# Fault Diagnosis of Rolling Bearing Based on Shift Invariant Sparse Feature and Optimized Support Vector Machine

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## Abstract

**:**

## 1. Introduction

## 2. Feature Extraction Using Shift Invariant K-SVD Algorithm

#### 2.1. Shift Invariant K-SVD Algorithm

#### 2.2. Shift Invariant Sparse Feature

_{1}norm, l

_{2}norm or maximum absolute value ${F}_{i}(i=1,2,\dots ,LK)$ of the sparse coefficient vector ${s}_{i}$ corresponding to the sub-dictionary ${D}_{i}$with p − q + 1 dictionary atoms is computed and thus LK-dimensional sparse feature $F=[{F}_{1},{F}_{2},\dots ,{F}_{LK}]$ can be obtained for each signal. Moreover, M(M ≥ 2) maximum absolute values ${F}_{i}(i=1,2,\dots ,LKM)$ of the sparse coefficient vector ${s}_{i}$ are also computed, which is denoted as M-Max and thus LKM-dimensional sparse feature $F=[{F}_{1},{F}_{2},\dots ,{F}_{LKM}]$ can be obtained for each signal. The LK-dimensional or LKM-dimensional sparse feature is named shift invariant sparse feature.

_{j}is $j(j=1,2,\dots ,L)$, the sub-dictionaries ${D}_{i}(i=K(j-1)+1,K(j-1)+2,\dots ,Kj)$ corresponding to the class j are more likely to be activated, i.e., solving the sparse coefficient using the whole over-complete dictionary D and then the non-zero terms in the sparse coefficients corresponding to the sub-dictionaries D

_{i}are most likely to appear in ${s}_{i}(i=K(j-1)+1,K(j-1)+2,\dots ,Kj)$, thus the l

_{1}norm, l

_{2}norm or M(M ≥ 1) maximum absolute values of the sparse coefficient vector ${s}_{i}$ corresponding to the sub-dictionaries D

_{i}are larger than the other sub-dictionaries. Therefore, the shift invariant sparse feature corresponding to different classes is distinguishable and can be employed as the input of classifier.

## 3. Classification with Optimized SVM

#### 3.1. Grid Search

#### 3.2. Genetic Algorithm

#### 3.3. Particle Swarm Optimization

_{i}. In the iteration process, the best value of the ith particle that indicates the local best is represented by pbest

_{i}, while the best particle that indicates the global best is represented by gbest. Firstly, the initialization of the particles is implemented by a random number in the specified range. For the kth iteration, the ith particle and its velocity are renewed as follows [35]:

_{1}and c

_{2}are acceleration coefficients, which represents the local and global search ability, respectively. r

_{1}and r

_{2}are random numbers uniformly distributed in [0, 1].

## 4. Bearing Fault Diagnosis Method Using Shift Invariant Sparse Feature and Optimized SVM

## 5. Experiment and Analysis

#### 5.1. Description of the Experiment

#### 5.2. Feature Extraction with Shift Invariant Sparse Feature

_{1}norm, l

_{2}norm or M-Max. Hence, 16-dimensional (l

_{1}norm, l

_{2}norm and Max), 32-dimensional (M = 2), or 48-dimensional (M = 3) feature vector is respectively acquired with regard to each sample.

_{1}norm, the shift invariant sparse feature can be obtained. The test sample randomly selected for four different states and the sum of shift invariant sparse feature of test samples from the same state are demonstrated in Figure 6 and Figure 7, respectively, where the sub-dictionary no. 1~4, 5~8, 9~12, 13~16 denotes normal, the fault of inner race, rolling element and outer race, respectively.

#### 5.3. Fault Diagnosis Using Shift Invariant Sparse Feature

#### 5.3.1. Diagnosis Result with Standard SVM

_{1}norm and l

_{2}norm, which are denoted as Max, 2-Max, 3-Max, L1, and L2, respectively, whose classification results are demonstrated in Table 1. Figure 8 respectively describes the detailed classification results corresponding to four classes.

_{1}norm) achieves the highest accuracy and thus l

_{1}norm is utilized in the subsequent classification task using optimized SVM. The accuracy of the feature extraction method based on Max (Maximum absolute values) is the lowest, which is due to that the Max method ignores a lot of important sparse feature information in the sparse codes. However, with the increase of M, the accuracy is improved. Figure 8 shows that on the whole, the rolling element fault acquired the worst result, which signifies that rolling element fault is very complicated and harder to recognize. For normal and outer race fault, the method based on L1 (l

_{1}norm) outperforms all the other methods.

#### 5.3.2. Influence of Parameter Set of Shift Invariant Sparse Feature

_{1}norm) and standard SVM, which means (c, g) are both set to 1, the influence of the parameter set of shift invariant sparse feature was discussed. In shift invariant K-SVD algorithm, the number of base functions K for each class has a great influence on the whole fault diagnosis method so different K varying from $\left\{2,3,\dots ,10\right\}$ was respectively conducted.

#### 5.3.3. Diagnosis Results Using Optimized SVM

^{−10}to 2

^{10}and 5-fold cross validation is used. As for grid search, the logarithms of c and g based on 2 are stepped with the step size 1. With regard to GA and PSO, the fitness represents the c ross-validation accuracy and the population size is 20, the max generations are 100. The other parameters of GA including crossover and mutation probability are set to 0.4 and 0.2, respectively, while the other parameters of PSO are: wv = 1, wp = 1, c1 = 1.5 and c2 = 1.7. The result of the grid search is demonstrated in Figure 12, while Figure 13 and Figure 14 demonstrate the fitness curves of GA and PSO, respectively. The figures show that the loops in the GA algorithm are terminated at the 50th generation and based on the training set, the best cross validation accuracies of the three methods corresponding to the best (c, g) are relatively high.

