## 4 The Description of Dataset and Problem in Life Prediction under Multiple Operating Conditions
### The Description of Problems in Life Prediction under Multiple Operating Conditions  
The traditional STM usually extracted the degradation model (degradation trend) of the monitoring signal to eliminate the influence of noise and other random fluctuations on the overall degradation trend of equipment. However, as shown in Figure 4a, for some monitoring signals, when the equipment was operating under multiple operating condi  
Figure 3: Correlation coefficient between the monitoring data of each sensor.  
tions, the monitoring signals were affected by different operating conditions. Further, the obvious degradation trend cannot be seen from the monitoring signal. Therefore, according to [25], the influence of operating conditions on the degradation trend of monitoring signal was eliminated by normalization of operating conditions as follows:  
It was assumed that each life cycle point \(x\) can be divided into one of the \(P\) operating regions \(O_{P}\), that was \(\left\{x\right\}^{(P)}=\left\{x_{i}|u_{i}\in O_{P}\right\}\), where \(u_{i}\) was the operating condition corresponding to each point \(x_{i}\). After dividing the operating conditions, the mean and variance of each operating condition were as follows:  
\[\tilde{x}^{(P)}=Mean\left(\left\{x\right\}^{(P)}\right) \tag{17}\]  
\[s^{(P)}=Std\left(\left\{x\right\}^{(P)}\right) \tag{18}\]  
Then the normalization of operating conditions \(\left\{y\right\}^{(P)}\) can be obtained as follows:  
\[\left\{y\right\}^{(P)}=\frac{\left\{x\right\}^{(P)}-\tilde{x}^{(P)}}{s^{(P)}} \tag{19}\]  
Finally, according to the time position of each point in the original life cycle, the normalized monitoring signal of the operating condition can be obtained, as shown in Figure 4b.  
Through the operating condition normalization, although the normalized monitoring signal of the operating condition can show an obvious degradation trend, the mean and variance of each operating condition needed to be known in the process of operating condition normalization. Therefore, this brought a problem, for the equipment in service, how to determine the mean and variance of each operating condition. Generally, there were two solutions: (1) the mean and variance were from each operating condition of service equipment in-service stage; (2) the mean and variance were from each operating condition of historical life cycle data. However, the mean and variance obtained by the two solutions were different from those of each operating condition after the complete failure of service equipment. As shown in Figure 5, the degradation model of the first half of Life Cycle 1 was obtained by the mean and variance obtained from the abovementioned two solutions respectively. As shown in Figure 5b,c, the degradation model of the first half of Life Cycle 1 obtained by the mean and variance which came from the abovementioned two solutions was obviously different from that of Life Cycle 1 obtained by the mean and variance of each operating condition of the whole life. This difference existed not only in shape but also in amplitude. Therefore, for the service equipment, the degradation model of monitoring data in-service stage cannot be accurately obtained under multiple operating  
Figure 4: The simulation data of turbofan engine. (**a**) Monitoring data of Sensor Channel 3. (**b**) Data after normalization of operating conditions.  
conditions, which was an important problem encountered in using STM for life prediction under multiple operating conditions.