Semi-differential analysis of irreversible voltammetric peaks

Voltammetric peaks obtained by simulation of electrochemical reactions under conditions of linear semi-infinite diffusion with an irreversible electron transfer process are analysed using a semi-differentiation procedure. Obtained semi-derivative peaks, separated or overlapped, are fitted with appropriate mathematical functions. The functions used for data fitting include a function describing symmetrical peaks, proposed by several authors for fitting irreversible semi-derivative peaks, and two alternative functions that express asymmetric shape of the irreversible semi-derivative signals. When applied to the overlapped irreversible semi-derivative peaks, the latter two functions allow calculating certain electrochemical parameters with a better accuracy as compared with the function derived for the symmetrical peaks.     

Analysis of peak asymmetry in chromatography

The knowledge of the symmetry of chromatographic peaks is extremely important regarding the digital signal processing. The significant deviation of the peak shape from the symmetrical peak makes hardly possible the acquisition of chromatographic signal information, such as the retention time, the peak area, the peak width at half peak height, the peak overlapping, etc. In the literature one can find many methods for the determination of the asymmetry factor. For example it is suitable to calculate the skewness from the third central moment. However in case of noisy baseline the value of the skewness oscillates highly depending on the number of points used for the mathematical calculation. In this work a new method is presented for the determination peak shape asymmetry. We order mathematical function to the chromatographic peaks by fitting, and then symmetrical curve is generated with the same peak maximum position and height, the peak width is fitted. The difference of the two functions is constituted and areas of the data differences are calculated, which are really characteristics of the peak asymmetry. Correlation between the area of the difference signal and the asymmetry factor is established. The method was applied for different types of chromatographic peak shapes and the results were interpreted.     

Quantitative information from x-ray diffraction of nylon-6 yarns. I. Development of a model for the analytical description of equatorial x-ray profiles

A wide-angle x-ray method has been developed by which quantitative structural information can be obtained on nylon-6 yarns. To this end, experimental equatorial diffractometer scans measured in transmission were fitted to a mathematical model describing the profiles as the envelopes of three bell-shaped functions. Four different models were investigated using, respectively, Gauss, Lorentz (Cauchy), Logistic, and Pearson-VII functions. The last model, which can be regarded as a generalized Lorentz function, gave the best fit. On the basis of a statistical analysis of the results of well-separated x-ray peaks, two parameters could be fixed. Another reduction of the number of parameters was achieved by interrelating the peak areas of the two outer reflections. These reductions widened the applicability of a computer program based on the aforementioned model to highly overlapping x-ray peaks. So the whole variety of x-ray scans, which can be obtained from nylon-6 yarns made under widely varying process conditions, can be well described. The fitting procedures provide unique solutions and hence objectively determined parameters.     

Rapid characterization of anthocyanins in red raspberry fruit by high-performance liquid chromatography coupled to single quadrupole mass spectrometry

This paper describes the results of a comparison of four peak functions in describing real chromatographic peaks. They are the empirically transformed Gaussian, polynomial modified Gaussian, generalized exponentially modified Gaussian and hybrid function of Gaussian and truncated exponential functions. Real chromatographic peaks of different shapes (fronting. symmetric, and tailing) are obtained by various separation conditions of reversed-phase liquid chromatography. They are then fitted to the peak functions via the Marquardt-Levenberg algorithm, a nonlinear least-squares curve-fitting procedure, by Microsoft Solver. The qualities of the fits are evaluated by the sum of the squares of the residuals. It is concluded in the study that the empirically transformed Gaussian function offers the highest flexibility (best fits) to all shapes of chromatographic peaks, including extremely asymmetric tailing peaks with a peak asymmetry of up to 8. The flexibility of this function should improve our ability to process chromatographic peaks such as deconvolution of overlapped peaks and smoothing noisy peaks for the determination of statistical moments.     

