# Failure Prediction of the Rotating Machinery Based on CEEMDAN-ApEn Feature and AR-UKF Model
## 2 Methodology
### Initial and Final Failure Prediction Stage  
Throughout the life cycle, the rotating machinery will degrade continuously over time, and gradually deteriorate from the healthy state to the failure state over a long period. Therefore, an appropriate threshold should to be selected to identify the beginning of the degradation process. In this paper, Chebyshev inequality is adopted to select the threshold of the initial failure.  
\[p\left\{\left|E-\mu_{k}\right|\right\}\geq\sigma_{k}^{2}\Big{/}\varepsilon_{ k}^{2}, \tag{18}\]  
where \(E\) is the CEEMDAN-ApEn feature sequence of the rotating machinery in the same state, \(\mu_{k}\) and \(\sigma_{k}\) are the mean and the standard deviation of \(E\), respectively, and \(\varepsilon_{k}\) is a real number.  
In theory, Chebyshev inequality shows that for an observation sequence of the same state with a probability distribution, all values in the sequence are close to the mean of the sequence. Therefore, the probability that all sequence values drops into interval \(\left|\mu_{k}-\varepsilon_{k},\mu_{k}+\varepsilon_{k}\right|\) is greater than \(1-\sigma_{k}^{2}/\varepsilon_{k}^{2}\). This theory can be illustrated by the following hypothesis testing methods:  
\[H_{0}:\left|E-\mu_{k}\right|<\varepsilon_{k}\quad H_{1}:\left|E-\mu_{k}\right| >\varepsilon_{k}. \tag{19}\]  
Denote \(E_{0}\) as a value of the CEEMDAN-ApEn feature in healthy state. When \(\left|E_{0}-\mu_{k}\right|\geq\varepsilon_{k}\), the first type of mistake will be made if we reject the hypothesis \(H_{0}\) and identify the rotating machinery as the failure state. According to the Chebyshev inequality, the probability of making above mistake is \(\alpha=\sigma_{k}^{2}/\varepsilon_{k}^{2}\). By contrast, suppose \(E_{1}\) is a value of the CEEMDAN-ApEn feature in failure state. When \(\left|E_{1}-\mu_{k}\right|<\varepsilon_{k}\), the second type of mistake will be made if we accept the hypothesis \(H_{0}\) and identify the rotating machinery as the healthy state, and the probability of making above mistake is \(\beta\). If \(\alpha\) is too large, the healthy state of the rotating machinery would be mistaken as the failure state. On the contrary, if \(\beta\) is too large, the failure state of the rotating machinery would be mistaken as the healthy state.  
In practical applications, priority should be given to the prevention of the first type of mistake. Therefore, we set \(\varepsilon_{k}=5\sigma_{k}\) by \(\alpha=4\%\), which means that the value of CEEMDAN-ApEn feature of the rotating machinery in the healthy state would fall into the interval \(\left[\mu_{k}-5\varepsilon_{k},\mu_{k}+5\varepsilon_{k}\right]\) with a confidence probability of \(96\%\). In other words, once the value of the CEEMDAN-ApEn feature exceeds the threshold \(\mu_{k}+5\varepsilon_{k}\), it should be considered that the rotating machinery is already in failure state. Therefore, \(\mu_{k}+5\varepsilon_{k}\) is the identification for the beginning of the degradation process, and prediction starting time is the time on which the value of RUL exceeds \(\mu_{k}+5\varepsilon_{k}\) for the first time.  
Final failure means the end of the rotating machinery. Therefore, it is necessary to determine a threshold to identify the time of the final failure. Based on the analysis of the references in this field [33; 34], the threshold of the final failure \(\zeta\) is usually selected according to the practice and experience. In details, the value of the health indicator shows a sudden change when a fault occurs, which can be used as the basis for the selection of \(\zeta\).