# A Deep Learning Approach to Detect Failures in Bridges Based on the Coherence of Signals
## 3 Multi-Input Machine Learning Modelling
### Iterative Models
By considering the set of measurements \(\mathbf{Y}_{j}=\left[y_{1j},\cdots,y_{nj}\right]\) available at the \(j^{th}\) timestamp, the iterative model can synthetically be represented by the system:  
\[\hat{y}_{ij}=\hat{f}_{i}(\mathbf{X}_{ij})\quad\text{for}\quad i=1,\cdots,n \tag{4}\]  
where \(\mathbf{X}_{ij}=\mathbf{Y}_{j}-\{y_{ij}\}\) represents the set of values measured by all the sensors except the \(i^{th}\) one, whereas \(\hat{f}_{i}\) is the regressor able to estimate the value of the \(i^{th}\) sensor from \(\mathbf{X}_{ij}\). Finally, the residuals can be computed with Equation (3). In this context, the residuals represent how distant the value measured by a sensor is from the expected one, and, therefore, the equation quantifies coherence among all sensors' readings. This approach reaches level 2 in the SHM hierarchical structure, namely the localisation of damage, as well as its assessment, even if only to a limited degree. In this regard, it is necessary to remark on an important aspect: the application of an iterative model to a new observation \(\mathbf{Y}_{\text{new}}\) will produce a vector of residuals: \(\mathbf{E}_{\text{new}}=\left[e_{1,\text{new}},\cdots,e_{n,\text{new}}\right]\), which must show low values in the case of a healthy structure. In addition, in the case of damage caused at the \(i^{th}\) sensor, the residual vector \(\mathbf{E}_{\text{new}}\) is expected to feature a \"spike\" in the \(i^{th}\) residual \(e_{i,\text{new}}\). However, since \(y_{i,\text{new}}\) is the input for the other sensors, an increase in the residuals will be seen in all of the other elements of the vector \(\mathbf{E}_{\text{new}}\). Therefore, the damage identification capability is maintained  
Figure 4: A model estimates the values of one group of sensors by taking all of the other groups as the input. Figure 3: A model (such as a neural network) takes the temperature as the input and gives an estimation of the readings of other sensors. as long as the residual in the \(i^{th}\) position of \(\mathbf{E}_{\text{new}}\) grows more than the others. However, since the method is fully data-driven, it is hard to define when and how this might happen. During the monitoring phase, some statistical considerations can be carried out on the vector of residuals \(\mathbf{E}\) to define the confidence level for raising warnings. It is worth mentioning that this algorithm has been tested on two other bridges that were monitored for about one and a half years. Iterative models were trained over a one-year time span and tested on the remaining period. In both structures, no anomaly conditions were recorded since the obtained residuals for all of the sensors maintained low values in time, comparable to the training loss. However, in this paper, the results related to these two bridges are not included because the authors want to focus on a test case in which a real _anomaly condition_ has been met and correctly identified. In fact, a flood is an exceptional event that has hardly been encountered in previous research works related to bridge monitoring. Due to the limited period of monitoring, it is not possible to be sure that the algorithm is robust to long-term changes. However, due to the fact that it evaluates the coherence of the different signals and, indirectly, the measurement of a single variable, it is realistic to suppose that it will be robust to long-term changes. If the algorithm finds a change in the system, it can be a preliminary warning that an inspection or, more generally, an evaluation of the bridge is required. If after the inspection, the bridge is considered healthy, it is possible to retrain the model and, therefore, to define a new nominal condition.