# Out-of-Roundness Wheel Damage Identification in Railway Vehicles Using AutoEncoder Models
## 2 Theoretical Conceptualization
### Shewhart \(T^{2}\) Control Chart  
The Shewhart (\(T^{2}\)) Control Chart is a visual statistical tool that monitors the stability of parameters over time. Its graphical representation, consisting of data points and horizontal \begin{table}
\begin{tabular}{p{42.7 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Model** & **Use in SHM** & **Advantages** & **Disadvantages** & **Key Features of SHM** \\ \hline \multirow{6}{*}{**SAE**} & - Feature extraction from vibration, acoustic, or strain signals. & - Promotes a compact, interpretable feature space. & - Requires careful tuning of sparsity penalties (too strong can lose key information, too weak can yield irrelevant representations). & - Particularly good for applications where explainability or interpretability of sensor features is desired. \\  & - Identifying relevant features indicative of advantage of advantage of advantage of advantage of advantage through sparse features. & - Relatively simple to implement and train. & - Not specifically tailored to spatial/temporal correlations. & - Works well in combination with other models for dimension reduction. \\ \hline \multirow{6}{*}{**VAE**} & - Anomaly detection by learning a probabilistic latent space. & - Provides a probabilistic framework. & - More complex to train due to KL divergence term and independent of damage scenarios. & - Enables thresholding on reconstruction probability (not just reconstruction error). \\  & - Estimating uncertainty in sensor signals. & - The latent space can capture underlying structural variability. & - Major produce smoother/less sharp reconstruction outputs, especially for high-dimensional signals/images. & - Helpful for tasks where uncertainty quantification is important in decision-making (e.g., high-stakes infrastructure). \\ \hline \multirow{6}{*}{**CAE**} & - Image-based SHM (e.g., thermographic images, structural surface crack detection). & - Lears translationally invariant features, good for imaging tasks. & - Less straightforward if data are purely 1D (e.g., single vibration signals) unless converted to 2D representations (spectrograms, wavelet transforms). & - Particularly well suited for visual inspection or 2D representations of structural signals (e.g., spectrogram-based methods). \\  & - Topentially applied to 2D/3D data (e.g., wavelets, damage scans). & - Often provides higher-fidelity reconstructions for visual data. & - Network can become large, requiring more computational resources. & - Commonly used in damage segmentation or crack detection tasks. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Feature analysis across different AutoEncoder implementations evaluated. Figure 3: Example of a CAE network structure (adapted from [PERSON] and [PERSON] (2023) [41]). control limits, allows deviations from the normal condition to be identified. When a point exceeds these limits, it indicates the presence of external factors affecting data variability, signaling an out-of-control condition. In addition to its early detection capability, \(T^{2}\) is widely recognized in the scientific community for its clear and intuitive interpretation, facilitating decision-making. Its applicability across various fields makes it a valuable tool for identifying structural changes in data, contributing to quality and reliability improvements in SHM systems [36]. In this study, the \(T^{2}\) statistic is employed to evaluate the integrity of railway wheels based on features extracted from AE models. This metric quantifies the distance of a new observation from the historical data mean, allowing the detection of anomalies that may indicate damage. The magnitude of \(T^{2}\) is directly related to data variability and correlation. By comparing the calculated value to a statistically established Upper Control Limit (UCL), it is possible to determine whether the structure operates within its normal conditions [36; 37].