## 5 Experimental Application: V\(\ddot{\text{\text{\char 2010}}}\)zea Nova Viaduct
#### Results  
The \(\widetilde{\epsilon}\) convergence curves (see Equations (2) and (3)) of the Varzea Nova viaduct are exhibited in Figure 11. Several numbers of internal codes were tested considering 30 different SAE models, in which the dimensionality of the problem reduces from 4096 time samples to the specified quantity of characteristics \(h\). The error evolution behavior for the viaduct signals is similar to the ones observed in the first example analyzed in this paper. Even so, the reconstruction errors of the present structure stabilize at higher values than in the numerical beam. While the error \(\widetilde{\epsilon}\) of the beam evaluation data reaches values around 0.5 for 100 internal codes, the error for the viaduct evaluation data stabilizes at values above 0.8. This is an expected result, since experimental data are normally more susceptible to noise. Besides, these viaduct plots also confirm the lower reconstruction errors for the training data. As with the beam analysis, a visual representation example of the SAE model reconstruction performance for 50 and 100 internal codes is shown in Figure 12. It is possible to observe that the reconstructed signal with 50 internal codes alludes to the original structural response. However, the improvement in the reconstruction of the signal obtained with 100 SAE characteristics is clearly visible. Table 6, as well as the confusion matrices of Figures 13 and 14, present the classification rates of the proposed damage detection methodology for 30 different SAE/SVM models developed with 50 and 100 internal codes respectively. They were assembled considering all evaluation subsets; thus, 4500 different examples were analyzed for accelerometer 1 (150 evaluating examples \(\times\) 30 models) and 4470 for accelerometer 2 (149 evaluating examples \(\times\) 30 models). According to the results, the SAE was efficient to characterize the viaduct vibration signals with both 50 or 100 internal codes. The classification rates were all above 85% with very low deviations (maximum value of 2.62%), highlighting the excellent performance achieved by the models with signals of accelerometer 1 reconstituted from 100 SAE features (total accuracy of 100% and zero standard deviation). In fact, the overall accuracy percentages also reflect the better signal reconstruction for 100 SAE internal codes verified in Figure 12. Moreover, it is possible to see that the worst classification rates are associated with signals from accelerometer 2. This outcome may be attributed to the position of accelerometer 2, which is less affected by the first natural frequency than of accelerometer 1 (placed exactly at the midspan). \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{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 & \(1.0\times 10^{-6}\) \\ Max. of training epochs & 1000 \\ Training error metric & Mean-squared error \\ \hline \multicolumn{2}{c}{**SVM Parameters**} \\ \hline Kernel function & RBF \\ Multiclass coding scheme & One-vs-one \\ \(\sigma\)—for 50 SAE internal codes & 0.3162 \\ \(\sigma\)—for 100 SAE internal codes & 0.3162 \\ C—for 50 SAE internal codes & 0.3162 \\ C—for 100 SAE internal codes & 0.0316 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: SAE and SVM parameters of the proposed strategy for the Varzea Nova viaduct. Figure 11: SAE internal codes \(\times\) SAE reconstruction error for the Várzea Nova viaduct. **(a)** Accelerometer \(1\). **(b)** Accelerometer \(2\). Figure 12: An original response of the Várzea Nova viaduct and its respective signals reconstructed by SAE. An interesting aspect could be noticed by interpreting the confusion matrices is the SAE/SVM capacity to correctly identify the \"after strengthening\" data.