## 3 Methodology
### Multi-Path Time Series Extractor
#### 3.2.2 Multi-Path Time Series Extractor  
This section will provide a detailed introduction on how to use the MPTSE module and other components to build an efficient fault diagnosis model. Each MPTSE is centered around the T-inception mentioned in the previous section, and the modules are organized using residual connections. This design concept originates from residual networks, whose core lies in introducing skip connections so that inputs can be directly passed to subsequent layers, helping gradients flow effectively in the network, thereby supporting the training of deeper network structures and preventing gradient vanishing or exploding problems. In addition, the model adopts a multi-layer stacking design, and the output of each layer will be used as the input of the next layer. Through this continuous processing, the model can extract and enhance fault-related features layer by layer. Normalization and activation layers are also introduced between each layer to further optimize the network's expressive and nonlinear processing capabilities. Through such an overall design, the model not only has strong data processing capabilities in theory but also achieves the expected fault detection effect in practical applications, greatly improving the accuracy and efficiency of rolling fault diagnosis. Specifically, for the bearing diagnostic signal sequence \(X_{1D}\) with a length of T, there is \(X_{1D}\in R^{T}\). Firstly, the original input is projected onto the depth feature \(X_{1D}^{0}\in R^{T\times d_{model}}\) through an embedding layer. The embedding operation is implemented using a combination of value embedding and position embedding methods. Similar to word embeddingin natural language processing, value embedding is implemented through a learnable linear layer, which aims to map the raw scalar data to a fixed size, higher dimensional representation space, converting the raw values of each time series into higher dimensional feature representations, where the dimension size is specified by the parameter \(d_{model}\). The use of one-dimensional convolutional layers for value embedding can be achieved through the following formula:  
\[X^{\prime}=WX+b \tag{17}\]  
where the input data are \(X\in R^{T\times 1}\), the weight matrix is \(W\in R^{1\times d_{model}}\), the bias term is \(b\in R^{d_{model}}\), and the embedded output result is \(X^{\prime}\in R^{T\times d_{model}}\). In addition, positional embedding is adopted to provide the model with positional information for each time point, helping the model understand the sequential relationships in the time series. Specifically, variations of sine and cosine functions are used to generate unique codes for each position [40], with the following formula  
\[PE_{pos,2i}=sin\bigg{(}\frac{pos}{10000^{\frac{2i}{d_{model}}}}\bigg{)} \tag{18}\]  
\[PE_{pos,2i+1}=cos\bigg{(}\frac{pos}{10000^{\frac{2i}{d_{model}}}}\bigg{)} \tag{19}\]  
where \(pos\) is the position index, \(i\) is the dimension index, and for each position, the encoding is composed of sine and cosine values from multiple dimensions. \(d_{model}\) is the dimension of the feature vector, and \(10000^{\frac{2i}{d_{model}}}\) is the scaling factor used to adjust the wave length in different dimensions to ensure that the model can capture position information in each dimension. Finally, the value embedding and position embedding are combined and then regularized by randomly setting some elements in the input vector to zero, where the proportion is specified by the dropout parameter. Regularization helps the model generalize and reduces the risk of overfitting through the above operations. For the \(l\)-th layer of the model, its input is denoted as \(X_{1D}^{l-1}\in R^{T\times d_{model}}\). The MPTSE is organized in the form of residuals; therefore:  
\[X_{1D}^{(l)}=\text{MPTSE}\Big{(}X_{1D}^{(l-1)}\Big{)}+X_{1D}^{(l-1)} \tag{20}\]  
For the \(l\)-th layer, the entire process consists of two parts: periodic two-dimensional transformation and feature capture, and aggregation of periodic representations.