Modeling of analytical peaks: peaks properties and basic peak functions

The general approach to the detailed characterizing of peak properties based on the use of characteristic points on a peak contour located at the certain levels is proposed. Three basic functions of peaks have been chosen which are submitted in normalized form on height and width. The ability to characterize a peaks shape of a certain analytical series is shown by an example of series of analytical peaks of thallium obtained by stripping voltammetry at linear and step potential sweep. Also the proposed shape parameters are studied for groups of stripping voltammetry and chromatographic peaks.     

色谱重叠峰分解的神经网络法

对于色谱图中两个相互重叠的峰, 本文提出了一种分解方法。首先在重叠峰的一阶导数曲线上取出五个无因次特征值, 然后用多层前传网来表达这五个特征值和重叠峰中子峰面积分率之间的关系。一系列实验的结果表明, 用神经网络方法所得子峰面积的准确度, 优于传统的垂线法及函数拟合法, 而且计算工作量较小, 可用于实时处理。     

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.     

A new mathematical function for describing electrophoretic peaks

A new model is proposed for characterizing skewed electrophoretic peaks, which is a combination of leading and trailing edge functions, empirically modified to get a rapid recovery of the baseline. The peak model is a sum of square roots and is called thereby "combined square roots (CSR) model". The flexibility of the model was checked on theoretical and experimental peaks with asymmetries in the range of 0-10 (expressed as the ratio of the distance between the center and the trailing edge, and the center and the leading edge of the chromatographic peak, measured at 10% of peak height). Excellent fits were found in all cases. The new model was compared with other three models that have shown good performance in modelling chromatographic peaks: the empirically transformed Gaussian, the parabolic Lorentzian-modified Gaussian, and the Haarhoff-van der Linde function. The latter model was proposed recently to describe electrophoretic peaks. The CSR model offered the highest flexibility to describe electrophoretic peak profiles, even those extremely asymmetrical with long tails. The new function has the advantage of using measurable parameters that allow the direct estimation of peak areas, which is useful for quantitative purposes.     

A new mathematical procedure to evaluate peaks in complex chromatograms

Automatic peak evaluation in chromatograms and subsequent quantification of compound concentrations is still a challenge in the analysis of complex samples containing hundreds or thousands of compounds. Although a number of software packages for peak evaluation exist, baseline definition and overlapping peaks of different shapes are the main reasons which prevent reliable automatic analysis of complex chromatograms. A new mathematical procedure is presented which uses peak shapes extracted from the chromatogram itself and modified by nonlinear (in fact, hyperbolic) stretching of the peak head and tail. With this approach, the peak parameters are position, height, scale of front, scale of tail, and smoothness of transition from front to tail scaling. This approach is found to give a substantially better fit than traditional analytically defined peak shapes. Together with a good peak finding heuristic and nonlinear optimization of parameters this allows a reliable automatic analysis of chromatograms with a large number of peaks, even with large groups of overlapping peaks. The analysis matches the quality of standard interactive methods, but still permits interactive refinement. This approach has been implemented and tested on a large set of data from chromatography of hydrocarbons in ambient air samples.     

Computerized detection and evaluation of peaks in survey spectra from photoelectron spectroscopy

The computer program described is written in Algol-60 and allows a fast data analysis of photoelectron peaks in survey spectra (0–1500 eV). Additional peak parameters such as area, background and peak width are also determined. The assumptions about peak shape are kept to a minimum. In spectra with good counting statistics, it is possible to detect peaks with peak/background ratio down to 0.01.     

Decomposition of overlapping plasmagram peaks by spectral subtraction

Low field atmospheric pressure Ion Mobility Spectroscopy (IMS) involves the careful analysis of plasmagrams with multiple peaks which can mask one another when they are closely spaced in drift time or corresponding reduced mobility. A typical signal processing approach to decomposing overlapped peaks would be to use an orthogonal decomposition technique, but unfortunately Gaussian-like functions are not orthogonal, so no unique decomposition can be guaranteed. However, each ion species in the drift tube will arrive at the Faraday plate with a known statistical distribution determined by the IMS instrument’s drift tube design, electric field strength, reagent gas flow and other instrument-specific factors such as the ion gate function. This paper presents a straightforward algorithm for decomposing plasmagrams into distinct peaks using a subtractive technique that independently estimates the statistical parameters of each peak, rejecting spurious peaks and electrical noise. The results show that for relatively short gate times, the plasmagram peaks are nearly Gaussian-shaped, but slightly fatter and asymmetric. We show that including of the gate rise and fall times is also significant in matching the plasmagram peak shape. We also show that the diffusion effects on resolution can be attributed to combinations of non-uniform ion distributions in the reaction chamber as well as detritus effects in the drift tube. Given the known peaks statistical parameters, one can then separate overlapping peaks using a straightforward spectral subtractive technique.     

