# Time-Shift Multi-scale Weighted Permutation Entropy and GWO-SVM Based Fault Diagnosis Approach for Rolling Bearing

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## Abstract

**:**

## 1. Introduction

## 2. Algorithm of TSMWPE

#### 2.1. MPE method

- (1)
- For a given maximum scale factor ${\tau}_{\mathrm{max}}$, the coarse-grained time series can be constructed from the original time series $\left\{x\right(i),i=1,2,\dots ,N\}$ by using formula (1)$${y}_{j}^{(\tau )}=\frac{1}{\tau}{\displaystyle \sum _{i=(j-1)\tau +1}^{j\tau}{x}_{i}},\hspace{1em}\hspace{1em}1\le j\le N/\tau $$
- (2)
- For the scale factor $\tau \ge 2$, permutation entropy of each coarse-grained time series is calculated. Finally, the entropy values of all scales are obtained and seen as a function of the scale factor.

#### 2.2. Algorithm of TSMPE

- (1)
- For a given time series $\left\{x\right(i),i=1,2,\cdots ,N\}$, there are$${y}_{k,\beta}=({x}_{k},{x}_{\beta +k},{x}_{2\beta +k},\dots ,{x}_{{\Delta}_{(\beta ,k)}\beta +k})$$
- (2)
- For scale factor $\tau \ge 2$, the PEs of each time-shift coarse-grained time series are calculated. The obtained different PEs of each time-shift coarse-grained time series are averaged by$$TSMPE(X,\tau ,m,\lambda )=\frac{1}{\tau}{\displaystyle \sum _{k=1}^{\tau}\mathrm{TS}PE({y}_{k,\beta}^{(\tau )},m,\lambda )}$$

#### 2.3. Algorithm of TSMWPE

- For the original time series $\left\{x\right(i),i=1,2,\cdots ,N\}$, the process of time-shift coarse-grained time series ${y}_{k,\beta}$ can be obtained by Equation (2).
- Each row in this matrix is regarded as a state vector and each state vector is mapped into $m!$ possible sorting mode ${\mathsf{\pi}}_{r}$, ${f}_{\omega}({\mathsf{\pi}}_{r})$ represents the frequency of the r-th permutation in the time series.$${f}_{\omega}{\mathsf{\pi}}_{r}={\displaystyle \sum _{s=1}^{S}f({\mathsf{\pi}}_{r}(s))}\cdot {\omega}_{r}(s),s=1,2,\cdots ,S$$
- The weighted relative probability of each state vector ${p}_{\omega}({\mathsf{\pi}}_{r})$ can be concluded by$${p}_{w}({\mathsf{\pi}}_{r})=\frac{{f}_{\omega}({\mathsf{\pi}}_{r})}{{\displaystyle {\sum}_{i=1}^{m!}{f}_{\omega}({\mathsf{\pi}}_{r})}}$$
- For $\tau $ time-shift coarse-grained time series, the weighted permutation entropy of each time-shift coarse-grained time series (TSMPE) can be defined as ${H}_{w}^{k}$ according to Shannon entropy as$${H}_{\omega}^{k}=-{\displaystyle \sum _{\Pi}^{}{p}_{\omega}({\mathsf{\pi}}_{r})\mathrm{ln}{p}_{\omega}}({\mathsf{\pi}}_{r})\hspace{1em}\hspace{1em}1\le k\le \tau $$
- Finally, $\tau $ ${H}_{w}^{k}$ are obtained and final TSMWPE of original time series is described as$$TSMWPE(x,\tau ,m,\lambda )=\frac{1}{\tau}{\displaystyle \sum _{k=1}^{\tau}{H}_{w}^{k}({y}_{k,\beta},m,\lambda )}$$

## 3. Analysis of Parameter Selection

#### 3.1. Selection of Parameter m

#### 3.2. Selection of Parameter $\lambda $

#### 3.3. Selection of Parameter N

#### 3.4. Stability Analysis

## 4. TSMWPE and GWO-SVM Based Fault Diagnosis Method for Rolling Bearing

#### 4.1. GWO-SVM

_{a}represents current location of wolves A, X

_{b}represents current location of wolves B, ${\mathrm{X}}_{c}$ represents current location of wolves C. ${C}_{1}$, ${C}_{2}$ and ${C}_{3}$ are random variables. $X(t)$ is the current location of the wolf species. The step lengths and directions of wolves E to wolves A, B and C are defined by formulas (13)–(15) and the final position of wolves E are defined by formulas (16).

