# Failure Prediction of the Rotating Machinery Based on CEEMDAN-ApEn Feature and AR-UKF Model
## 3 Experiment and Results
### RUL Prediction  
The RUL prediction of the bearing starts from PST and ends at the data point where the value of the CEEMDAN-ApEn feature exceeds the threshold of the final failure for the first time. To illustrate the good performance of the proposed AR-UKF model, AR-PF model is also constructed. With the combination of AR model and particle filtering (PF) algorithm, AR-PF model has been adopted to resolve RUL prediction problem in many fields [19; 37; 38; 39; 40].  
At the very beginning, a number of \(n=40\) data points from 538 to 577 is adopted to set up the AR model, and UKF is used to perform RUL prediction of the bearing as described in Figure 2. Because the predicted value of the CEEMDAN-ApEn feature exceeds the threshold of the final failure (1.3251) at the 124 th data point, the predicted RUL at data point 578 is thus 1240 min (124 \(\times\) 10 min) because the interval between neighbor data points is 10 min.  
By repeating above process at each data point, a RUL curve can be achieved as shown in Figure 8, wherein the horizontal ordinate represents current time for prediction, and the vertical ordinate represents the corresponding real and predicted RUL. Obviously, the predicted RUL values of the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Features** & \(Corr(F,T)\) & \(Mon(F)\) & \(Rob(F)\) & \(J\) \\ \hline
**Kurtosis** & 0.5589 & 0.0356 & 0.9573 & 0.4221 \\
**RMS** & 0.6596 & 0.0945 & 0.9837 & 0.4743 \\
**EEMD-EMTR** & 0.7266 & 0.0193 & 0.9748 & 0.4474 \\
**ENTR** & 0.7431 & 0.0051 & 0.8315 & 0.4006 \\
**ApEn** & 0.7509 & 0.0234 & 0.9666 & 0.4518 \\
**CEEMDAN-ApEn** & 0.8321 & 0.0641 & 0.9802 & 0.4925 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The performance of all kinds of features. RMS = root mean square.  
proposed AR-UKF can follow the developing trend of the real RUL values accurately. By contrast, the predicted RUL values of the AR-PF model are more fluctuate, which means the uncertainty of prediction results.  
To quantify the performance of the proposed AR-UKF model, results about the predicted RUL are assessed by the average error (AE) and root-mean-square error (RMSE), which are expressed as follows.  
\[AE=\frac{1}{N}\sum_{i=1}^{N}\big{|}RUL_{p}\left(i\right)-RUL_{r}\left(i\right) \big{|}, \tag{43}\]  
\[RMSE=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\big{(}RUL_{p}\left(i\right)-RUL_{r}\left( i\right)\big{)}^{2}}, \tag{44}\]  
where \(RUL_{p}\) and \(RUL_{r}\) are the predicted and the real RUL values, respectively, and \(N\) is the number of predicted points. The results are listed in Table 3.  
As shown in Table 3, it is obvious that the values of AE and RMSE of the proposed AR-UKF model are less than that of the AR-PF model. Therefore, the proposed AR-UKF model can achieve RUL prediction task of the bearing accurately.  
Figure 8: The RUL prediction curve of different models.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Metrics** & **AR-PF** & **AR-UKF** \\ \hline \(AE\) **(unit: hour)** & 0.2240 & 0.0093 \\ \(RMSE\) **(unit: hour)** & 0.2959 & 0.0394 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Quantitively evaluation results of different models. PF = particle filtering.