## 4 Selection of the Mother Wavelet
In this way, it is proved that there is a considerable increase in the mean energy from measure 525, which almost corresponds with the beginning of class 1. 4. The DEV value for each mother wavelet is calculated (this parameter was defined in previous work [25]). It measures the difference in energy between two signal conditions: in this case, the healthy condition (i.e., class 0) and another condition taken at a higher number of cycles (i.e., class 1). A higher DEV value indicates a greater energy contrast between both states, meaning the wavelet is more effective at highlighting differences associated with the progression of damage. This is particularly relevant in the context of crack detection, since the key to early and reliable identification of structural changes lies in the ability to discriminate subtle variations in signal features. A wavelet that maximizes the DEV value ensures greater sensitivity to damage, improving the model's ability to differentiate between normal and anomalous conditions. Therefore, selecting the mother wavelet with the highest DEV is essential for optimizing the performance of the monitoring system (Equation (6)). \[DEV=\sqrt{\frac{\sum_{i=1}^{n}\left(P_{i}-O_{i}\right)^{2}}{n}} \tag{6}\]  
where \(n\) is the number of packets, \(i\) is the packet number, \(P_{i}\) is the mean energy of packet \(i\) for the instantaneous condition (defective), and \(O_{i}\) is the mean energy of packet \(i\) for the reference condition (healthy). 5. According to point 4, the mother wavelets (MW) with the highest DEV values will be preselected (\(DEVmax\)). These values will be used for each MW \(DEV(n)\) to calculate the variation in the DEV (\(\mathrm{DEV}(\%)\)) with Equation (7). Those MW with less than 2% variation in the DEV value with respect to the one with the highest DEV value will also be preselected. The choice of the 2% threshold is based on its successful application in previous studies [25], where it provided an effective compromise between sensitivity and selectivity in the context of wavelet-based signal characterization. In this work, the same criterion was adopted to retain mother wavelets whose DEV values were within 2% of the maximum observed value. This threshold ensures that only wavelets with comparable discriminatory power are selected, while preventing the inclusion of marginally relevant ones. Although heuristic in nature, the 2% value was also verified in the present study through preliminary sensitivity analysis, showing that variations in the threshold (e.g., 1% or 5%) had negligible influence on the ranking of top-performing wavelets, but resulted either in unnecessary narrowing or broadening of the candidate set. Therefore, the use of the 2% threshold is not arbitrary but reflects a practical and validated trade-off suitable for the current application. The DEV evolution for all mother wavelets is shown in Figure 6a. In Figure 6b, the DEV variation (%) values are shown. \[DEV(\%)=\frac{DEVmax-DEV(n)}{DEVmax}\cdot 100 \tag{7}\]  
Figure 4: Considered classes for mother wavelet selection. According to the previous criterion, for [PERSON] family, the highest DEV is obtained with Daubechies 10 (Figure 6a) and below that order there is none of said family with a variation of less than 2% (Figure 6b), so only the db10 is preselected. For Symlets family, the highest DEV is obtained with sym9 (Figure 6a), and sym6, sym8 and sym10 have a DEV value with less than 2% variation (Figure 6b), so these mother wavelets are preselected for this family. About the [PERSON] family, coiflet 5 (10 vanishing moments) and 4 (8 vanishing moments) are preselected for the same reasoning. The packet energies obtained using the preselected mother wavelets are used as input for an intelligent classification system built with MATLAB R2024a(tm)'s \"Classification Learner\" app [54]. This tool allows training and comparing multiple classification models under the same conditions.