# A Data Loss Recovery Technique Using EMD-BiGRU Algorithm for Structural Health Monitoring
## 2 Proposed EMD-BiGRU Architecture
### Bidirectional Gated Recurrent Unit  
Recurrent neural network (RNN) is mainly used to extract features from the time sequence. Because its neurons can accept information not only from other neurons but also from themselves, RNN has characters of memorability and parameter sharing, which are powerful in handling non-linear features of monitoring data, such as vibration data, response data. Due to the problem of RNN gradient disappearing and weak backpropagation, many researchers have studied long short time memory (LSTM) that can extract the dependency information between long and short time sequence data [30].  
With the development of deep learning algorithms, GRU, as a variant of the LSTM network model, have few parameters that can improve the trained speed of the model and predict missing data with suitable high accuracy [31]. In addition, the structure of the GRU algorithm is similar to LSTM, which is consisted of update and reset gates. The update gate represents the degree of influence from the output information of the hidden neuron layer at the previous moment on the hidden layer at the current moment. The larger value of the update gate represents a larger influence. Thus, GRU can extract the time-sequence extract features in the order from front to back.  
Among them, the update gate represents the degree of influence from the output information of the hidden neuron layer at the previous moment on the hidden layer at the current moment. The reset gate represents the neglecting information degree of the hidden neuron layer at the previous moment. The larger reset gate represents that less information is neglected. The specific structure of GRU is shown in Figure 2.  
Figure 1: EMD decomposition of acceleration data for a three-story building structure.  
The following equation can calculate the hidden layer of the GRU:  
\[r_{t}=\sigma(W_{r}x_{t}+U_{r}h_{t-1}) \tag{2}\]  
\[z_{t}=\sigma(W_{z}x_{t}+U_{z}h_{t-1}) \tag{3}\]  
\[\widetilde{h}_{t}=\tanh(r_{t}\circ Uh_{t-1}+Wx_{t}) \tag{4}\]  
\[h_{t}=(1-z_{t})\circ\widetilde{h}_{t}+z_{t}\circ h_{t-1} \tag{5}\]  
where \(r\) and \(z_{t}\) is the reset gate and the update gate, respectively. \(\widetilde{h}_{t}\) is a candidate activation vector. \(\sigma\) and \(tanh\) are sigmoid function and hyperbolic tangent function, respectively. \(\circ\) denotes Hadamard product. \(W_{r}\), \(U_{r}\), \(W_{z}\), \(U_{z}\), \(W\), \(U\) are the weight matrix.  
Considering that vibration data at the current moment is associated with the previous and future moment, Bidirectional GRU (BiGRU) extract two-directions feature of vibration data. However, unidirectional LSTM and GRU can only access the above but not the below [32], which is shown in Figure 3.  
The formula for BiGRU is shown below:  
\[\overset{\rightarrow}{h}_{t}=GRU\bigg{(}x_{t},\overset{\rightarrow}{h}_{t -1}\bigg{)} \tag{6}\]  
Figure 3: BiGRU structural diagram.  
Figure 2: GRU structure diagram.  
\[\overset{\leftarrow}{h}_{t}=GRU\left(x_{t},\overset{\leftarrow}{h}_{t-1}\right) \tag{7}\]  
\[h_{t}=\alpha_{t}\overset{\rightarrow}{h}_{t}+\beta_{t}\overset{\leftarrow}{h}_ {t}+b_{t} \tag{8}\]  
where \(\alpha_{t}\) is the weight of the forward propagating at time \(t\). \(\beta_{t}\) is the weight of the backward propagating. \(b_{t}\) is the bias vector corresponding to the hidden layer state at time \(t\).