Uncertainty Analysis of Decomposition Level Choice in Wavelet Threshold De-Noising
Abstract
:1. Introduction
2. Uncertainty Analysis in Choosing Decomposition Level
2.1. Complexity of Noises
2.2. The Improved Method of Choosing DL
- (1)
- For the noisy series X analyzed, we first calculate the theoretical maximum M of DL by Equation (1), and normalize it by Equation (4):
- (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 and D(WEE) by Equation (2) and Equation (3) respectively, based on which we obtain the curves of WEE and D(WEE) of X.
- (3)
- According to the practical situations and experiences, we choose an appropriate probability distribution to generate “normalized” noise with the same length as that of X. Then we determine the curves of WEE and D(WEE) of noise by doing Monte-Carlo test, and also estimate the confidence intervals of them with a proper significance level (e.g., 5%).
- (4)
- Finally, we compare the value of WEE, especially D(WEE), of the noisy series X with those of noise with the increasing of DLs. Once the values of D(WEE) and WEE of X are obviously different from those of noise and exceed the confidence interval under certain DL*, the best DL can be chosen as DL* less 1. Besides, if the values of WEE and D(WEE) of X are close to those of noise under all DLs, the noisy series X can be regarded as a random series.
3. Discussion of the Applicability of the Improved Method
3.1. Synthetic Series Analysis
Characters | S1 | S2 | S3 | ||||||
---|---|---|---|---|---|---|---|---|---|
S11 | S12 | S13 | S21 | S22 | S23 | S31 | S32 | S33 | |
R1 | 0.666 | 0.326 | 0.083 | 0.714 | 0.212 | 0.062 | 0.681 | 0.234 | 0.051 |
R2 | 0.663 | 0.343 | 0.067 | 0.710 | 0.216 | 0.025 | 0.671 | 0.260 | 0.006 |
True SNR | 7.117 | −7.440 | −25.855 | 8.352 | −12.175 | −32.097 | 6.751 | −12.972 | −32.192 |
3.2. Observed Series Analysis
4. Conclusions
Acknowledgements
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Sang, Y.-F.; Wang, D.; Wu, J.-C. Uncertainty Analysis of Decomposition Level Choice in Wavelet Threshold De-Noising. Entropy 2010, 12, 2386-2396. https://doi.org/10.3390/e12122386
Sang Y-F, Wang D, Wu J-C. Uncertainty Analysis of Decomposition Level Choice in Wavelet Threshold De-Noising. Entropy. 2010; 12(12):2386-2396. https://doi.org/10.3390/e12122386
Chicago/Turabian StyleSang, Yan-Fang, Dong Wang, and Ji-Chun Wu. 2010. "Uncertainty Analysis of Decomposition Level Choice in Wavelet Threshold De-Noising" Entropy 12, no. 12: 2386-2396. https://doi.org/10.3390/e12122386