# A Novel Feature Extraction Method for Ship-Radiated Noise Based on Variational Mode Decomposition and Multi-Scale Permutation Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Basic Theory

#### 2.1. VMD Method

- (1)
- Initialize $\{{\widehat{u}}_{k}^{1}\}$, $\left\{{w}_{k}^{1}\right\}$, ${\widehat{\lambda}}^{1}$ , and n = 0.
- (2)
- Updated the value of $\{{\widehat{u}}_{k}^{n+1}\}$ $\left\{{w}_{k}^{n+1}\right\}$ and ${\widehat{\lambda}}^{n+1}$ according to Equations (4)–(6).
- (3)
- Judge whether or not ${u}_{k}$ meets the convergence condition in Equation (9):$${{\displaystyle {\sum}_{k}\Vert {\widehat{u}}_{k}^{n+1}-{\widehat{u}}_{k}^{n}\Vert}}_{2}^{2}/{\Vert {\widehat{u}}_{k}^{n}\Vert}_{2}^{2}<e$$

#### 2.2. Analysis of the Simulation Signal Based on VMD

#### 2.3. PE Method

#### 2.4. MPE Method

## 3. Feature Extraction Method Based on VMD and MPE

## 4. Analysis of Simulation Signal Based on VMD and MPE

#### 4.1. The VMD of Simulation Signal

#### 4.2. The MPE of IMF with the Highest Energy

## 5. Feature Extraction of Ship-Radiated Noise Based on VMD and MPE

#### 5.1. The VMD of Ship-Radiated Noise

#### 5.2. Feature Extraction of Ship-Radiated Noise

#### 5.3. Classification of Ship-Radiated Noise

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Siddagangaiah, S.; Li, Y.A.; Guo, X.J.; Yang, K.D. On the dynamics of ocean ambient noise: Two decades later. Chaos
**2015**, 25, 103117. [Google Scholar] [CrossRef] [PubMed] - Tucker, J.D.; Azimi-Sadjadi, M.R. Coherence-based underwater target detection from multiple disparate sonar platforms. IEEE J. Ocean Eng.
**2011**, 36, 37–51. [Google Scholar] [CrossRef] - Wang, S.G.; Zeng, X.Y. Robust underwater noise targets classification using auditory inspired time-frequency analysis. Appl. Acoust.
**2014**, 78, 68–76. [Google Scholar] [CrossRef] - Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.A.; Yen, N.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond.
**1998**, 454, 903–995. [Google Scholar] [CrossRef] - Wu, Z.; Huang, N.E. A study of the characteristics of white noise using the empirical mode decomposition method. Proc. R. Soc. Lond.
**2004**, 460, 1597–1611. [Google Scholar] [CrossRef] - Wu, Z.; Huang, N.E. Ensemble empirical mode decomposition: A noise-assisted data analysis method. Adv. Adapt. Data Anal.
**2009**, 1, 1–41. [Google Scholar] [CrossRef] - Dragomiretskiy, K.; Zosso, D. Variational mode decomposition. IEEE Trans. Signal Process.
**2014**, 62, 531–544. [Google Scholar] [CrossRef] - Wang, Y.X.; Liu, F.Y.; Jiang, Z.S.; He, S.L.; Mo, Q.Y. Complex variational mode decomposition for signal processing applications. Mech. Syst. Signal Process.
**2017**, 86, 75–85. [Google Scholar] [CrossRef] - Lei, Y.; He, Z.; Zi, Y. Application of the EEMD method to rotor fault diagnosis of rotating machinery. Mech. Syst. Signal Process.
**2009**, 23, 1327–1338. [Google Scholar] [CrossRef] - Wang, Y.X.; Markert, R.; Xiang, J.W.; Zheng, W.G. Research on variational mode decomposition and its application in detecting rub-impact fault of the rotor system. Mech. Syst. Signal Process.
**2015**, 60–61, 243–251. [Google Scholar] [CrossRef] - Shih, M.T.; Doctor, F.; Fan, S.Z.; Jen, K.K.; Shieh, J.S. Instantaneous 3D EEG Signal Analysis Based on Empirical Mode Decomposition and the Hilbert-Huang Transform Applied to Depth of Anaesthesia. Entropy
**2015**, 17, 928–949. [Google Scholar] [CrossRef] - Xie, P; Yang, F.; Li, X. Functional coupling analyses of electroencephalogram and electromyogram based on variational mode decomposition-transfer entropy. Acta Phys. Sin.
**2016**, 65, 118701. [Google Scholar] - Xue, C.F.; Hou, W.; Zhao, J.H.; Wang, S.G. The application of ensemble empirical mode decomposition method in multiscale analysis of region precipitation and its response to the climate change. Acta Phys. Sin.
**2013**, 62, 109203. [Google Scholar] - Yang, L. An empirical mode decomposition approach to feature extraction of ship-radiated noise. In Proceedings of the 4th IEEE Conference on Industrial Electronics and Applications, Xi’an, China, 25–27 May 2009; pp. 3682–3686. [Google Scholar]
- Yang, H.; Li, Y.; Li, G. Energy analysis of ship-radiated noise based on ensemble empirical mode decomposition. J. Vib. Shock