#### 4.2. The Proposed Fault Diagnosis Approach

- (1)
- Let the rolling bearing contains K class work conditions, N sets of samples are collected for each state. TSMWPE is computed for all samples of each state of rolling bearing in M scales. The TSMWPE values obtained are used as the sample feature information to form the original feature vector matrix ${\mathrm{R}}^{K\times N\times M}$.
- (2)
- For each state of rolling bearing, N samples are collected and I samples are selected from the N ones as training samples to form a feature training set (${\mathrm{R}}^{K\times I\times M}$) and the rest (N−I) ones are seen as testing samples to form the testing feature set (${\mathrm{R}}^{K\times (N-I)\times M}$).
- (3)
- The training model feature set is employed to train the GWO-SVM based multi-classifier.
- (4)
- The testing sample feature set is inputting to the trained multi-classifier for prediction. The fault categories and severity of rolling bearing are judged according to the output of GWO-SVM multi-fault classifier. The flowchart of proposed method of fault diagnosis is shown in Figure 8.

#### 4.3. Experimental Analysis of Rolling Bearing

#### 4.3.1. Case 1

#### 4.3.2. Case 2

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Time domain waveform and power spectrum of WGN and 1/f noise. (

**a**) Time domain waveform of WGN, (

**b**) Power spectrum of WGN, (

**c**) Time domain waveform of 1/f noise and (

**d**) Power spectrum of 1/f noise.

**Figure 3.**Comparisons of TSMWPE, TSMPE and MPE under different embedding dimensions (

**a**) 1/f noise and (

**b**) WGN.

**Figure 6.**The mean standard deviation of MPE/TSMPE/TSMWPE under the same parameter (

**a**) 1/f noise and (

**b**) WGN.

**Figure 10.**The waveforms of vibration signals of rolling bearings for case 1. (

**a**) BE1; (

**b**) BE2; (

**c**) IR1; (

**d**) IR2; (

**e**) OR1; (

**f**) OR2; (

**g**) Norm.

**Figure 11.**MPE, TSMPE and TSMWPE of different states of rolling bearings for case 1 (

**a**) MPE (

**b**) TSMPE and (

**c**) TSMWPE.

**Figure 14.**Waveforms of vibration signal of rolling bearing from case 2. (

**a**) BA1; (

**b**) IR1; (

**c**) IR2; (

**d**) OR1; (

**e**) OR2; (

**f**) Norm.

**Figure 15.**MPE, TSMPE and TSMWPE of different states for rolling bearings for case 2. (

**a**) MPE; (

**b**) TSMPE; and (

**c**) TSMWPE.

**Figure 16.**Comparison of identification accuracy for different number of features. (GWO-SVM and SVM).

Abbreviation | Fault Location | Fault Diameter (mm) |
---|---|---|

BE1 | Ball element | 0.1778 |

BE2 | Ball element | 0.5334 |

IR1 | Inner race | 0.1778 |

IR2 | Inner race | 0.5334 |

OR1 | Outer race | 0.1778 |

OR2 | Outer race | 0.5334 |

Norm | Normal bearing | 0 |

Number of Used Features | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Best c | 2.7 | 19.3 | 81.1 | 12.7 | 4.9 | 14.5 | 90.4 | 34.5 | 3.6 | 52.8 |

Best g | 67.8 | 14.1 | 35.4 | 26.7 | 33.7 | 5.4 | 46.8 | 92.7 | 33.7 | 16.9 |

Number of used features | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Best c | 58.4 | 48.3 | 91.5 | 6.2 | 83.5 | 48.2 | 68.2 | 91.9 | 22.1 | 5.1 |

Best g | 1.1 | 20.0 | 0.6 | 8.1 | 19.9 | 15.2 | 17.6 | 0 | 24.5 | 0 |

Number of Used Features | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Best c | 72.6 | 64.1 | 70.6 | 1.0 | 55.3 | 48.7 | 7.9 | 66.6 | 43.4 | 49.0 |

Best g | 100 | 15.9 | 14.0 | 57.6 | 46.6 | 77.0 | 64.8 | 10.4 | 20.5 | 2.9 |

Number of used features | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Best c | 34.1 | 58.1 | 63.6 | 97.2 | 40.0 | 52.8 | 93.7 | 14.0 | 79.9 | 61.5 |

