## 3 Basics for Implementation of the Machine-Learning Algorithm
### Damage-Identification Results
#### 3.3.1 Damage Detection  
The dataset organized in Table 4 was used for training the detection ANN; it was composed of 560 observation samples, including 81 features each. The FFT in the dataset were affected by damages that varied in terms of position and dimension. Note that 280 examples of the same undamaged condition were sampled from the noise PDF in order to produce a balanced database.  
The PCA was applied to reduce the input-space dimensionality, thus allowing a more efficient ANN training. In Figure 18, the variance associated with each principal component of the detection dataset is presented, in which it is possible to see, by considering only the first 10 principal components, that it was possible to consider 99% of the total variance included in the dataset. As an example, six FFTs observed in presence of different damage extents were compared with their PCA reconstructions in Figure 19, specifically including just 10 out of 81 principal components. It is clear that a negligible error was made when considering the reduced input space when including the first 10 principal components only.  
Figure 17: SHM architecture.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Range of** & **Units of** \\  & **Values** & **Measure** \\ \hline
**Damage radial** & [12 24 36] & mm \\
**position** & [0 0.2 0.35 0.48 0.63 1] & Equivalent pitted area \\ \hline \hline \end{tabular}
\end{table}
Table 3: Training dataset composition.  
Figure 19: A comparison among the original FFT and the reconstructed FFT with 10 principal components: (**a**) Undamaged, (**b**) damage = 0.2, (**c**) damage = 0.35, (**d**) damage = 0.48, (**e**) damage = 0.63, (**f**) damage = 1.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \# Observations &
\begin{tabular}{c} **Equivalent** \\ **Pitted Area** \\ \end{tabular} & **Classification** \\ \hline
280 & 0 & Undamaged \\ \hline
56 & 0.2 & Damaged \\ \hline
56 & 0.35 & Damaged \\ \hline
56 & 0.48 & Damaged \\ \hline
56 & 0.63 & Damaged \\ \hline
56 & 1 & Damaged \\ \hline \hline \end{tabular}
\end{table}
Table 4: Training-set composition for damage detection.  
Figure 18: Variance associated with each feature after PCA on the detection dataset and cumulative sum.  
The classification ANN architecture included 10 input nodes, one binary output node, and 20 hidden nodes, with the latter defined based on trial-and-error procedure; however, as a compromise was made between training accuracy and generalization performance on test data never seen during training, to avoid overfitting. The ANN performances were synthetized with confusion matrices as shown in Figure 20, in which the similar performances in the training, validation, and test sets suggest the ANN possessed a high generalization capability, and no overfitting error occurred. Overall, less than 10% of misclassified examples were obtained.  
Since the detection performance was tested in the presence of damages of increasing extent, thus with different influences on the simulated signal, it was interesting to verify the detection performance (in terms of the number of misclassified samples) as a function of the damage dimension. It is clear from Table 5 that the probability for missed detection reasonably increased for smaller damages, while the detection performance improved with increasing damage entity.