# Bearing Fault Diagnosis Using Refined Composite Generalized Multiscale Dispersion Entropy-Based Skewness and Variance and Multiclass FCM-ANFIS

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generalized Refined Composite Multi-Scale Dispersion Entropy

#### 2.1. Dispersion Entropy

**y**is mapped to an integer between 1 and c (Equation (2)):

^{th}member of the classified series ${z}^{c}$.

#### 2.2. Generalized Multiscale Dispersion Entropy

^{th}-moment-based generalized MDispEns are displayed as GMDispEn

_{n}. They are implemented as follows:

**x**with the scale $\tau $ and the n

^{th}moment, is constructed [19]:

_{2}, based on the second moment (variance):

_{3}, based on the third moment (skewness):

#### 2.3. Generalized Refined Composite Multi-Scale Dispersion Entropy

^{th}-moment-based RCGMDispEn (RCGMDispEn

_{n}) with a scale factor of $\tau $, $\tau $ different time series are constructed by coarse-graining based on the first and higher momenta in order and with different starting points. The relative frequency of the dispersion patterns is calculated from every $\tau $ time series. The k

^{th}coarse-grained time series ${x}_{k}^{n,(\tau )}=\left\{{x}_{k,1}^{n,(\tau )},{x}_{k,2}^{n,(\tau )},\dots \right\}$ from the series $x$ is obtained based on the n

^{th}moment and the scale $\tau $ as follows:

_{n}is defined as follows:

## 3. Multiclass Adaptive Neuro-Fuzzy Classifier

#### 3.1. Adaptive Neuro-Fuzzy Inference System (ANFIS)

^{th}rule and j

^{th}feature, and ${x}_{sj}$ denotes the j

^{th}feature of the s

^{th}sample. The parameters ${c}_{ij}$ and ${\sigma}_{ij}$ respectively represent the center and width of the Gaussian function.

^{th}rule for the sample ${x}_{c}$, is obtained as follows:

^{th}node ($\overline{{\theta}_{ic}}$) is determined as follows:

#### 3.2. Fuzzy C-Means

^{th}data point, and ${c}_{j}$ is the center of the j

^{th}cluster. ${\mu}_{ij}$ is the membership degree of ${x}_{i}$ with respect to the j

^{th}cluster, and $m$ represents the fuzziness parameter. $\Vert .\Vert $ denotes the Euclidean distance.

#### 3.3. Multiclass FCM-ANFIS

^{th}FCM-ANFIS examined the possibility of assigning the class k to the inputs, and the target was considered to be 1 for the class k and zero for the rest of the classes. The final class is the one whose FCM-ANFIS has the largest output:

## 4. Analysis of a Simulated Bearing Signal

_{2}, GMDispEn

_{3}, RCMDispEn, RCGMDispEn

_{2}, and RCGMDispEn

_{3,}were calculated for the simulated signals, with the results displayed in Figure 3. In this figure, p-values smaller than 0.05 are identified with asterisks. According to Figure 3, RCMDispEn, RCGMDispEn

_{2}, and RCGMDispEn

_{3}possess higher fault distinguishing capability than MDispEn, GMDispEn

_{2}, and GMDispEn

_{3}, respectively, and their results have a smaller standard deviation. Distinguishing abilities of the bearing faults using the generalized methods are also displayed.

_{2}, GMDispEn

_{3}, RCGMDispEn

_{2}, and RCGMDispEn

_{3}algorithms effectively show the differences between the healthy and the faulty conditions, similar to MDispEn and RCMDispEn. RCMDispEn, RCGMDispEn

_{2}, and RCGMDispEn

_{3}have larger size effects and better fault separation capability than MDispEn, GMDispEn

_{2}, and GMDispEn

_{3}, respectively.

## 5. Analysis of the Experiments

#### 5.1. Analysis of the Vibration Signals Acquired from the Case Western Reserve University Dataset

_{2}, GMDispEn

_{3}, RCMDispEn, RCGMDispEn

_{2}, and RCGMDispEn

_{3}were calculated for all the signals, and their values were used in 20 scales as features for fault detection and classification. A binary vector was used as the target vector for every bearing condition. This binary vector had a length of 10 because 10 conditions were being studied. This research employed 10 FCM-ANFIS, each of which detected one element in the target vector.

