ShaoXueguang, Yu Zhengliang
(Department of Chemistry, University of Science and Technology of China, Hefei, Anhui, 230026, China)

Abstract Based on the wavelet compression and immune algorithm (IA), a novel algorithm for fast resolution of two-dimensional multicomponent overlapping chromatogram is proposed. Due to the characteristic of the linear property of the wavelet transform (WT), the overlapping chromatogram (antigen) can be compressed by WT before it is input into the immune network, the standard chromatogram of each component (antibodies) is also compressed to the same scheme. Especially, for speeding up the computation, the two-dimensional data matrix is arranged into one-dimensional vector form before it is compressed. After the compressed information of each component was extracted by IA, the chromatogram can be reconstructed by the inverse WT algorithm and re-arranged back into matrix form. It was proven that the result is almost the same with the result from IA, but the calculation speed is much faster. At the same time, satisfactory quantitative result can also be obtained.

1. INTRODUCTION
   Along with the development of modern chemical instrumentation, multicomponent two-dimensional data matrices can be easily obtained. For the aim of resolving the multicomponent overlapping matrices, several methods, such as chemical factor analysis (CFA),^[1,2] wavelet transform (WT) ^[3-5] and immune algorithm (IA) ^[6-8] have been proposed. In our previous works,^[6-10] it has been proven that the IA is an efficient tool for the resolution of overlapping multicomponent analytical signals. Multicomponent overlapping chromatogram can be easily resolved by an IA and the calculation speed is faster than conventional least-square method.^[7] However, when the size of data set is large and there are parameters need to be optimized, the consumed time of the computation is too long to be feasible in practical uses, e.g., in GA-IA^[10] method, because the IA procedure is invoked repeatedly. One useful way to speed up the calculation is to compress the raw experimental data. There have been many efficient tools for analytical data compression, such as binary coding method, Adams and Black algorithm, Fourier transform, chemical factor analysis, and wavelet transform. ^[11-14]
   In this paper, both the matrices of the standard chromatogram (antibodies) and the multicomponent overlapping chromatogram (antigen) are converted into one-dimensional vector form and compressed by wavelet transform at first, then perform the resolution by using an IA. It was found that the calculation speed can be improved by the conversion and compression. It can provide a fast preprocessing tool to the GA-IA method.
2. ALGORITHM
   The principle and application of IA has been reported in our previous works.^[6-10] The essence of an IA is that, taking the signal of the multicomponent mixture as antigen and the signals of the standard samples as antibodies, the information of each component in multicomponent overlapping signal can be extracted by a process of recognition, iterative elimination etc. The calculation process of an IA can be simply described by the following formulae:
   (1)
   (2)
   (3)
   where T is the weight of input layer, k is the number of iteration, V is the overlapping chromatographic signal (antigen), V_0i is the standard chromatographic signal of the ith component with known concentration (antibodies), c_i is the relative concentration of the ith component, and V_F is the feedback vector or matrix denoting the eliminated antigen. It can be seen that when dc^(k) approaches to zero, V_F will be the information of each component in the overlapping chromatographic signal. In many cases, due to the variation caused by the experimental reproducibility etc., parameters of V_0i, such as the position and the shape of the peaks may need to be optimized. The optimization will be a time-consumed procedure when the data number of V and V_0i is large. Therefore, an efficient way to compress the V and theV_0i will be necessary for speeding up the algorithm.
   The wavelet transform has been proven to be an efficient technique for analytical data compression.^[15,16] The dual localization characteristic in both frequency and time domains, the linearity, and the existence of fast algorithm make the WT an ideal candidate for preprocessing data for the IA. In this paper, the multi-resolution signal decomposition (MRSD) algorithm^[17,18] is used.
   Based on the algorithm of the IA and WT compression, a fast algorithm for resolution of multicomponent overlapping chromatogram is proposed. The flowchart can be described in Figure 1, including the following steps:
   
   Fig.1 The flowchart of the proposed algorithm
   (1) Input the overlapping 2-D chromatographic data matrix as antigen.
   (2) Estimate the possible components from the original chromatogram.
   (3) Input the standard 2-D chromatographic data matrices as antibodies.
   (4) Arrange all 2-D data matrices of the antigen and antibodies into one-dimensional vectors along with the chromatographic orientation.
   (5) Apply the wavelet compression to the antigen vector, i.e., perform WT on the antigen vector, then suppress those coefficients whose value is less than a threshold. The value of the threshold is determined by a predefined compression ratio that is determined by trial and inspection of the reconstruction error. The remained coefficients will be taken as the antigen for further calculation.
   (6) Compress the antibody vectors by performing WT with the same parameters as in the last step and remaining those coefficients at the same position with the compressed antigen.
   (7) Extract the compressed information of each component from the compressed antigen by the IA mentioned above, where the compressed antigen is taken as V and the compressed antibodies are taken as V_0i. In this step, the compressed information of each component can be extracted from the compressed antigen.
   (8) Reconstruct the extracted information of each component by the inverse WT algorithm to obtain a full extracted chromatographic data in vector form.
   (9) Finally, re-arrange the extracted data vector back into matrix form.

