# 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

_{f(t)}is the length n of series f(t). For the generated noises with the same length of 1,000, the calculated M is 9. Detailed wavelet coefficients of noises are analyzed here to provide useful suggestions for choosing DLs.

_{i}of the DLs from 1 to 9 and apply dyadic DWT to noises, then reconstruct the sub-signal under each level. Finally, we calculate the value of WEE using Equation (2):

_{j}(t) is the sub-signal under DL j. The WEE is defined according to the concept of Shannon entropy [25], which quantifies series’ complexity. Then, we also calculate the differential coefficients of WEE using Equation (3):

**Figure 1.**Values of wavelet energy entropy (WEE) and differential coefficient of WEE (D(WEE)) of various noises, and the corresponding 95% confidence interval (CI) obtained by using different decomposition levels (DLs).

#### 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):$$X=\frac{(X-\overline{X})}{\sigma (X)}$$
- (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.

**Figure 2.**Steps of choosing decomposition level (DL) in the process of wavelet threshold de-noising by using the improved method.

## 3. Discussion of the Applicability of the Improved Method

#### 3.1. Synthetic Series Analysis

**Table 1.**The lag-1 autocorrelation coefficient (R

_{1}), lag-2 autocorrelation coefficient (R

_{2}) and the signal-to-noise ratio (SNR) of three types of synthetic series used in this paper.

Characters | S1 | S2 | S3 | ||||||
---|---|---|---|---|---|---|---|---|---|

S11 | S12 | S13 | S21 | S22 | S23 | S31 | S32 | S33 | |

R_{1} | 0.666 | 0.326 | 0.083 | 0.714 | 0.212 | 0.062 | 0.681 | 0.234 | 0.051 |

R_{2} | 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 |

**Figure 3.**Values of wavelet energy entropy (WEE) and differential coefficient of WEE (D(WEE)) of three types of synthetic series obtained by using different decomposition levels (DLs), where the 95% confidence intervals (CIs) of noise’s WEE and D(WEE) are considered.

_{1}values of them are as little as 0.083, 0.062 and 0.051, respectively; therefore in the authors’ opinion, these series can be regarded as random series and their real signals need not be identified again.

#### 3.2. Observed Series Analysis

**Figure 5.**Values of wavelet energy entropy (WEE) and differential coefficient of WEE (D(WEE)) of three observed series obtained by using different decomposition levels (DLs), where the 95% confidence intervals (CIs) of noise’s WEE and D(WEE) are considered.

**Figure 6.**De-noising results of three observed series by using the chosen decomposition levels (DLs) (upper), and their wavelet variance curves (lower).

## 4. Conclusions

## Acknowledgements

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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

Sang, 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