System peaks in capillary zone electrophoresis of anions with negative voltage polarity and counter-electroosmotic flow

The system peaks that often appear on electropherograms in anion separation by CE with indirect spectrophotometric detection, negative voltage polarity and cathodic EOF are studied. The system peaks are shown to correspond to the zones with the changed concentration of the BGE constituents; they appear while the zone of each analyte anion passes through the outlet end of the capillary and are transported to the detector by EOF. An equation is suggested for predicting migration times of the system peaks with an error of 1%. The ratios of the system peak area to the analyte peak area are found to amount to 20%. It is shown that it is possible to avoid overlapping of the system peaks and analyte peaks by controlling the EOF velocity owing to hydrodynamic pressure. Using the mathematical simulation of CE shows that the system peaks and baseline shift can result from changing the transference numbers of the BGE ions and analyte ions at the capillary edge. The cases when the system peak may be incorrectly identified as the peak of analyte ion are considered. In order to avoid such errors, some practical recommendations are given.     

Temporally convoluted Gaussian equations for chromatographic peaks

Both spatial and temporal peaks that are produced by the discrete parcel model can be mathematically approximated by Gaussian functions, but the transformation from a spatial pattern to a temporal image requires a convolution treatment. A first-order convolution is given for temporal peaks under a linear isotherm, whereas a second-order convolution is proposed for those under non-linear isotherms. Numerical tests show that the peak shapes generated by the proposed temporally convoluted Gaussian equations (TCG) match perfectly with those obtained by the discrete parcel model. Although the full TCG equation may be quite complicated, it can be made easier by a recursion calculation technique, and a group of peak curves can be plotted simultaneously on computer worksheet. The results also suggest that the temporal distortion effect should be predominately considered, in addition to those known-to-exist spatial effects, for explaining the peak asymmetry.     

Surface plasmon resonances of silver triangle nanoplates: graphic assignments of resonance modes and linear fittings of resonance peaks

The extinction spectra of five silver equilateral triangle plates with a fixed thickness of 10 nm and side lengths of 50, 100, 150, 200 ,and 250 nm, respectively, have been simulated by the discrete dipole approximation (DDA) method in which a geometric object of interest is meshed and represented by a lattice of spatial dipoles. Irradiated by an incident plane wave with a given propagation direction and polarization state, each triangle nanoplate presents three surface plasmon resonance (SPR) peaks in the range of 300 to 1200 nm. At a given peak, every complex spatial oscillatory vector derived by DDA (corresponding to a certain dipole in the meshed target) is orthogonally resolved into three basic oscillations. Each basic component can be subsequently expressed by two parameters, amplitude (P) and phase angle (varphi). The distributions of six such physical parameters of all the dipoles in the selected cross plane of the target are illustrated colorfully in plots as a graphic characterization and assignment of the SPR modes. The graphic method is applied to reveal the local fine features of SPR modes. And it provides direct evidence for classifying SPR peaks which belong to different triangle nanoplates and appear at different wavelengths. Three SPR modes are recognized graphically and the wavelengths of SPR peaks are found to have linear relationships with the side lengths of the triangle nanoplates.     

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.     

Parametric studies of matched filters to enhance the signal-to-noise ratios of LC-MS-MS peaks

