# A Fault-Signal-Based Generalizing Remaining Useful Life Prognostics Method for Wheel Hub Bearings
## 4 Validation
### Analysis of Experimental Results  
As a comparison, the RUL of wheel hub bearings is prognosticated utilizing traditional grey theory [27] with original signal (GM). The original signal is described in Equation (13). For the amplitude feature, the threshold is the maximum amplitude value of the original signal when the damage happens. Here, the amplitude threshold value is taken as 1.4. The RUL prognostics results are shown in Figure 5 and Table 1. The accuracies of RUL prognostics are 0.6429, 0.5571, 0.5642, and 0.6214 based on 25%, 50%, 75%, and 100% of the original data, as shown in Figure 5a-d. It is found that the prognostics results are random. The accuracy reduces as the original data increases for some prognostics results, which indicate that the traditional grey theory cannot prognosticate the RUL of wheel hub bearings with the original signal in a complex working environment. For the energy feature with Equation (9), the threshold value is taken as 0.12. The RUL prognostics results are shown in Figure 6 and Table 2. The accuracies of RUL prognostics are 0.6648, 0.5215, 0.5279, and 0.6759 based on 25%, 50%, 75%, and 100% of the original data, as shown in Figure 6a-d. It is found that the prognostics results are also random. Therefore, the prognostics results indicate that the traditional grey theory cannot prognosticate the RUL of wheel hub bearings with the original signal in a complex working environment either. For the kurtosis feature with Equation (10), the threshold value is taken as 1.2. The RUL prognostics results are shown in Figure 7 and Table 3. The accuracies of RUL prognostics are 0.5026, 0.3511, 0.6586, and 0.7264 based on 25%, 50%, 75%, and 100% of the original data, as shown in Figure 7a-d. It is found that the prognostics results are also random. The prognostics results indicate that the traditional grey theory also cannot prognosticate the RUL of wheel hub bearings with the original signal in a complex working environment. can prognosticate the RUL of wheel hub bearings with the accumulative signal in a complex and variable working environment. However, the accuracy of RUL prognostics is very poor in the early stage with original data less than 50%. Therefore, the traditional grey theory has a poor generalization with accumulative signal. For the energy feature, the RUL prognostics results are shown in Figure 9 and Table 2. The accuracies of RUL prognostics are 0.4357, 0.7040, 0.8910, and 0.9998 based on 25%, 50%, 75%, and 100% of the accumulative data, as shown in Figure 9a-d. Similarly, the accuracy increases as the original data increases. To some extent, the traditional grey theory can prognosticate the RUL of wheel hub bearings with the accumulative signal in a complex and variable working environment. The accuracy of RUL prognostics is also very poor in the early stage. This verifies that the traditional grey theory also has a poor generalization with the accumulative signal. Figure 6: The RUL prognostics results with the energy of the original signal utilizing grey theory: (**a**) prognostics result based on 25% of the original data; (**b**) prognostics result based on 50% of the original data; (**c**) prognostics result based on 75% of the original data; (**d**) prognostics result based on 100% of the original data. Figure 8: The RUL prognostics results with the amplitude of the accumulative signal utilizing grey theory: (**a**) prognostics result based on 25% of the accumulative data; (**b**) prognostics result based on 50% of the accumulative data; (**c**) prognostics result based on 75% of the accumulative data; (**d**) prognostics result based on 100% of the accumulative data. Figure 7: The RUL prognostics results with kurtosis of the original signal utilizing grey theory: (**a**) prognostics result based on 25% of the original data; (**b**) prognostics result based on 50% of the original data; (**c**) prognostics result based on 75% of the original data; (**d**) prognostics result based on 100% of the original data.