**2015**, 34, 55–59. [Google Scholar] - Li, Y.; Li, Y.; Chen, X. Ships’ radiated noise feature extraction based on EEMD. J. Vib. Shock
**2017**, 36, 114–119. [Google Scholar] - Zhang, Z.; Liu, C.; Liu, B. Ship noise spectrum analysis based on HHT. In Proceedings of the 2010 IEEE 10th International Conference on Signal Processing (ICSP), Beijing, China, 24–28 October 2010; pp. 2411–2414. [Google Scholar]
- Liu, Y.Y.; Yang, G.L.; Li, M.; Yin, H.L. Variational mode decomposition denoising combined the detrended fluctuation analysis. Signal Process.
**2016**, 125, 349–364. [Google Scholar] [CrossRef] - Wang, Y.; Marker, R. Filter bank property of variational mode decomposition and its applications. Signal Process.
**2016**, 120, 509–521. [Google Scholar] [CrossRef] - Zanin, M.; Zunino, L.; Rosso, O.A.; Papo, D. Permutation entropy and its main biomedical and econophysics applications: A review. Entropy
**2012**, 14, 1553–1577. [Google Scholar] [CrossRef] - Wu, S.D.; Wu, P.H.; Wu, C.W.; Ding, J.J.; Wang, C.C. Bearing fault diagnosis based on multiscale permutation entropy and support vector machine. Entropy
**2012**, 14, 1343–1356. [Google Scholar] [CrossRef] - Gao, Y.; Villecco, F.; Li, M.; Song, W. Multi-Scale permutation entropy based on improved LMD and HMM for rolling bearing diagnosis. Entropy
**2017**, 19, 176. [Google Scholar] [CrossRef] - Sharma, R.; Pachori, R.B.; Acharya, U.R. Application of entropy measures on intrinsic mode functions for automated identification of focal electroencephalogram signals. Entropy
**2015**, 17, 669–691. [Google Scholar] [CrossRef] - Tripathy, R.K.; Sharma, L.N.; Dandapat, S. Detection of shockable ventricular arrhythmia using variational mode decomposition. J. Med. Syst.
**2016**, 40, 1–13. [Google Scholar] [CrossRef] [PubMed] - Shumbayawonda, E.; Fernández, A.; Hughes, M.P.; Abásolo, D. Permutation entropy for the characterisation of brain activity recorded with magnetoencephalograms in healthy ageing. Entropy
**2017**, 19, 141. [Google Scholar] [CrossRef] - Hsieh, N.K.; Lin, W.; Young, H.T. High-speed spindle fault diagnosis with the empirical mode decomposition and multiscale entropy method. Entropy
**2015**, 17, 2170–2183. [Google Scholar] [CrossRef] - Zhang, X.Y.; Liang, Y.T.; Zhou, J.Z.; Zang, Y. A novel bearing fault diagnosis model integrated permutation entropy, ensemble empirical mode decomposition and optimized SVM. Measurement
**2015**, 69, 164–179. [Google Scholar] [CrossRef] - Tang, G.J.; Wang, X.L.; He, Y.L.; Liu, S.K. Rolling bearing fault diagnosis based on variational mode decomposition and permutation entropy. In Proceedings of the 13th International Conference on Ubiquitous Robots and Ambient Intelligence (URAI), Xi’an, China, 19–22 August 2016; pp. 626–631. [Google Scholar]
- Cheng, G.; Chen, X.H.; Li, P.; Liu, H.G. Study on planetary gear fault diagnosis based on entropy feature fusion of ensemble empirical mode decomposition. Measurement
**2016**, 91, 140–154. [Google Scholar] [CrossRef] - Villecco, F.; Pellegrino, A. Entropic measure of epistemic uncertainties in multibody system models by axiomatic design. Entropy
**2017**, 19, 291. [Google Scholar] [CrossRef] - Li, Y.X.; Li, Y.A.; Chen, Z.; Chen, X. Feature extraction of ship-radiated noise based on permutation entropy of the intrinsic mode function with the highest energy. Entropy
**2016**, 18, 393. [Google Scholar] [CrossRef]

