# Particle Swarm Optimization Fractional Slope Entropy: A New Time Series Complexity Indicator for Bearing Fault Diagnosis

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## Abstract

**:**

## 1. Introduction

## 2. Algorithms

#### 2.1. Slope Entropy Algorithm

**Step****1:**- set an embedding dimension m, which can divide the time series into $k=n-m+1$ subsequences, where m is greater than two and much less than $n$. The disintegrate form is as follows:

**Step****2:**- subtract the latter of the two adjacent elements in all the subsequences obtained in Step 1 from the former to obtain k new sequences. The new form is as follows:

**Step****3:**- lead into the two threshold parameters $\eta $ and $\epsilon $ of SlEn, where $0<\epsilon <\eta $, and compare all elements in the sequences obtained from Step 2 with the positive and negative values of these two threshold parameters. The positive and negative values of these two threshold parameters $-\eta ,-\epsilon ,\epsilon $ and $\eta $ serve as the dividing lines, they divide the number field into five modules $-2,-1,0,1$, and $2$. If ${t}_{k}<-\mathsf{\eta}$, the module is $-2$; if $-\mathsf{\eta}<{t}_{k}<-\mathsf{\epsilon}$, the module is $-1$; if $-\mathsf{\epsilon}<{t}_{k}<\mathsf{\epsilon}$, the module is $0$; if $\mathsf{\epsilon}<{t}_{k}<\mathsf{\eta}$, the module is $1$; if $-\mathsf{\epsilon}<{t}_{k}<\mathsf{\epsilon}$, the module is $0$; if ${t}_{k}>\mathsf{\eta}$, the module is $2$. The intuitive module division principle is shown by the coordinate axis in Figure 1 below:

**Step****4:**- the number of modules is 5, so all types of the sequences ${E}_{k}$ are counted as $j={5}^{m-1}$. Such as when m is 3, there will be at most 25 types of ${E}_{k}$, which are $\left\{-2,-2\right\}$, $\left\{-2,-1\right\}$, …, $\left\{0,0\right\}$, $\left\{0,1\right\}$, …, $\left\{2,1\right\}$, $\left\{2,2\right\}$. The number of each type records as ${r}_{1},{r}_{2},\dots ,{r}_{j}$, and the frequency of each type is calculated as follows:

**Step****5:**- based on the classical Shannon entropy, the formula of SlEn is defined as follows:

#### 2.2. Fractional Slope Entropy Algorithm

**Step****1:**- Shannon entropy is the first entropy to consider fractional calculus, and its generalized expression is as follows:

**Step****2:**- extract the fractional order information of order $\alpha $ from Equation (6):

**Step****3:**- combine the fractional order with SlEn, which is to replace $-\mathrm{ln}{R}_{j}$ with Equation (7). Therefore, the formula of FrSlEn is defined as follows:

#### 2.3. Particle Swarm Optimization and Algorithm Process

## 3. Proposed Feature Extraction Methods

**Step****1:**- the 10 kinds of bearing signals are normalized, which can make the signals neat and regular, the threshold parameters $\eta $ and $\epsilon $ less than 1, where $\epsilon $ is less than 0.2 in most cases.
**Step****2:**- the five kinds of single features of these 10 kinds of normalized bearing signals are extracted separately under seven different fractional orders.
**Step****3:**- the distribution of the features is obtained and the hybrid degrees between the feature points are observed.
**Step****4:**- these features are classified into one of the 10 bearing signals by K-Nearest Neighbor (KNN).
**Step****5:**- the classification accuracies of the features are calculated.

## 4. Single Feature Extraction

#### 4.1. Bearing Signals

#### 4.2. Feature Distribution

#### 4.3. Classification Effect Verification

## 5. Double Feature Extraction

#### 5.1. Feature Distribution

#### 5.2. Classification Effect Verification

## 6. Conclusions

- (1)
- As an algorithm proposed in 2019, SlEn has not been proposed any improved algorithm. It is proposed for the first time to combine the concept of fractional information with SlEn, and get an improved algorithm of SlEn named FrSlEn.
- (2)
- In order to solve the influence of the two threshold parameters of SlEn on feature significance, PSO is selected to optimize the two threshold parameters, which assists FrSlEn to make the extracted features more significant.
- (3)
- In the experiment of single feature extraction, under any values of α, the classification accuracies of PSO-FrSlEn are the highest. The classification accuracies of PSO-FrSlEn are higher than that of PSO-SlEn, where 88% is the highest classification accuracy of PSO-FrSlEn under α = −0.3. The highest classification accuracy of PSO-FrSlEn is at least 5.33% higher than FrPE, FrWPE, FrDE, and FrFDE.
- (4)
- In the experiment of double feature extraction, the classification accuracies of PSO-FrSlEn under five double feature combinations are 100%. The highest classification accuracies of FrPE, FrWPE, FrDE, and FrFDE are at least 4% less than PSO-FrSlEn, where the highest classification accuracy of FrWPE is 27.33% less than PSO-FrSlEn.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