On the equations describing chromatographic peaks and the problem of the deconvolution of overlapped peaks

The problem of the appropriate choice of the function that describes a chromatographic peak is examined in combination with the deconvolution of overlapped peaks by means of the non-linear least-squares method. It is shown that the majority of the functions proposed in the literature to describe chromatographic peaks are not suitable for this purpose. Only the polynomial modified Gaussian function can describe almost every peak but it is mathematically incorrect unless it is redefined properly. Two new functions are proposed and discussed. It is also shown that the deconvolution of an overlapping peak can be done with high accuracy using a non-linear least-squares procedure, like Microsoft Solver, but this target is attained only if we use as fitted parameters the position of the peak maximum and the peak area (or height) of every component in the unresolved chromatographic peak. In case we use as fitted parameters all the parameters that describe each single peak enclosed in the multi-component peak, then Solver leads to better fits, which though do not correspond to the best deconvolution of the peak. Finally, it is found that Solver gives much better results than those of modern methods, like the immune and genetic algorithms.     

Multi‐scale Gaussian/Haar wavelet strategies coupled with sub‐window factor analysis for an accurate alignment in nontargeted metabolic profiling to enhance herbal origin discrimination capability

Metabolic dataset can provide an overview of different herbal origin, which is conducted by some statistical procedures. Such results often deviate to a certain degree, due to peaks shifts in chromatographic signals. In order to solve this problem, an improved algorithm of combining sub‐window factor analysis with the mass spectrum information is proposed. The algorithm uses a peak detection approach derived either from multi‐scale Gaussian function or Haar wavelet to locate the peaks with different application scope; the candidate drift points at each peak are estimated by Fast Fourier transform cross correlation; Specifically, the best drift points at each candidate peaks are confirmed by sub‐window factor analysis and mass spectrum information in nontargeted metabolic profiling. Finally, the peak regions were aligned against a reference chromatogram, and the non‐peak regions were used linear interpolation. The chromatographic signals of 30 Bupleurum samples were aligned as an illustration of this algorithm, and they could be well distinguished using some statistical procedures. The result demonstrates that the presented method is stronger than other mass‐spectra based algorithms, when facing the alignment of some co‐eluted peaks.     

Kinetic response of an oxygen-selective electrode evaluated for the quantification of oxygen

An evaluation of the transient response of a membrane-based amperometric electrode for the quantification of oxygen in aqueous and gaseous phases is described. Data collected during the early part of the responses are used to compute the steady-state signals that would be measured if the responses were monitored to steady state. Both first-order and variable-order models were used successfully to fit data for aqueous samples; only the variable-order model worked successfully with data for gaseous samples. Because the shapes of response curves did not permit steady-state signals to be measured, comparisons are based on measured values of current at fixed times in each response. The slopes of computed vs. measured signals varied from 0.90 to 1.03 for fits of data over 3–5 half-lives with both models. Calibration graphs of computed current vs. concentrations were linear for similar fitting ranges and the least-squares parameters were similar to those for current measured at different fixed points in time.     

Theoretical study of the accuracy of the elution by characteristic points method for bi-langmuir isotherms

The bi-Langmuir equation has recently been proven essential to describe chiral chromatographic surfaces and we therefore investigated the accuracy of the elution by characteristic points method (ECP) for estimation of bi-Langmuir isotherm parameters. The ECP calculations was done on elution profiles generated by the equilibrium-dispersive model of chromatography for five different sets of bi-Langmuir parameters. The ECP method generates two different errors; (i) the error of the ECP calculated isotherm and (ii) the model error of the fitting to the ECP isotherm. Both errors decreased with increasing column efficiency. Moreover, the model error was strongly affected by the weight of the bi-Langmuir function fitted. For some bi-Langmuir compositions the error of the ECP calculated isotherm is too large even at high column efficiencies. Guidelines will be given on surface types to be avoided and on column efficiencies and loading factors required for adequate parameter estimations with ECP.     

Non-linear least-squares fitting with microsoft excel solver and related routines in HPLC modelling of retention

Summary Two problems related to non-linear regression, the evaluation of the best set of fitting parameters and the reliability of the methods used for the estimation of the standard errors of these parameters, are examined. It is shown that a non-linear curve fitting routine, like the Microsoft Excel Solver, may give more than one solution for the same data set and a simple Monte Carlo routine is described for the evaluation of the bestfit. For standard errors, the reliability of two procedures based on the conventional curvature matrix method, four Jackknife techniques and the bootstrap method are examined by comparing their results to those obtained from a Monte Carlo simulation of the experimental data. It is shown that a fitting parameter may follow a nonnormal distribution when the equation to be fitted is complicated, even if the errors on the data are normally distributed. In this case only Monte Carlo methods of data simulation can give accurate information about the standard errors and the confidence intervals of these parameters.     

