## 2 Methodology
### Assessing the WCDPSO-KELM Model  
The Kernel parameters and penalty coefficient of KELM are optimized by the WCDPSO algorithms, and the WCDPSO-KELM model is built to predict the residual life of rolling bearings.  
The specific assessing steps of the WCDPSO-KELM model are described as follows.  
Step 1: Acquiring whole life-cycle data of rolling bearings.  
Step 2: The sample data are processed, different features are extracted using different methods, and the performance degradation index is selected from different features.  
Step 3: The performance degradation index of each feature space is divided into the training dataset and the test dataset; the training dataset is input to train the WCDPSO-KELM model, then obtain the performance degradation prediction model.  
Step 4: The testing dataset is input to the trained model to test its performance.  
A flowchart of the performance degradation prediction model based on WCDPSO-KELM is shown in Figure 3.  
Mean Absolute Percentage Error (MAPE) is used as the assessment index to analyze the prediction performance of the WCDPSO-KELM model; output error is used to assess the validity of the prediction model. The specific formula is described as follows.  
\[X_{MAPE}=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{\hat{y}_{i}-y_{i}}{y_{i}}\right| \times 100\% \tag{16}\]  
where \(N\) is the number of testing datasets, \(y_{i}\) is the actual output, and \(\hat{y}_{i}\) is the prediction output.  
The performance of the WCDPSO-KELM model is compared with that of other models such as the PSO-KELM model, the WPSO-KELM model, the CPSO-KELM model and the DPSO-KELM model. In order to reduce errors and avoid randomness, each prediction model runs 20 times to calculate the average value and standard deviation. The MAPE values of various models are listed in Table 1 and shown in Figure 4.  
_Appl. Sci._**2022**, _12_, 4676  
The WCDPSO-KELM model has a smaller average MAPE value and smaller standard deviation than other models. The experimental results show that the WCDPSO-KELM model has the smallest error, so it has higher prediction accuracy.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{8}{c}{**MAPE**} & \multirow{2}{*}{**Average**} & \multirow{2}{*}{**Standard**} \\  & & & & & & & & & & & **Deviation** \\ \hline PSO- & 0.0738 & 0.0707 & 0.0729 & 0.0741 & 0.0688 & 0.0699 & 0.0705 & 0.0681 & 0.0732 & 0.0728 & \\ KELM & 0.0719 & 0.0726 & 0.0722 & 0.0701 & 0.0708 & 0.0733 & 0.0764 & 0.0727 & 0.0687 & 0.0713 & \\ \hline WPSO- & 0.0686 & 0.0670 & 0.0727 & 0.0723 & 0.0738 & 0.0682 & 0.0715 & 0.0684 & 0.0714 & 0.0717 & \\ KELM & 0.0738 & 0.0703 & 0.0692 & 0.0671 & 0.0718 & 0.0700 & 0.0720 & 0.0687 & 0.0698 & 0.0696 & 0.0704 & \\ \hline CPSO- & 0.0693 & 0.0724 & 0.0692 & 0.0703 & 0.0689 & 0.0732 & 0.0728 & 0.0701 & 0.0714 & 0.0706 & \\ KELM & 0.0751 & 0.0725 & 0.0744 & 0.0727 & 0.0709 & 0.0690 & 0.0723 & 0.0704 & 0.0712 & 0.0741 & \\ \hline DPSO- & 0.0723 & 0.0679 & 0.0664 & 0.0706 & 0.0706 & 0.0726 & 0.0664 & 0.0694 & 0.0685 & 0.0677 & \\ KELM & 0.0700 & 0.0688 & 0.0730 & 0.0729 & 0.0698 & 0.0707 & 0.0696 & 0.0678 & 0.0705 & 0.0738 & \\ \hline WCDPSO- & 0.0687 & 0.0688 & 0.0671 & 0.0692 & 0.0675 & 0.0680 & 0.0682 & 0.0673 & 0.0660 & 0.0689 & \\ KELM & 0.0698 & 0.0704 & 0.0691 & 0.0701 & 0.0658 & 0.0697 & 0.0708 & 0.0693 & 0.0685 & 0.0689 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: MAPE value of different models.  
Figure 3: Flowchart of prediction model based on WCDPSO-KELM method.