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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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min (fj) = (p + 1) x 2-jfc = (p + 1) x 2-j2^ = /cut = 2j^ (5.45)
i.e.,
p,h=2'WS -1 (5-46)
By such an approach, trials sometimes, may still be needed to determinate the parameter pth in practical applications. It is clear that only those wp with j = J and p around pth, instead of all the wp, need be examined. (Note: The best-basis selection step is not necessary if we use the scale threshold in smoothing.)
Example 5.7: Denoising and Smoothing of Simulated and Experimental Chromatograms Using WPT. Figure 5.19 shows the denoised and the smoothed results of the simulated chromatogram in Figure 5.14, curve (b) using the WPT. The denoised curve (a) in Figure 5.19 is obtained with Symmlet4 filter and J = 10 by hard thresholding. The best basis was selected using Coifman-Wickerhauser entropy method, and the threshold value e = 0.008 x [max(w) - min(w)] was determined by trial and error. The smoothed curve (b) in Figure 5.19 is obtained with the same filter and J = 4. jth = 4 and pth = 0 were used as the scale threshold. Both the denoising and the smoothing give us a satisfactory result. If we further compare the denoised and the smoothed results, we can find that the smoothed result is superior to the denoised result.
data denoising and smoothing
181
0 200 400 600 800 1000
Data Point
Figure 5.19. Comparison of the denoised (a) and smoothed (b) results by WPT of the simulated noisy chromatogram.
Computational Details of Example 5.7
Denoising: Figures 5.19 (a):
1. Generate the original signal with 1024 data points using the Gaussian equation.
2. Make a wavelet filter—Symmlet4.
3. Normalize the data to noise level 1.
4. Set decomposition level D = 10 (dyadic length of the signal).
5. Perform WPT to obtain the WP coefficients.
6. Find best basis according to the entropy criteria.
7. Apply hard thresholding to the WP coefficients of the best basis.
8. Perform inverse WPT with the WP coefficients after thresholding to obtain the denoised signal.
9. Display Figure 5.19, curve (a).
Smoothing: Figure 5.19 (b):
1. Generate the original signal with 1024 data points using the Gaussian equation.
2. Make a wavelet filter—Symmlet4.
3. Set scale threshold Jth = 4 and Pth = 0, and set decomposition level J = Jth.
4. Perform WPT to obtain the WP coefficients.
5. Perform inverse WPT with WP coefficients within the scale threshold to obtain the smoothed signal.
6. Display Figure 5.19, curve (b).
182
application of wavelet transform in chemistry
2 4 6 8 10 12
Retention Time / min
Figure 5.20. Plots of the smoothed chromatograms by WPT with scale threshold of jth = 7 and pth = 2 (a), 3 (b), 4 (c), 5 (d), 6 (e), and 7 (f).
As mentioned above, if we perform smoothing of the experimental chromatogram in Figure 5.17 using the improved algorithm of WPT by scale threshold jth = 4 and pth = 0, the smoothed result should be same as the result in Figure 5.18, curve (b). If we want to smooth out the small fluctuation remained in that curve, we can perform a further smoothing using the WPT by scale threshold jth = 7 and pth < 7, because WPT further decomposes the w4 into w0 ~ w7 and the fcut for jth = 4 and pth = 0,1 x 2-4 fc, is equal to that for jth = 7 and pth = 7,8 x 2-7 fc.
Figure 5.20 shows the smoothed results by scale threshold jth = 7 and pth = 2 ~ 7. It is evident that the threshold jth = 7 and pth = 7 gives the same result with that in Figure 5.18 (b). Going from curve (f) to curve (a), it is clear that the curves become increasingly smooth with the decrease of pth. Therefore, we can conveniently choose a curve as our smoothed result.
5.2.4. Comparison between Wavelet Transform and Conventional Methods
In order to compare the WT or WPT denoising and smoothing with the conventional methods, the simulated and the experimental chromatograms are smoothed by moving-average, Savitsky-Golay, and FFT filtering methods, respectively. Figures 5.21 and 5.22 show their results. The smoothing window orfilterwidthforthethree methods is respectively 25,13, and 13points for the simulated chromatogram in Figures 5.21 and 25, 17, and 17 points for the experimental chromatogram in Figure 5.22. By comparing these
baseline/background removal
183
I • A\ jf\\ /A'? i\ // Vi // \
ii/i ill A' ?' An '/A i \; / / \ i,1 ; I; \1 \V/\
!ilk'.W ju /A AAA / f j 1 ' ? : ///a\ \\Jj fA -yiii \ aa\
-7; 7-7;; /1 KJi ' i .1 ^ ; i ;? A /// ' / A—A V ^ ; AA\ \/v\ ? 7
(d)
(s) J ;i' I \ y
0 200 400 600 800 1000
Data Point
Figure 5.21. A simulated noisy chromatogram (SNR = 20) (a) and the smoothed results with the use of moving-average (b), Savitsky-Golay (c), and FFT (d) filtering.
curves with those by WT or WPT, it can be found that, for the simulated signal, all the methods can give similar results, but for the experimental signal, WT and WPT give more satisfactory results.
5.3. BASELINE/BACKGROUND REMOVAL
In many cases, the baseline drift or background in an analytical signal is just like the noise, which often increases the difficulties in further processing. The baseline drift mainly induces errors in the determination
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