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Resolving of Overlapping Asymmetrical Chromatographic Peaks by Using Wavelet-Transform and Gram-Charlier Peak Model

The paper describes a method for resolving overlapping asymmetric peaks that make up a chromatogram. The presented method uses the Gram-Charlier model in the form of the first three terms of the Gram-Charlier series as a basis. Using the wavelet transform, the parameters of this model are determined, which is used to describe a single or overlapping chromatographic peak. Hermitian wavelets of the first four orders are used in the computation of the wavelet transform. To speed up the computation of multiple wavelet transforms, the possibility of coding a signal using the Chebyshev-Hermite functions is considered in order to further restore the set of wavelet transforms simultaneously. According to the presented method, the parameters of the peaks are determined by analytical expressions without using the numerical approximation of the chromatogram by the peak model, which avoids the disadvantages of the numerical approach. The resulting method is used to resolve overlapping asymmetric peaks. The advantage of the method over others is shown by calculating the area of each of the resolved peaks.


Introduction
Analytical instrumentation is an intensively developing branch of measurement technology designed to study the composition and properties of substances. The development of this branch goes along the path of improving the characteristics of analytical devices and introducing new methods for analyzing measurement results. Among the tasks of analytical instrumentation functional transformation, filtering, detection and resolving overlapping chromatographic peaks, the elimination of the distorting effect of the device's spread function can be allocated [1].
The improvement of computational processing technologies for measuring information for signals of complex configuration is associated with the development of applied mathematical methods that underlie these technologies, with the introduction of new software and algorithmic data processing tools [2]. Thus, the actual problem is the development of theoretical and applied approaches that allow synthesizing compact and fast computational algorithms for estimating the desired signal, aimed at realizing all the potential possibilities of analytical devices.
Wavelet analysis is one of the most powerful and flexible means of research and digital processing of signals. For chromatographic signals it can be used for denoising, compression, spectral analysis [3], detecting apexes positions of overlapping peaks in low resolution chromatogram, resolving overlapping symmetrical peaks [4] and others.
The wavelet transform is implemented by integrating of a signal multiplied with a shifted and scaled wavelet, which makes calculations laborious if wavelets of complex shape are used or calculating of wavelet transform with many different wavelets are needed. To simplify the calculations, the signal can be encoded in the basis of the Chebyshev-Hermite functions (also named Hermite functions in some references) and the wavelet transforms can be restored using the algorithm described in the paper [5]. This encoding is based on decomposing the signal into a series by Chebyshev-Hermite functions [5][6][7]. These functions are based on Chebyshev-Hermite polynomials (also named Hermite polynomials in some references) and both of them are widely used in different branches of computational science: for design of filter banks [8], to determine optical parameters of laser beam [9], to key factors recognition of biometrical images [10] and other areas [11][12][13]. Simultaneous localization of these functions in both frequency and time space makes the method sufficiently resistant to instrumental errors [14][15]. In addition, encoding-decoding in the Chebyshev-Hermite function basis has a smoothing property [8].
At the moment, when processing signals from analytical instruments, there is no unified approach to the resolve overlapping chromatographic peaks, which is further complicated by the presence of asymmetry [16], and therefore the issue of developing new computational algorithms aimed at solving this problem is urgent. This paper described the method for resolving overlapping asymmetric chromatographic peaks by calculating its wavelet transforms of various orders.

Asymmetrical peak model selection
The first task that needs to be solved when developing a method for resolving overlapping chromatographic peaks is the choice of a model for these peaks. For simplicity, symmetric Gaussian function is often chosen as a model, and because of its symmetry the methods developed on its basis can't provide high accuracy of processing real chromatograms consisting of asymmetric peaks.
The most well-known models were considered: exponential modified Gaussian function [17], generalized exponential function [18], Fraser-Suzuki model [19], lognormal distribution function [20], Gram-Charlier series [21]. All these models except the last can be used in signal processing only by using numerical methods, such as the nonlinear least squares method. Such methods are more labor intensive and their accuracy deteriorates with decreasing SNR, increasing the number of overlapping peaks, and with incorrectly chosen initial conditions. It is not possible to analytically determine the parameters of these models, but for the Gram-Charlier model, the parameters of the peaks can be determined analytically through the wavelet transform without the described disadvantages [4], therefore this model was chosen for the developed method.
For practical use only the first three terms of (1) are usually taken: This expression has as parameters the amplitude A, a shift along the time axis , asymmetry (apex displacement within the localization region), and kurtosis (apex sharpening). In contrast to the Gaussian function in model (2) at the position of the apex is not equal to and at or at the apex height is not equal to A.