Chromatographic parameters of reference signals employed in matched filter methods have been studied using numerical experiments to improve the signal-to-noise (S/N) ratios of small liquid chromatography (LC) peaks obtained with electrospray tandem mass spectrometers (MS-MS). These parameters include the width, shape, and S/N ratios of chromatographic peaks used as the reference signal profiles. Our results show the effect of reference peak widths on improving the S/N ratio of chromatographic peaks; the influence of reference peak shapes is negligible. To verify simulation results, various reference signals, including analyte peaks of high concentration standards, internal standard peaks, and artificial Gaussian peaks of different widths, have been employed to enhance signal peaks on real liquid chromatography-tandem mass spectrometry (LC-MS-MS) chromatograms via matched filter methods. Our experimental results demonstrate that the S/N ratio enhancement of chromatographic peaks agree with the simulation predictions. These findings, therefore, suggest that regardless of peak shape, a well-smooth peak with a width close to that of the analyte peak is an adequate reference signal, when matched filter methods are used to improve LC-MS-MS chromatograms. Nevertheless, all methods processed LC-MS-MS peaks in this study do not achieve the ideal improvement ratios estimated with simulation results. We attribute this deficiency to spike-like noise, which have considerable low frequency components riding on LC-MS-MS chromatograms. Matched filtering, which works as a low-pass filter in the frequency domain, cannot effectively eliminate low frequency flicker noise contributed by these spikes. In addition, simple median filtering does not provide adequate improvement despite being able to smooth out most spikes in the chromatograms.     

Recognizing chromatographic peaks with pattern recognition methods : Part 1. Development of a k-nearest-neighbour technique

Problems in automated peak recognition in chromatography are discussed. An algorithm based on the k-nearest neighbour technique is proposed. Recognition of a peak is done by comparing it with a predefined profile function (normally a Gaussian peak profile). The profile and a part of the chromatogram are both interpreted as points in a multi-dimensional pattern space. The distance between the two points gives the value of the peak recognition function. The effects of different properties of chromatographic peaks (i.e., peak width, peak height and noise) and of the profile parameter (i.e., dimension of the pattern space, shape and width of the function, and characteristics of the distance measure) are evaluated. The method has excellent properties for recognizing peaks with low signal/noise (S/N) ratios; an example with S/N = 1 is shown. Changing peak widths and drifting baselines have little effect on the recognition ability. Difficulties with changing peak heights can be compensated by range scaling. Problems occur when two peaks are not sufficiently separated.     

一种变压器油色谱峰识别算法的设计

在变压器油色谱峰识别领域,传统的一阶导数法需要斜率阈值来实现色谱峰识别,因此自动化程度低及容易失真。针对这些缺点,本文在一阶导数法的基础上进行了改进,将迭代移动平均及归一化分析技术应用到色谱峰识别中,通过对信号曲线及方波曲线多次迭代移动平均确定最优的归一化识峰参数,结合色谱峰的绝对保留时间及识峰窗口实现对色谱峰的准确识别。实验结果表明：该算法可以准确识别色谱峰,对噪声、色谱峰的峰宽及峰形变化不敏感,具有很强的自适应性,满足变压器油中气体在线监测装置的现场使用要求。     

Simulation of Asymmetric Peak-Shaped Analytical Signals by the Frame Representation of Their Shape Using Stripping Voltammetry as an Example

To simulate peak-shaped analytical signals, the shape characteristics and geometrical properties of four asymmetric basic functions (the derivative of the logistic function, the Gaussian function, the Cauchy function modified by Fraser and Suzuki, and the Poisson function with the Stromberg correction) are systematically studied. The frame representation, which was proposed earlier by the two authors of this paper, is used to characterize the peak shape. Some regularities are found in peak properties expressed in terms of the frame parameters and considered as functions of the coefficient of skewness b. Peak shapes are compared using two approaches to obtaining dimensionless frame parameters: one using relative parameters and another using parameters of the unit frame. It is shown that the frame representation of the peak shape can be used in the phenomenological simulation of metal peaks in stripping voltammetry. The practical use of information on the shape of the analytical signal in the solution of various problems of analytical chemistry is discussed.     

A preliminary study of non-EMG peaks

Summary It is shown that part of non-EMG peaks are gamma peaks. The describing function of a gamma peak is derived. The peak area equation and the formation mechanism of a gamma peak are discussed.     

2nd-order spline wavelet convolution method in resolving chemical overlapped peaks

A 2nd-order spline wavelet convolution method in resolving overlapped peaks is developed. It determines the number of peaks, peak positions and width through wavelet's convolution, then uses spline function to construct the resoluter, which is used to resolve overlapped peaks. Theoretical proof is given, and the selections of wavelets and parameters are discussed. It is proven that baseline separation can be achieved after processed, the relative errors of peak position and area are less than 0.2% and 4.0% respectively. It can be directly applied to seriously overlapped signals, noisy signals and multi-component signals, and the results are satisfactory. It is a novel effective method for resolution.