are well described by the eighth-order polynomial over the inverse of the line separation. The shifts for asymmetrical Gaussian, Lorentzian, and Voigt doublets were calculated numerically. The qualitative patterns were obtained and some abnormal phenomena were revealed.
Evaluation of the Uncertainty in Spectral Peak Location Case Study: Symmetrical Lines
2014
The dependences of the relative peak shifts of Gaussian and Lorentzian symmetrical doublets on the peak separation were evaluated theoretically. It has been shown that these dependences are well described by the eighth-order polynomial over the inverse of the line separation. The shifts for asymmetrical Gaussian, Lorentzian, and Voigt doublets were calculated numerically. The qualitative patterns were obtained and some abnormal phenomena were revealed.
Reconstruction of chromatographic peaks using the exponentially modified Gaussian function
Journal of Chemometrics, 2011
A method of noise filtering based on confidence interval evaluation is described. In the case of the approximation of a function, measured with error by a polynomial or other functions that allow estimation of the confidence interval, a minimal confidence interval is used as a criterion for the selection of the proper parameters of the approximating function. In the case of the polynomial approximation optimized parameters include the degree of the polynomial, the number of points (window) used for the approximation, and the position of the window center with respect to the approximated point. The Method is demonstrated using generated and measured chromatograms. The special considerations on confidence interval evaluation and quality of polynomial fit using noise properties of the 2 data array are discussed. The Method provides the lowest possible confidence interval for every data point.
Journal of Magnetic Resonance (1969), 1991
The analysis of multidimensional NMR spectra has been a challenging problem since the earliest two-dimensional experiments were reported as stacked plots (I). The first step in analysis involves obtaining a list of chemical-shift coordinates for the cross peaks. Initially, simple programs were used to generate contour plots of twodimensional NMR data, which were analyzed manually. As the utility and versatility of multidimensional NMR spectroscopy grew, several automated methods of peak picking have been developed. This Communication describes a new peak-picking algorithm which is based on contour diagrams and designed for the automated interpretation of higher dimensional 3D and 4D spectra.