## 2 Framework of RUL Prediction Based on TCNMABG
### Multi-Domain Mixed Feature Extraction and Selection  
#### 2.3.1 Multi-Domain Mixed Feature Extraction  
The vibration signals can reflect the real-time status information and degradation trend of the equipment. To fully characterize the time change characteristics, frequency components, energy distribution, and other information of the signal. In this paper, dimensional time domain features, dimensionless time domain features, frequency domain features, entropy features, and spectral kurtosis features of signals are extracted respectively to form a multi-domain mixed feature set, and the equipment degradation process is described from multiple dimensions to predict the remaining useful life of rolling bearing equipment. The dimensional time domain feature is sensitive to the signal characteristics of the equipment running state, but it is also susceptible to changes in load, speed, and other working conditions. The dimensionless time domain features are less affected by environmental disturbance but less sensitive to fault information. Therefore, the comprehensive use of dimensionless and dimensionless time domain features for feature extraction can give full play to their respective advantages so as to monitor the operating state of the equipment more effectively and obtain the degradation information of the equipment. The 17 time-domain feature calculation formulas extracted in this paper are shown in Table 1.  
Frequency domain features can reflect the distribution of different frequency components in the signal, the degree of energy concentration, and the change of frequency. The calculation formulas of 5 frequency domain features extracted in this paper are shown in Table 2.  
The entropy feature can effectively measure the uniformity of the signal and the complexity of the probability distribution and can quantify the confusion and uncertainty of the signal, which helps understand the information and randomness contained in the signal. In this paper, three entropy feature indicators (F23, F24, F25) are extracted, including power spectrum entropy, singular spectrum entropy, and energy entropy. The spectral kurtosis feature can effectively identify the transient shock and its distribution in the frequency band from the signal containing background noise and can deal with the non-stationary signal well. This article extracts four features related to spectral kurtosis (F26, F27, F28, F29), including mean value, standard deviation, skewness, and kurtosis of spectral kurtosis. Therefore, a total of 29 feature indexes are extracted from the vibration signal after noise reduction to form a multi-domain mixed feature set. To better select appropriate and effective degradation features from the constructed multi-domain mixed feature set, a comprehensive evaluation index is developed by the weighted sum of correlation and monotonicity, which can better consider the correlation of time-series features and monotonicity in the degradation process. The comprehensive evaluation index can be evaluated by Equation (2). \begin{table}
\begin{tabular}{c c c c c c} \hline
**Features** & **Meaning** & **Calculation Formula** & **Features** & **Meaning** & **Calculation Formula** \\ \hline F1 & Max & \(x_{\text{max}}=\max\{|x_{i}|\}\) & F2 & Min & \(x_{\text{min}}=\min\{|x_{i}|\}\) \\ F3 & Mean & \(\overline{x}=\frac{1}{N}\sum\limits_{i=1}^{N}x_{i}^{2}\) & F4 & Median & \(x_{\text{med}}=X\left(\frac{N+1}{2}\right)\) \\ F5 & Peak-to-peak value & \(x_{pp}=x_{\text{max}}-x_{\text{min}}\) & F6 & Average & \\  & & & & & verified \\ F7 & Variance & \(x_{\text{var}}=\frac{1}{N-1}\sum\limits_{i=1}^{N}(x_{i}-\overline{x})^{2}\) & F8 & Standard & \\  & & & & deviation & \(\sigma=\sqrt{\frac{1}{N!