## 2 Methodology
### One-Dimensional Convolutional Neural Network  
The LeNet-5 model proposed by LeCun in 1998 was the first convolutional neural network (CNN), and its structure mainly contained input and output layers, convolutional, pooling, and fully connected layers. CNNs have strong self-learning, parallel-processing, and fault-tolerance capabilities, are characterized by a low risk of overfitting and high model training efficiency compared with traditional neural networks, and they have been widely used in image processing. The main difference between one-dimensional convolutional neural networks (1 DCNN) and CNNs is the difference in input data dimension, so 1 DCNNs have a greater advantage in one-dimensional signal processing. Due to the rolling bearing vibration signal being a one-dimensional time-series signal, the 1 DCNN structure is used in this paper for model training. Figure 1 shows the basic structure of the 1D convolutional neural network. The role of the one-dimensional convolutional layer is to perform the feature extraction of the input signal [29]. The formula for the convolutional operation performed by current convolutional layer I is shown below [29]:  
\[x_{j}^{l}=f\left[\sum_{i\in M}\left(x_{i}^{l-1}*k_{ij}^{l}\right)+b_{j}^{l}\right] \tag{18}\]  
In the above equation, \(x_{j}^{l}\) is the output mapping of the \(j\)-th neuron of the current (\(l\)) layer; \(x_{i}^{l-1}\) is the output mapping on the previous layer, which is also the input of the current (\(l\)) layer; \(k\) is the convolutional kernel; \(j\) is the number of convolutional kernels; \(M\) is the number of channels of input \(x_{i}^{l-1}\); \(b\) is the deviation of the corresponding kernel; \({}^{*}\) is the convolutional operator; \(f\) is the activation function, generally the ReLU function. The activation function is PReLU in this paper to solve the problem of zero gradients of the ReLU activation function, the formula of which is as follows:  
\[a_{j}^{l}=f\left(x_{j}^{l}\right)=\max\left(cx_{j^{l}}^{l},x_{j}^{l}\right) \tag{19}\]  
Parameter c is learnable and generally takes a number between 0 and 1. In this paper, parameter c is set to 0.25. The role of the activation layer is to non-linearly map the output of the previous layer, and the ReLU activation function has strong convergence  
Figure 1: The 1 DCNN structure. to overcome the problem of gradient disappearance. However, when the value of the input activation function is always negative, then the gradient through this point in the backpropagation process is a constant of 0 that cannot update the bias parameters and the corresponding weight. Therefore, the ReLU activation function has neuronal decay, which can be effectively solved by the PReLU activation function. The function of the pooling layer is to downsample the increased amount of feature sequence data extracted from the convolutional layer to prevent the occurrence of overfitting by reducing the quantity of data computation in the neural network. This usually uses maximal pooling for downsampling; the formula of maximal pooling is as follows:  
\[y_{i}^{l}=\mathit{maxpooling}\left(x_{j}^{l},\text{scale},\text{stride}\right) \tag{20}\]  
In the above equation, \(y_{i}^{l}\) represents the output of the i-th neuron of the current (\(l\)) layer; \(\mathit{maxpooling}(x)\) is the downsampling function, which serves to take the maximal value of a certain range; scale is the scale of pooling; and stride is the step of pooling. The dropout layer further prevents the occurrence of the overfitting phenomenon. Using the dropout function can reduce the sensitivity of the neural network to small data changes and effectively improve the accuracy of data processing.