# Chemometrics from basick to wavelet transform - Chau F.T

ISBN 0-471-20242-8

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spectrum. From the result, it is obvious that the background is removed. The solid line in Figure 5.24c shows the result obtained by the conventional cubic spline curve fitting after many trials. By comparing the two curves, it can be seen that their main shapes are almost the same, but the result by the WT method is superior to that of the conventional method in both shape and noise level in the high-fr region.

Method B can also be used for removing the background in this example. Figure 5.26 shows the decomposed approximations obtained by using

baseline/background removal

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Figure 5.24. An experimental EXAFS spectrum of Cu sample (a), the converted spectrum in k space (b), and background-removed results obtained by conventional (c-solid) and WT (c-dot) methods.

Equations (5.32) and (5.33) of the improved algorithm. It can be seen that the oscillation signal is gradually removed from c0 to c4, but the smoothed background remains. Therefore, if we subtract c4 from c0, the oscillation part can be obtained, which is shown in Figure 5.27, curve (a) by the dotted line. The solid line shows the results obtained by the conventional cubic spline interpolation method for comparison. Comparing with the two curves in Figure 5.27, curve (a), we can also find that there is no significant difference between them except that the results obtained by the WT method are superior to those obtained by the conventional method in both shape and noise level in the high-k region.

188

application of wavelet transform in chemistry

8 16 24 32 40 48 56 64

4 8 12 16 20 24 28 32

A /x. _ _ «y

2 4 6 8 10 12 14 16

i—<—i—'—i—'—i—<—i—<—i—<—i—<—i

1 2 3 4 5 6 7 8

1 ' 2 ' 3 ' 4 ' 5 ' 6 ' 7 ' 8

Data Point

Figure 5.25. Plots of the decomposed approximation (c4) and details (d1, d2, d3, d4) obtained by applying WT to the k-space EXAFS spectrum with the MRSD algorithm.

Computational Details of Example 5.8

Method A:

1. Load experimental spectrum (Fig. 5.24a) and the spectrum in k space, (Fig. 5.24b).

2. Extend the spectral data to an integer power of 2.

3. Make a wavelet filter—Daubechies4.

4. Set resolution level J = 4 (8 - 4).

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5. Perform forward WT to obtain the wavelet coefficients.

6. Display Figure 5.25.

7. Set approximation coefficients with zeros, and construct the signal by applying inverse WT.

8. Display Figure 5.24

Method B:

1. Load experimental spectrum (Fig. 5.24a).

2. Extend the spectral data for avoiding the edge effect.

3. Make a wavelet filter—Daubechies4.

4. Set resolution level J = 5.

5. Perform forward WT to obtain the c and d components with the improved algorithm.

6. Display Figure 5.26.

7. Subtract c4 from the experimental spectrum.

8. Convert the subtracted result to k space.

9. Display Figure 5.27, curve (a).

As mentioned above, the aim of analyzing the EXAFS spectrum is to obtain the structural parameters such as N and r in Equation (5.47). In order to obtain the structural parameters from the EXAFS oscillation, Fourier filtering and least-square fitting can be performed. Figure 5.27, curve (b) shows the filtered results for the first coordination shell from the EXAFS signals in Figure 5.27, curve (a), and Table 5.3 compares the structural parameters obtained by least-square fitting of three Cu samples. The last column in the table shows a comparison of the fitted errors, which is

Figure 5.26. Plots of the approximations obtained by applying WT to the experimental EXAFS spectrum with the improved algorithm.

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application of wavelet transform in chemistry

4 6 8 10 12 14 (k)

Figure 5.27. Background-removed results by conventional (solid line) and WT (dotted line) methods (a), and their filtered results (b).

calculated by

1 N

E = M^Xal- Xxp)2 (5.48)

N i=1

where N is the number of points in the spectra and Xc'al and Xxp are, respectively, the fitted and the experimental values. For the sake of comparison between the two methods, both of the experimental and the fitted spectra were normalized when calculating the fitted error.

From Figure 5.27, curve (b), it is clear that the result by WT method is superior to that of the cubic spline method. From Table 5.3 it can be seen that, except for the coordination distance r, which is very close to the results of the two methods, all the other three parameters obtained by the wavelet transform method are larger than the results of cubic spline method. But they are reasonable. The fitted errors are also improved by the WT method. Table 5.3 also shows that the reproducibility of the three results

Table 5.3. Comparison of Structural Parameters and Fitted Errors Obtained by Least-Squares-Fitting from Background-Removed Spectra with WT and Spline Method Respectively

Spectrum Method N r a X Fitted Error

I WT 12 2.50 0.112 6.5 0.0026

Spline 8 2.52 0.077 4.4 0.0074

II WT 12 2.49 0.109 6.3 0.0029

Spline 9 2.53 0.082 4.5 0.0027

III WT 12 2.51 0.113 6.3 0.0011

Spline 9 2.52 0.084 4.5 0.0056

baseline/background removal

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