## 4 Selection of the Mother Wavelet
In WPT, increasing the decomposition level leads to finer frequency resolution, but also to a rapid increase in the number of data (e.g., \(2^{n}\) data for level \(n\)), which can result in redundant information, overfitting, or unnecessary computational load. At level 3, the signal is decomposed into 8 frequency bands, which is typically sufficient to isolate transient high-frequency components associated with crack initiation events in AE signals. Preliminary comparisons with levels 2 and 4 showed that level 2 lacked sufficient frequency resolution to distinguish subtle differences in AE energy distributions, while level 4 increased feature dimensionality without significantly improving discrimination performance. Therefore, level 3 was selected as a practical and effective compromise for this application. It is worth noting that, for Symlets and Daubechies families, the order of the mother wavelet coincides with the number of vanishing moments, while, in the Coiflets family, the number of vanishing moments is double the order, which means that Coif4, for example, has 8 vanishing moments, whereas Daubechies 8 has 8 vanishing moments. For the mother wavelet selection, the following steps have been carried out:  
1. First, two different conditions of the axle must be selected (Figure 4). The first class or condition, from now on called \"class 0\", should be considered at the very beginning of the test, since it is known that the axle has the healthiest condition and there should be no crack initiation yet. The second one, from now on called \"class 1\", can be considered before the load increase so that it can be analyzed under the same test conditions and that sufficient time has passed for susceptible changes to occur. This selection will be justified with a subsequent change point analysis (CPA) in point 3. During the fretting fatigue test, a continuous acoustic emission (AE) monitoring system recorded a large number of raw signals throughout the cycling process. However, many of these signals corresponded to background noise, frictional contact, or spurious sources unrelated to material degradation. To ensure the relevance and quality of the data analyzed, a manual preselection process was applied and only some of the signals displaying a clear burst-type waveform, characteristic of AE events linked to crack initiation or structural activity, were retained. As a result, a curated dataset of 822 waveforms was obtained: 528 from the healthy condition (class 0), collected during the first 100,000 cycles, and 294 from the early damage condition (class 1), acquired over the subsequent 10,000 cycles. The larger number of waveforms in the healthy class reflects the need for a more comprehensive baseline characterization of the undamaged system, which is essential for assessing deviations associated with incipient damage. Figure 4 shows both selected classes, the number of cycles of each class and the stress conditions applied during the test. 2. WPT with a decomposition level of 3 is applied, with different mother wavelets and the mean energy of the eight obtained packets are calculated, which means that the frequency range is divided into eight equal parts. Considering the sampling frequency shown in Table 1, each packet has a resolution of 3.125 \(\times\) 10\({}^{5}\) Hz (2.5 \(\times\) 10\({}^{6}\) Hz/8). 3. A change point analysis (CPA) is performed [53] to check the previous selection of classes 0 and 1. This algorithm allows to find noticeable trend changes in the data, according to statistical parameters such as the mean, as in this case (packets energy). It also sets when that change happens. If the energy of all packets is represented for all the selected signals (Figure 5), where the energy of packet 1 (blue) and packet 2 (orange) stand out from the rest and, therefore, contain most of the energy, it can be seen that the selection of the data is well carried out, because there is an increase in the mean energy with class 1.