## 3 Uncertainty-Controlled Remaining Useful Life Prediction with a Data Augmentation Strategy
### Wiener-LSTM Bearing RUL Prediction Model  
#### 3.2.1 Forward Propagation of the LSTM \(\phi\)  
The LSTM network is established as the RUL predictor in the proposed method, which is one of the variants of the Recurrent Convolutional Neural (RNN). The gradient vanishing and explosion problems of the RNN can be addressed by the LSTM network memory cell. Figure 3 shows the memory neuron structure of the LSTM network. The memory neuron consists of three kinds of gate units called forget gate \(f_{t}\), input gate \(i_{t}\), and output gate \(O_{t}\). The forward propagation process of the automatic predictor \(\phi\) is introduced in detail now.  
The forget gate is used to determine what information should be discarded. The input of the forget gate is the output of the hidden state in last time \(h_{t-1}\) and the current time input data \(x_{t}\). The \(x_{t}\) is the signal of the monitored machine at time \(t\). The important information of the last time would be reserved and the redundant information would be discarded. The calculation process of the forget gate can be followed as  
\[f_{t}=\sigma\Big{(}W_{f}\cdot h_{t-1}+W_{f}\cdot x_{t}+b_{f}\Big{)}, \tag{12}\]  
where \(W_{f}\) and \(b_{f}\) are the weights and bias of forget gate. \(\sigma()\) is the sigmoid activation function.  
Two steps are involved in this part. The first step is the calculation of the input gate. The second step is adding the candidate values \(\tilde{C}_{t}\) to the memory cell \(C_{t}\). \(C_{t-1}\) is the old state value of the memory cell at time \(t-1\), \(C_{t}\) is the new state value of cell at time \(t\) which is updated by the process as  
\[\begin{split}& i_{t}=\sigma(W_{i}\cdot h_{t-1}+W_{i}\cdot x_{t}+b_{i}),\\ &\tilde{C}_{t}=\tanh(W_{i}\cdot h_{t-1}+W_{i}\cdot x_{t}+b_{C}), \\ & C_{t}=f_{t}*C_{t-1}+i_{t}*\tilde{C}_{t},\end{split} \tag{13}\]  
Figure 2: **(a)** The RMS of the whole life data of bearing (**b**) Degradation model fitting of the degradation stage. Two stages are included in the run-to-failure data of bearing. The RMS of the bearing shows a steady state in the health stage. After Failure Occurrence Time (FOT), the RMS of the bearing shows an exponential degradation in the degradation stages.  
where \(W_{i}\) and \(b_{i}\) are the weights and bias of input gate. \(i_{t}\) is the output of the input gate. \({}^{\star}\) denotes dot product.  
The output gate is used to determine what information should be exported as  
\[\begin{split}& O_{t}=\sigma(W_{O}\cdot h_{t-1}+W_{O}\cdot x_{t}+b_{ O}),\\ & h_{t}=O_{t}*\tanh(C_{t}),\end{split} \tag{14}\]  
where \(W_{O}\) and \(b_{O}\) are the weights and bias of the output gate. \(h_{t}\) denotes the output of the LSTM cell at time \(t\). The LSTM layer is normally constructed with the dense layer, the output \(h_{t}\) of the LSTM is the input of the dense layer. The output of the dense layer is the predicted RUL, which can be calculated as  
\[\alpha_{t}=\sigma(W_{d}\cdot h_{t}(x_{t})+b_{d}), \tag{15}\]  
where \(W_{d}\) and \(b_{d}\) are the weights and bias of the dense layer. All the parameters of the LSTM layer and dense layer, such as \(W_{f}\), \(b_{f}\), \(W_{i}\), \(b_{i}\), and so on, are recorded as \(\Theta\) for convenience.