_{2}, and RCGMDispEn

_{3}performed better at classification than MDispEn, GMDispEn

_{2}, and GMDispEn

_{3}, respectively. Moreover, the simultaneous use of RCMDispEn, RCGMDispEn

_{2}, and RCGMDispEn

_{3}as the classifier inputs produced the most accurate classification. Table 4 represents the confusion matrix of the best performance using these inputs.

#### 5.2. Analysis of the Signals Acquired from the PHMAP 2021 Data Challenge Dataset

_{2}, RCGMDispEn

_{2}, GMDispEn

_{3}, and RCGMDispEn

_{3}were calculated for all the signals, and their values were used in 20 scales as features for fault detection and classification. For each condition, 120, 30, and 150 samples were used for training, validation, and testing, respectively. These data were classified 20 times using multiclass FCM-ANFIS. The results are displayed in Figure 5 and Table 5. As can be seen, the highest accuracy was achieved by the combined use of RCMDispEn, RCGMDispEn

_{2}, and RCGMDispEn

_{3}as inputs. However, the mean accuracy of RCMDispEn and RCGMDispEn

_{2}as simultaneous inputs was greater than that of other inputs. These results confirm the proposal of this paper regarding the use of generalized multiscale entropies with multiscale entropies to improve the results. The best classification results are displayed in Table 6.

#### 5.3. Analysis of Vibration Signals Acquired from the Paderborn University Dataset

_{2}, RCGMDispEn

_{2}, GMDispEn

_{3}, and RCGMDispEn

_{3}were calculated for all the signals, and their values were used in 20 scales as features for fault detection and classification. For each condition, 120, 30, and 150 samples were used for training, validation, and testing, respectively. These data were classified 20 times using multiclass FCM-ANFIS. The results, displayed in Figure 6 and Table 9, confirm the suggestion made by the present study. Specifically, the highest classification accuracy corresponds to the features extracted by the combination of RCMDispEn, RCGMDispEn

_{2}, and RCGMDispEn

_{3}. Moreover, the smallest classification accuracy corresponds to the features extracted by RCMDispEn, RCGMDispEn

_{2}, and RCGMDispEn, separately.

## 6. Conclusions

_{3}and RCGMDispEn

_{2}are more capable in separating bearing fault conditions compared to GMDispEn

_{3}and GMDispEn

_{2}. Moreover, the combined use of RCGMDE, RCGMDE

_{2}, and RCGMDE

_{3}produces better results than using one or two of these approaches in bearing fault diagnosis. The authors suggest investigating the potential of simultaneously using generalized multiscale and multiscale algorithms in other fields.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Simulated signals corresponding to the healthy and faulty bearings (

**a**) without noise; (

**b**) with noise.

**Figure 3.**Comparison of 50 independent simulated signals corresponding to healthy and faulty bearings at 20 scales. (

**a**) MDispEn; (

**b**) RCMDispEn; (

**c**) GMDispEn

_{2}; (

**d**) RCGMDispEn

_{2}; (

**e**) GMDispEn

_{3}; (

**f**) RCGMDispEn

_{3}.

**Figure 4.**Classification accuracies of ball bearing fault diagnosis using ten different methods from the CWRU dataset.

**Figure 5.**Results of classifying fault conditions: (1) high looseness of V-belt, (2) faulty bearing, and (3) fault-free condition using multiclass FCM-ANFIS with different inputs.

**Figure 6.**Results of classification of bearing fault signals using multiclass FCM-ANFIS with different inputs.

**Table 1.**Hedges’ g effect size of MDispEn, RCMDispEn, GMDispEn

_{2}, RCGMDispEn

_{2}, GMDispEn

_{3}, and RCGMDispEn

_{3}in 20 scales on 50 independent healthy and faulty bearing signals.

