# Failure Prediction of the Rotating Machinery Based on CEEMDAN-ApEn Feature and AR-UKF Model
## 3 Experiment and Results
### Fault Feature Extraction and Selection
\[e\left(t_{k}\right)=e_{T}\left(t_{k}\right)+e_{R}\left(t_{k}\right), \tag{37}\]  
where \(e_{T}\left(t_{k}\right)\) is the stationary trend part, and \(e_{R}\left(t_{k}\right)\) is the random margin part. The correlation between \(E\) and \(T\) is remarked as \(Corr\left(E,T\right)\), the monotonicity and robustness of \(E\) is marked as \(Mon\left(E\right)\) and \(Rob\left(E\right)\), respectively. \[Corr\left(E,T\right)=\frac{\left|K\underset{k}{\sum}e_{T}\left(k\right)t_{k}- K\underset{k}{\sum}e_{T}\left(k\right)t_{k}\underset{k}{\sum}t_{k}\right|}{ \sqrt{\left[K\underset{k}{\sum}e_{T}\left(k\right)^{2}-\left(\underset{k}{ \sum}e_{T}\left(k\right)\right)^{2}\right]\left[K\underset{k}{\sum}\left(t_{k} \right)^{2}-\left(\underset{k}{\sum}t_{k}\right)^{2}\right]}}, \tag{38}\]  
\[Mon\left(E\right)=\frac{1}{K-1}\left|\underset{k}{\sum}\delta\left(e_{T}\left( k+1\right)-e_{T}\left(k\right)\right)-\underset{k}{\sum}\delta\left(e_{T}\left(k \right)-e_{T}\left(k+1\right)\right)\right|, \tag{39}\]  
\[Rob\left(E\right)=\frac{1}{K}\underset{k}{\sum}\exp\left(-\left|\frac{e_{R} \left(k\right)}{e\left(k\right)}\right|\right), \tag{40}\]  
where \(\delta\left(\bullet\right)\) is a simple unit step function, which can be expressed as follows. \[\delta\left(x\right)=\left\{ \begin{array}{ll}1,&x\geq 0\\ 0,&x<0\end{array} .\right. \tag{41}\]  
It is obvious that above three metrics vary in the interval [0,1]. However, a single metric can only measure certain aspect of the feature for failure prediction. To test the performance of the proposed feature comprehensively, a weighted combination metric should be constructed to fuse above three metrics, which is specifically expressed as:  
\[ \begin{array}{l}J=\omega_{1}Corr\left(E,T\right)+\omega_{2}Mon\left(E \right)+\omega_{3}Rob\left(E\right)\\ s.t.=\left\{\begin{array}{ll}\omega_{i}>0\\ \underset{i}{\sum}\omega_{i}=1\end{array},&i=1,2,3\end{array} \right.,\end{array} \tag{42}\]where \(\omega_{i}\) represents the importance weight of \(i\)-th metric. Because the monotonic trend has the greatest impact on the results of the RUL prediction, the importance weights are thus set to 0.2, 0.5, and 0.3 for correlation, monotonic, and robustness, respectively [35; 36]. To illustrate the performance of the proposed feature, a comparison between other features (Kurtosis, RMS, ENTR [19], ApEn, and EEMD-EMTR [20]) and the proposed CEEMDAN-ApEn feature is carried out, and the results are shown in Table 2. As it shown, all above features can achieve good performance on robustness because their values are close to 1. Among them, RMS and the proposed CEEMDAN-ApEn features outperform others, which means they are robust to outliers. Furthermore, RMS achieves the best performance on monotonicity because it adopts an average strategy to reduce the random errors. It is worth mentioning that the proposed CEEMDAN-ApEn feature has a higher value on monotonicity; thus, it is applicable for continuous degradation tracking of the bearing. It is remarkable that the proposed CEEMDAN-ApEn feature achieves the best performance on correlation. Therefore, it can be utilized to perform RUL prediction of the bearing. In summary, the proposed CEEMDAN-ApEn feature has the highest value on the weighted combination metric, which means it can keep excellent balance among all influence factors of the failure prediction.