# A Hybrid Degradation Modeling and Prognostic Method for the Multi-Modal System
## 2 Degradation Modeling and Prognostic Hybrid
### Latest-Term Degradation Assessment Based on Local Features
#### 2.2.1 Construction of Local Aging Features  
Constructing local aging features of these complex signal series is significant for obtaining useful aging information. In order to predict latest stage degradation, the overlapped time window is used to decompose the time series. The size of the time window is \(L\). Two kinds of features, statistical features and energy features, are extracted, as shown in Figure 3.  
The statistical features are used to represent the mathematical rules of sequence, which are the average value \(v_{1}\), variance \(v_{2}\) and skewness \(v_{3}\). For multi-modal systems, the ensemble empirical mode decomposition (EEMD) algorithm is used to decompose the original signal to intrinsic mode function (IMF) components based on the local characteristic time scale of the raw signal. Among them, the white Gaussian noise with uniform distribution is added in the time-frequency space to reduce the mixing degree of IMF component model and achieve signal continuity in different frequency regions. The algorithm's steps are as follows.  
Firstly, the number of empirical mode decomposition executions is \(N\) and the amplitude coefficient of white noise. In execution epoch \(n\), the white Gaussian noise series \(g(t)\) is added to the signal data \(s(t)\), as the \(s_{n}^{noise}=s(t)+g_{n}(t)\). In order to obtain IMF components of \(s_{n}^{noise}\), the mean of the upper and lower envelope \(m(t)\) is obtained between the local maximum and minimum of \(s_{n}^{noise}\). Let \(l(t)=s_{n}^{noise}-m(t)\); if \(l_{n}(t)\) satisfies the condition of intrinsic mode function component, it means that it is the first IMF. When it does not, replace \(s_{n}^{noise}\) with \(l_{n}(t)\) and repeat calculation \(l_{n}(t)\) until the IMF conditions are satisfied. The \(l_{n}(t)\) from \(s_{n}^{noise}\) is subtracted to obtain the remainder. The process of extracting an IMF component is repeated until all components are separated from the \(s_{n}^{noise}\). The remainder becomes a monotonic function \(r_{n}(t)\), Equation (11):  
\[s_{n}^{noise}=\sum_{z=1}^{Z}s_{nz}^{imf}(t)+r_{n}(t), \tag{11}\]  
where \(Z\) is the number of IMFs which contains the local characteristics of original signal. In each empirical mode's decomposition execution time, a different white noise series is added to \(s(t)\). After that, we calculate the mean value of all IMFs in \(N\) times as in Equation (12):  
\[s_{z}^{imf}(t)=\frac{1}{N}\sum_{n=1}^{N}s_{nz}^{imf}(t). \tag{12}\]  
\(s_{z}^{imf}(t)\) is the \(z\)th IMF based on signal decomposition. After the signal is decomposed, each IMF component represents a stable signal of different frequency. The frequency contains important aging information too. The energy variation of each frequency band could show the aging state of the system.  
Figure 3: The process of local feature extraction based on a sample from NASA’s Prognostics Repository. The local features include statistical features (average value \(v_{1}\), variance \(v_{2}\) and skewness \(v_{3}\)) and the energy features (\(e_{0},e_{1}\ldots e_{z}\)) which are calculated based on intrinsic mode function (IMF) components.  
Therefore, the energy entropies of different IMF components are extracted as the local aging features, which can be calculated as \(e_{s^{\text{inf}}(t)}=\sum\limits_{t=1}^{L}\left|s^{\text{inf}}(t)\right|^{2}\).