# Fractional Order Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis

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## Abstract

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## 1. Introduction

## 2. Fractional Order Fuzzy Dispersion Entropy

#### 2.1. ${\mathit{FuzzDE}}_{\mathsf{\alpha}}$

#### 2.2. Parameter Selection

## 3. Experiments on Simulated Signals

#### 3.1. Noise Signals Experiment

#### 3.2. Chirp Signal Experiment

#### 3.3. MIX Signal Experiment

## 4. Experiments on Bearing Fault Diagnosis

#### 4.1. Analysis of Experiment Data

#### 4.2. Single Feature Extraction and Classification

#### 4.3. Double Features Extraction and Classification

#### 4.4. Triple Features Extraction and Classification

## 5. Conclusions

- Fractional order calculation is introduced on the basis of fuzzy dispersion entropy (FuzzDE), and a new entropy called fractional order FDE (${\mathrm{FuzzDE}}_{\alpha}$) is proposed. Simulated experiments have shown that compared with $\mathrm{FuzzDE}$, ${\mathrm{FuzzDE}}_{\alpha}$ can provide more features of greater sensitivity to changes in the dynamics of the time series.
- ${\mathrm{FuzzDE}}_{\alpha}$ is combined with ${\mathrm{DE}}_{\alpha}$, ${\mathrm{PE}}_{\alpha}$ as well as ${\mathrm{FDE}}_{\alpha}$ to present a mixed features extraction method. For ten classes of bearing signals, the proposed mixed features fault diagnosis method achieves 100% recognition rate at only triple features.
- Regardless of how many features are selected, the ${\mathrm{FuzzDE}}_{\alpha}$ proposed in this paper is the most effective in fault diagnosis compared to the other three fractional order entropies, where ${\mathrm{FuzzDE}}_{\alpha =-0.1}$ appears a total of 11 times in the combination of the triple features with the recognition rate of 100%

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

FuzzDE | Fuzzy Dispersion Entropy |

${\mathrm{FuzzDE}}_{\alpha}$ | Fractional order fuzzy dispersion entropy |

PE | Permutation entropy |

${\mathrm{PE}}_{\alpha}$ | Fractional order permutation entropy |

DE | Dispersion entropy |

${\mathrm{DE}}_{\alpha}$ | Fractional order dispersion entropy |

FDE | Fluctuation-based dispersion entropy |

${\mathrm{FDE}}_{\alpha}$ | Fractional order fluctuation-based dispersion entropy |

NCDF | Normal cumulative distribution function |

SE | Sample entropy |

FRDE | Fluctuation-based reverse dispersion entropy |

RDE | Reverse dispersion entropy |

FuzzEn | Fuzzy entropy |

RCMDE | Refined composite multiscale dispersion entropy |

${\mathrm{GRCMFDE}}_{\alpha}$ | Generalized refined composite multiscale fluctuation-based fractional dispersion entropy |

LM | Linear mapping |

TANSIG | Tangent sigmoid |

LOGSIG | Logarithm sigmoid |

SORT | Sorting method |

## References

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**Figure 2.**Means and standard deviations of different class number $c$ at different fractional orders. (

**a**) $c$ = 2; (

**b**) $c$ = 3; (

**c**) $c$ = 4; (

**d**) $c$ = 5.

**Figure 3.**Means and standard deviations of different embedding dimensions $m$ at different fractional orders. (

**a**) $m$ = 3; (

**b**) $m$ = 4; (

**c**) $m$ = 5; (

**d**) $m$ = 6.

**Figure 4.**Means and standard deviations of different mapping approaches at different fractional orders. (

**a**) LM; (

**b**) NCDF; (

**c**) LOGISG; (

**d**) TANSIG; (

**e**) SORT.

**Figure 5.**Means and standard deviations of different fractional order entropies under noise signals.

**Figure 10.**Distribution of fractional entropy features of different classes of bearing signals. (

**a**) $\alpha $ = −0.2; (

**b**) $\alpha $ = −0.1; (

**c**) $\alpha $ = 0; (

**d**) $\alpha $ = 0.1; (

**e**) $\alpha $ = 0.2.

**Figure 11.**Distribution of the highest classification recognition rate of mixed double features. (

**a**) ${\mathrm{FuzzDE}}_{\alpha =0.1}$. & ${\mathrm{FDE}}_{\alpha =0.1}$; (

**b**)${\mathrm{FuzzDE}}_{\alpha =-0.1}$ & ${\mathrm{FDE}}_{\alpha =0.1}$; (

**c**) ${\mathrm{FuzzDE}}_{\alpha =0.1}$ & ${\mathrm{FDE}}_{\alpha =-0.1}$.

**Figure 12.**Distribution of the mixed triple features at 100% recognition rate (${\mathrm{FuzzDE}}_{\alpha =-0.1}$ , ${\mathrm{PE}}_{\alpha =-0.2}$, ${\mathrm{FDE}}_{\alpha =0.1}$).

