## 3 Validation
### Hybrid Ceramic Bearing RUL Prediction Results  
A total of 849 time steps were extracted from the bearing data in which all the data up until time step 792 were used. As seen from Figure 6, a rather large dip in the RMS values occurred from time steps 720 through 735. Features collected from those points were simply removed from the data and treated as outliers. \(L=1\) and \(L=10\) were used to predict 5 min and 50 min into the future for the bearing data, respectively. The predicted  
In Figures 7 and 8, the green color represents the average predicted RUL values across the bootstrapped samples. The red error bars represent the 95% bounds. The predicted results show for both \(L=1\) and \(L=10\) that it can accurately predict the true RUL, and as the bearing approaches the point of failure, the accuracy of the predictions tends to increase. The confidence bounds for the \(L=10\) predicts the RUL of the bearing slightly early when compared to the \(L=1\) predictions and exhibits a greater variance. The common error metrics used in Table 3 are the root mean squared error (RMSE), \(\alpha-\lambda\) metric [37; 38], and the mean absolute percentage error (MAPE). The MAPE, RMSE, and \(\alpha-\lambda\) metric are defined by the following equations:  
\[MAPE=\left(\frac{1}{n}\sum_{t=1}^{n}\frac{A_{t}-F_{t}}{A_{t}}\right)*100\% \tag{33}\] \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{\(L\)} & **DBN Learning Rate** & **DBN Epochs** & **Hidden Layer Structure** & **FNN Learning Rate** & **FNN Epochs** \\ \hline
1 & 0.002 & 74 & [146; 53] & 0.0017 & 176 \\
10 & 0.0023 & 82 & [120; 54] & 0.0014 & 92 \\ \hline \end{tabular}
\end{table} 
Table 4: Hyperparameters of the deep belief network (DBN). Figure 8: Plot of bearing \(\widehat{RUL_{t}}\) values with \(L=10\). \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{4}{c}{**Combined DBN and Particle Filter-Based Approach**} \\ \hline \(L\) & **RMSE** & **MAPE** & \(\boldsymbol{\alpha-\lambda}\) **Accuracy (\(\boldsymbol{\alpha=10\%}\))** \\ \hline
1 & 2.04 & 7.33\% & 0.80 \\
10 & 3.52 & 8.68\% & 0.61 \\ \hline \multicolumn{4}{c}{**Particle Filter-Based Approach**} \\ \hline \(L\) & **RMSE** & **MAPE** & \(\boldsymbol{\alpha-\lambda}\) **Accuracy (\(\boldsymbol{\alpha=10\%}\))** \\ \hline
1 & 2.53 & 7.47\% & 0.71 \\
10 & 3.65 & 8.73\% & 0.53 \\ \hline \end{tabular}
\end{table} 
Table 3: Root mean squared error (RMSE) and mean absolute percentage error (MAPE) results. \[RMSE\ =\sqrt{\frac{1}{n}\sum_{t=1}^{n}\left(A_{t}-F_{t}\right)^{2}} \tag{34}\]  
\[\left(\alpha-\lambda\right)_{t_{\lambda}}= \begin{cases}1&if\ (1-\alpha)RUL\ t_{ \lambda}\leq\widehat{RUL\ t_{\lambda}}\leq(1+\alpha)RUL\ t_{\lambda}\\ &0\ otherwise\end{cases}  \tag{35}\]  
In Equations (33) and (34), \(A_{t}\) denotes the actual value, \(F_{t}\) represents the predicted value, and \(n\) equals the number of points to be predicted. In Equation (35), \(\alpha\) represents a user specified bound level, \(RUL\ t_{\lambda}\) represents the actual RUL at time \(t_{\lambda}\), and \(\widehat{RUL\ t_{\lambda}}\) represents the predicted RUL at time \(t_{\lambda}\). \(\lambda\) is given as a percentage of RUL for a given equipment (i.e., \(t_{\lambda=50\%}\) represents the time index where half the RUL is left). The reported \(\alpha-\lambda\) in Table 3 represents the average \(\alpha-\lambda\) metric across the all testing samples. In the \(\alpha-\lambda\) metric, results closer to one indicate better performance. The loss functions used to train the state transition models for \(L=1\) and \(L=10\) were the MSE (\(RMSE\)2) and the MAPE respectively. Satisfactory hyperparameters for building up Equations (22) and (27) for both \(L=1\) and \(L=10\) were set as [105; 59] for the hidden layer structure using 122 epochs with a learning rate of 0.00085. The hyperparameters for the reconstructed mono-dimensional measurement data were set as [142; 87] for the hidden layer structure using a small learning rate of 0.0094 and 212 epochs.