# Out-of-Roundness Wheel Damage Identification in Railway Vehicles Using AutoEncoder Models
## 5 Application of the Proposed SHM Methodology to Numerical Data  
### Analysis of Reconstruction Results  
The original vibrational response and its respective reconstruction made by VAE, SAE, and CAE models for the first accelerometer of each simulated scenario are presented in Figures 11-13, respectively. The results of the other accelerometers were similar to those presented and, thus, were omitted. The optimal hyperparameters for the AutoEncoders, determined using Optuna, are listed in Table 5. They are applied in both the training and validation of the models. Visualization of the results obtained using [PERSON]'s \(T^{2}\) Control Chart (Figures 14-16) confirms the effectiveness of the AutoEncoders in detecting anomalies. Analysis of the data from accelerometers 1 to 4 reveals a clear separation among classes, indicating that the models can discriminate between different operating conditions. This separability is particularly evident for the SAE and CAE models, corroborating the results of the qualitative analysis of the reconstructions. Figure 14: Shewhart \(T^{2}\) Control Charts for the analysis of the CAE model. (**a**) Accelerometer 1; (**b**) accelerometer 2; (**c**) accelerometer 3; (**d**) accelerometer 4. Figure 13: Original and reconstructed signals by the VAE model for accelerometer 1: (**a**) training of damage class 0; (**b**) validation of damage class 0; (**c**) testing of damage class 1; (**d**) testing of damage class 2. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Hyperparameter** & **VAE** & **SAE** & **CAE** \\ \hline learning\_rate & 0.00000104471 & 0.000690141 & 0.000688171 \\ epochs & 140 & 306 & 283 \\ batch\_size & 12 & 32 & 5 \\ original\_dim & 3916 & 3916 & 3916 \\ intermediate\_dim & 2170 & - & - \\ latent\_dim & 214 & 345 & 157 \\ optimizer & adam & adam & adam \\ lambda\_sparse & - & 0.00003972 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Optimized hyperparameters used in AutoEncoder models. The analysis of Figures 14 and 15 reveals a good agreement between the \(T^{2}\) results obtained by the CAE and SAE models in both training and validation classes. This similarity indicates a generalization capability of the models, i.e., their ability to accurately classify data not used during training. The VAE model, as illustrated in Figure 16 and corroborated by Figure 19, exhibits mixed results. In some cases, a less evident separation between validation and monitoring data is observed, as indicated in Figure 16b,c. In other cases, it is possible to identify an overlap between damage classes 1 and 2, as evidenced in Figures 16a and 19a. Furthermore, the model suggests a convergence of damage classes 1 and 2 with the training and validation data, as demonstrated in Figures 16d and 19d. These mixed behaviors and inconsistencies highlight the difficulty of the VAE in capturing the intrinsic characteristics of the data and, consequently, in generalizing to new datasets. Except for the VAE model, all proposed methodologies demonstrated sensitivity to structural changes, as evidenced by crossing the Upper Control Limit in the monitoring classes. The VAE model, in none of the six evaluated scenarios, exhibited sensitivity to structural changes, as previously discussed, showing varied behaviors and being outperformed in 100% of the cases by the SAE methodology and in 83.33% of the cases by the CAE methodology. However, the comparative analysis of the results indicates that the  
Figure 16: Shewhart \(T^{2}\) Control Charts for the analysis of the VAE model. (**a**) Accelerometer 1; (**b**) accelerometer 2; (**c**) accelerometer 3; (**d**) accelerometer 4. Figure 15: Shewhart \(T^{2}\) Control Charts for the analysis of the SAE model. (**a**) Accelerometer 1; (**b**) accelerometer 2; (**c**) accelerometer 3; (**d**) accelerometer 4. SAE methodology stands out in precisely quantifying these changes. When comparing the performances of the CAE and SAE models, the latter proved to be more effective in quantifying the magnitude of structural changes.