## 3 Prediction of Pump Amplitude Based on the Support Vector Regression Algorithm
By employing quadratic programming methods, the optimal solution \(\left(\alpha_{i},\alpha_{i}^{*}\right)\) for Equation (11) can be obtained, resulting in the decision function of the SVR:  
\[f(x)=\sum\limits_{i=1}^{N}\left(\alpha_{i}-\alpha_{i}^{*}\right)K(x_{i},x_{j})+b \tag{11}\]  
Due to the advantages of high accuracy and low computational complexity, the radial basis function (RBF) is employed for training and prediction. It is defined as:  
\[K(x_{i},x_{j})=\exp(-\gamma\big{\|}x_{i}-x_{j}\big{\|}^{2}) \tag{12}\]  
where \(\gamma\) is the undetermined kernel parameter. The prediction results are evaluated using the mean relative error (MRE), mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), and coefficient of determination (R\({}^{2}\)). The amplitude prediction process is illustrated in Figure 14, and the specific steps are as follows:  
Step 1: Construct the training and testing sets. From the given n data samples, select N data samples as the training set. The remaining (n - N) test samples are used as the testing set. Step 2: Normalize the data samples. The'mapminmax' function is used to normalize the data samples to the range of \([-1,1]\) to improve the convergence speed and accuracy of the SVR. Step 3: Parameter optimization. The Gaussian radial basis kernel function is used for training and prediction. The grid search method combined with ten-fold cross-validation is employed to find the optimal penalty factor \(c\) and kernel parameter \(\gamma\). The training set is randomly divided into ten subsets, with nine subsets as the training set and one subset as the validation set. The training and validation sets iterate alternately for ten rounds. In each iteration, the'svmtrain' function is used to train the prediction model, and the RMSE \(e_{i}\) of the prediction model is recorded. The average value \(E_{i}=\frac{1}{10}\sum\limits_{i=1}^{10}e_{i}\) is calculated. Then, change the parameters, traverse the grid to train the model for m rounds, and obtain the average RMSE E\({}_{\text{m}}\) of each round of training. Obtain the minimum value of E\({}_{\text{m}}\) and determine the optimal parameters of the model. Step 4: Find out the predicted results. Train the predictive model with the optimal parameters and use the model to predict the output values for the (n \(-\) N) test samples, obtaining the model's output values. The parameter settings for the SVR algorithm are presented in Table 5. The input data for the amplitude prediction based on the SVR algorithm is shown in Appendix A. The results of the amplitude prediction based on the SVR algorithm are displayed in Figure 15, demonstrating a close match between the actual and predicted values. This indicates a high level of reliability in the prediction results. Figure 14: Amplitude prediction process based on the SVR algorithm. \begin{table}
\begin{tabular}{c c} \hline \hline
**Values of Penalty Factor C** & \([-10,0.5,10]\) \\ \hline Values of kernel parameter \(\gamma\) & \([-10,0.5,10]\) \\ Number of cross-validation folds v & 5 \\ Training set & 50 \\ Testing set & 10 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: SVR algorithm parameter settings.