## 3 Case Study
According to the characteristics of the vibration signals, the mileage, i.e., the X-axis of Figure 5, was selected as a generalized time, which would be used in our proposed model. Meanwhile, the RMS of original vibration signals was extracted as the degradation indicator. To obtain the RMS, the recorded mileages were separated into equally distributed sliding intervals, each of which was 200 km in length. In this way, the first RMS value was calculated based on the amplitudes between 0 and 200 km, and the second RMS value was calculated based on the amplitudes between 100 and 300 km, and so on. Figure 7 shows the extracted RMS values of the two wheel treads. There were in total 389 extracted data points for the first wheel tread and 456 data points for the second wheel tread. We found that the extracted RMS values were relatively stable with some fluctuations at the beginning, but present obvious increasing trends within the following monitoring process. Therefore, we detected the PCT of each wheel tread and divided the degradation path into normal operation phase and rapid degradation phase. Results are also presented in Figure 7. It is observed that the RMS was sensitive to the incipient fault of wheel treads. Except this, we found that the second wheel tread performed a two-phase degradation behavior within its rapid degradation phase. Combined with its failure mode, we though that this was reasonable as the health states of the wheel tread could be different from each other before and after the occurrence of stripping. In this view, the changing point of the two-state degradation should be the time of stripping, and when the balance of rotary wheel tread was broken, more severe vibrations accelerated its degradation. For convenience, the changing point was denoted as \(t_{stripping}\) in the following. To carry out RUL prediction, we need to determine the failure threshold of wheel treads. As mentioned above, the two wheel treads were considered to be failed in practical applications. Thus, we selected the mean of the final RMS values of the two wheel treads as the threshold, i.e., \(d=35\). In addition, \(\varphi(t;\theta)=\theta t^{\theta-1}\) was selected for the 1-st wheel tread and \(\varphi(t;\theta)=\theta\) was selected for the 2-nd wheel tread because the extracted RMS values always increase with the running mileage in a power-law form (or a linear form) for the wheel treads whose failure mode is out-of-round (or stripping) according to our previous engineering experience. Under these assumptions, we illustrated our proposed prediction model and corresponding parameter updating algorithm based on RMS values of the rapid degradation phase. Figure 6: Illustration of the two failure modes of wheel tread. (**a**) Out-of-round and (**b**) stripping. To verify the rationality of the proposed model, we separated the degradation paths of the two wheel treads into 10 equal intervals respectively and calculated degradation rate and degradation diffusion of each interval. According to the results, a correlation test between degradation rate and degradation diffusion was carried out. Figure 8 shows the test results, which mean that there really existed a linear relationship between the drift parameters and the square of diffusion parameters, as mentioned in our proposed model. Then, based on the parameter initialization method, we initialized the unknown parameters by using the first twenty RMS values from each changing point including two PCTs and a stripping point. Results are shown in Table 1. Next, the model parameters were updated continuously based on KF and EM algorithm with the increasing of RMS values. The parameter updating procedure is presented in Figure 9. It is observed that all parameters were not estimated accurately at the beginning of the predicting process. However, when enough RMS values were put into parameter estimation, these parameters gradually converged to fixed values. Note that, for the second wheel tread whose failure mode was stripping, the parameter updating procedure included two phases.