## 6. Conclusions

_{1}norm achieves the highest classification accuracy. The influence of the parameter in shift invariant sparse feature, namely the number of basis functions is also discussed, which shows that the number of basis functions should be set comprehensively considering the diagnosis precision, the computing, and memory consumption. As for optimized SVM, the classification results indicate that parameter optimization is very essential for SVM and optimized SVM using the methods of grid search, GA, or PSO can dramatically improve the classification ability of SVM. With respect to the three methods, although PSO owns the longest running time, it obtains the highest classification accuracy. In future work, combining other effective shift invariant dictionary learning methods to obtain superior sparse features of bearing fault will be explored. For the optimized SVM, improved optimization methods based on GA or PSO can be considered to further enhance the optimization ability.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notations

c | penalization factor in SVM |

c_{1} | acceleration coefficient that represents the local search ability |

c_{2} | acceleration coefficient that represents the global search ability |

d_{k} | the kth basis function |

d | basis function |

D | over-complete dictionary |

F | shift invariant sparse feature |

g | the width of RBF kernel in SVM using RBF kernel |

gbest | the best particle that indicates the global best |

j | class label |

K | basis function number |

L | class number of signals |

M | the number of maximum absolute values |

N | population size in PSO |

p | the length of the long signal x |

pbest_{i} | the best value of the ith particle that indicates the local best |

p_{i} | the ith particle |

q | the length of the basis function |

r | residual signal |

r_{1} | random number uniformly distributed in [0, 1] |

r_{2} | random number uniformly distributed in [0, 1] |

s | sparse coefficient corresponding to the long signal |

S_{k,τ} | the sparse coefficient corresponding to the dictionary atom after basis function ${d}_{k}$ is translated to time τ and extended |

t | iteration number |

T | sparsity prior |

T_{τ} | shift operator |

${T}_{\tau}^{\ast}$ | the operator corresponding to ${T}_{\tau}$, which can extract a segment with the same length q as the basis function ${d}_{\kappa}$ from the long signal and the segment starts at time τ |

v_{i} | velocity of the ith particle |

wv | elastic coefficient for velocity update |

wp | elastic coefficient for particle update |

x | long signal |

X | training set |

${\widehat{x}}_{\kappa}$ | the signal with no contribution from other basis functions ${d}_{k}(k\ne \kappa )$ |

σ_{κ} | the set of non-zero elements |

ε | tolerance error |

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**Figure 3.**Vibration signals corresponding to different running statuses: (

**a**) Normal; (

**b**) inner race fault; (

**c**) rolling element fault; (

**d**) outer race fault.

**Figure 4.**The learned basis functions corresponding to four running states: (

**a1**–

**a4**) normal; (

**b1**–

**b4**) inner race fault; (

**c1**–

**c4**) rolling element fault; (

**d1**–

**d4**) outer race fault.

**Figure 5.**Shift invariant sparse coefficients with respect to different running statuses: (

**a**) normal; (

**b**) inner race fault; (

**c**) rolling element fault; (

**d**) outer race fault.

**Figure 6.**Shift invariant sparse feature of a test sample with respect to different running statuses: (

**a**) normal; (

**b**) inner race fault; (

**c**) rolling element fault; (

**d**) outer race fault.

**Figure 7.**Sum of shift invariant sparse feature of all test samples with respect to different running statuses: (

**a**) normal; (

**b**) inner race fault; (

**c**) rolling element fault; (

**d**) outer race fault.

Max | 2-Max | 3-Max | L1 | L2 |
---|---|---|---|---|

89.7 | 90.0 | 90.3 | 93.3 | 93.0 |

Default | Grid Search | GA | PSO | |
---|---|---|---|---|

Normal | 97.3 | 98.7 | 98.0 | 98.0 |

IRF | 94.7 | 96.7 | 97.3 | 96.7 |

REF | 88.7 | 92.0 | 90.7 | 93.3 |

ORF | 91.3 | 96.7 | 96.7 | 97.3 |

Average | 93.0 | 96.0 | 95.7 | 96.3 |

Time/s | 0.1976 | 57.3465 | 81.7135 | 112.4451 |

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**MDPI and ACS Style**

Yuan, H.; Wu, N.; Chen, X.; Wang, Y.
Fault Diagnosis of Rolling Bearing Based on Shift Invariant Sparse Feature and Optimized Support Vector Machine. *Machines* **2021**, *9*, 98.
https://doi.org/10.3390/machines9050098

**AMA Style**

Yuan H, Wu N, Chen X, Wang Y.
Fault Diagnosis of Rolling Bearing Based on Shift Invariant Sparse Feature and Optimized Support Vector Machine. *Machines*. 2021; 9(5):98.
https://doi.org/10.3390/machines9050098

**Chicago/Turabian Style**

Yuan, Haodong, Nailong Wu, Xinyuan Chen, and Yueying Wang.
2021. "Fault Diagnosis of Rolling Bearing Based on Shift Invariant Sparse Feature and Optimized Support Vector Machine" *Machines* 9, no. 5: 98.
https://doi.org/10.3390/machines9050098