Best g | 2.8 | 36.8 | 13.5 | 7.0 | 10.2 | 16.9 | 17.6 | 5.5 | 6.9 | 1.3 |

Number of Used Features | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Best c | 36.4 | 78.2 | 29.6 | 94.4 | 3.8 | 34.1 | 84.5 | 3.1 | 62.4 | 37.9 |

Best g | 97.1 | 38.7 | 70.0 | 55.2 | 14.9 | 9.1 | 13.9 | 8.2 | 0.0 | 16.3 |

Number of used features | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Best c | 17.9 | 31.6 | 72.2 | 0.0 | 55.8 | 89.0 | 73.4 | 14.6 | 92.4 | 2.3 |

Best g | 11.4 | 8.9 | 13.5 | 0.0 | 0.0 | 0.0 | 0.3 | 0.2 | 6.7 | 0.0 |

Abbreviation | Fault Location | Fault Diameter (mm) |
---|---|---|

BE1 | Ball element | 0.6 |

IR1 | Inner race | 0.2 |

IR2 | Inner race | 0.6 |

OR1 | Outer race | 0.2 |

OR2 | Outer race | 0.6 |

Norm | Normal bearing | 0 |

Number of Used Features | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Best c | 42.9 | 97.3 | 19.1 | 29.8 | 53.6 | 86.7 | 39.3 | 41.1 | 67.0 | 15.8 |

Best g | 54.1 | 96.8 | 87.3 | 26.5 | 62.9 | 34.9 | 3.3 | 60.7 | 3.1 | 33.7 |

Number of used features | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Best c | 17.3 | 45.3 | 24.9 | 45.2 | 98.5 | 58.2 | 29.7 | 40.1 | 33.7 | 97.8 |

Best g | 82.6 | 51.5 | 32.5 | 2.6 | 59.8 | 64.4 | 85.5 | 38.6 | 33.6 | 21.2 |

Number of Used Features | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Best c | 78.4 | 16.8 | 100 | 15.3 | 15 | 68.6 | 36.7 | 96.2 | 68.1 | 89.1 |

Best g | 31.2 | 13.7 | 23.8 | 29.4 | 38.4 | 0.0 | 30.8 | 0.9 | 89.6 | 88.5 |

Number of used features | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Best c | 69.8 | 98.4 | 68.4 | 31.4 | 24.6 | 58.7 | 91.5 | 90.9 | 15.7 | 37.9 |

Best g | 61.6 | 26.8 | 83.8 | 83.0 | 33.1 | 42.1 | 85.5 | 36.3 | 75.1 | 89.3 |

Number of Used Features | 1 | 22 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Best c | 29.8 | 18.3 | 4.6 | 63.8 | 48.9 | 0.3 | 0.0 | 48.6 | 43.3 | 57.4 |

Best g | 95.0 | 40.0 | 3.1 | 6.1 | 18.4 | 2.5 | 4.8 | 9.0 | 1.7 | 19.5 |

Number of used features | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Best c | 58.4 | 13.6 | 100.0 | 25.7 | 17.4 | 70.0 | 60.0 | 77.4 | 79.1 | 55.1 |

Best g | 0.0 | 4.2 | 7.7 | 1.7 | 0.0 | 4.7 | 3.6 | 9.9 | 2.1 | 3.7 |

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## Share and Cite

**MDPI and ACS Style**

Dong, Z.; Zheng, J.; Huang, S.; Pan, H.; Liu, Q.
Time-Shift Multi-scale Weighted Permutation Entropy and GWO-SVM Based Fault Diagnosis Approach for Rolling Bearing. *Entropy* **2019**, *21*, 621.
https://doi.org/10.3390/e21060621

**AMA Style**

Dong Z, Zheng J, Huang S, Pan H, Liu Q.
Time-Shift Multi-scale Weighted Permutation Entropy and GWO-SVM Based Fault Diagnosis Approach for Rolling Bearing. *Entropy*. 2019; 21(6):621.
https://doi.org/10.3390/e21060621

**Chicago/Turabian Style**

Dong, Zhilin, Jinde Zheng, Siqi Huang, Haiyang Pan, and Qingyun Liu.
2019. "Time-Shift Multi-scale Weighted Permutation Entropy and GWO-SVM Based Fault Diagnosis Approach for Rolling Bearing" *Entropy* 21, no. 6: 621.
https://doi.org/10.3390/e21060621