## 4 Numerical Application: Simply Supported Beam
#### Results  
Since the damage identification of the proposed strategy is based on the features extracted by the SAE, the first stage of the analysis examines how many SAE internal codes are necessary for a good representation of the dynamic structural responses. As previously mentioned, the SAE training is performed by reconstructing its input data. Therefore, it is reasonable that the difference between the original and the reconstructed signals may indicate whether the extracted characteristics contain relevant information about the structural response or not. In the present paper, this difference is denoted as reconstruction error \(\epsilon\), and is individually calculated by Equation (2):  
\[\epsilon_{j}=\frac{\sum_{i=1}^{M}\left|x_{ji}-\hat{x}_{ji}\right|}{\sum_{i=1}^ {M}\left|x_{ji}\right|}, \tag{2}\]  
where \(M\) is the total length of the sampled signal, and \(x_{ji}\) and \(\hat{x}_{ji}\) are the \(i\)th element of the \(j\)th original signal vector \(x\) and the \(j\)th SAE reconstructed signal vector \(\hat{\textbf{x}}\), respectively (\(j=1,\ \ldots,\ J,\ J\) being the length of the corresponding analyzed dataset). For the complete set of dynamic signals, the reconstruction error is the mean of all errors \(\epsilon_{j}\), as given in Equation (3):  
\[\widetilde{\epsilon}_{j}=\frac{1}{J}\sum_{j=1}^{J}\epsilon_{j}. \tag{3}\]  
Figure 5 shows a box plot evolution of the error \(\widetilde{\epsilon}\) as the amount of SAE features increases for all selected points of the numerical beam example. Each box was created by considering 30 different SAE models that reduce the problem's dimensionality from 5001-time samples to the indicated number of internal codes (vector _h_). The thirty different SAE models are auto-encoder models developed with the same parameters but varying in the random seed. The boxes associated with the blue dashed line indicate the reconstruction performance of the 30 models for data used during the training phase. The boxes related to \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{3}{c}{**SAE Parameters**} \\ \hline Sparsity proportion (\(\rho\)) & 0.050 \\ Sparsity regularization (\(\beta\)) & 4.000 \\ Weight regularization (\(\lambda\)) & 0.001 \\ Encoder/decoder activation functions & Logarithmic sigmoid/linear \\ Optimization method & Scaled Conjugate Gradient \\ Gradient maximum value & 1.00E-6 \\ Max. of training epochs & 1000 \\ Training error metric & Mean-squared error \\ \hline \multicolumn{3}{c}{**SVM Parameters**} \\ \hline Kernel function & RBF \\ Multiclass coding scheme & One-vs-one \\ \(\sigma\)—for 50 SAE internal codes & 1.0000 \\ \(\sigma\)—for 100 SAE internal codes & 1.5000 \\ \(\boldsymbol{C}\)—for 50 SAE internal codes & 0.3162 \\ \(\boldsymbol{C}\)—for 100 SAE internal codes & 0.3162 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: SAE and SVM parameters of the proposed strategy for the numerical beam example. the red dashed line, on the other hand, correspond to the reconstruction errors of the same models when applying unseen data (evaluation phase). Figure 5: SAE internal codes \(\times\) SAE reconstruction error for the numerical beam. (**a**) Measurement point Ac1. (**b**) Measurement point Ac2. (**c**) Measurement point Ac3. (**d**) Measurement point Ac4. In Figure 5, it is possible to observe that the plotted curves are quite similar for all simulated accelerometer channels. In all cases, the median of training and evaluation errors tends to have a linear behavior as the number of internal codes increases. For 100 SAE characteristics (internal codes), \(\widetilde{e}\) achieves values around 0.25 and 0.50 for training and evaluation data, respectively. Moreover, although not represented in the graphs the same linear behavior was verified for SAE models, with more than 100 internal codes (from 100 to 500). Obviously, the greater the number of SAE features, the better the signal reconstruction and the smaller the errors \(\widetilde{e}\).