# A Method for Constructing a Synthesis Health Index for Metro Vehicle Wheelsets
## 4 Instance Verification
### Dimension Reduction  
First, feature selection was performed on the dynamic detection data of subway vehicle wheelsets. Given that features should exhibit a \"predictable\" trend, greater weights were assigned to monotonicity and trend. These two features directly reflect how the characteristics change over time, so they were considered more important in predicting the health of the wheelset. Therefore, monotonicity (\(\omega_{1}\)) and trend (\(\omega_{2}\)) were assigned higher weights of 0.4 each, while robustness (\(\omega_{3}\)) was given a lower weight of 0.2. This weighting scheme was based on the idea that monotonicity and trend provide a more direct description of the feature's time-based changes.  
To further test the impact of different weight configurations, another set of weights (\(\omega_{1}=0.2,\ \ \omega_{2}=0.2,\ \ \omega_{3}=0.6\)) was applied, placing more emphasis on robustness. This configuration assumes that features exhibiting greater stability and less fluctuation under varying conditions contribute more to the composite health index.  
Although changing the weight configuration affects the ranking of features, the final selected features remained consistent regardless of the weight configuration. Whether using the default weights (\(\omega_{1}=0.4,\ \ \omega_{2}=0.4,\ \ \omega_{3}=0.2\)) or the alternative weights (\(\omega_{1}=0.2,\ \ \omega_{2}=0.2,\ \ \omega_{3}=0.6\)), the top features selected for the construction of the SHI were the same, with only slight variations in their order.  
The results for both weight configurations are as follows (see Tables 1 and 2).  
Although different weight configurations influence the ranking of the features, the final selection of features remains unchanged. Specifically, it is evident that there is a significant decline in the \(FC\) values from the \(Qr\) value to the wheel diameter difference within the same carriage, and the \(FC\) values for the wheel diameter differences in the same carriage are notably lower. This indicates that these features do not adequately reflect the degradation state of the wheelset. Therefore, the top six features from the table were selected to construct the SHI.  
After standardizing the data obtained from the feature selection process, KPCA was employed for dimensionality reduction. Figure 3 illustrates the contribution rates of different principal components at various dimensions. By setting the cumulative contribution rate threshold to 95%, it is evident that PC3 provides a relatively high contribution rate, while PC4 contributes less than 5%. Consequently, reducing the dimensionality to 3 is considered an appropriate choice.  
After dimensionality reduction, the distribution of the wheelset data can be visualized, and the iForest algorithm was applied to remove outliers, as illustrated in Figure 4.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Feature** & **FC(X)** \\ \hline
**1** & **Flaene height** & **0.8209** \\
**2** & **Wheel diameter** & **0.8040** \\
**3** & **Radial runout** & **0.7725** \\
**4** & **Inner distance** & **0.7586** \\
**5** & **Flaene thickness** & **0.7151** \\
**6** & **QR** & **0.7085** \\
7 & Wheel diameter difference & 0.6011 \\
8 & Wheel diameter difference & 0.5543 \\
9 & Wheel diameter difference & 0.5182 \\
10 & Scratch depth & 0.5009 \\
11 & Scratch depth & 0.49465 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Feature selection index values for different features of subway vehicle wheelsets (high robustness).  
Figure 3: Contribution rates for different dimensions based on KPCA reduction. (**a**) Contribution rate for PC3. (**b**) Contribution rate for PC4.