Wavelet transform with Hermitian wavelets
As it is well known wavelet transform of signal is determined by the expression: There is a known method for resolving overlapping symmetric chromatographic peaks given by the Gaussian function, built on the wavelet transform with Hermitian wavelets [4]. Since model (2) at and turns into a Gaussian function, the mentioned method can be considered as a special use case of the Gram-Charlier model, and for asymmetric peaks it is necessary to find a general solution.
Hermitian wavelets are a series of wavelets obtained from the product of an exponent and the Hermite polynomial of order m:

Resolving asymmetrical peaks by wavelet transform
3.1. Method for resolving overlapping asymmetrical chromatographic peaks by using wavelettransform and Gram-Charlier peak model By applying a wavelet transform with Hermitian wavelets to the Gram-Charlier model at and the following expressions can be obtained: is wavelet transform with Hermitian wavelet. To simplify the calculation of four orders of the wavelet transform at once, the previously mentioned method of reconstructing wavelet transforms from the signal coding coefficients in the basis of the Chebyshev-Hermite functions are used [5]. The wavelet transform is calculated at , since in this case the resulting expressions are greatly simplified. From expressions (3)-(6), estimates of the parameters of the Gram-Charlier model can be obtained: To calculate the parameter estimates from expressions (7) -(10), it is necessary to accurately determine , which cannot be done analytically by wavelet or other transform. To solve this problem, it is necessary to take into account two circumstances: 1) the local maxima of even wavelet transforms with Hermitian wavelets coincides with the position of the maxima of the overlapped peaks [4]; 2) for , the apex is shifted to the left of , and for , it shifts to the right [18]. Consequently, it is possible to determine numerically, assuming the position of the maximum as the initial value and using the sum of the squares of the differences between the values of the chromatogram and the peak calculated by (2) with the substitution of (7) -(10) and the current estimate of as recovery accuracy. In this paper the dichotomy method is used to search optimal .
The presented method of resolving overlapped asymmetrical chromatographic peaks can be described by the following sequence of actions applied to signal : • calculate the (3)-(6) for the fragment in which the last peak is localized (from the minimum between the tops of the last two peaks to the end of the signal); • using the dichotomy method, find by assessing the quality of restoration by the fragment selected at the previous step; • restore the last peak according to model (2) and estimates (7) -(10), calculate the difference signal ; • repeat the previous steps replacing with , respectively changing the localization area. The last step is repeated for all peaks remaining in the signal after subtracting the last one. If the reconstruction result is unsatisfactory, the described procedures should be repeated (until the desired accuracy is obtained) for individual difference signals -in the case of two overlapped peaks, this  It is suggested to start with the last peak, since the studied real signals have an asymmetry with a gamma greater than zero, which corresponds to the mu position to the right of the peak maximum. In such signals, the last peak is usually the least superimposed on other peaks in the region where mu is located (from the top to the end). Since the wavelet transforms (3) -(6) are calculated for a single signal at the point mu, the absence of overlaps with other peaks in this region allows one to determine the peak parameters with high accuracy using (7) - (10). Since the method is implemented iteratively, this feature can be ignored, but more iterations may be required to obtain optimal results.

Resolving peaks by presented method
To test the performance of the presented method, a synthesized signal was investigated, consisting of the sum of three overlapped peaks, the parameters of each of which are reliably known. To resolve the peaks, the signal was processed according to the presented method and by wavelet method with Gaussian peak model described in [4]. The comparison results are shown in Figure 1. The final goal of any method of chromatographic signals processing is to reliably determine the informative parameters -the area and position of the apex of each peak. To assess the reliability of the resolving results by the proposed method, the area was calculated for the obtained peaks. For comparison, the area of the peaks was also calculated using the wavelet method with Gaussian peak model and classical methods -perpendicular drop and triangulation [23]. Table 1 shows the relative errors of the calculated areas. The position of the apexes when using all methods was determined without error, therefore, it is not indicated in the table.

Result discussion and conclusion
As can be seen from Figure 1, approximation by the Gaussian function does not allow to reliably reconstruct the shape of asymmetric peaks. According to the results presented in Table 1, the presented method has the smallest error among all those considered. These results show the possibility of using the presented method when processing chromatograms containing asymmetric peaks. The advantage of the method is the increased accuracy of processing asymmetric chromatograms, as well as the elimination of the disadvantages of numerical methods for approximating overlapping peaks.