After all above calculation, the chromatograms of each component can be resolved from the multicomponent overlapping data matrix. From the theory of the IA, c_i is the concentration of each component relative to the concentration of the standard sample, i.e., if the concentration of the standard sample is known, then the concentration of each component in the mixture can be calculated by the parameter c_i. Therefore, this method can give us both the resolution and the quantitative determination simultaneously.

3. EXPERIMENTAL
   The experimental data sets were measured on an HPLC system comprising a Spectrasystem FL2000 (Spectra-Physics, USA) with the spectra Focus multi-wavelength UV detector (Spectra-Physics) and a Spectrasystem workstation. The column was packed with 10mm ODS silica (250 mm×5 mm, Shimadzu). The mobile phase was 0.25mol/L (pH~3.5) lactic acid (A.R.) with 0.01mol/L dodecyl sulfonic acid sodium. The color developing reagent of post column was 1.0×10^-4mol/L arsenazoⅢ (Fluka Chemie AG). The flow rate of the mobile phase was 1.0mL/min. The flow rate of the color developing reagent was 1.0mL/min. The temperature of the column was 20°C. The detection wavelength was from 580nm to 720nm. The interval of sampling time is 0.005min.

Table 1 Composition of the samples (unit: mg/ml)

Sample No.	Er	Tm	Yb
1	0.2000	0.1999	0.2001
2	0	0.1499	0.2001
3	0	0.1999	0.2001

Table 1 shows the composition of the three samples, which are mixed by Yb, Tm and Er, respectively. Figure 2 and Figure 3 (a) (b) (c) show the two-dimensional chromatogram of the mixture sample 1 (antigen) and the standard chromatograms of Yb, Tm and Er (antibodies) obtained by the experiment.

Fig.2 The experimental multicomponent overlapping chromatogram (antigen) of the sample No.1

Fig.3 The standard chromatograms of single component (antibodies)
(a) Er (b) Tm (c) Yb

4. RESULTS AND DISCUSSION
   4.1 Selection of the wavelet basis and the decomposition level
   Figure 4 shows the coefficients obtained by the wavelet transform of the chromatogram (arrange in vector form) in Figure 2 at level 7 using the Symmlet5 (L=10) wavelet basis. It can be found that the information of the chromatographic signal is mainly concentrated on only a few of the coefficients. Removing the smaller coefficient will not affect the total information. In order to obtain the optimal wavelet basis and decomposition level, reconstruction error obtained by Haar, Daubechies (L=4-20), Coiflets (L=4-20), Symmlets (L=6-30) at different decomposition level, where L is the length of filter, was investigated with the data sets in Figures 2 and 3. Table 2 summarized some of the results when the compression ratio is 1/58. The reconstruction error is calculated by
   (4)
   where, X is the original signal, X_R is the reconstructed signal from remained coefficients, n is the size of the original data set.
   
   Fig.4 The wavelet coefficients obtained by WT of the chromatogram in Figure 2

From Table 2, it can be found that the variation of reconstructed error is almost the same for all the four data sets. For every wavelet basis the minimal reconstruction errors generally appear at decomposition level 6 or 7. Comparing the reconstructed errors between different wavelet basis, it can be found that Symmlet 4, 5, 6 give smaller results. Therefore, Symmlet5 at decomposition level 7 is adopted in the following studies.

Table 2 Reconstruction errors by different wavelet basis at different decomposition level

Wavelet basis	Data	Decomposition level
		4	5	6	7	8
Haar	Mix.	309.601	194.953	177.447	171.839	168.520
	Er	94.252	78.993	74.375	73.626	74.169
	Tm	69.741	58.817	55.637	55.588	56.266
	Yb	69.070	58.046	53.986	52.874	51.045
Db4	Mix.	276.538	114.605	92.823	88.633	95.276
	Er	68.043	43.651	37.795	37.023	40.095
	Tm	51.618	34.362	31.237	32.520	33.644
	Yb	53.556	37.539	32.522	30.989	32.992
Db8	Mix.	277.929	109.555	89.389	88.817	99.135
	Er	68.060	42.529	38.665	40.783	46.580
	Tm	51.735	34.138	32.182	35.101	38.922
	Yb	53.529	37.544	33.170	33.769	37.752
Sym3	Mix.	276.373	120.943	98.082	92.855	94.883
	Er	68.186	44.532	37.727	38.435	39.737
	Tm	51.816	35.000	31.922	32.380	34.307
	Yb	53.856	38.130	33.074	31.258	32.033
Sym4	Mix.	275.467	115.130	92.764	85.817	87.348
	Er	67.950	43.506	37.304	37.294	40.324
	Tm	51.568	34.594	30.649	31.270	32.425
	Yb	53.539	37.624	32.082	30.208	30.807
Sym5	Mix.	279.138	112.817	89.699	84.277	84.930
	Er	68.870	44.143	38.136	38.141	39.722
	Tm	52.365	35.039	31.369	31.726	31.908
	Yb	54.374	38.296	32.159	29.435	29.962
Sym6	Mix.	275.515	110.207	87.794	82.778	86.383
	Er	67.938	42.396	36.546	37.034	39.834
	Tm	51.502	33.856	30.895	31.911	33.564
	Yb	53.468	37.125	32.402	30.549	31.978
Sym7	Mix.	279.568	110.628	87.745	84.261	87.822
	Er	68.977	43.888	37.681	38.089	40.615
	Tm	52.404	34.883	31.168	31.642	34.493
	Yb	54.392	38.508	32.354	30.697	31.047
Coif2	Mix.	276.228	113.440	90.970	84.528	83.760
	Er	68.185	43.589	37.174	37.834	39.003
	Tm	51.813	34.415	30.914	31.251	31.713
	Yb	53.780	37.451	32.044	29.921	30.360
Coif4	Mix.	277.145	107.641	86.738	83.806	87.135
	Er	68.182	42.591	37.072	37.435	41.578
	Tm	51.813	33.722	31.162	33.030	35.006
	Yb	53.752	36.947	32.133	31.387	33.073

