## 3 Methodology
### Multi-Path Time Series Extractor
#### 3.2.2 Multi-Path Time Series Extractor
The process of periodic two-dimensional transformation and feature capture first requires converting the original signal into a two-dimensional format. Specifically, according to the method mentioned in Section 3.1, estimate the period length and frequency index of \(X_{1D}^{(l-1)}\), and reshape the data based on this to obtain a set of two-dimensional tensors, namely:  
\[\{A_{1},\cdots,A_{k}\},\{f_{1},\cdots,f_{k}\},\{p_{1},\cdots,p_{k}\}=\text{ Period}\Big{(}X_{1D}^{(l-1)}\Big{)} \tag{21}\]  
\[X_{2D}^{(l,i)}=Reshape_{p_{i},f_{i}}(Padding(X_{1D}^{(l-1)}) \tag{22}\]  
Among them, \(Period(\cdot)\) is used to calculate the amplitude \(\{A_{1},\cdots,A_{k}\}\), frequency \(\{f_{1},\cdots,f_{k}\}\), and period \(\{p_{1},\cdots,p_{k}\}\), which correspond to the \(i\)-th two-dimensional tensor \(X_{2D}^{(l,i)}\) of \(p_{i}\) and \(f_{i}\), where \(X_{2D}^{(l,i)}\in R^{p_{i}\times f_{i}\times d_{model}}\), for \(i\in 1,\cdots,k\). Subsequently, the core module T-inception(\(\cdot\)) mentioned in Section 3.2.1 is used to extract features from the two-dimensional tensor, namely:  
\[\hat{X}_{2D}^{(l,i)}=\text{T-Inception}\Big{(}X_{2D}^{(l,i)}\Big{)},i\in\{1, \cdots,k\} \tag{23}\]Finally, in order to facilitate subsequent classification operations, it is necessary to convert the learned two-dimensional representation \(\hat{X}_{2D}^{(Lj)}\) back to one-dimensional space. Specifically, \(Reshape(\cdot)\) is used to reshape the two-dimensional data into a matrix of size \(1\times(p_{i}\times f_{i})\), and \(Cut(\cdot)\) is used for truncation operation. The purpose is to restore the time series that has been previously filled and transformed to its original length \(T\). Therefore, the transformed one-dimensional space representation is:  
\[\hat{X}_{\text{ID}}^{(Lj)}=\text{Cut}\bigg{(}\text{Reshape}_{1,(p_{i}\times f_{ i})}\bigg{(}\hat{X}_{2\text{D}}^{(Lj)}\bigg{)}\bigg{)} \tag{24}\]  
where the \(i\)-th one-dimensional signal \(X_{1D}^{(Lj)}\) obtained has \(\hat{X}_{1D}^{(Lj)}\in R^{T\times d_{model}}\). After performing two-dimensional transformation and recognition, it is necessary to fuse k different one-dimensional representations \(\hat{X}_{1D}^{(l,1)},\cdots,X_{1D}^{(l,k)}\) for the next layer. In addition, since the amplitudes \(A_{1},...,A_{k}\) can reflect the relative importance of the selected frequency and period, they are used as weights for each transformed two-dimensional tensor. In order to emphasize the most critical features in the signal and ensure consistency and stability when processing data at different scales, the softmax method is used to normalize the amplitude and convert it into weights representing relative importance. This enables the model to more effectively identify and utilize this key information, namely:  
\[\hat{A}_{f_{1}}^{(l-1)},\cdots,\hat{A}_{f_{k}}^{(l-1)}=Softmax\Big{(}A_{f_{1}} ^{(l-1)},\cdots,A_{f_{k}}^{(l-1)}\Big{)} \tag{25}\]  
Finally, based on amplitude aggregation one-dimensional representation, the output \(X_{1D}^{(l)}\) obtained from the \(l\)-th layer is:  
\[X_{\text{ID}}^{(l)}=\sum_{i=1}^{\kappa}\hat{A}_{f_{i}}^{(l-1)}\times\hat{X}_{ \text{ID}}^{(Lj)} \tag{26}\]  
where \(k\) is the selected number of cycles. Due to the fact that multiple highly structured two-dimensional tensors contain variations within and between periods, the MPTSE can simultaneously capture multi-scale temporal two-dimensional variations. Therefore, compared to directly training with the original one-dimensional sequence, this model can achieve more effective representation learning. Figure 4 shows the model design. \[PE_{pos,2i}=sin\bigg{(}\frac{pos}{10000^{\frac{2i}{d_{model}}}}\bigg{)} \tag{18}\]0 Model integral design.