Scale | Methods | |||||
---|---|---|---|---|---|---|

MDispEn | RCMDispEn | GMDispEn_{2} | RCGMDispEn_{2} | GMDispEn_{3} | RCGMDispEn_{3} | |

1 | 0.6493 | 0.6493 | - | - | - | - |

2 | 0.2235 | 0.4915 | 2.2364 | 2.2569 | - | - |

3 | 1.1242 | 1.7345 | 5.0136 | 5.5878 | 3.2119 | 6.9364 |

4 | 1.1183 | 2.4674 | 10.8933 | 10.7749 | 7.9872 | 14.1039 |

5 | 1.0230 | 2.9302 | 16.6406 | 17.7200 | 6.6340 | 12.5567 |

6 | 1.0875 | 2.0787 | 15.7878 | 17.2561 | 4.6298 | 12.3608 |

7 | 0.9397 | 2.0842 | 11.0137 | 14.0667 | 5.9228 | 9.5684 |

8 | 1.0843 | 1.4531 | 1.8651 | 12.4676 | 4.3129 | 8.9234 |

9 | 0.2716 | 0.7559 | 10.4116 | 12.0355 | 3.8295 | 8.1357 |

10 | 0.0886 | 0.2386 | 9.0303 | 11.0007 | 3.4057 | 6.2796 |

11 | 0.3981 | 0.4373 | 8.7524 | 10.5785 | 2.7631 | 4.1878 |

12 | 0.6708 | 0.9714 | 7.4537 | 10.0266 | 2.0409 | 2.8769 |

13 | 1.4599 | 1.6187 | 7.3757 | 9.5024 | 1.8816 | 2.0764 |

14 | 0.9879 | 2.0674 | 6.8795 | 8.7588 | 0.5327 | 1.2610 |

15 | 1.9579 | 2.7560 | 6.7228 | 8.8263 | 0.3114 | 0.3482 |

16 | 1.7376 | 3.1014 | 6.3236 | 8.5332 | 0.0566 | 0.2229 |

17 | 3.2926 | 3.3136 | 6.1438 | 7.9683 | 0.3684 | 0.9012 |

18 | 1.9379 | 3.6472 | 4.9962 | 7.4363 | 1.1923 | 1.5027 |

19 | 3.2548 | 3.7864 | 4.4918 | 6.8664 | 1.8531 | 2.0079 |

20 | 2.7194 | 4.0228 | 4.2644 | 6.0740 | 0.7243 | 2.4493 |

Bearing Condition | Defect Size (mm) | Label of Classification |
---|---|---|

Normal | 0 | 1 |

Rolling element Fault | 0.1778 | 2 |

Rolling element Fault | 0.3556 | 3 |

Rolling element Fault | 0.5334 | 4 |

Inner race Fault | 0.1778 | 5 |

Inner race Fault | 0.3556 | 6 |

Inner race Fault | 0.5334 | 7 |

Outer race Fault | 0.1778 | 8 |

Outer race Fault | 0.5334 | 9 |

**Table 3.**The classification results of ball bearing faults using ten different inputs from the CWRU dataset.

Accuracy (%) | |||
---|---|---|---|

Features | Min | Mean | Max |

MDispEn | 88.7654 | 90.5679 | 91.7284 |

GMDispEn_{2} | 89.1358 | 90.1728 | 91.2346 |

GMDispEn_{3} | 91.8519 | 93.1605 | 94.3210 |

RCMDispEn | 93.9506 | 95.1420 | 96.0494 |

RCGMDispEn_{2} | 91.6049 | 92.3457 | 93.4568 |

RCGMDispEn_{3} | 95.6790 | 96.4198 | 97.0370 |

RCMDispEn+ RCGMDispEn_{2} | 97.2840 | 98.1790 | 99.0123 |

RCMDispEn+ RCGMDispEn_{3} | 98.0247 | 98.2531 | 98.2716 |

RCGMDispEn_{2}+ RCGMDispEn_{3} | 94.3210 | 95.8765 | 97.2840 |

RCMDispEn+RCGMDispEn_{2}+ RCGMDispEn_{3} | 99.0123 | 99.1235 | 99.1358 |

**Table 4.**Confusion matrix of the testing set of the multiclass FCM-ANFIS using RCMDispEn, RCMDispEn, and RCMDispEn as the input.

True Label | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Sensitivity | ||

Predicted Label | 1 | 90 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 |

2 | 0 | 90 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 | |

3 | 0 | 0 | 88 | 1 | 0 | 0 | 0 | 0 | 0 | 98.87 | |

4 | 0 | 0 | 0 | 87 | 0 | 2 | 0 | 0 | 0 | 97.75 | |

5 | 0 | 0 | 0 | 0 | 90 | 0 | 0 | 0 | 0 | 100 | |

6 | 0 | 0 | 0 | 1 | 0 | 88 | 0 | 0 | 0 | 98.88 | |

7 | 0 | 0 | 0 | 1 | 0 | 0 | 90 | 0 | 0 | 98.90 | |

8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 90 | 0 | 100 | |

9 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 90 | 97.83 | |

Precision | 100 | 100 | 97.78 | 96.67 | 100 | 97.78 | 100 | 100 | 100 | AC * = 99.13 |

**Table 5.**Results of classifying fault conditions: (1) high looseness of V-belt, (2) faulty bearing, and (3) fault-free condition using multiclass FCM-ANFIS with different inputs.

