Entropy-Based Method of Choosing the Decomposition Level in Wavelet Threshold De-noising
<p>The lag-1 autocorrelation coefficient <span class="html-italic">R<sub>1</sub></span> and energy <span class="html-italic">E</span> of sub-signals of various noises under different decomposition levels (DLs).</p> "> Figure 2
<p>Values of wavelet energy entropy (WEE) of various noises when analyzed by using different decomposition levels (DLs).</p> "> Figure 3
<p>Steps of choosing suitable decomposition level (DL) in the process of wavelet threshold de-noising by using the proposed method.</p> "> Figure 4
<p>Three synthetic series data used in this paper.</p> "> Figure 5
<p>Values of WEE of three synthetic series and the corresponding derivation coefficients when analyzed by using different decomposition levels (DLs).</p> "> Figure 6
<p>Wavelet variance curves of the de-noised synthetic series obtained by using different decomposition levels (DLs).</p> "> Figure 7
<p>De-noising results of the three synthetic series by using the chosen decomposition levels (DLs).</p> "> Figure 8
<p>Values of WEE of the two observed series and the corresponding derivation coefficients when analyzed by using different decomposition levels (DLs).</p> "> Figure 9
<p>De-noising results of the two observed series by using the chosen decomposition levels (DLs) (upper), histograms of the separated noise (mid) and the wavelet variance curves of the de-noised series and observed series data (lower).</p> ">
Abstract
:1. Introduction
2. The Proposed Method of Choosing the Decomposition Level
2.1. Wavelet Threshold De-noising
2.2. Autocorrelations and Energy Distributions of Noises
Type | Expression | Parameters |
---|---|---|
Normal | μ, σ | |
Lognormal | μy, σy | |
Pearson-III | α, β, a0 |
Type | Index | Decomposition level (DL) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
Normal | R1 | −0.611 | 0.331 | 0.806 | 0.948 | 0.987 | 0.995 | 0.997 | 0.999 | 0.999 |
E | 499.59 | 249.87 | 125.29 | 62.64 | 30.95 | 17.30 | 12.21 | 7.24 | 5.86 | |
Lognormal | R1 | −0.612 | 0.332 | 0.806 | 0.948 | 0.987 | 0.996 | 0.998 | 0.999 | 0.999 |
E | 499.50 | 249.42 | 125.15 | 62.70 | 30.91 | 17.24 | 11.94 | 7.09 | 5.70 | |
Pearson-III | R1 | −0.611 | 0.331 | 0.806 | 0.948 | 0.987 | 0.996 | 0.997 | 0.999 | 0.999 |
E | 499.45 | 250.21 | 124.89 | 62.61 | 30.97 | 17.44 | 12.14 | 7.32 | 5.83 |
2.3. The Method Proposed
- (1)
- For the noisy series X analyzed, we first calculate the theoretical maximum M of DL by Equation (2), and normalize it by Equation (5):
- (2)
- Then, we apply dyadic DWT to X by using each value of the DLs from 1 to M, and calculate the values of WEE by Equation (4), based on which we obtain the WEE curve of X.
- (3)
- According to the practical situations and experiences, we choose an appropriate probability distribution to generate “normalized” noise series with the same length as that of X. Then we determine the WEE curve of noise by doing Monte-Carlo tests.
- (4)
- Finally, we compare the values of WEE of the noisy series X with those of noise with the increasing of DLs. Once the value of WEE of X is obviously different from that of noise under certain DL*, the best DL can be chosen as DL* less 1. Besides, the differential coefficient of WEE in Equation (6) can be used together to compare the WEEs of noisy series X and noise, because it is an extreme value under the DL* and can reflect their difference more clearly;
3. Case Studies
3.1. Synthetic Series Analysis
Series | Index | Real value | Decomposition level (DL) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |||
SS1 | SNR | -9.562 | 4.844 | -2.121 | -5.543 | -7.466 | -8.310 | -9.383 | -9.581 | -9.530 | -9.545 |
MSE | 0.469 | 0.236 | 0.121 | 0.065 | 0.040 | 0.013 | 0.005 | 0.004 | 0.010 | ||
Rxy | 0.665 | 0.782 | 0.870 | 0.922 | 0.949 | 0.983 | 0.993 | 0.995 | 0.987 | ||
SS2 | SNR | -4.012 | 7.054 | 0.954 | -1.833 | -3.983 | -18.58 | -19.18 | -19.67 | -25.38 | -29.27 |
MSE | 1.708 | 0.839 | 0.417 | 0.387 | 1.500 | 1.481 | 1.458 | 1.59 | 1.611 | ||
Rxy | 0.760 | 0.858 | 0.921 | 0.921 | 0.640 | 0.650 | 0.661 | 0.660 | 0.706 | ||
SS3 | SNR | -3.365 | 7.867 | 1.811 | -1.048 | -1.792 | -4.525 | -10.46 | -17.82 | -19.45 | -19.53 |
MSE | 0.305 | 0.146 | 0.067 | 0.046 | 0.074 | 0.158 | 0.191 | 0.211 | 0.224 | ||
Rxy | 0.775 | 0.871 | 0.934 | 0.953 | 0.918 | 0.817 | 0.817 | 0.794 | 0.768 |
Series | Decomposition level (DL) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
SS1 | -0.646 | 0.326 | 0.804 | 0.940 | 0.990 | 0.998 | 0.998 | 0.996 | 1.000 | |
SS2 | -0.579 | 0.326 | 0.810 | 0.949 | 0.990 | 0.996 | 0.999 | 0.999 | 0.998 | |
SS3 | -0.642 | 0.341 | 0.814 | 0.937 | 0.987 | 0.994 | 0.996 | 0.999 | 0.998 |
3.2. Observed Series Analysis
Series | Decomposition level (DL) | ||||||
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | |
RS1 | 30.904 | 26.768 | 20.049 | 18.200 | 17.297 | 16.705 | 16.285 |
RS2 | 30.908 | 50.205 | 48.944 | 48.707 | 48.622 | 48.550 |
Series | Decomposition level (DL) | ||||||
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | |
RS1 | -0.562 | 0.431 | 0.856 | 0.962 | 0.980 | 0.995 | 0.999 |
RS2 | -0.567 | 0.424 | 0.844 | 0.951 | 0.990 | 0.986 |
4. Conclusions
Acknowledgements
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Sang, Y.-F.; Wang, D.; Wu, J.-C. Entropy-Based Method of Choosing the Decomposition Level in Wavelet Threshold De-noising. Entropy 2010, 12, 1499-1513. https://doi.org/10.3390/e12061499
Sang Y-F, Wang D, Wu J-C. Entropy-Based Method of Choosing the Decomposition Level in Wavelet Threshold De-noising. Entropy. 2010; 12(6):1499-1513. https://doi.org/10.3390/e12061499
Chicago/Turabian StyleSang, Yan-Fang, Dong Wang, and Ji-Chun Wu. 2010. "Entropy-Based Method of Choosing the Decomposition Level in Wavelet Threshold De-noising" Entropy 12, no. 6: 1499-1513. https://doi.org/10.3390/e12061499