## 2 Proposed Method
### Calibration Module  
After feature extraction, the key step is data calibration. The calibration module combines the GRU and FECAM, followed by a fully connected layer to output the calibrated values. LSTM [28] is used for analyzing time series data, addressing the gradient explosion and vanishing problems in RNNs [29]. The GRU [30], an improvement on LSTM, incorporates only a reset gate and an update gate. While both offer similar prediction accuracy, the GRU converges faster and is more computationally efficient. The reset gate integrates current input with past information, while the update gate preserves memory across time steps. The GRU neural network structure is shown in Figure 4, with the mathematical formulation provided in Equations (4)-(7). \[r_{t}=\sigma(W_{r}x_{t}+U_{r}h_{t-1}+b_{r}) \tag{4}\]  
\[z_{t}=\sigma(W_{z}x_{t}+U_{z}h_{t-1}+b_{z}) \tag{5}\]  
\[\bar{h}_{t}=tanh(W_{h}x_{t}+U_{h}(r_{t}\odot h_{t-1})+b_{h}) \tag{6}\]  
\[h_{t}=(z_{t}\odot h_{t-1})+(1-z_{t})\odot\bar{h}_{t} \tag{7}\]  
In the equations: \(z_{t}\) is the update gate; \(r_{t}\) is the reset gate; \(h_{t}\) is the hidden layer output; \(\bar{h}_{t}\) is the combination of input \(x_{t}\) and previous hidden state \(h_{t-1}\); \(W_{r}\), \(W_{z}\), \(W_{h}\), \(U_{r}\), \(U_{z}\), \(U_{h}\) are trainable matrices; \(b_{r}\), \(b_{z}\), \(b_{h}\) are the biases; \(\sigma\) is the sigmoid function; and tanh is the activation function. The GRU excels in modeling time-domain data, capturing long-term dependencies in time series, but it overlooks frequency-domain information, which can reveal features that are hard to detect in the time domain. Frequency analysis can uncover valuable insights, especially when noise obscures signals, making it difficult to distinguish patterns in the time domain. By modeling in both time and frequency domains, periodic features can be captured, providing a more comprehensive feature representation and improving overall model performance. Figure 3: ResNet structure. To address the limitations of time-domain modeling, frequency information is introduced via Discrete Cosine Transform (DCT) [31]. The Frequency-domain Enhanced Channel Attention Mechanism (FECAM) [32] leverages frequency-domain modeling to assign channel weights, learning the significance of different frequencies for each channel. Frequency analysis is a natural complement to time series analysis, yet most models overlook its impact, failing to capture inherent features. Traditional methods based on Fourier Transform (FT) [33] and Inverse Fourier Transform (IFT) [34] often introduce high-frequency noise due to the Gibbs phenomenon. In contrast, the FECAM avoids these issues by capturing more time-series information through DCT, which extracts additional frequency components and automatically learns the importance of each channel while suppressing irrelevant features. This enhances the learning of inter-channel dependencies through frequency-domain modeling. The structure of the FECAM is shown in Figure 5.  
The FECAM divides the input feature map into n subsets along the channel dimension, denoted as \([v_{0},v_{1},\ldots,v_{n-1}]\), where \(V^{i}=\mathbb{R}^{1\times L}\), \(i\in\{0,1,\ldots,n-1\}\) and \(n=N_{v}\). Each subset is then processed with Discrete Cosine Transform (DCT) frequency components from low to high, with each channel undergoing the same frequency processing. \[Freq^{i}=DCT_{j}(V^{i})=\sum_{j=0}^{j=L_{S}-1}(V^{i}_{;i})B^{j}_{l} \tag{8}\]  
For \(i\in\{0,1,\ldots,N_{v}-1\}\) and \(j\in\{0,1,\ldots,L_{S-1}\}\), where j is the 1D index of the frequency component in \(V^{i}\), and \(Freq^{i}\in\mathbb{R}^{L}\) is the L-dimensional vector after Discrete Cosine Transform (DCT). The entire channel vector is obtained through stacking these components. \[Freq=stack([Freq^{0},Freq^{1},\cdots,Freq^{n-1}]) \tag{9}\]  
Figure 4: The network structure of the GRU model. Figure 5: Structure diagram of FECAM. where \(Freq\in\mathbb{R}^{C\times L}\) represents the attention vector for \(V\in\mathbb{R}^{C\times L}\).