# A Deep Learning Approach to Detect Failures in Bridges Based on the Coherence of Signals
## 5 Results  
The selection of the periods for training and testing has been corroborated by Figures 8 and 9. In these two figures, the readings of two tiltmeter (T2 and T8) averages per day are plotted against the temperature. Then, the data points are grouped according to months. In Figure 8, the readings of T2 present a transition between January 2021 and April 2021. Then, it looks as if the data points stabilised around a regression line. A similar situation can be observed in Figure 9. The tiltmeter T8 features a longer transition phase, from January 2021 to August 2021, before it settles down around a new regression line. Apparently, it took to the bridge between three and nine months to become stable in a new equilibrium position after the October 2020 flood. This consideration justifies a partition of the two years of measurements available between the training set (from December 2021 to December 2022), which is used to validate the models as well, and the test set (from December 2020 to November 2021), although, usually, the collection of the training set comes chronologically before the test one. Thus, the following procedure was applied to evaluate the two models (iterative and autoencoder) and establish whether they are able to recognise and quantify the effect of the flood. 1. The _iterative model_ and the _autoencoder_ introduced in Section 4 are trained on the training set. Then, both the training set and the test set are passed to the two models. The estimated values are then used to compute the residuals (Equations (3) and (5)). 2. For each sensor, a moving average and a moving standard deviation are applied with a one-week window to evaluate the distributions of residuals in time. 3. For each month, the mean absolute error (MAE) of the scaled residuals is computed; that is, Equations (3) and (5) are applied before the true values, and the estimations are scaled back to the original range. This allows the evaluation of how the sensors detect unusual behaviours on a comparable scale. The values obtained during step 2 are computed for both the iterative model and the autoencoder. The results for sensor T1 are plotted in Figures 10 and 11. In red are the mean values, whereas the area in orange refers to a double confidence level (\(\pm\)2 std). From the figures, one might notice that not only the dispersion of the data for the iterative model is lower, but also the results at the end of 2022 are more consistent compared to those of the autoencoder. In fact, the residual for the AU model has an unexpected deviation at the end of 2022, which does not have any physical reason behind it. For other sensors, such as T2, the residuals produced by the two models are surprisingly similar (Figures 12 and 13). Likewise, this happens to T3, T4, T7, T8, T9, T12 and T13. For other sensors, such as T10, the iterative model seems to succeed in recognising the flood effects (Figure 14) whereas the autoencoder totally failed (Figure 15). Figure 11: Autoencoder: T1 residuals. Figure 10: Iterative model: T1 residuals. Figure 14: Iterative model: T10 residuals. Figure 12: Iterative model: T2 residuals. Figure 13: Autoencoder: T2 residuals. Figure 15: Autoencoder: T10 residuals. Consistently with step 3, a barplot has been produced for every month and model. Since the residuals are computed before scaling back to the original measurement units, the distributions associated with every sensor (in the training set) have a mean of zero and a variance of one, and, consequently, the results can be compared. For instance, Figures 16 and 17 show the bar plots of MAE for the first month available after the flood (December 2020). The iterative model estimates that strong variations are located at sensors T1, T8 and T9, whereas the situation is practically nominal in T2, T4 and T13, for instance. On the contrary, the autoencoder is less sensitive overall to variations since the scale is even lower. By looking at the later months, the MAE of the residuals rapidly decreases. The month of June is reported in Figures 18 and 19.