## 3 Model Structure
### Tcn-Lstm  
The Temporal Convolutional Neural Network-Long Short-Term Memory (TCN-LSTM) model epitomizes a cutting-edge hybrid neural network architecture, adeptly merging the benefits of Temporal Convolutional Networks (TCNs) and Long Short-Term Memory (LSTM). TCNs, by harnessing their unique convolutional structure for parallel processing of data series, significantly bolster a model's proficiency in capturing long-term data dependencies, thus addressing time-series prediction challenges. It consists of a one-dimensional full convolutional network and several stacked expansive randomized convolutional networks. This characteristic renders TCN exceptionally suitable for analyzing and predicting sequence data with intricate time-dependent structures. Meanwhile, LSTM, a distinct class of Recurrent Neural Networks, addresses the short-term memory challenges traditional RNNs face with lengthy data sequences. LSTM efficiently orchestrates long-term information storage and retrieval via a gating mechanism, enabling sustained historical memory and agile capture of short-term sequence dependencies. This capability is pivotal in comprehending and forecasting time series of variable lengths and intricate patterns. Merging TCN's long-term dependency-capturing prowess with LSTM's adept management of short-term dependencies, the TCN-LSTM model emerges as a potent and versatile solution for time-series prediction. Fusing the strengths of both networks, the model optimizes time-series data processing depth and breadth and amplifies its adaptability and precision for intricate sequence prediction tasks. The TCN algorithm [36] unfolds in two primary phases: convolution operations and residual linking. Its convolution operations utilize dilated convolution to expand the receptive field by interspersing zeros between convolution kernels. The mathematical expression for the dilated convolution employed in the TCN model is given by Equation (7):  
\[Y[t]=\sum_{i=0}^{K-1}w[i]\cdot X[t-d\cdot i] \tag{7}\]  
where \(Y[t]\) represents the output at time t, \(X[t]\) is the input signal, \(W[i]\) denotes the weight at the \(i\)th position of the convolution kernel, \(k\) is the size of the kernel, and \(d\) represents the dilation factor, which determines the spacing between the weights in the convolution operation. The residual block in the TCN model comprises two convolutional layers coupled with a residual connection. In this, 1D-FCN is a one-dimensional fully convolutional network that is responsible for processing the input data and slides the convolution kernel over the time step or sequence length of the input data to extract temporal or sequential features. The computation process within the residual block unfolds as follows:  
1. The input \(x\) is processed through a dilated convolutional layer, resulting in the output \(y\). 2. The output \(y\) is then added to the input \(x\) to form the residual block's output. The mathematical expression for the residual block is the following Equation (8): \[Output=x+Conv1D(x)\] (8)  
Long Short-Term Memory (LSTM) represents a specialized Recurrent Neural Network (RNN) variant designed to learn long-term temporal dependencies. The essence of LSTM is its cell state, akin to a conveyor belt running through the network, which is impacted by a minimal number of linear transformations, thereby ensuring a consistent flow of information. Additionally, LSTM manages the addition or removal of information to/from the cell state through an ingeniously designed gate mechanism. The mathematical expression for the core function in the LSTM is given in the following Equation (9):  
\[ \begin{split}\widetilde{c}_{t}&=tanh(W_{c}x_{t}+U_{c}h _{t-1}+b_{c})\\ c_{t}&=f_{t}\odot c_{t-1}+i_{t}\odot\widetilde{c}_{t} \end{split}  \tag{9}\]  
In the context of LSTM, \(\odot\) denotes the element-wise multiplication operation. \(\widetilde{c}_{t}\) denotes the candidate cell state, which represents the result of processing the new information at the current moment \(t\).