# An Improved Method Based on EEMD-LSTM to Predict Missing Measured Data of Structural Sensors
## 2 Theoretical Background of the Models
### Long Short-Term Memory  
As a variant of recurrent neural networks, proposed by [PERSON] [8], it has been widely used in time series prediction problems. With the advantage of mapping all historical data to each prediction and, thus, remembering the long-term temporal correlation of time series, LSTM has made fantastic progress in the field of time series such as speech recognition.  
The difference between an RNN and DNN is the connection between the hidden layers. The hidden state of the previous time step in an RNN can be used as input for the current time step so that features can be transferred forward or backward. The main component of the LSTM network is the memory block, which introduces a memory mechanism based on RNN, i.e., by providing three special adaptive multiplicative gates to decide whether to retain or remove temporal information.  
As can be seen from Figure 2, three gate units--an input gate, output gate, and forget gate--were used in the LSTM unit to control the information transfer. Usually, the time series data of each time step and the previous hidden state are used as the input for the LSTM unit. The input gate, \(i_{t}\), can control the current moment of the candidate status, \(\widetilde{c}_{t}\), which needs to save the information; the forget gate, \(f_{t}\), can control the internal state, \(c_{t-1}\), of the last moment, which needs to discard information; the output gate, \(o_{t}\), can control the internal state's, \(c_{t}\), output information to the external state, \(h_{t}\), at the current moment.  
Figure 1: The procedure for EEMD.  
The computational process inside the cyclic cell structure in an LSTM network is as follows:  
First, calculate the input gate, the output gate, the forget gate, and the candidate state using the external state at the last moment, \(h_{t-1}\), and the current moment, \(x_{t}\).  
Second, update the memory cell, \(c_{t}\), by combining the forget gate, \(f_{t}\), and the input gate, \(i_{t}\).  
Finally, combine the output gate, \(o_{t}\), to transfer the internal state information to the external state, \(h_{t}\).  
The LSTM will accumulate important feature information for the final prediction results during the processing of the time series data and make full use of the historical data through the back propagation process, corresponding to the LSTM update equation as follows:  
\[i_{t}=\sigma(W_{i}x_{t}+U_{i}h_{t-1}+b_{i}) \tag{1}\]  
\[f_{t}=\sigma(W_{f}x_{t}+U_{f}h_{t-1}+b_{f}) \tag{2}\]  
\[o_{t}=\sigma(W_{o}x_{t}+U_{o}h_{t-1}+b_{o}) \tag{3}\]  
\[c_{t}=f_{t}\odot c_{t-1}+i_{t}\odot\widetilde{c}_{t} \tag{4}\]  
\[h_{t}=o_{t}\odot\tanh(c_{t}) \tag{5}\]  
\[\widetilde{c}_{t}=\tanh(W_{c}x_{t}+U_{c}h_{t-1}+b_{c}) \tag{6}\]  
where \(x_{t}\) is the input vector at moment \(t\); \(i_{t}\), \(o_{t}\), \(f_{t}\), and \(\sigma\) are the input gate, output gate, forget gate, and sigmoid function, respectively; \(W\), \(b\), and \(h\) are the weight matrix, bias matrix, and hidden state vector, respectively; \(\widetilde{c}_{t}\) and \(c_{t}\) are the input memory cell vector and cell state vector, respectively.