# Fault Diagnosis of Rolling Element Bearings with a Two-Step Scheme Based on Permutation Entropy and Random Forests

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## Abstract

**:**

## 1. Introduction

## 2. Background Knowledge

#### 2.1. Permutation Entropy (PE)

_{p}(m) attains the maximum value $\mathrm{ln}(m!)$, when all the symbol sequences have the same probability distribution as ${P}_{l}=1/m!$. For convenience, the permutation entropy can be standardized as:

_{P}ranges from 0 to 1. A smaller H

_{p}value indicates that the time sequence is much more regular, and conversely, the larger the H

_{p}value, the more random the time series is.

#### 2.2. Variational Mode Decomposition (VMD)

^{2}-norm of the gradient, so these composition modes are called band-limited intrinsic mode functions (BLIMFs). The solution of VMD that is constructed as a constrained variational problem can be expressed as follows:

Algorithm 1 VMD Realization |

Initialize $\left\{{u}_{k}^{1}\right\},\left\{{\omega}_{k}^{1}\right\},{\lambda}^{1},n=0$, boolean = ture while Boolean n = n+1 for k = 1:K
$$\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{u}_{k}^{n+1}=\underset{{u}_{k}}{\mathrm{arg}\mathrm{min}}L(\{{u}_{i<k}^{n+1}\},\{{u}_{i\ge k}^{n}\},\{{\omega}_{i}^{n}\},{\lambda}^{n})$$
end for k = 1:K
$$\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{\omega}_{k}^{n+1}=\underset{{\omega}_{k}}{\mathrm{arg}\mathrm{min}}L(\{{u}_{i}^{n+1}\},\{{\omega}_{i<k}^{n+1}\},\{{\omega}_{i\ge k}^{n}\},{\lambda}^{n})$$
End
$$\text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{\lambda}^{n+1}={\lambda}^{n}+\tau (f-{\displaystyle {\sum}_{k}^{K}{u}_{k}^{n+1}})$$
if ${\sum}_{k}^{K}({\Vert {u}_{k}^{n+1}-{u}_{k}^{n}\Vert}_{2}^{2}/{\Vert {u}_{k}^{n}\Vert}_{2}^{2})<\epsilon$ boolean = false end |

#### 2.3. Random Forests (RF)

#### 2.3.1. Fundamental Principle

#### 2.3.2. Importance Evaluation of Characteristics Based on Out-of-Bag (OOB) Estimation

_{k}obtained from the k-th decision tree were employed to evaluate the model performance and then the corresponding classification error rate can be obtained. Complying with the same pattern, the classification error rates of all decision trees can be obtained and the average value of all classification error rates is considered as the generalization error of random forests, as well as the classification performance estimation of the model.

## 3. The Proposed Fault Diagnosis Model

#### 3.1. Features Extraction Based on PE and VMD

_{i}of one given signal x

_{i}can be extracted to realize the health judgment.

_{i}, obtain a collection of BLIMF components $\left\{{u}_{k}\right\}=\{{u}_{1},\cdots ,{u}_{K}\}$ by VMD. During signal decomposition, the quadratic penalty $\alpha $ and the bandwidth $\tau $ are respectively set to the default values of 2000 and 0.01.

_{i}, denoted as $MP{E}_{i}=\{MP{E}_{i1},\cdots ,MP{E}_{iK}\}$, where MPE

_{ij}is the permutation entropy value of the j-th BLIMF.

#### 3.2. Fault Detection Based on the Statistical Classification Model

#### 3.3. Random Forests-Based Fault Identification

_{TV}, it may be assumed that there are faults in bearings. Considering the type diversity and characteristic complexity of bearing faults, this work proposed an intelligence diagnostic model based on VMD-PE and RF. To further enhance the diagnostic performance of the model, the original VMD-PE features were reelected by using OOB estimation, and then the refined VMD-PE features are considered as the inputted data for training model. The implementation of the second step is described as follows.

_{1}denotes the diagnosis error rate of the first step, and E

_{2}be the error rate of the fault identification step. Hence, the final diagnosis accuracy $\eta $ of the proposed model can be calculated as:

## 4. Experimental Results and Analysis

_{TV}. Then, in the second step, the MPE feature sets with fault condition were split into two groups: 25% and 75%. 25% was employed as training data to train the RF-based fault identification model and the remaining 75% was used to test model.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Time domain waveforms and the corresponding envelope spectrums of the vibration signals under different working conditions.

**Figure 7.**Decomposed results obtained by variational mode decomposition (VMD) and envelope spectrums of the corresponding band- limited intrinsic mode function (BLIMF) components.

**Figure 8.**Dissimilarity and aggregation of the VMD-PE distributions under different fault conditions.

**Figure 9.**Importance evaluation of multiscale permutation entropy (MPE) features based on out-of-bag (OOB) estimation.

Type | Inner Race Fault | Outer Race Fault | Ball Fault | Normal | |
---|---|---|---|---|---|

Size (cm) | |||||

0.0178 | √ | √ | √ | √ | |

0.0356 | √ | √ | √ | ||

0.0533 | √ | √ | √ | ||

0.0711 | √ | -- | √ | ||

“√” indicates the working condition is under consideration. |

Case 1 | Case 2 | |||
---|---|---|---|---|

Accuracy (%) | Cost Time (s) | Accuracy (%) | Cost Time (s) | |

MPE-ELM ^{1} | 94.11 ± 1.11 | 0.006 | 96.15 ± 1.45 | 0.007 |

MPE-SVM | 96.76 ± 0.86 | 5.452 | 97.83 ± 1.01 | 5.029 |

MPE-RF | 98.44 ± 0.67 | 0.075 | 99.09 ± 0.67 | 0.074 |

^{1}MPE, multiscale permutation entropy; ELM, extreme learning machine; SVM, support vector machine; RF, random forests.

**Table 3.**Diagnosis results obtained by the proposed method, the two-step method with no features refinement and the traditional one-step method. OOB, out-of-bag.

Case 1 | Case 2 | |||||
---|---|---|---|---|---|---|

E_{1} (H = 350) | E_{2} (h = 330) | η | E_{1} (H = 350) | E_{2} (h = 330) | η | |

Two-step+ OOB | 0% | 1.52% | 98.57% | 0% | 0.30% | 99.997% |

Two-step | 0% | 1.56% | 98.57% | 0% | 0.91% | 99.14% |

One-step | - | - | 97.50% | - | - | 98.89% |

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**MDPI and ACS Style**

Xue, X.; Li, C.; Cao, S.; Sun, J.; Liu, L.
Fault Diagnosis of Rolling Element Bearings with a Two-Step Scheme Based on Permutation Entropy and Random Forests. *Entropy* **2019**, *21*, 96.
https://doi.org/10.3390/e21010096

**AMA Style**

Xue X, Li C, Cao S, Sun J, Liu L.
Fault Diagnosis of Rolling Element Bearings with a Two-Step Scheme Based on Permutation Entropy and Random Forests. *Entropy*. 2019; 21(1):96.
https://doi.org/10.3390/e21010096

**Chicago/Turabian Style**

Xue, Xiaoming, Chaoshun Li, Suqun Cao, Jinchao Sun, and Liyan Liu.
2019. "Fault Diagnosis of Rolling Element Bearings with a Two-Step Scheme Based on Permutation Entropy and Random Forests" *Entropy* 21, no. 1: 96.
https://doi.org/10.3390/e21010096