**Figure 1.**The simulation signals and the decomposition result of EMD, EEMD, and VMD. (

**a**) Simulation signals; (

**b**) EMD result; (

**c**) EEMD result; and (

**d**) VMD result.

**Figure 4.**The simulation signals and the decomposition result of EMD, EEMD, and VMD. (

**a**) Simulation signals; (

**b**) EMD result; (

**c**) EEMD result; and (

**d**) VMD result.

**Figure 5.**The time-domain waveform for three types of ship-radiated noise. (

**a**) The first type of ship-radiated noise; (

**b**) the second type of ship-radiated noise; and (

**c**) the third type of ship-radiated noise.

**Figure 6.**The results of VMD for three types of ship-radiated noise. (

**a**) The first type of ship-radiated noise; (

**b**) the second type of ship-radiated noise; and (

**c**) the third type of ship-radiated noise.

EMD | EEMD | VMD | ||||
---|---|---|---|---|---|---|

${f}_{4}(t)$ | — | — | IMF2 | 0.8947 | IMF1 | 0.9833 |

${f}_{3}(t)$ | IMF1 | 0.9538 | IMF5 | 0.9928 | IMF2 | 1 |

${f}_{2}(t)$ | IMF2 | 0.9188 | IMF6 | 0.9731 | IMF3 | 1 |

${f}_{1}(t)$ | IMF4 | 0.8274 | IMF8 | 0.9664 | IMF4 | 1 |

EMD | EEMD | VMD | |
---|---|---|---|

S1 | IMF5 | IMF5 | IMF8 |

S2 | IMF4 | IMF4 | IMF7 |

S3 | IMF3 | IMF3 | IMF6 |

S1 | EMD | EEMD | VMD | |
---|---|---|---|---|

MPE (scale = 1) | 0.4478 | 0.4869 | 0.4578 | 0.4504 |

MPE (scale = 2) | 0.4809 | 0.5584 | 0.5122 | 0.4947 |

The First Type | The Second Type | The Third Type | |
---|---|---|---|

IMF(level) | 8 | 8 | 6 |

Types of Ships | Train Sample | Test Sample | Overall Correctness (%) | ||
---|---|---|---|---|---|

Number | Correctness (%) | Number | Correctness (%) | ||

First type | 20 | 45 | 30 | 30 | 78.67 |

Second type | 20 | 100 | 30 | 100 | |

Third type | 20 | 100 | 30 | 100 |

Types of Ships | Train Sample | Test Sample | Overall Correctness (%) | ||
---|---|---|---|---|---|

Number | Correctness (%) | Number | Correctness (%) | ||

First type | 20 | 80 | 30 | 83.33 | 94 |

Second type | 20 | 100 | 30 | 100 | |

Third type | 20 | 100 | 30 | 100 |

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**MDPI and ACS Style**

Li, Y.; Li, Y.; Chen, X.; Yu, J.
A Novel Feature Extraction Method for Ship-Radiated Noise Based on Variational Mode Decomposition and Multi-Scale Permutation Entropy. *Entropy* **2017**, *19*, 342.
https://doi.org/10.3390/e19070342

**AMA Style**

Li Y, Li Y, Chen X, Yu J.
A Novel Feature Extraction Method for Ship-Radiated Noise Based on Variational Mode Decomposition and Multi-Scale Permutation Entropy. *Entropy*. 2017; 19(7):342.
https://doi.org/10.3390/e19070342

**Chicago/Turabian Style**

Li, Yuxing, Yaan Li, Xiao Chen, and Jing Yu.
2017. "A Novel Feature Extraction Method for Ship-Radiated Noise Based on Variational Mode Decomposition and Multi-Scale Permutation Entropy" *Entropy* 19, no. 7: 342.
https://doi.org/10.3390/e19070342