# A Deep Learning Approach to Detect Failures in Bridges Based on the Coherence of Signals
## 3 Multi-Input Machine Learning Modelling
### Iterative Models  
The dataset collected from a structure is unsupervised most of the time, namely a dataset with no specific labels (numerical or categorical) associated with the dataset objects. Most of the unsupervised approaches rely on the detection of the dataset modifications. However, in this way, the monitoring system is only able to identify whether something has changed, without actually _localising_ where the modification took place [20]. A possible approach to overcome these limitations could be to turn the unsupervised problem into a _supervised_ one. For example, some health monitoring algorithms are based on the use of environmental variables (such as temperatures) as the input of a regressor, which must estimate the features extracted from the sensors. This allows the normalisation of the signals with respect to the environmental effects, before applying an anomaly detection algorithm that is usually based on statistics [21; 22]. If the physical variables measured are _static_, which means they measure low-frequency variations only, this procedure can be applied directly to the sensors' readings, without any particular feature extraction. Among the possible machine learning methods for regression, _neural networks_ are among the most flexible ones. The layers of neural networks can be selected and adjusted to take in input and output tensors of any size. It is then possible to divide the sensors mounted on the structure into groups according to the type of physical quantity that they measure (tiltmeter, strain gauges, thermometers, etc.) or their location (first span of the bridge, second span, and so on) and then train a neural network so that it can use one group from the input to estimate the values of the sensors of another group. Then, the residuals \(e_{ij}\) can be computed as:  
\[e_{ij}=y_{ij}-\hat{y}_{ij} \tag{3}\]  
where \(y_{ij}\) and \(\hat{y}_{ij}\) refer to the measured value and the value estimated by the neural network, respectively, while the subscript \(i\) refers to the \(i^{th}\) sensor considered and \(j\) to the \(j^{th}\) timestamp. The residuals can then be used as an indicator of the structure's health condition. In this way, the _normalisation of the signals with respect to the environmental effect_ can just be seen as a particular case of this procedure, where the inputs of the neural network are the _temperatures_ and the output of a group (or multiple groups) of sensors, as shown in Figure 3. By following this procedure, the _residuals_ obtained represent all the disturbances that cannot be attributed to temperatures. However, one might also focus on one single group and establish the effect of all of the other groups of sensors (Figure 4). The physical meaning of residuals is now different compared to the case where only the temperatures are used as the input. Now the model takes into account the readings of sensors of different natures. Therefore, the residuals are proportional to the _level of coherence_ of one group of sensors with respect to the others. By iterating the process for all groups of sensors, given \(G\) groups, every iteration consists of the training of \(G\) models, so that each of them estimates the value of one group of sensors taking into account the other \(G-1\). The logical extension of this procedure is to consider only the groups made of _one single_ sensor (leave-one-out strategy). If \(N\) sensors are mounted on the structure, this strategy involves the training of \(N\) neural networks, so that at every iteration the neural network estimates the value of one single sensor by taking the other \(N-1\) ones as the input. Eventually, a _comprehensive model_ made of \(N\) neural networks is associated with the structure. For simplicity, we call this \"comprehensive\" model an _iterative model_. It is worth mentioning that the iterative model can be made of other machine learning regressive models rather than neural networks, since the output is, in this case, a single continuous variable. [23; 24].