# 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
#### 2.2.2 Latest-Term Degradation Assessment Based on Local Features and LightGBM  
In present research, the gradient boosting decision tree has good performance when predicting short sequences, but the computational complexity needs to be improved. In this paper, the improved gradient boosting decision tree, the light gradient boosting machine, is used to predict the remaining useful life of latest stage series based on local features. LightGBM used the leaf-wise growth with depth-limiting strategy instead of the traditional level-wise strategy to train decision trees to reduce errors and improve accuracy. The leaf-wise strategy is used to find the leaf with the largest split gain from all current leaves, and then split each time to reduce more errors and obtain higher accuracy at the same split time. To ensure high efficiency and prevent over-fitting, the maximum depth limit is added to the leaf-wise strategy.  
For the local features \((v_{1},v_{2},v_{3},e_{0},e_{1}\ldots e_{z})\), \(J\) decision trees are trained to predict the residuals of the prior models. If \(\varphi_{j}\) is the learning function of the \(j\)th decision tree and the \(\hat{y}_{i}^{(j)}\) is the prediction of for \(i\)th sample at the \(j\)th iteration; the process can be expressed as Equation (13):  
\[\begin{array}{l}\hat{y}_{i}^{(0)}=0\\ \hat{y}_{i}^{(1)}=\varphi_{1}(x_{i})=\hat{y}_{i}^{(0)}+\varphi_{1}(x_{i})\\ \hat{y}_{i}^{(2)}=\varphi_{1}(x_{i})+\varphi_{2}(x_{i})=\hat{y}_{i}^{(1)}+ \varphi_{2}(x_{i})\\ \cdot\quad\cdot\quad\cdot\\ \hat{y}_{i}^{(J)}=\sum\limits_{j=1}^{J}\varphi_{j}(x_{i})=\hat{y}_{i}^{(J-1)}+ \varphi_{j}(x_{i})\end{array} \tag{13}\]  
In each iteration, the current model \(\hat{y}_{i}^{(j)}\) is reserved and the learning residual function \(\varphi\) is learned by minimizing the objective function, as in Equation (14):  
\[\ell^{(J)}=\sum\limits_{i=1}^{p}\textit{loss}(y_{i},\hat{y}_{i}^{(J)})+\sum \limits_{j=1}^{J}\Omega(\varphi_{j}), \tag{14}\]  
The function consists of two parts. One is a loss function regarding the difference between the prediction \(\hat{y}_{i}^{(J)}\) and the actual \(y_{i}\). The other is the regular term \(\sum\limits_{j=1}^{J}\Omega(\varphi_{j})\) that penalizes the complexity of model, as in Equation (15):  
\[\Omega(\varphi_{j})=\mu Z+\frac{1}{2}\tau\sum\limits_{z=1}^{Z}\omega_{z}^{2}, \tag{15}\]  
where \(\mu\) is the penalty parameter for the number of leaves \(Z\), \(\omega\) is the weight of a leaf and \(\tau\) is the penalty parameter. Considering that the samples with larger gradients play a more important role in calculating information gain, the gradient-based one-side sampling (GOSS) strategy [31] is used to segment the data. Firstly, the local feature dataset is sorted in descending order based on absolute values of gradients. Then, data instances with large gradients are retained, and samples with small gradients are randomly sampled while splitting out the set according to the estimated variance gain.  
\[\hat{y}_{i}=\sum\limits_{j=1}^{J}\varphi_{j}(x_{i}),\varphi_{j}\in S_{\textit{ tree}}. \tag{16}\]  
As Equation (16) shows, predictions of all decision trees are added to obtain the latest stage prediction. \(S_{\textit{tree}}\) is a space that contains all possible structures of trees.