# A Method for Constructing a Synthesis Health Index for Metro Vehicle Wheelsets
## 4 Instance Verification
### Health Status Classification  
After outlier removal, the number of data points used for classification learning was 16,903. HC was employed to generate the dendrogram shown in Figure 5a, and the inconsistency coefficients between each level were calculated with a depth of 2. The inconsistency coefficients for classifications of \(n=2\), \(n=3\), and \(n=6\) were 2.07, 1.68, and 1.49, respectively. Based on these results, dividing the training data into three classes emerged as the optimal solution. This classification aligns well with the different health states of subway vehicle wheelsets: healthy, suboptimal health, and rapid degradation. While dividing the data into two categories might suggest a binary health classification (healthy or failed), it would fail to provide a comprehensive representation of the wheelset's health condition. Therefore, the decision to classify the data into three distinct states offers a more nuanced and accurate understanding of the wheelset's health. The distribution of the classified data points is illustrated in Figure 5b.  
Figure 4: Visualization of data after dimension reduction for wheelsets. (**a**) Original data after wheelset dimensionality reduction. (**b**) Outlier-removed data after wheelset dimensionality reduction.  
In the figure, the orange color represents the first cluster of hierarchical clustering, the blue color represents the second cluster, and the red color represents the third cluster. Based on the numbering of subway vehicle wheels, the starting and ending points of each wheel's lifespan can be identified in the diagram, allowing for the determination of the health status of different categories. As illustrated in Figure 5b, the classes represented by dark gray, gray, and light gray correspond to healthy, suboptimal health, and rapidly  
Figure 5: Training data classification learning results. (**a**) Hierarchical clustering dendrogram. (**b**) Data classification corresponding to different health states.  
deteriorating states, respectively. Furthermore, the geometric centers, member counts, and radii of the different classes can be statistically derived, as presented in Table 3.  
After obtaining the geometric center of the healthy class (Cluster 1), further subdivision of this class was performed, resulting in the dendrogram shown in Figure 6. The inconsistency coefficients for subdividing the healthy class into two subclasses and three subclasses were 1.13 and 1.11, respectively, indicating a minimal difference. However, subdividing the healthy class into three subclasses provides a more accurate baseline. Therefore, the healthy class was further divided into three subclasses, labeled SUB1, SUB2, and SUB3.  
In the figure, the orange color represents the first cluster of hierarchical clustering, the blue color represents the second cluster, and the green color represents the third cluster.  
After subdivision, the healthy class contained a total of 1547 data points. The detailed information about the subclasses of the healthy class is presented in Table 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Class Code/Color** & **Cluster 1/Green** & **Cluster 2/Yellow** & **Cluster 3/Red** \\ \hline \hline \multirow{2}{*}{Health status} & Healthy & Suboptimal health & Rapid degradation \\ \hline \hline \multirow{2}{*}{Center} & PC1 & \(-\)0.1693 & 0.0400 & 0.1998 \\  & PC2 & 0.0158 & \(-\)0.0278 & 0.0266 \\  & PC3 & \(-\)0.0171 & 0.0429 & \(-\)0.0526 \\ \hline \hline \multicolumn{2}{c}{Number of members} & 6100 & 7045 & 3758 \\ \hline \hline \multicolumn{2}{c}{Radius} & 0.3088 & 0.3281 & 0.2765 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The information of the different categories after classification.  
Figure 6: Further subdivision of the healthy category.