# Stationary Wavelet-Fourier Entropy and Kernel Extreme Learning for Bearing Multi-Fault Diagnosis

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## Abstract

**:**

## 1. Introduction

## 2. Shannon Entropy Measures

#### 2.1. Stationary Wavelet Packet Fourier Entropy

#### 2.2. Stationary Wavelet Packet Permutation Entropy

- Step 1:
- Create a set of m-dimensional vectors ${W}_{i}^{m}$ as follows:$${W}_{i}^{m}=[w(i),w(i+1),\dots ,w(i+m-1)],\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,N-m+1$$
- Step 2:
- Each vector ${W}_{i}^{m}$ is sorted in ascending order with permutation pattern $\pi $ as follows:$$\begin{array}{cc}\hfill {W}_{i}^{m}& =[w(i+{j}_{1}-1)\le w(i+{j}_{2}-1),\le \dots \le ,w(i+{j}_{m}-1)]\hfill \end{array}$$$$\begin{array}{cc}\hfill \pi & =[{j}_{1},{j}_{2},\dots ,{j}_{m}]\hfill \end{array}$$
- Step 3:
- Calculate the probability of occurrence for each permutation pattern $\pi $ as follows:$$p(\pi )=\frac{\mathbf{Number}\left(\right)open="\{"\; close="\}">i|i=1,2,\dots ,N-m+1;{W}_{i}^{m}\phantom{\rule{1.em}{0ex}}\mathbf{has}\mathbf{type}\phantom{\rule{1.em}{0ex}}\pi}{}N-m+1$$
- Step 4:
- Calculate the normalized SWPPE of the ${i}^{\mathrm{th}}$ wavelet sub-band signal $w(n)$ using Equation (6):$$SWPPE[w(n)]=\frac{-1}{log\left(\right)open="("\; close=")">m!}log\left(\right)open="("\; close=")">p\left(\right)open="("\; close=")">{\pi}_{j}$$

#### 2.3. Stationary Wavelet Packet Dispersion Entropy

- Step 1:
- The wavelet sub-band signal $\{w(n)\}$ is normalized between zero and one using the normal cumulative distribution function as follows:$$y(n)=\frac{1}{\sigma \sqrt{2\pi}}{\int}_{-\infty}^{w(n)}exp\left(\right)open="["\; close="]">\frac{-{(t-\mu )}^{2}}{2{\sigma}^{2}}$$
- Step 2:
- The normalized signal $y(n)$ is mapped into c classes with integer indices from 1–c using the equation as follows:$${z}^{c}(n)=round\left(\right)open="("\; close=")">c\xb7y(n)+0.5$$
- Step 3:
- Create multiple m-dimensional vectors ${z}_{i}^{c,m}$ as follows:$${z}_{i}^{c,m}=[{z}^{c}(i),{z}^{c}(i+1),\dots ,{z}^{c}(i+m-1)],\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,N-m+1$$
- Step 4:
- Each embedding vector ${z}_{i}^{c,m}$ is mapped into a dispersion pattern ${\pi}_{{v}_{0},{v}_{1},\dots ,{v}_{m-1}}$, where ${z}^{c}(i)={v}_{0},{z}^{c}(i+1)={v}_{1},\dots ,{z}^{c}(i+(m-1)={v}_{m-1}$. Thus, the number of possible dispersion patterns is equal to ${c}^{m}$.
- Step 5:
- Calculate the probability of occurrence for each permutation pattern ${\pi}_{{v}_{0},{v}_{1},\dots ,{v}_{m-1}}$ as follows:$$p({\pi}_{{v}_{0},{v}_{1},\dots ,{v}_{m-1}})=\frac{\mathbf{Number}\left(\right)open="\{"\; close="\}">i|i=1,2,\dots ,N-m+1;{z}_{i}^{c,m}\phantom{\rule{1.em}{0ex}}\mathbf{has}\mathbf{type}\phantom{\rule{1.em}{0ex}}{\pi}_{{v}_{0},{v}_{1},\dots ,{v}_{m-1}}}{}N-m+1$$
- Step 6:
- Calculate the normalized SWPDE of the ${i}^{\mathrm{th}}$ wavelet sub-band signal $w(n)$ using Equation (11):$$SWPDE[w(n)]=\frac{-1}{log\left(\right)open="("\; close=")">{c}^{m}}$$

