## 2 Framework of RUL Prediction Based on TCNMABG
### Signal Acquisition and Data Preprocessing
#### 2.2.2 Signal Noise Reduction Method Based on DBOVMD  
When using the variational mode decomposition (VMD) method for signal noise reduction, the number of modal decomposition \(k\) and the penalty factor \(\alpha\) are crucial to the performance and noise reduction effect of the algorithm. Choosing an appropriate \(k\) value can retain effective information when decomposing the signal structure, but a high  
Figure 1: Framework of RUL prediction based on TCNMABG for rolling bearing. \(k\) value may introduce unnecessary details or noise. The penalty factor \(\alpha\) regulates the sparsity and smoothness of the modal function. Too large a value may lead to excessive smoothness, while too small a value may make the decomposition too sparse. Therefore, the correct selection of \(k\) and \(\alpha\) is crucial for balancing the retention of effective signals and the suppression of noise. By adjusting these two parameters, optimizing the VMD algorithm to adapt to different signal characteristics and noise reduction requirements is a key step in improving the noise reduction effect and algorithm performance. In this paper, a signal denoising algorithm based on Dung beetle optimization VMD (DBOVMD) is used to find the best parameter combination \(k\) and \(\alpha\) to reduce the noise of the original data. This method solves the complex optimization problem by simulating the path-planning process of dung beetles when they search for food, with fast convergence speed and high precision. The fitness function in the DBOVMD algorithm has a significant impact on the parameter optimization results. Envelope entropy can be used to analyze the time-varying characteristics, dynamic behavior, and nonlinear characteristics of time series signals, effectively representing the sparse characteristics of the original signal, especially suitable for non-stationary signals. Therefore, envelope entropy is chosen as the fitness function for the DBOVMD algorithm. When there is more noise and less feature information in the IMF components, the envelope entropy value is larger; otherwise, the envelope entropy value is smaller. The detailed DBOVMD process is summarized as follows, which is shown in Figure 2.  
Figure 2: Flowchart of DBOVMD algorithm. (1) Initialize the key parameters of the DBO algorithm [32], including dung beetle population size, iteration number, and other parameters \(k\) and \(\alpha\). (2) Perform VMD based on the determined parameter combination. (3) Divide the population randomly into four types, including ball-rolling dung beetles, breeding dung beetles, foraging dung beetles, and stealing dung beetles. (4) Calculate the fitness values of all the dung beetle positions by envelope entropy, which is shown in Equation (1). \[\left\{ \begin{aligned} E_{p}&=-\sum\limits_{j=1}^{N}p_{ j}\text{lg}p_{j}\\ p_{j}&=a(j)/\sum\limits_{j=1}^{N}a(j)\end{aligned} \right. \tag{1}\]  
where \(a(j)\) is the envelope signals obtained by Hilbert demodulation of \(k\) modal components decomposed by VMD. \(p_{j}\) is the probability distribution sequence obtained by normalizing \(a(j)\). \(N\) is the number of sampling points, and the envelope entropy \(E_{p}\) can be computed by evaluating the entropy of the probability distribution sequence \(p_{j}\). (5) Update the positions of ball-rolling dung beetles, breeding dung beetles, foraging dung beetles, and stealing dung beetles by different strategies. (6) Calculate the fitness values of the updated population. (7) Update and save the optimal solution as the current population with the best fitness value when it is better than the previous optimal solution. (8) If the current iteration exceeds the maximum number of iterations, output the optimal parameter combination; otherwise, return to Step 2. According to the optimal parameter combination of \(k\) and \(\alpha\), the vibration signal is decomposed into modes of different frequencies, and each internal modal component IMF can be evaluated based on the correlation coefficient between the IMF and the raw signals.