## 5 Case Study
### Feature Extraction  
A series of response history dynamic analyses is performed to obtain the structural vibrations in terms of the acceleration time history of the top of the column (since it is basically a 1 degree of freedom, DOF, system). Then, these signals are analyzed and processed to extract features (usually scalar quantities) that can quantify the health of the structure. These features can be extracted either in the time domain or in the frequency domain. The features analyzed are reported below. Each time series is stored with a health code that is later used for artificial intelligence (AI) training. Figure 3: Scheme of the FE model. In this paper, the following features are extracted and analyzed:  
1. peak amplitude
2. peak frequency
3. damping coefficient
4. peak band
5. natural frequency  
For dynamic identification, the frequency domain decomposition (FDD) algorithm [34] is used, where the search for the resonance peaks is automated using a local maximum search algorithm. Once the features were extracted, they were plotted on graphs so that their quality could be quantified. An excellent feature allows easy discrimination of the healthy or unhealthy state of the structure. The algorithm, one-way analysis of variance (ANOVA), was used to quantify the goodness-of-fit of the extracted features. This is to determine if the data from different groups have a common mean. ANOVA thus makes it possible to find out if different groups of an independent variable have different effects on the response variable. The one-way ANOVA is a simple case of a linear model. Such a model has the form:  
\[y_{ij}\ =\ \alpha_{j}\ +\ e_{ij} \tag{1}\]  
where:  
\(y_{ij}\) is an observation: \(i\) represents the observation number, and \(j\) represents a different group of the predictor variables \(y\);  
\(\alpha_{i}\) represents the population mean for the \(j\)-th group;  
\(e_{ij}\) is the random error, independent and having a normal distribution with zero mean and constant variance \(e_{ij}\sim N\left(0,\sigma^{2}\right)\). ANOVA assumes that all distributions are normal or with small deviations from this assumption. In this study, this assumption is satisfied. In this section, we examine how the AI responds to training, including damage with multiple classes. Specifically, the goal is to predict a corrosion thickness with a sensitivity of 1 mm and a maximum of 5 mm. Figure 5 shows the training scheme used; 600 response history analyses were performed by integrating as many stochastic white noise processes, where 1 mm of thickness is removed every 100 analyses to simulate the layer lost to corrosion. _FaultCode 0_ corresponds to the healthy state, while from _FaultCode 1_, the removal of 1 mm of thickness was started; in this way, the _FaultCode_ identifier also corresponds to mm of corrosion loss. To best simulate the typical conditions of a structure, the mass lumped at the top of the column was introduced stochastically (while it was kept constant in the single analysis to keep the mass assumptions invariant during the \(i\)-th record). A plot of the introduced mass change is shown in Figure 6. This condition represents a challenge for AI prediction, as the AI must distinguish and recognize the change in features due to a change in stiffness (in this case due to corrosion) from a change in mass. Figure 4: Gaussian stochastic processes. Experience and testing have shown that even training AI with the same parameters and features results in different AI with different performance. Therefore, the question arises: what are the parameters that can best be used to train AI with the same features? An optimization problem is then configured for the AI training parameters, properly called hyperparameters hereafter, with the goal of minimizing the error function in classification. The problem takes the form of a huge and incredibly complex multidimensional optimization problem, and it also has a stochastic objective function, since the outcome varies each time an AI is re-trained with the same parameters.