## 6 Results
### Results from the Numerical Simulations  
Regarding the first numerical case study (the Fossano bell tower), cases 00 to 04--i.e., the baseline and the story-uniform damage cases--are visually compared in Figure 9 to prove the algorithms capability to discern simpler configurations and widely extended damage from the normality model. Figure 10 shows the results for cases 05 to 10 (realistic damage scenarios). To estimate the robustness of each MFCC-based variant to false alarms, cases 15 to 18 (i.e., the scenarios with small fluctuations of the Young's modulus respect to the baseline) are presented in Figure 11. Apart from the Teager-Kaiser Energy Cepstral Coefficients, the standard definition of MFCCs is also shown for comparison. Table 7 reports all results. In all figures, the black dashed line represents the defined threshold; bars exceeding it are coloured in red and they stand for the realisations labelled as damaged, while blue bars indicate values of DI under the prescribed threshold. From a statistical pattern recognition standpoint, this means that the cepstral features that correspond to that specific realisation have not departed considerably enough from their respective values for the reference baseline and are therefore labelled as undamaged. As can be inferred from the first two columns of Table 7, the GTCCs, the Logarithmic Scale CCs, and the three Mel-Modified Scales (ECO-, MO-, and LO-MMFCCs) have all relatively low type 1 errors. The Damage Index values inside any given damage case are quite similar; that is a valuable proof of the robustness of the method. Bark-scale based CCs wrongly labelled as damaged 60% of the undamaged inputs coming from the same scenario used for the training. The TECCs were the most performing feature; in Figure 9.a one can clearly see that two out of the three mislabeled realisations for TECCs fall very short of the threshold. The third column in Table 7 (cases 05-14) shows that all algorithms work effectively in terms of damage detection. The classic definition of the Mel-Scale, the Bark-Scale, and the linear Scale nevertheless present some relatively larger fluctuations along the ten realisations of the same damage scenario with different inputs. The Log-Scale and the Mel-modified Scales provide more stable results, with a flat trend inside each damage scenario. This seems to indicate that a kind of nonlinear scale performs better than the constantly spaced filters. The filter shape might be even more incisive, as GTCCs and TECCs are all spaced according to the Bark Critical Bands yet perform better than triangular filters-based options both in terms of type I and II errors. Most of the algorithms recognise substantial gaps (Figures 8 and 9) between different scenarios, which reflect the difference of damage extension; this seems to indicate a possibility to also exploit them for instances of damage severity assessment. Finally, the last column of Table 7 (cases 15 to 18) evidently portrays how only the TECCs correctly label all the proposed cases as unmodified respect to the baseline; the other features proved to be quite unsatisfactory, especially the Mel-Scale and Bark-Scale. That seems to once more validate the assumption that the straight application of the original definition of the mel-scale, tuned for speech processing aims, is not well-performing and proper corrections are needed. In particular, the scenarios with a variation in the material properties of \(\pm\)0.25% are most often correctly labelled as healthy; otherwise, the \(\pm\)1.00% deviation of the Young's modulus provides DI values above the threshold. This phenomenon could be justified by the fact that even a relatively small difference between the pristine and modified mechanical properties of the model's material cause a greater variation in the power spectrum of the structure, which is also reflected in the power and is wrongly seen as damage-induced. This aspect is not helpful in the damage detection process; the TKE operator seems to be more resilient to these slight changes.