## 2 Background Theory and Proposed Framework
### Brief Description of LSTM Network
\(\tanh(\cdot)\in(-1,\;1)\) is a hyperbolic tangent function, as shown in Equation (5):  
\[\tanh(x)= \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} \tag{5}\]  
Combined with the output information of the forget gate and the input gate, the unit state of the previous time step is updated, as shown in Equation (6):  
\[\mathbf{c}_{t}= \mathbf{f}_{t}\otimes\mathbf{c}_{t-1}+\mathbf{i}_{t}\otimes\widetilde{\mathbf{c}}_{t} \tag{6}\]  
where \(\mathbf{c}_{t-1}\in\mathbb{R}^{N_{h}\times 1}\) and \(\mathbf{c}_{t}\in\mathbb{R}^{N_{h}\times 1}\) are the unit states at \(t-1\) and \(t\), respectively. \(\otimes\) is the Hadamard product, which expresses the mathematical operation of the multiplication of the elements corresponding to the position of the vector. \(\mathbf{f}_{t}\otimes\mathbf{c}_{t-1}\) is the unit state of  
Figure 1: The diagram of LSTM unit structure. selectively passing or ignoring the previous time step. \(\mathbf{i}_{t}\otimes\mathbf{\widetilde{c}}_{t}\) adds new information for the current time step. The function of the output gate is to control the proportion of the current unit state that needs to be output to the current output value, as shown in Equations (7) and (8):  
\[\mathbf{o}_{t} =\sigma(\mathbf{W}_{\mathbf{o}}\cdot[\mathbf{h}_{t-1};\,\mathbf{x}_{t}]+\mathbf{b}_{ \mathbf{o}}) \tag{7}\] \[\mathbf{h}_{t} =\mathbf{o}_{t}\otimes\tanh(\mathbf{c}_{t}) \tag{8}\]  
where \(\mathbf{o}_{t}\in\mathbb{R}^{N_{\mathbf{h}}\times 1}\) is the output of the output gate at time \(t\) and \(\mathbf{h}_{t}\in\mathbb{R}^{N_{\mathbf{h}}\times 1}\) is the hidden layer state at time \(t\). \(\mathbf{W}_{\mathbf{o}}\in\mathbb{R}^{N_{\mathbf{h}}\times(N_{\mathbf{h}}+N_{\mathbf{x}})}\) and \(\mathbf{b}_{\mathbf{o}}\in\mathbb{R}^{N_{\mathbf{h}}\times 1}\) are the weight coefficient matrix and bias vector of the output gate, respectively. The mean squared error (MSE) between the predicted and measured data is usually adopted as the loss function when constructing a deep learning framework, as shown in Equation (9):  
\[\mathbf{f}_{t}=\sigma(\mathbf{W}_{f}[\mathbf{h}_{t-1};\;\mathbf{x}_{t}]+\mathbf{b}_{f}) \tag{1}\]0  
where \(\mathbf{W}_{\mathbf{a}}\) and \(\mathbf{b}_{\mathbf{a}}\) (\(\mathbf{a}\,=\,f,\mathbf{i},\mathbf{c},\mathbf{o}\), etc.) are the weight coefficient matrix and bias value vector of each value, respectively. \(\mathbf{D}=\{(\mathbf{X}_{\mathbf{I}},\,Y_{i})\,|\,i=1,2,\ldots,\,N_{\mathbf{D}}\}\) represents the training set data. \(\mathbf{X}_{i}\in\mathbb{R}^{N_{\mathbf{x}}\times M_{i}}\) and \(\mathbf{Y}_{i}\in\mathbb{R}^{N_{\mathbf{y}}\times M_{i}}\) are the input data and output data, respectively, of the \(i_{\text{th}}\) group data sample in the dataset. \(N_{\mathbf{y}}\) is the dimension of the output data. \(M_{i}\) is the number of time steps of the \(i_{\text{th}}\) group data sample, and \(N_{\mathbf{D}}\) is the number of data samples. \(\hat{Y}_{i}(\mathbf{W}_{\mathbf{a}},\,\mathbf{b}_{\mathbf{a}},\,\mathbf{X}_{i})\) are the neural network prediction output results of given network parameters and input data. For the stacked LSTM network, the unknown parameters to be trained in each layer include the weight matrix and bias vector of the four activation functions, the number of which can be calculated according to Equation (10):  
\[\mathbf{f}_{t}=\sigma(\mathbf{W}_{f}[\mathbf{h}_{t-1};\;\mathbf{x}_{t}]+\mathbf{b}_{f}) \tag{1}\]1  
Combining the advantages of AdaGrad and RMSProp optimization algorithms, Adam optimizer [19] is utilized to train the LSTM network model. The optimization goal is shown in Equation (11):  
\[\mathbf{f}_{t}=\sigma(\mathbf{W}_{f}[\mathbf{h}_{t-1};\;\mathbf{x}_{t}]+\mathbf{b}_{f}) \tag{1}\]2  
where \(\hat{\mathbf{W}}_{\mathbf{a}}\) and \(\hat{\mathbf{b}}_{\mathbf{a}}\) represent the optimal weight coefficient matrix and the optimal bias value vector, respectively.