#### 2.4. Stationary Wavelet Packet Transform

## 3. Bearing Fault Diagnosis Algorithm

#### 3.1. Proposed Diagnosis Algorithm

- Step 1:
- Divide the discrete time raw vibration signal into multiple non-overlapped signals of N data points.
- Step 2:
- Step 3:
- Create a D-dimensional features vector based on multi-scale wavelet Shannon entropy as follows:$${u}_{k}=[1/{E}_{1},1/{E}_{2},\dots ,1/{E}_{i},\dots ,1/{E}_{D}]$$
- Step 4:
- Normalize the features matrix Z as follows:$${z}_{i}=\frac{{u}_{i}-{u}_{i,min}}{{u}_{i,max}-{u}_{i,min}}\phantom{\rule{1.em}{0ex}}i=1,2,\dots ,D$$
- Step 5:
- Create the KELM classifier based on both the features matrix Z and the k-fold cross-validation method.

#### 3.2. Kernel-ELM Classifier

#### 3.3. Experimental Setup

## 4. Experiments and Results

#### 4.1. Case 1: Fan-End Bearing

#### 4.2. Case 2: Drive-End Bearing

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Experimental setup [33].

**Figure 2.**Stationary wavelet packet Fourier entropy (SWPFE)-kernel extreme learning machine (KELM) diagnosis result with five-fold CV during the testing phase for the fan-end bearing: (

**a**) average accuracy and (

**b**) F-score.

**Figure 3.**SWPPE-KELM diagnosis result with five-fold CV during the testing phase for the fan-end bearing: (

**a**) average accuracy and (

**b**) F-score.

**Figure 4.**SWPDE-KELM diagnosis result with five-fold CV during the testing phase for the fan-end bearing: (

**a**) average accuracy and (

**b**) F-score.

**Figure 5.**SWPFE-KELM diagnosis result with five-fold CV during the testing phase for the drive-end bearing: (

**a**) average accuracy and (

**b**) F-score.

**Figure 6.**SWPPE-KELM diagnosis result with five-fold CV during the testing phase for the drive-end bearing: (

**a**) average accuracy and (

**b**) F-score.

**Figure 7.**SWPDE-KELM diagnosis result with five-fold CV during the testing phase for the drive-end bearing: (

**a**) average accuracy and (

**b**) F-score.

**Table 1.**Structure of bearing datasets. NB, normal bearing; ORF, outer race fault; IRF, inner race fault; BF, ball fault.

Fault Types | Speed (r/min) | Load (hp) | Fault Diameter (mils) | Samples Numbers | Class Label ${}^{1}$ | Class Label ${}^{2}$ |
---|---|---|---|---|---|---|

NB | 1797-1730 | 0-3 | 0 | 200 | 1 | 1 |

ORF | 1797-1730 | 0-3 | 7 | 200 | 2 | 2 |

14 | 200 | 3 | 3 | |||

21 | 200 | 4 | 4 | |||

IRF | 1797-1730 | 0-3 | 7 | 200 | 5 | 5 |

14 | 200 | 6 | 6 | |||

21 | 200 | 7 | 7 | |||

BF | 1797-1730 | 0-3 | 7 | 200 | 8 | 8 |

14 | 200 | 9 | 9 | |||

21 | 200 | 10 | 10 |

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**MDPI and ACS Style**

Rodriguez, N.; Barba, L.; Alvarez, P.; Cabrera-Guerrero, G.
Stationary Wavelet-Fourier Entropy and Kernel Extreme Learning for Bearing Multi-Fault Diagnosis. *Entropy* **2019**, *21*, 540.
https://doi.org/10.3390/e21060540

**AMA Style**

Rodriguez N, Barba L, Alvarez P, Cabrera-Guerrero G.
Stationary Wavelet-Fourier Entropy and Kernel Extreme Learning for Bearing Multi-Fault Diagnosis. *Entropy*. 2019; 21(6):540.
https://doi.org/10.3390/e21060540

**Chicago/Turabian Style**

Rodriguez, Nibaldo, Lida Barba, Pablo Alvarez, and Guillermo Cabrera-Guerrero.
2019. "Stationary Wavelet-Fourier Entropy and Kernel Extreme Learning for Bearing Multi-Fault Diagnosis" *Entropy* 21, no. 6: 540.
https://doi.org/10.3390/e21060540