## 6 Results
### Results from the Numerical Simulations
By summing up all of the damage scenarios of this first case, it turns out that of all features, the TECCs are the only one always correctly labelling the cases with realistic damage patterns (05-14) and the ones with global, small fluctuations of \(E\), unrelated to damage (15-18). Indeed, the TECCs fall short of a perfect score for only three mislabeled realisations (out of 10) in case 01, which was also the most demanding scenario due to the proximity of the damage to the fixed end. Figure 9: Damage indexes for TECCs (**a**) and MFCCs (**b**); first numerical case study (Fossano bell tower), scenarios 00–04 (story-level damage). Figure 11: _Cont._  
Figure 10: Damage indexes for TECCs (**a**) and MFCCs (**b**); first numerical case study (Fossano bell tower), scenarios 05–14 (realistic damage). Table 8 reports the results for the second case study (the highly flexible prototype wing); Figure 12 graphically illustrates them. It can be inferred that, in this case, the exact detection of damage was more difficult. MFCCs and similar variants proved to be able to spot everywhere the damage but at the cost of not actually discerning it from the two cases with small perturbations. The TECCs, on the other hand, did not perform perfectly, yet they avoided all but one false alarm and were able to discern the damaged conditions decently in the inspected scenarios. If one does not take into account Case 07, which presents an imperceptible decrease of 0.03 Hz and went completely unnoticed by the trained algorithm, the type 2 error percentage drops to 7.50%, with two errors in Case 04, no one in Case 05, and a single mislabeled element in Case 06. Figure 11: Damage indexes for TECCs (**a**) and MFCCs (**b**); first numerical case study (Fossano bell tower), scenarios 15–18 (no damage). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Case 00** & **Cases 01–04** & **Cases 05–14** & **Cases 15–18** \\ \hline  & **Type 1 Errors [\%]** & **Type 2 Errors [\%]** & **Type 2 Errors [\%]** & **Type 1 Errors [\%]** \\ \hline MFCCs & 60.00 & 0.00 & 0.00 & 45.00 \\ \(f_{\text{cut-off}}=\frac{1}{4}\ f_{\text{s}}\) & 30.00 & 0.00 & 0.00 & 45.00 \\ \(f_{\text{cut-off}}=\frac{1}{8}\ f_{\text{s}}\) & 30.00 & 0.00 & 0.00 & 45.00 \\ Log-Scale & 20.00 & 0.00 & 0.00 & 65.00 \\ Linear & 60.00 & 0.00 & 0.00 & 45.00 \\ ECO-MMFCCs & 20.00 & 0.00 & 0.00 & 62.50 \\ MO-MMFCCs & 10.00 & 0.00 & 0.00 & 62.50 \\ LO-MMFCCs & 20.00 & 0.00 & 0.00 & 65.00 \\ BFCCs & 60.00 & 0.00 & 0.00 & 47.50 \\ GTCCs & 20.00 & 0.00 & 0.00 & 60.00 \\ TECCs & 0.00 & 7.50 & 0.00 & 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results for the first numerical case study (bell tower). As in the previous case of the Fossano bell tower, this seems to point out the larger robustness of the TECCs feature to small changes, assumed here as statistical variations of the structural parameters and not directly linked to damage occurrence. A possible explanation comes from the TKE Operator being notably sensitive to frequency and amplitude changes, while much more robust than MFCCs to noise and minimal variations in the field of speech recognition accuracy. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Cases 00–02** & **Cases 03–07** \\ \cline{2-3}  & **Type 1 Errors [\%]** & **Type 2 Errors [\%]** \\ \hline MFCCs & 66.67 & 0.00 \\ \(f_{\text{cut-off}}=\frac{1}{4}\int_{s}\) & 66.67 & 0.00 \\ \(f_{\text{cut-off}}=\frac{8}{8}\int_{s}\) & 66.67 & 0.00 \\ Log-Scale & 70.00 & 0.00 \\ Linear & 66.67 & 0.00 \\ ECO-MMFCCs & 66.67 & 0.00 \\ MO-MMFCCs & 66.67 & 0.00 \\ LO-MMFCCs & 66.67 & 0.00 \\ BFCCs & 66.67 & 0.00 \\ GTCCs & 66.67 & 0.00 \\ TECCs & 3.33 & 26.00 (*) \\ \hline \hline \end{tabular} (*) 7.50\% on Cases from 03 to 06.  
\end{table} 
Table 8: Results for the second numerical case study (HAR wing). Figure 12: Damage indexes for TECCs (**a**) and MFCCs (**b**); second numerical case study (highly flexible prototype wing). In interpreting the data, one must consider the different definition of 'damage' between the two numerical and the experimental case studies.