PE | Permutation entropy |

WPE | Weighted permutation entropy |

DE | Dispersion entropy |

FDE | Fluctuation dispersion entropy |

SlEn | Slope entropy |

PSO-SlEn | Particle swarm optimization slope entropy |

FrPE | Fractional permutation entropy |

FrWPE | Fractional weighted permutation entropy |

FrDE | Fractional dispersion entropy |

FrFDE | Fractional fluctuation dispersion entropy |

FrSlEn | Fractional slope entropy |

PSO-FrSlEn | Particle swarm optimization fractional slope entropy |

$\alpha $ | Fractional order |

$m$ | Embedding dimension |

$\tau $ | Time lag |

$c$ | Number of classes |

NCDF | Normal cumulative distribution function |

$\eta $ | Large threshold |

$\epsilon $ | Small threshold |

N-100 | Normal signals |

IR-108 | Inner race fault signals (fault diameter size: 0.007 inch) |

B-121 | Ball fault signals (fault diameter size: 0.007 inch) |

OR-133 | Outer race fault signals (fault diameter size: 0.007 inch) |

IR-172 | Inner race fault signals (fault diameter size: 0.014 inch) |

B-188 | Ball fault signals (fault diameter size: 0.014 inch) |

OR-200 | Outer race fault signals (fault diameter size: 0.014 inch) |

IR-212 | Inner race fault signals (fault diameter size: 0.021 inch) |

B-225 | Ball fault signals (fault diameter size: 0.021 inch) |

OR-237 | Outer race fault signals (fault diameter size: 0.021 inch) |

KNN | K-Nearest Neighbor |

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**Figure 4.**The normalized 10 bearing signals: (

**a**) N-100; (

**b**) IR-108; (

**c**) B-121; (

**d**) OR-133; (

**e**) IR-172; (

**f**) B-188; (

**g**) OR-200; (

**h**) IR-212; (

**i**) B-225; (

**j**) OR-237.

**Figure 5.**Single feature distribution of PSO-FrSlEn: (

**a**) $\alpha =-0.3$; (

**b**) $\alpha =0.3$; (

**c**) $\alpha =-0.2$; (

**d**) $\alpha =0.2$; (

**e**) $\alpha =-0.1$; (

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

**g**) $\alpha =0$.

**Figure 6.**Classification results and distribution of PSO-FrSlEn: (

**a**) $\alpha =-0.3$; (

**b**) $\alpha =0.3$; (

**c**) $\alpha =-0.2$; (

**d**) $\alpha =0.2$; (

**e**) $\alpha =-0.1$; (

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

**g**) $\alpha =0$.

**Figure 8.**Double feature distribution of the nine highest classification accuracies: (

**a**) FrPE, $\alpha =-0.3\&0$; (

**b**) FrWPE,$\alpha =-0.30.3$; (

**c**) FrDE, $\alpha =0.1\&0.3$; (

**d**) FrFDE,$\alpha =-0.3-0.1$; (

**e**) PSO-FrSlEn, $\alpha =-0.3\&0.2$; (

**f**) PSO-FrSlEn, $\alpha =-0.2\&0.2$; (

**g**) PSO-FrSlEn, $\alpha =-0.1\&0.1$; (

**h**) PSO-FrSlEn, $\alpha =-0.1\&0.2$; (

**i**) PSO-FrSlEn,$\alpha =00.1$.

Figure | FrPE Accuracy (%) | FrWPE Accuracy (%) | FrDE Accuracy (%) | FrFDE Accuracy (%) | PSO-FrSlEn Accuracy (%) |
---|---|---|---|---|---|

−0.3 | 64.67 | 46.67 | 82.67 | 77.33 | 88 |

−0.2 | 78.67 | 60.67 | 81.33 | 73.33 | 84 |

−0.1 | 76.67 | 66.67 | 80.67 | 79.33 | 83.33 |

0 | 76.67 | 69.33 | 69.33 | 79.33 | 81.33 |

0.1 | 75.33 | 69.33 | 80 | 79.33 | 86 |

0.2 | 75.33 | 69.33 | 82 | 80.67 | 85.33 |

0.3 | 66 | 72.76 | 82.67 | 80 | 83.33 |

Entropy | Fractional Order Combinations | Accuracy (%) |
---|---|---|

FrPE | $-0.3\&0$ | 78.67 |

FrWPE | $-0.3\&0.3$ | 72.67 |

FrDE | $0.1\&0.3$ | 96 |

FrFDE | $-0.3\&-0.1$ | 89.33 |

PSO-FrSlEn | $-0.3\&0.2$ | 100 |

$-0.2\&0.2$ | 100 | |

$-0.1\&0.1$ | 100 | |

$-0.1\&0.2$ | 100 | |

$0\&0.1$ | 100 |

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**MDPI and ACS Style**

Li, Y.; Mu, L.; Gao, P.
Particle Swarm Optimization Fractional Slope Entropy: A New Time Series Complexity Indicator for Bearing Fault Diagnosis. *Fractal Fract.* **2022**, *6*, 345.
https://doi.org/10.3390/fractalfract6070345

**AMA Style**

Li Y, Mu L, Gao P.
Particle Swarm Optimization Fractional Slope Entropy: A New Time Series Complexity Indicator for Bearing Fault Diagnosis. *Fractal and Fractional*. 2022; 6(7):345.
https://doi.org/10.3390/fractalfract6070345

**Chicago/Turabian Style**

Li, Yuxing, Lingxia Mu, and Peiyuan Gao.
2022. "Particle Swarm Optimization Fractional Slope Entropy: A New Time Series Complexity Indicator for Bearing Fault Diagnosis" *Fractal and Fractional* 6, no. 7: 345.
https://doi.org/10.3390/fractalfract6070345