# A Method for Constructing a Synthesis Health Index for Metro Vehicle Wheelsets
## 3 The Construction Process of the Synthesis Health Index for Metro Vehicle Wheelsets
### Dimension Reduction of Metro Vehicle Wheelset Data
#### 3.1.1 Feature Selection of Metro Vehicle Wheelset Data  
Common feature selection methods can be categorized into three groups: filtering, wrapper, and embedded methods. The filtering method evaluates and selects features based solely on the intrinsic properties of the data, independent of any subsequent training models. In comparison to the wrapper and embedded methods, utilizing filtering for feature selection in metro vehicle wheelset profile data does not necessitate multiple rounds of model training, thereby enhancing computational efficiency.  
Choosing an appropriate feature evaluation criterion is the key aspect of the filtering method. This paper combines monotonicity, trend, and robustness for feature selection.  
1. Monotonicity  
The calculation method for assessing the monotonicity of metro vehicle wheelset features is presented in Equation (1):  
\[M_{X}=\frac{1}{n-1}\Bigg{|}\sum_{x\in X}\left(\frac{d}{dx}>0\right)-\sum_{x \in X}\left(\frac{d}{dx}<0\right)\Bigg{|} \tag{1}\]  
In this equation, \(X=\left(x_{i}\right)_{i=1:n}\) represents the time series of a specific feature for the metro vehicle wheelset, \(x_{i}\) denotes the value of this feature at time \(i\), and \(n\) indicates the length of the feature sequence. \(\frac{d}{dx}=x_{i+1}-x_{i}\) represents the derivative of this sequence. \(M_{X}\) is used to measure the absolute difference between the positive and negative derivatives in the sequence \(X\), with a value range of [0, 1]. A larger value indicates a stronger monotonicity.
2. Trend  
Trend is assessed by the correlation coefficient between the features of metro vehicle wheelsets and time. Commonly used correlation coefficients include the Pearson and Spearman correlation coefficients.  
Metro vehicle wheelsets frequently exhibit nonlinear degradation trends and significant fluctuations, rendering the Pearson correlation coefficient unsuitable. The Spearman correlation coefficient, a non-parametric method, considers the ranking of variables within their respective sequences rather than their actual values. Consequently, it is sensitive to nonlinear relationships and less influenced by outliers. This paper employs the Spearman correlation coefficient as the trend indicator:  
\[S_{X,T}=\frac{\frac{1}{\bar{n}}\sum_{i=1}^{n}\left(R(x_{i})-\overline{R(x)} \right)\times\left(R(t_{i})-\overline{R(t)}\right)}{\sqrt{\left(\frac{1}{\bar {n}}\sum_{1}^{n}\left(R(x_{i})-\overline{R(x)}\right)^{2}\right)\times\left( \frac{1}{\bar{n}}\sum_{1}^{n}\left(R(t_{i})-\overline{R(t)}\right)^{2}\right)}} \tag{2}\]  
In this equation, \(R(x_{i})\) and \(R(t_{i})\) represent the ranks of \(x_{i}\) and \(t_{i}\) within their respective data sequences, while \(\overline{R(x)}\) and \(\overline{R(t)}\) denote the average ranks of sequences \(X\) and time \(T\). The value of \(S_{X,T}\) ranges from \(-1\) to \(1\); when \(S_{X,T}\) approaches either \(1\) or \(-1\), it indicates a strong positive or negative correlation between \(X\) and \(T\).
3. Robustness  
The robustness of metro vehicle wheelset features is defined as follows [20]:  
\[R_{X}=\frac{\sum_{i=1}^{n}\exp\left(-\Big{|}\frac{r_{x_{i}}}{x_{i}}\Big{|} \right)}{n} \tag{3}\]  
In this equation, \(r_{x_{i}}=x_{i}-\hat{x_{i}}\) represents the residual of SHI at \(x_{i}\), while \(\hat{x_{i}}\) denotes the average trend value of SHI at \(x_{i}\), typically obtained through smoothing.  
By integrating monotonicity, trend, and robustness, the criteria for feature selection in metro vehicle wheelsets are established:  
\[F(X)=\omega_{1}M_{X}+\omega_{2}S_{X,T}+\omega_{3}R_{X} \tag{4}\]  
In this equation, \(\omega>0\), \(\sum_{i=1}^{3}\omega_{1}=1\). When the value of \(FC\) increases, it indicates that the feature should be selected more strongly.