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

ISBN 0-471-20242-8

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m=1 1L

dj,k = ^ gj,mcj-1,k-m (5.33)

v2 m=1

where the hj and gj are obtained by inserting 2j-1 - 1 zeros into the every adjacent element of h and g in Equation (5.1) and (5.2). Subsequently, the length of the filter L will be doubled. By this algorithm, both cy = {Cj,k}

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application of wavelet transform in chemistry

and dj = {dj,k} keep the same length with cy_1 = {Cj_1,k}. Therefore, we can plot Cj and dj, respectively, to inspect their frequency. Furthermore, the algorithm can be used to decompose the signals c0 with any length without using the special techniques such as coefficient position retaining (CPR) method.

The corresponding reconstruction algorithm can be described by

Example 5.5: Data Smoothing of a Simulated Chromatogram. Figure

5.15 shows the Cj and dy obtained by Equations (5.32) and (5.33) with a Daubechies4 (L = 4) filter and J = 5 from the simulated signal of curve

(b) in Figure 5.14. It is clear that the d1 ~ d4 are attributed to the high-frequency components (noise). Therefore, the smoothed results can be obtained by reconstruction by Equation (5.34) where d1 ~ d4 is set to zeros, which is shown in Figure 5.16. By comparison of the smoothed result with the simulated signal in Figure 5.16, it can be seen that there is almost no distortion after the smoothing. The RMS between the two curves is only 0.0028, which is smaller than any of the denoising methods discussed above.

L

L

0 300 600 900 0

Data Point

300 600 900

Data Point

Figure 5.15. Plot of the wavelet coefficients obtained by WT with the improved algorithm.

data denoising and smoothing

175

0 200 400 ' 600 ' 800 1000

Data Point

Figure 5.16. Comparison of the simulated clean chromatogram (a) and the smoothed result by WT with the improved algorithm (b).

Computational Details of Example 5.5

1. Generate the noisy signal with 1024 data points [Fig. 5.14

(a),(b)] using the Gaussian equation.

2. Make a wavelet filter—Daubechies4.

3. Set resolution level J = 5.

4. Perform WT to obtain the c and d components with the improved

algorithm.

5. Display Figure 5.15.

6. Perform smoothing by replacing the d1 - d4 with zeros and

constructing the signal with inverse WT.

7. Display Figure 5.16.

8. Display the RMS between the original signal and the smoothed

signal.

We can also propose a method to estimate an approximate scale threshold for deciding the coefficients at which scale should be set to zero using the characteristics of Equations (5.32) and (5.33) and the concept of the Nyquist critical frequency in sampling theory. The Nyquist critical frequency is defined by

fc = A (5.35)

2As

where As is sampling time interval. For a signal with the Nyquist critical frequency fc, the frequency of the Cj and dj calculated by Equations (5.32)

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application of wavelet transform in chemistry

and (5.33) will match the following inequality:

(2~Jfc < d < +1fc

[fcj > 2-ifc

(5.36)

Then, if we take the peak of an analytical signal (e.g., a chromatogram) as a periodic signal, following the definition of the Nyquist critical frequency fc, the frequency of the peak can be defined by

f=W <5-37)

where Wb denotes the width of the peak. Because the aim of the smoothing is to remove high-frequency fluctuation and retain the chromatographic signal, the frequency of the coefficients to be removed, fcut, should be

much higher than that of the analytical peak:

fp = < fcut = min (fd.) = 2-ifc = 2-j(5.38)

p 2Wb 1 c 2AS

If we define the fcut by

=W (539)

where Wth denotes the estimated maximal width of noise, then rearrange the Equation (5.38) and perform a logarithm on both sides, we obtain

(5.40)

Wth ^ Wb

Therefore, if we denote the j in Equation (5.40) as yth, we have

i* = log2 (^) (5.41)

Wth ^ Wb

We can estimate an approximate scale threshold jth by Equation (5.41); all the detail coefficients at the scale lower than the jth should be set to zero in the reconstruction calculation. The difficulty in using the method is that the parameter Wth must be estimated by experience.

Example 5.6: Data Smoothing of an Experimental Chromatogram.

Figure 5.17 shows an experimental chromatogram measured by an reverse-phase HPLC and postcolumn reaction detection with arsenazo III. The sample is composed of six rare-earth ions (Lu, Yb, Tm, Er, Ho, and

data denoising and smoothing

177

“I-----1-----1-----1-----1------1----1-----1-----1-----1-----1

2 4 6 8 10 12

Retention Time / min

Figure 5.17. Experimental chromatogram with great level noise of a mixed rare-earth solution sample.

Yb). The chromatogram recorded between 1.505 and 11.811 min sampled every 0.005 min is shown. The data length is 2048. The noise is caused by the strong absorption of the postcolumn reaction agent and the pulse of the mobile phase. Therefore, the noise is an oscillation as shown in the enlarged part of the figure. It is impossible to denoise such a chromatogram by other commonly used denoising methods. But we can easily smooth the chromatogram by using the WT smoothing method mentioned above.

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