# Failure Prediction of the Rotating Machinery Based on CEEMDAN-ApEn Feature and AR-UKF Model
## 3 Experiment and Results
### Fault Feature Extraction and Selection  
As mentioned in Section 2.2, CEEMDAN is applied to decompose the vibration signals \(X\) into a series of IMFs and a residual. It is well known that an eligible IMF contains helpful information of the original signals, which means they are strongly related. By contrast, an ineligible IMF has randomness and fuzziness, which means it is weakly related with the original signals. Considering that MIC can measure the strengthen of the correlation, it is thus adopted to determine the effectiveness of IMFs. Moreover, an appropriate threshold \(\gamma\) is introduced to select all eligible IMFs as follows. \[\gamma=\beta\times M\left(X,\Omega\right), \tag{36}\]  
where \(\Omega\) is Gaussian distributed with zero mean and unit variance, and \(M\left(X,\Omega\right)\) is the MIC value between the original signals \(X\) and the random vector \(\Omega\). It should be noted that \(\beta\) is a scaling factor used to overcome the random factor. If the weak correlation between the IMF and the original signal is ignored, a relatively large \(\beta\) can be chosen. Otherwise, a small \(\beta\) is adopted. After testing the performance for various values of \(\beta\), we found \(\beta=1.5\) is appropriate for the selection of eligible IMFs in our work. Figure 4 shows the original signals of the first data file, and all IMFs achieved by CEEMDAN are shown in Figure 5. With the definition of the threshold \(\gamma\), the IMF with MIC value larger than \(\gamma\) is selected as the eligible one. For the first data file, Table 1 shows the MIC values of all IMFs achieved by CEEMDAN. The averaged MIC value between the original signals \(X\) and the random vector \(\Omega\) is \(M\left(X,\Omega\right)=0.0479\); thus, the threshold is set to \(\gamma=1.5\times 0.0479=0.07185\). Figure 4: The original signals of the first data file. Figure 3: Bearing test-to-failure experiment system. As shown in Table 1, IMFs numbered as 1, 2, 3, and 4 are selected to reconstruct the signals because their MIC values are greater than \(\gamma\). Figure 6 shows the reconstructed signals with eligible IMFs. Then, the approximate entropy (ApEn) of the reconstructed signals is calculated as the health indicator of the rotating machinery. In other words, CEEMDAN-ApEn is the new proposed feature for failure prediction. Figure 7 shows the degradation curve based on the proposed CEEMDAN-ApEn feature. the curve shows an upward trend, which means the gradual degradation of performance. At the final period, sharp fluctuation of the curve represents that the bearing appears a failure, and that is the end of the bearing. In summary, the proposed CEEMDAN-ApEn feature can reveal the entire life cycle of the bearing. To describe the degradation process of the rotating machinery accurately, an effective feature should satisfy the following three conditions [35,36]. (1) The feature should vary with degradation process gradually, thus maintaining a certain correlation with time. (2) The feature should show a consistent increasing or decreasing characteristics throughout the life cycle. (3) The feature should be robust to outliers. Therefore, correlation, monotonicity and robustness are three metrics used to evaluate the performance of the proposed feature. Among them, the correlation metric is adopted to measure the correlation strength between the feature sequence and the time series, the monotonicity metric can describe the apparent degree of the feature sequence's continuous increasing or decreasing, and the robustness metric is used to reveal the feature sequence's tolerance to noise interference or abnormal signals. Suppose \(E=\left[e\left(1\right),e\left(2\right),...,e\left(K\right)\right]\) is the feature sequence, and \(T=\left[t_{1},t_{2},...,t_{K}\right]\) is a time series. \(e\left(t_{k}\right)\) represents the feature value at time \(t_{k}\), and \(K\) represents the total time length. The feature sequence is divided into two parts by the moving average method as follows.