# 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
Although both models show an increase in \(T^{2}\) values as damage severity increases, the SAE provides a clearer separation between damage classes 1 and 2, indicating higher sensitivity to subtle variations in the data. Furthermore, among the six accelerometers evaluated, the SAE methodology did not present any overlap of the whiskers in the box-plots for monitoring data, whereas the CAE methodology exhibited whisker overlap for accelerometer 4, as shown in Figure 14d. These results reveal a 100% accuracy rate for the SAE model compared to 83.33% for the CAE model. For a more detailed visualization of the \(T^{2}\) value distributions and to facilitate model comparisons, box-plots were constructed (Figures 17-19). These graphs enable the identification of the variability in \(T^{2}\) values for each model and scenario, as well as highlighting potential outliers. The analysis of the box-plots supports the previous conclusions, emphasizing the superiority of the SAE methodology in detecting and quantifying structural changes. When submitting an AutoEncoder trained with data representative of a healthy structural state to new data from the same state, it is expected that the latent representations will exhibit small variations around a central value, as evidenced by \(T^{2}\) values close to the mean. This observation reflects the model's ability to capture the intrinsic characteristics of the healthy state. However, when the model is exposed to data from a damaged state, the latent representations tend to deviate significantly from the learned pattern, manifesting in high \(T^{2}\) values. This divergence results from the model's inability to adequately reconstruct anomalous data, revealing the presence of damage or anomalies in the structure. The results obtained in this study support the theoretical expectations, with the exception of the VAE. Specifically, for the SAE and CAE models, damage classes 1 and 2, associated with structural damage in railway wheels, exhibited significantly higher \(T^{2}\) values, indicating a clear correlation between the severity of the damage and the distance of the latent representations from the mean. This trend demonstrates the effectiveness of AutoEncoders in quantifying structural damage severity, with SAE and CAE showing greater sensitivity to changes in data characteristics. The comparative analysis between the CAE and SAE models revealed good performance in discriminating between different damage classes, with no overlap between the whiskers of the box-plots corresponding to distinct classes. However, the SAE model proved to be more sensitive to damage gradations, showing greater separation between damage classes 1 and 2. This observation suggests that SAE may be more suitable for accurately quantifying degradation stages. In contrast to the results obtained for the CAE and SAE models, the VAE faced challenges in distinguishing between different levels of irregularities, as shown in Figure 19. The significant overlap between the box-plots corresponding to validation data and damage class 1, along with the deviation between training and validation data, indicates that the model failed to effectively capture the distinctive features of these classes. Although the results for accelerometers 2 and 3 were relatively acceptable, the same does not apply to accelerometers 1 and 4, as their box-plots showed significant overlap. This limitation may be attributed to various factors, such as problem complexity, model architecture, and even the range of hyperparameters used in optimization performed by Optuna.