# A Method for Constructing a Synthesis Health Index for Metro Vehicle Wheelsets
## 4 Instance Verification
### Comprehensive Health Factor Calculation and Assessment  
Using the geometric center of subclass SUB1 of the wheelset healthy class (\(-0.1647\), \(0.0095\), \(0.0707\)) as the baseline, the SHI for all training samples is calculated using Equation (15). The SHI for the testing samples can be derived based on similarity, as illustrated in Figure 7. The SHI is a one-dimensional indicator that varies over time; a larger value signifies a worse health status. As time progresses, the increasing trend of the SHI corresponds with the deterioration trend of the wheelset. The lines in the figure are colored differently to represent the SHI curves for all training samples. Two randomly selected samples from the training set are taken as examples: the sixth wheel of car 2, referred to as wheel #226, and the second wheel of car 6, referred to as wheel #652. Figure 8 illustrates the variation trends of the SHI and health status for these two wheels. As the working time increases, the condition of the wheelset gradually transitions from healthy to suboptimal health, eventually entering the rapid deterioration stage, with the corresponding SHI continuously increasing. The health status labels may exhibit some fluctuations. Taking wheel #226 as an example, after entering a sub-health state, it occasionally returns to a healthy state at certain time points. Similar scenarios also occur during the transition from sub-health to rapid degradation. This is because when classifying states based on HC clustering, perfect classification cannot always be achieved, leading to imperfect transitions of states over time. Due to the fluctuations in the health status labels, when a particular state persists for a week, i.e., when the state label remains stable for seven consecutive data points, it is considered that the wheelset has entered that state. Using the two samples mentioned above as examples, the monotonicity, trendiness, robustness, and recognizability of the SHI are calculated, as presented in Table 5. \begin{table}
\begin{tabular}{c c c c} \hline
**Class Code/Color** & **Cluster 1/Green** & **Cluster 2/Yellow** & **Cluster 3/Red** \\ \hline PC1 & \(-0.1647\) & \(-0.1425\) & \(-0.1882\) \\ Center & PC2 & \(0.0095\) & \(0.1469\) & \(-0.0612\) \\ PC3 & \(0.707\) & \(-0.0509\) & \(-0.0445\) \\ \hline Number of members & \(1547\) & \(1731\) & \(2822\) \\ \hline \end{tabular}
\end{table} 
Table 4: Information on subcategories after further classification of the healthy category. Figure 7: SHI curves for all training samples. The SHI constructed in this paper demonstrates high trendiness and recognizability, along with reasonably high robustness; however, its monotonicity is notably low. This low monotonicity is attributed to significant fluctuations in the original sample data, which adversely affect the constructed SHI's ability to maintain a consistent trend. The considerable fluctuations in the SHI also impact the prediction accuracy of the RUL. Consequently, it is common in the literature to apply smoothing techniques to the constructed SHI. In this paper, the Exponential Weighted Moving Average (EWMA) method is utilized to smooth all SHIs. The calculation formula for the EWMA model is as follows [23]:  
\[S_{t}=\lambda\times Y_{t}+(1-\lambda)\times S_{t-1} \tag{16}\]  
Figure 8: Health status and SHI of random samples in the training set. (**a**) Wheel #226 health status and SHI. (**b**) Wheel #652 health status and SHI. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sample** & **Monotonicity** & **Trendiness** & **Robustness** & **Identifiability** \\ \hline \#226 & 0.0840 & 0.9875 & 0.5753 & 0.9204 \\ \#652 & 0.1146 & 0.9750 & 0.5662 & 0.8828 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Performance evaluation of SHI for random samples from the training set. In the equation, \(S_{t}\) and \(S_{t-1}\) represent the smoothed values at times \(t\) and \(t-1\), respectively, while \(\lambda\) denotes the smoothing coefficient.