# A Hierarchical Decision Fusion Diagnosis Method for Rolling Bearings
## 3 Case Study
### Diagnostic Results of Rolling Bearings
#### 3.3.3 Fault Severity Estimation of Rolling Bearings  
After the fault location diagnosis of the rolling bearing is obtained in the fault isolation layer, the degree of fault is investigated. (1) Fault severity estimation results based on single time-domain features  
For the same rolling bearing vibration signal sample, the hierarchical diagnosis based on different time-domain features will produce different diagnosis results. This is because the experimental data cannot achieve absolute idealization, it is easy to be interfered with by environmental factors or human factors in the actual experiment, and the selection of data samples also has a certain randomness. According to the test sample set in Table 4, the fault of a bearing may be between two fault degrees, which leads to the hierarchical diagnosis based on different time-domain features and gives different diagnosis results, as shown in Figures 10-12. Among them, the black points represent the expected output of the test sample and the other different-colored points represent the diagnosis results of the rolling bearing under different single time-domain features. If the dots of other colors coincide with the black dots, the diagnosis is correct; if not, the diagnosis is misdiagnosed or missed. Figure 11: Diagnosis results of ball fault severity estimation based on the BPNN_BA network. Figure 10: Diagnosis results of inner race fault severity estimation based on the BPNN_IR network. (3) The diagnosis result analysis of the fault severity estimation layer  
According to the diagnosis results in Figures 10 and 13, the diagnosis results of inner race fault severity under each single feature and fusion decision are obtained, as shown in Figure 16.  
It can be seen from the diagram that the diagnostic effect of the wavelet packet reconstruction signal under other single time-domain features is higher than 90%, except for the kurtosis feature (feature 6). In particular, features 1-5, 7 and 8 have an excellent diagnostic effect on the fault severity estimation of the inner race, and the fault diagnosis rate is more than 99%. The result of decision fusion diagnosis under each time-domain feature (number 12) is a fault diagnosis rate of 100% of the wavelet packet reconstructed signal, which is not lower than that of the wavelet packet reconstructed signal under a single feature. This shows that the decision fusion diagnosis method improves the reliability of fault diagnosis of the rolling bearing inner ring to some extent. According to the diagnosis results in Figures 11 and 14, the diagnosis results of ball fault degree under each single feature and fusion decision are obtained, as shown in Figure 17.  
It can be seen from the diagram that there is a big gap in the diagnostic accuracy of ball fault degree based on different single features. Among them, feature 4, feature 5 and feature 7 of the wavelet packet reconstruction signal have a good diagnostic effect on rolling body fault degree, and fault diagnosis accuracy is more than 90%. The diagnostic accuracy based on kurtosis (feature 6) and waveform factor (feature 8) is obviously lower than that based on other single features. The diagnosis results based on each single feature are fused (number 12), and the fault diagnosis accuracy of wavelet packet reconstruction signal is 98.57%. Figure 16: Diagnosis result of fault severity of inner race under different features. 1: _Ma_; 2: _Mi_; 3: \(P_{k}\); 4: \(A_{v}\); 5: _st_; 6: \(k_{u}\); 7: _rn_; 8: \(S\); 9: \(C\); 10: \(l\); 11: \(l\); 12: decision fusion. Figure 17: Diagnosis result of fault severity of ball under different features. 1: _Ma_; 2: _Mi_; 3: \(P_{k}\); 4: \(A_{v}\); 5: _st_; 6: \(k_{u}\); 7: _rn_; 8: \(S\); 9: \(C\); 10: \(l\); 11: \(L\); 12: decision fusion.