Multiscale Distribution Entropy and t-Distributed Stochastic Neighbor Embedding-Based Fault Diagnosis of Rolling Bearings
2. Algorithms of DE and MDE
2.1. Definition of DistEn
2.2. DistEn Parameter Selection
2.3. Multiscale Distribution Entropy
3. Comparison Analysis of MSE and MDE
4. The Proposed Fault Diagnosis Approach and Its Applications
4.1. t-SNE Algorithm
4.2.1. Basic Concepts and Frameworks of KVPMCD
4.2.2. Kriging Model-Based KVPMCD Method
- For g class classification problem, n training samples are collected and each sample number is . The feature X =  is extracted from all training samples and the size of each feature is respectively.
- The feature (i = 1, 2, …, p) of the kth (1 ≤ k ≤ g) training sample is selected as the predicted variable and the remaining p-1 feature (j ≠ i) is seen as predictive variables.
- Let the regression model type z = 1 (1 ≤ z ≤ Z) (zero, one and two order polynomial. Three models are marked as 1, 2, and 3, respectively). The model category of the relevant models is h = 1 (1 ≤ h ≤ H) (exponential, generalized exponential, Gaussian, linear, spherical, cubic, spline, respectively, is marked as 1, 2, 3, 4, 5, 6 and 7), and then a mathematical model is established.
- Set h = h + 1 and z = z + 1, respectively, until h = H, z = Z. The combination of predictive variables is common to H × Z species. Therefore, = H × Z mathematical equations can be established.
- equations can be established for each feature set . The feature of each training sample in the kth class can be obtained. The predicted value of the feature can be obtained by the Kriging model.
- To calculate the prediction error square sum of the variable prediction model respectively, where v represents the vth training sample and . The variable prediction model corresponding to the minimum value of is selected as the variable prediction model of the feature in the kth class training sample. Then save the corresponding model parameters and predictive variables.
- Let k = k + 1, repeat steps (3)~(6) until k = g. At this point, in the case the variable prediction model are established for all the feature of g categories respectively, where k (k = 1, 2, …, g) denotes category label and i (i = 1, 2, …, p) represents the feature. These variable predictive models form a VPM matrix with size of g × p.
- All training samples are constructed as a test sample set to perform a return classification test for each VPM matrix. The regression model type and the relevant model type corresponding to the VPM matrix with the highest correct classification rate are selected as the type of the best variable prediction model.
- The feature X =  are extracted for the selected test sample set. For all the feature values of the test sample, respectively, the variable prediction model is employed to predict it, and the predicted value is obtained.
- The squared sum of the predicted errors is calculated for all features in the same category. And the minimum is used as the discriminant function to classify the test samples.
4.3. The Proposed Fault Diagnosis Method
- Assume that the states of rolling bearing contain K classes and each state is collected by N groups. MDE of each vibration signal is computed with parameters m = 2, δ = 1, M = 512 and the maximum scale factor . eigenvalues were obtained to represent the fault information of the vibration signals of rolling bearing in each group and the feature vector matrix is constituted, which can adequately digs out the characteristics information of different complex time series.
- t-SNE is used to reduce the dimension of feature vector matrix and a low dimensional sensitive feature set RN×i can be obtained, where N represents the number of samples and i represent the dimensions after dimensionality reduction.
- Training samples are composed of the selected 1/2 N group of each state randomly, the rest as test samples. The training samples are input to the KVPMCD based multi-classifier for training. The predictive model . is established, where k (k = 1, 2, …, g) represents different categories, i (i = 1, 2, …, p) represents different characteristic values.
- The outputs of classifier are used to diagnose the fault types of rolling bearing.
4.4. Experimental Data Analysis
Conflicts of Interest
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|Methods||Accuracy Rate (%)|
|DistEn + SVM||69.64|
|MDE (the first 8 scales) + SVM||82.14|
|MDE (all 20 scales) + KVPMCD||94.64|
|MDE (three DEs in 1, 8 and 15 scales) + KVPMCD||89.29|
|MDE + PCA + KVPMCD||96.43|
|MDE + t-SNE + KVPMCD||100|
|MDE + t-SNE + VPMCD||98.21|
|MDE + t-SNE + SVM||87.50|
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Tu, D.; Zheng, J.; Jiang, Z.; Pan, H. Multiscale Distribution Entropy and t-Distributed Stochastic Neighbor Embedding-Based Fault Diagnosis of Rolling Bearings. Entropy 2018, 20, 360. https://doi.org/10.3390/e20050360
Tu D, Zheng J, Jiang Z, Pan H. Multiscale Distribution Entropy and t-Distributed Stochastic Neighbor Embedding-Based Fault Diagnosis of Rolling Bearings. Entropy. 2018; 20(5):360. https://doi.org/10.3390/e20050360Chicago/Turabian Style
Tu, Deyu, Jinde Zheng, Zhanwei Jiang, and Haiyang Pan. 2018. "Multiscale Distribution Entropy and t-Distributed Stochastic Neighbor Embedding-Based Fault Diagnosis of Rolling Bearings" Entropy 20, no. 5: 360. https://doi.org/10.3390/e20050360