Fig.5 The remained wavelet coefficients after compression and the extracted results (wavelet basis: Symmlet 5, decomposition level: 7)

4.2 Resolved result by the proposed algorithm
In order to resolve the overlapping chromatogram by the proposed algorithm, both the antigen (the multicomponent overlapping chromatogram) and the antibodies (the standard chromatograms of each component) were compressed with Symmlet5 wavelet basis at decomposition level 7. The number of data point is reduced from 52200 to 930. The solid line in Figure 5 shows the compressed result of the overlapping chromatogram in Figure 2. The dot lines show the resolved result by the IA. In order to see clearly, three different regions are enlarged in Figure 6, in which (a) (b) (c) are corresponding to the data points in the range of 140~190, 550~600, 660~690 respectively. It can be seen that the IA can give a very good resolution of the compressed wavelet coefficients.

Fig.6 The enlargement of Figure 5
(a) 140~190 data point (b) 550~600 data point (c) 660~690 data point

Fig.7 The reconstructed 2-D chromatograms
(a) Er (b) Tm (c) Yb

The reconstructed chromatograms from the resolved coefficients in Figure 5 are shown in Figure 7. (a) (b) (c) are corresponding to the reconstructed chromatogram of each component respectively. In comparison with the standard chromatogram of each component in Figure 3, it can be seen that the overlapping chromatogram is well resolved and the chromatogram of each component can be well obtained. The residual is shown in Figure 8, the intensity of the residual is very small compared with that of the overlapping chromatograms or the reconstructed chromatograms, which indicates that almost all the information contained in the overlapping chromatogram was extracted. The little error is mainly caused by the irreproducibility of the experiment. Read more about us.

Fig.8 The reconstructed residual information

4.3 Comparison of the proposed algorithm with immune algorithm and WT-IA
In our previous works, a WT-IA useing a two-dimensional wavelet compression algorithm was proposed for the sake of improving the calculation speed. In order to investigate the efficiency of the proposed algorithm, the consumed time and the residual after resolution are compared with those of the IA and WT-IA, where the value of the residual is the summation of every data point of the residual matrix as in Figure 8. The results are listed in Table 3. It can be seen clearly that the speed of the proposed algorithm is 2.48 times faster than that of the IA, 2.16 times faster than that of WT-IA. The residual is also smaller than that of IA and WT-IA.

Table 3 Comparison of conventional IA, WT-IA and the proposed algorithm*

No. Run	Consumed time (s)			Residual (×102)
	Conv. IA	WT-IA	Proposed Algorithm	Conv. IA	WT-IA	Proposed Algorithm
1	18.56	16.37	7.47	2.1011	2.2326	2.0946
2	18.84	16.21	7.52
3	18.84	16.36	7.58
4	18.89	16.37	7.63
5	18.51	16.37	7.58
Aver.	18.72	16.34	7.56

* Program runs on Pentium(r)/233MHz/Memory 64M.

Table 4 Quantitative results by the proposed algorithm

Sample		Added Conc. (mg/ml)	Calculated conc. (mg/ml)	Recovery (%)
1	Er	0.2000	0.1977	98.85
	Tm	0.1999	0.1914	95.75
	Yb	0.2001	0.2070	103.45
2	Er	0	0	0
	Tm	0.1499	0.1498	99.93
	Yb	0.2001	0.2046	102.25
3	Er	0	0	0
	Tm	0.1999	0.1966	98.35
	Yb	0.2001	0.2041	102.00

4.4 Quantitative determination using the proposed algorithm
In order to investigate the ability of the proposed algorithm for the quantitative determination, the three samples listed in Table 2 were analyzed and the results were listed in Table 4. It can be seen that all the recoveries are between 100± 5% with the minimum being 95.72% and maximum being 103.44%. The results are satisfactory.

5. CONCLUSION
   Based on the wavelet compression and immune algorithm, a fast algorithm for resolution of 2-D multicomponent overlapping chromatogram is proposed. By application of the method in resolution and quantitative determination of multicomponent 2-D overlapping chromatograms, it has been proven that this method is fast in calculation speed and accurate in quantitative calculation. Therefore, the proposed algorithm may be an alternative effective method for resolution of multicomponent 2-D overlapping chromatogram.