Accuracy (%) | |||
---|---|---|---|

Features | Min | Mean | Max |

RCMDispEn | 93.3333 | 94.4667 | 96.8889 |

RCMDispEn + RCGMDispEn_{2} | 96.6667 | 97.3222 | 97.7778 |

RCMDispEn + RCGMDispEn_{3} | 92.2222 | 93.3778 | 97.1111 |

RCMDispEn + RCGMDispEn_{2} + RCGMDispEn_{3} | 95.5556 | 96.4889 | 98.8889 |

**Table 6.**Most accurate classification of three fault conditions: (1) high looseness of V-belt, (2) faulty bearing, and (3) fault-free condition using RCMDispEn, RCMDispEn

_{2}, and RCMDispEn

_{3}as inputs.

True Condition | |||||
---|---|---|---|---|---|

Belt Looseness High | Bearing Fault | Normal | Sensitivity (%) | ||

Predicted condition | Belt Looseness High | 148 | 0 | 0 | 100 |

Bearing fault | 0 | 150 | 3 | 98.04 | |

Normal | 2 | 0 | 147 | 98.66 | |

Precision (%) | 98.67 | 100 | 98 | AC * = 98.89 |

No. | Rotational Speed [rpm] | Load Torque [Nm] | Radial Force [N] |
---|---|---|---|

1 | 1500 | 0.7 | 1000 |

2 | 1500 | 0.1 | 1000 |

3 | 1500 | 0.7 | 400 |

Type of Bearing | |||
---|---|---|---|

Healthy | Outer Ring Damage | Inner Ring Damage | |

Bearing Code | KI04 | KA04 | K001 |

KI14 | KA15 | K002 | |

KI16 | KA16 | K003 | |

KI18 | KA22 | K004 | |

KI21 | KA30 | K005 |

**Table 9.**Classification results of bearing fault conditions: (1) inner race damage, (2) outer race damage, and (3) healthy.

Accuracy (%) | |||
---|---|---|---|

Features | Min | Mean | Max |

RCMDispEn | 97.5556 | 98.21111 | 98.6667 |

RCGMDispEn_{2} | 90.6667 | 91.34445 | 91.7778 |

RCGMDispEn_{3} | 86.6667 | 89.62222 | 92.2222 |

RCMDispEn + RCGMDispEn_{2} | 98.4444 | 99.07778 | 100 |

RCMDispEn + RCGMDispEn_{3} | 97.3333 | 97.6000 | 98.6667 |

RCMDispEn + RCGMDispEn_{2} + RCGMDispEn_{3} | 98.8889 | 99.27778 | 100 |

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**MDPI and ACS Style**

Rostaghi, M.; Khatibi, M.M.; Ashory, M.R.; Azami, H.
Bearing Fault Diagnosis Using Refined Composite Generalized Multiscale Dispersion Entropy-Based Skewness and Variance and Multiclass FCM-ANFIS. *Entropy* **2021**, *23*, 1510.
https://doi.org/10.3390/e23111510

**AMA Style**

Rostaghi M, Khatibi MM, Ashory MR, Azami H.
Bearing Fault Diagnosis Using Refined Composite Generalized Multiscale Dispersion Entropy-Based Skewness and Variance and Multiclass FCM-ANFIS. *Entropy*. 2021; 23(11):1510.
https://doi.org/10.3390/e23111510

**Chicago/Turabian Style**

Rostaghi, Mostafa, Mohammad Mahdi Khatibi, Mohammad Reza Ashory, and Hamed Azami.
2021. "Bearing Fault Diagnosis Using Refined Composite Generalized Multiscale Dispersion Entropy-Based Skewness and Variance and Multiclass FCM-ANFIS" *Entropy* 23, no. 11: 1510.
https://doi.org/10.3390/e23111510