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Chemometrics from basick to wavelet transform - Chau F.T

Chau F.T Chemometrics from basick to wavelet transform - Wiley publishing , 2004. - 333 p.
ISBN 0-471-20242-8
Download (direct link): chemometricsfrombasics2004.pdf
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Computational Details of Example 5.12
1. Load the experimental chromatogram [Fig. 5.45 (c0)].
2. Extend the chromatographic data for avoiding the edge effect.
3. Make a wavelet filter—Symmlet4.
4. Set resolution level J = 6.
5. Perform WT to obtain the c and d components with the improved algorithm.
6. Display Figure 5.45.
Because this method can extract the high-resolution information from a low-resolution or overlapping analytical signal, a method for determination of the component number in overlapping chromatograms was proposed. Figure 5.46 shows four experimental chromatograms with different intensities. It is impossible to obtain the correct component number from such chromatograms. Figure 5.47 shows the d5 component of the four chromatograms in Figure 5.46 obtained with Symmlet wavelet filters (L = 4). The dotted line in the figure indicates the position of zero in the magnitude axis. From Figure 5.47, it is clear that there are five components in the chromatograms, and all four chromatograms of different magnitude give us the same result.
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application of wavelet transform in chemistry
4 ' 5 ' 6 ' 7
Retention Time / min
Figure 5.46. Four experimental chromatograms with different amplitudes [from Chemometr. Intel!. Lab. Syst. 43:147-155 (1998)].
Because the WT is a linear transform, the high-resolution information should be used for quantitative calculation. An example of quantitative determination of the components in overlapping chromatographic peaks was published in Analytical Chemistry, [69:1722-1725 (1997)]. Figure 5.48 shows the chromatograms of five mixed samples of benzene, methyl benzene, and ethyl benzene. It is difficult to perform quantitative calculation
4 ' 5 ' 6 ' 7
Retention Time(min)
Figure 5.47. The detail component d5 obtained from the four chromatograms in Figure 5.46 by WT decomposition [from Chemometr. Intel!. Lab. Syst. 43:147-155 (1998)].
resolution enhancement
215
120000
100000
80000
>, 60000

c
a)
I 40000
20000 0
-20000
1.2 1.6 2.0 2.4 2.8 3.2
Retention Time / min
Figure 5.48. Experimental chromatograms of 5 three-component samples [from Anal. Chem. 69:1722-1725 (1997)].
200000
150000
100000
50000
-50000
-100000
-150000
2.0 2.4 2.8
Retention Time / min
3.2
Figure 5.49. Wavelet coefficients d3 obtained from the five chromatograms in Figure 5.48 by WT decomposition [from Anal. Chem. 69:1722-1725 (1997)].
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application of wavelet transform in chemistry
250000
200000
150000
&
100000
a>
50000
0
-50000
1.
2
Figure 5.50. Baseline-corrected wavelet coefficients after subtracting the estimated baseline by linking the minimum point at both sides of the peak in Figure 5.49 [from Anal. Chem. 69:1722-1725 (1997)].
of the components by directly using the chromatograms because of overlapping of the three peaks. Figure 5.49 shows the d3 component obtained by WT decomposition with Haar wavelet. It is clear that the information of the three peaks is resolved. In order to calculate the peak area, we can estimate a baseline by simply linking the minimum point at both sides of a peak. Figure 5.50 shows the results after subtracting such a baseline. Figure 5.51 shows the relationship between the area and the concentration. It can be seen that a very good linearity of the signals in the wavelet coefficients is kept.
Example 5.13: Resolution Enhancement of an Overlapping NMR Spectrum Using Method B. In Figure 5.52, spectrum (a) shows a simulated NMR spectrum by the Lorentzian equation in (5.77). From left to right, the peaks are doublet, triplet, quartet, and quintet. Spectrum (b) in the figure shows the reconstructed spectrum by multiplying the d-i and d2 by k1 = k2 = 55. Figure 5.53 shows the detail coefficients d! to d4 obtained by WT decomposition of spectrum (a) with Symmlet (L = 4) wavelet filters. From Figure 5.53 it can be seen that, except for discrete detail d4, di through d3 all represent the resolved information of the peaks in the overlapping spectrum, but from di to d3 the resolution decreases. Therefore, if we amplify the details d1 and d2, and then perform reconstruction, the
2 1.6 2.0 2.4 2.8 3.
Retention Time / min
resolution enhancement
217
Concentration / pl-ml'1
Figure 5.51. Calibration curves obtained from the peak area in Figure 5.50 and concentrations of the three components [from Anal. Chem. 69:1722-1725 (1997)].
resolution of the reconstructed spectrum will be improved. From spectrum (b) in Figure 5.52, it is clear that all four groups of peak are well resolved.
Selection of the details undergoing amplification is generally by visual inspection on the decomposed details as shown in Figure 5.53. It will be
O^O CL5 l!o 15 2.0 2^5 3^0 3^5
Chemical Shift / ppm
Figure 5.52. A simulated NMR spectrum (a) and the reconstructed spectrum (b) by multiplying the d1 and d2 by k1 = k2 = 55.
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application of wavelet transform in chemistry
(dj
(ds)
— AA aAa ^
I---'----------1---------1-1-'-1-1-1-'-1-----1-1-'---1
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