Statistically sound calibration curves for chromatographic methods involving negative response data

A statistical procedure is presented for calibrating chromatographic methods that generate negative chromatographic peaks. The technique is illustrated via data collected during the ion-chromatographic analysis of the ammonium ion in methansulfonic-acid solutions. A peak-integration protocol is explained and subsequent regression data are presented. For the concentration range of 1 to 5 ppb, a quadratic model could be fitted to the calibration data, using ordinary least squares as the fitting technique. The precision was approximately ± 0.5 ppb, at 95% confidence.     

Theoretical study of the accuracy of the pulse method,frontal analysis,and frontal analysis by characteristic points for the determination of single component adsorption isotherms

The adsorption isotherms of selected compounds are our main source of information on the mechanisms of adsorption processes. Thus, the selection of the methods used to determine adsorption isotherm data and to evaluate the errors made is critical. Three chromatographic methods were evaluated, frontal analysis (FA), frontal analysis by characteristic point (FACP), and the pulse or perturbation method (PM), and their accuracies were compared. Using the equilibrium-dispersive (ED) model of chromatography, breakthrough curves of single components were generated corresponding to three different adsorption isotherm models: the Langmuir, the bi-Langmuir, and the Moreau isotherms. For each breakthrough curve, the best conventional procedures of each method (FA, FACP, PM) were used to calculate the corresponding data point, using typical values of the parameters of each isotherm model, for four different values of the column efficiency (N=500$N=500$, 1000, 2000, and 10,000). Then, the data points were fitted to each isotherm model and the corresponding isotherm parameters were compared to those of the initial isotherm model. When isotherm data are derived with a chromatographic method, they may suffer from two types of errors: (1) the errors made in deriving the experimental data points from the chromatographic records; (2) the errors made in selecting an incorrect isotherm model and fitting to it the experimental data. Both errors decrease significantly with increasing column efficiency with FA and FACP, but not with PM.     

Comprehensive kinetic studies on isothermal and non-isothermal reactions of some aluminium compounds

Residual differences after model fitting were investigated in both isothermal and non-isothermal kinetics in order to make numerical comparisons between several models and various parameter-estimating methods. Data from two independent experimental series were evaluated. A large data set, collected earlier under isothermal conditions from decompositions and hydrothermal reactions of aluminium hydroxides and oxides, was processed first. It showed that mechanical activation of the starting gibbsite affected reactivity of samples in several subsequent reactions for all model equations tried. The relative residual deviation concept is introduced, and statistics were applied to find a model that fits a certain reaction in most of the cases. In the second study, the sulphate decomposition step of aluminium sulphate octadecahydrate was investigated. TG curves were measured using a constant heating rate. Dynamic models were fitted by three mathematical methods, including a new general purpose one. Fitting ability of the methods with various complexity were compared on the basis of residual deviations obtained after integration of the model equations. As well as evaluating the best fit, this new parameter-estimating method provides a statistical analysis of the reliability of the whole model fitting process.     

Automatic program for peak detection and deconvolution of multi-overlapped chromatographic signals part II: peak model and deconvolution algorithms

Several interlinked algorithms for peak deconvolution by non-linear regression are presented. These procedures, together with the peak detection methods outlined in Part I, have allowed the implementation of an automatic method able to process multi-overlapped signals, requiring little user interaction. A criterion based on the evaluation of the multivariate selectivity of the chromatographic signal is used to auto-select the most efficient deconvolution procedure for each chromatographic situation. In this way, non-optimal local solutions are avoided in cases of high overlap, and short computation times are obtained in situations of high resolution. A new algorithm, fitting both the original signal and the second derivatives is proved to avoid local optima in intermediate coelution situations. This allows achieving the global optimum without the need of background knowledge by the user. A previously reported peak model, a Gaussian with a polynomial standard deviation whose complexity can be modulated to enhance the fitting quality, was applied. However, the original formulation was modified to account baseline outside the peak region. Also, the optimal model complexity was auto-selected via error propagation theory. The method is able to process simultaneously several related chromatograms. The software was tested with both simulated and experimental chromatograms obtained with monolithic silica columns.