Class | Label | Fault Size (mm) | Selected Data |
---|---|---|---|

Normal | NORM | 0 | 100_normal_3 |

Inner race fault | IR1 | 0.1778 | 108_IR007_3 |

Balling element fault | BE1 | 0.1778 | 121_B007_3 |

Outer race fault | OR1 | 0.1778 | 133_OR007@6_3 |

Inner race fault | IR2 | 0.3556 | 172_IR014_3 |

Balling element fault | BE2 | 0.3556 | 188_B014_3 |

Outer race fault | OR2 | 0.3556 | 200_OR014@6_3 |

Inner race fault | IR3 | 0.5334 | 212_IR021_3 |

Balling element fault | BE3 | 0.5334 | 225_B021_3 |

Outer race fault | OR3 | 0.5334 | 237_OR021@6_3 |

Entropy | Recognition Rates (%) | ||||
---|---|---|---|---|---|

$\mathbf{\alpha}=-0.2$ | $\mathbf{\alpha}=-0.1$ | $\mathbf{\alpha}=0$ | $\mathbf{\alpha}=0.1$ | $\mathbf{\alpha}=0.2$ | |

${\mathrm{FuzzDE}}_{\alpha}$ | 82.8 | 81.6 | 74 | 68.4 | 67.6 |

${\mathrm{DE}}_{\alpha}$ | 76.4 | 79.6 | 76.4 | 71.2 | 66.0 |

${\mathrm{PE}}_{\alpha}$ | 59.6 | 56.8 | 58.4 | 56.8 | 54.0 |

${\mathrm{FDE}}_{\alpha}$ | 79.2 | 82.8 | 78.4 | 77.6 | 80.4 |

**Table 3.**Highest classification recognition rates for each feature extraction method (double features).

Feature Extraction Methods | Combinations | Recognition Rate (%) |
---|---|---|

${\mathrm{FuzzDE}}_{\alpha}$-based | ${\mathrm{FuzzDE}}_{\alpha =0}{\mathrm{FuzzDE}}_{\alpha =0.2}$ | 91.6 |

${\mathrm{DE}}_{\alpha}$-based | ${\mathrm{DE}}_{\alpha =0}{\mathrm{DE}}_{\alpha =0.2}$ | 88.4 |

${\mathrm{PE}}_{\alpha}$-based | ${\mathrm{PE}}_{\alpha =-0.2}{\mathrm{PE}}_{\alpha =0.1}$ | 58.4 |

${\mathrm{FDE}}_{\alpha}$-based | ${\mathrm{FDE}}_{\alpha =-0.1}{\mathrm{FDE}}_{\alpha =0}$ | 90.0 |

Proposed method | ${\mathrm{FuzzDE}}_{\alpha =0.1}$& ${\mathrm{FDE}}_{\alpha =0.1}$ (1 of 3) | 99.6 |

**Table 4.**Highest classification recognition rates for each feature extraction method (triple features).

Feature Extraction Methods | Combinations | Recognition Rate (%) |
---|---|---|

${\mathrm{FuzzDE}}_{\alpha}$-based | ${\mathrm{FuzzDE}}_{\alpha =-0.1}{\mathrm{FuzzDE}}_{\alpha =0}{\mathrm{FuzzDE}}_{\alpha =0.2}$ | 92 |

${\mathrm{DE}}_{\alpha}$-based | ${\mathrm{DE}}_{\alpha =-0.1}{\mathrm{DE}}_{\alpha =0}{\mathrm{DE}}_{\alpha =0.2}$ | 92 |

${\mathrm{PE}}_{\alpha}$-based | ${\mathrm{PE}}_{\alpha =-0.2}{\mathrm{PE}}_{\alpha =0}{\mathrm{PE}}_{\alpha =0.1}$ | 58 |

${\mathrm{FDE}}_{\alpha}$-based | ${\mathrm{FDE}}_{\alpha =-0.2}{\mathrm{FDE}}_{\alpha =-0.1}{\mathrm{FDE}}_{\alpha =0}$ | 91.6 |

Proposed method | ${\mathrm{FuzzDE}}_{\alpha =-0.1}{\mathrm{PE}}_{\alpha =-0.2}{\mathrm{FuzzDE}}_{\alpha =0.1}$ (1 of 15) | 100 |

**Table 5.**Number of occurrences of each feature in the combination of the mixed triple features with 100% recognition rate.

Feature | Appear Times |
---|---|

${\mathrm{FuzzDE}}_{\alpha =-0.1}$ | $11$ |

${\mathrm{FuzzDE}}_{\alpha =-0.2}$ | $4$ |

${\mathrm{PE}}_{\alpha =-0.2}$ | $2$ |

${\mathrm{PE}}_{\alpha =-0.1}$ | $3$ |

${\mathrm{PE}}_{\alpha =0}$ | $3$ |

${\mathrm{PE}}_{\alpha =0.1}$ | $4$ |

${\mathrm{PE}}_{\alpha =0.2}$ | $3$ |

${\mathrm{FDE}}_{\alpha =-0.1}$ | $6$ |

${\mathrm{FDE}}_{\alpha =0}$ | $5$ |

${\mathrm{FDE}}_{\alpha =0.1}$ | $4$ |

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**MDPI and ACS Style**

Li, Y.; Tang, B.; Geng, B.; Jiao, S.
Fractional Order Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis. *Fractal Fract.* **2022**, *6*, 544.
https://doi.org/10.3390/fractalfract6100544

**AMA Style**

Li Y, Tang B, Geng B, Jiao S.
Fractional Order Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis. *Fractal and Fractional*. 2022; 6(10):544.
https://doi.org/10.3390/fractalfract6100544

**Chicago/Turabian Style**

Li, Yuxing, Bingzhao Tang, Bo Geng, and Shangbin Jiao.
2022. "Fractional Order Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis" *Fractal and Fractional* 6, no. 10: 544.
https://doi.org/10.3390/fractalfract6100544