## 2 Materials and Methods
### Sequential Features Generation
Power bandwidth: Identifies the frequency range within which a certain percentage of the signal's power is concentrated, offering insights into the spread of the signal's energy;
6. Peak amplitude: Captures the maximum amplitude within the frequency spectrum, pointing to the dominant frequency component;
7. Peak location: Specifies the frequency at which the peak amplitude occurs, aiding in the identification of the principal frequency component. For each of the 11 signals and the seven features proposed above, the extraction process produced a matrix of size \(100\times 77\), resulting in a total of 77 potential sequential features from the frequency domain. In the time-frequency domain, the MeanEnvelopeEnergy feature was generated to analyze tool wear in milling processes. This variable captures the mean energy of the upper and lower envelopes of each intrinsic mode function (IMF), providing insight into the tool's condition during operation [41; 42; 43; 44]. The generation of the MeanEnvelopeEnergy involved computing the mean envelope energy of the IMFs derived from the input time-domain signals. The empirical mode decomposition (EMD) method was applied to decompose the signals into their IMFs. This is a critical step in capturing the intrinsic oscillatory modes embedded within the data. The MeanEnvelopeEnergy was then calculated by averaging the energy of the upper and lower envelopes for each IMF, thus encapsulating the signal's energy distribution in a comprehensive manner. For each of the 11 signals under investigation, the process produced a 10-element vector, resulting in a total of 110 potential features (\(100\times 110\) matrix) (MeanEnvelopeEnergy) for subsequent analysis. This rich feature set served as the foundation for the development of predictive models aimed at enhancing tool condition monitoring in milling processes, ultimately contributing to the optimization of manufacturing operations. The practical implications of using the MeanEnvelopeEnergy are significant, providing a detailed understanding of tool wear dynamics. By integrating these features into deep learning algorithms, the study aims to achieve high-accuracy predictions of tool wear, thereby facilitating timely maintenance interventions and reducing the likelihood of tool failure. This approach not only promises to elevate the precision of tool condition monitoring but also contributes significantly to the sustainability and efficiency of manufacturing workflows. The mathematical formulation for the MeanEnvelopeEnergy feature is defined as follows:  
\[MeanEnvelopeEnergy=\frac{1}{N}\sum_{n=1}^{N}\left(\frac{1}{M_{n}}\sum_{m=1}^{ M_{n}}\sqrt{U_{m}^{n}(t)^{2}+L_{m}^{n}(t)^{2}}\right) \tag{1}\]  
where  
* \(N\) is the total number of IMFs considered for each signal;
* \(M_{n}\) represents the number of discrete time points in the \(n\)-th IMF;
* \(U_{m}^{n}(t)\) and \(L_{m}^{n}(t)\) denote the upper and lower envelope values at time \(t\) for the \(m\)-th time point in the \(n\)-th IMF, respectively. This formula encapsulates the essence of the MeanEnvelopeEnergy feature by averaging the energies of the upper and lower envelopes across all IMFs and time points. The resulting feature provides a compact yet informative representation of the signal's energy characteristics, pivotal for assessing tool wear in the milling processes. In the process of generating sequential features, a total of 308 features were produced, represented in the form of a matrix with dimensions \(100\times 308\). These features can be categorized into three distinct domains, each contributing a specific set of features:  
1. Time domain features: A total of 121 features were generated within the time domain, organized into a matrix of dimensions \(100\times 121\). These features are primarily derived from the time-based analysis of the sequential data;
2. Frequency domain features: In the frequency domain, 77 features were identified and encapsulated within a matrix of \(100\times 77\).