## 4 Time-Domain and Time-Frequency-Domain Feature Extraction Based on the LMD Method
### Time-Domain Feature Extraction  
When the track structure is in the deformed state, the waveform of the vibration signal definitely changes [59]. The time-domain analysis method analyzes the system intuitively and accurately in the time domain and can be used to conduct structural damage identification studies by calculating the time-domain characteristic parameters of the vibration signal to reflect the relevant information of the vibration signal. It has been shown that the hierarchical function obtained by the adaptive decomposition method carries more damage information and is more sensitive to damage after decomposition [60]. Therefore, this paper employs the peak-to-peak acceleration, variance, root mean square, shape factor, crest factor, and skewness of the characteristic PFs [52] as time-domain feature indicators. Time-domain feature analysis is performed separately for the characteristic PFs and then the average value is taken as the feature vector. Table 1 lists the specific information of the time-domain features.  
The amplitude of the vibration can only describe the instantaneous value of the vibration magnitude and does not include the time duration of vibration, and its root mean square value considering the time duration can characterize the magnitude of the vibration signal energy. The numerator in time-domain feature indicators of peak-to-peak acceleration, shape factor, and crest factor are the maximum value of vibration amplitude, highlighting the role of large amplitude. In addition, shape factor and crest factor can detect the presence of shocks in the vibration response signal. Variance can effectively characterize the extent to which a random variable deviates from its mean and, to a certain extent, can reflect the stability of the signal. Skewness can reflect the distribution characteristics of the vibration signal waveform. Due to the limitation of length, this paper only shows the scatter plot of the skewness of the track slab in normal and deformed states, and the results are shown in Figure 5. In this paper, 100 data were randomly selected from the sample pool of normal and deformed states, respectively, for comparative analysis. The black boxes and red dots in the figure represent the skewness in normal and deformed states, respectively. It can be clearly seen that when the track slab is in the deformation state, the vibration waveform asymmetry increases. In the range of coordinates both less or greater than 0, the skewness of the damage condition is greater than that in the normal state. This phenomenon indicates that the vibration waveform asymmetry increases when track slab deformation occurs and the skewness index increases. This is consistent with the existing literature research results [61].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Expression** & **Description** \\ \hline Peak-to-Peak & \(V_{P-P}=\) & The difference between the maximum and minimum values. \\  & \(\max(PF_{q})-\min(PF_{q})\) & The average of the sum of squares of the difference between the data and the mean value. \\ \hline  & \(S^{2}=\frac{1}{F}\sum_{q=1}^{q}\left(PF_{q}-\overline{PF}\right)^{2}\) & The average of the sum of squares of the difference between the data and the mean value. \\ \hline Root mean square & \(PF_{rms}=\sqrt{\frac{1}{F}\sum_{q=1}^{q}\left(PF_{q}\right)^{2}}\) & Indicates the magnitude of the energy of the vibration signal. \\ \hline  & \(S^{2}\) & Refers to a value that is affected by the shape of waveforms. \\ \hline  & \(S^{2}=\max(PF_{q})/PF_{rms}\) & Detects the existence of shocks. \\ \hline  & \(S^{2}=\frac{1}{F}\sum_{q=1}^{q}\left(PF_{q}-\overline{PF}\right)^{3}\) & Measures the skewness of the data distribution \\ \hline \hline \end{tabular} (Note: \(PF_{q}\) represents the \(q\)th characteristic PE, \(\overline{PF}\) represents the mean value of characteristic PFs, and \(r\) represents the number of characteristic PFs.)  
\end{table}
Table 1: Summary of time domain features.