## 3 Uncertainty-Controlled Remaining Useful Life Prediction with a Data Augmentation Strategy
### Wiener-LSTM Bearing RUL Prediction Model
#### 3.2.2 The Wiener-LSTM Model with Joint Optimization Loss Function  
In order to control the uncertainty of the increment of the prediction result, the uncertainty of the predicted RUL is modeled by the Wiener process. The object of the training of the LSTM network is shown in Equation (5) now. The possibility function of \(r_{t}\), \(\alpha_{t}\), and \(\alpha_{s}\) should be acquired to solve the parameters in Equation (5) The possibility of joint prediction for two times is recorded as \(P(r_{t},\alpha_{t},\alpha_{s})\). The joint equations can be get to solve the possibility of joint prediction as  
\[\left\{ \begin{array}{l}\phi(\mathbf{x}_{0:t};\Theta)=\alpha_{t}\\ \phi(\mathbf{x}_{0:s};\Theta)=\alpha_{s}\\ P(r_{t},\alpha_{t},\alpha_{s})=P(\alpha_{s})P(\alpha_{t}\mid r_{s})P(r_{t}\mid \alpha_{t}),\end{array} \right. \tag{16}\]  
Figure 3: The structure of the LSTM network cell. The cell consists of three kinds of gates units called forget date, input gate, and output gate. \(f_{t}\) is the output of the forget gate, \(i_{t}\) is the output of the input gate, \(O_{t}\) is the output of the output gate. \(x_{t}\) is the input of the LSTM cell at time \(t\), \(h_{t}\) is the output of the LSTM cell. \(c_{t}\) is the state of the LSTM at time \(t\). where the \(\alpha_{t}\) and \(\alpha_{s}\) at different times are dependent because of the Markov property of the stochastic process. The possibility of the joint prediction \(P(r_{t},\alpha_{t},\alpha_{s})\) can be transformed as three possibility as shown in Equation (16). \(r_{t}\) denotes the real RUL at time \(t\), the \(\alpha_{t}\), and \(\alpha_{s}\) denotes the RUL predicted by the predictor at time \(t\) and \(s\) (\(t>s\)). The modeling of \(P(\alpha_{t}\mid\alpha_{s})\), \(P(r_{t}\mid\alpha_{t})\) and \(P(\alpha_{s})\) is given in [35] As for two predicted RUL \(\alpha_{t}\) and \(\alpha_{s}\) at two times \(t>s\), the prediction increment is assumed to follow the normal distribution  
\[\alpha_{t}-\alpha_{s}\sim\mathcal{N}(s-t,(t-s)/c), \tag{17}\]  
where the mean of the normal distribution is \(s-t\), the variance of the normal distribution is \((t-s)/c\). The \(P(\alpha_{s})\) is assumed to follow the normal distribution [35]  
\[\alpha_{s}\sim\mathcal{N}(r_{s},r_{s}/c). \tag{18}\]  
The \(P(r_{t}\mid\alpha_{t})\) follows the Inverse Gaussian distribution [39]  
\[IG(\alpha_{s},c)=\left[\frac{c\alpha_{s}^{2}}{2\pi\chi^{3}}\right]^{1/2}\exp \left\{\frac{-c(x-\alpha_{s})^{2}}{2x}\right\}, \tag{19}\]  
where \(IG\) denotes the Inverse Gaussian distribution. The means of the distribution is \(\alpha_{s}\), the variance of the distributed is \(\alpha_{s}/c\). Thus, inspired by the joint likelihood possibility proposed in Equation (16), the loss function is reconstructed to guide the uncertainty control and accuracy improvement. As for a run-to-failure sample, the failure time of the machine is recorded as \(\tau\). \(\alpha_{0},\alpha_{1},\alpha_{2}.\ldots\ldots\). denotes the predicted RUL at different times. If the historical predictions are all considered in the joint likelihood possibility function, which can be represented as  
\[\left\{ \begin{array}{l}\phi(\mathbf{x}_{0};\Theta)=\alpha_{0}\\ \phi(\mathbf{x}_{0:1};\Theta)=\alpha_{1}\\ \ldots\ldots\\ \phi(\mathbf{x}_{0:\tau};\Theta)=\alpha_{\tau}\\ P(r_{\tau},\alpha_{\tau},\alpha_{\tau-1}\ldots\alpha_{0})=P(\alpha_{0})\prod_{ i=1}^{\tau}P(\alpha_{i}\mid\alpha_{i-1})P(r_{\tau}\mid\alpha_{\tau}).\end{array} \right. \tag{20}\]  
All the samples should be considered in the joint likelihood function as the new loss function of the Wiener-LSTM network. Then the joint likelihood function can be maximized automatically by several iterations of the LSTM network. The new loss function can be expressed as  
\[Loss_{new}(\mathbf{x}_{0:\tau}^{n},\mathbf{r}_{0:t}^{n};\Theta,c)=-\frac{2}{ N}\sum_{n=1}^{N}\left(P(\alpha_{0}^{n})\prod_{i=1}^{\tau}P(\alpha_{i}^{n}\mid \alpha_{i-1}^{n})P(r_{\tau}^{n}\mid\alpha_{\tau}^{n})\right), \tag{21}\]  
where \(n\) denotes the number of the running-to-failure samples.