# Optimization Study of Driver Crash Injuries Considering the Body NWH Performance
## 6 Analysis of Elman Forecast Results  
The data used for model fitting were collected from numerous simulation experiments. To ensure data quality, a data cleaning process was applied to remove individual outliers and anomalies. After cleaning, the data underwent normalization to prevent issues arising from excessively large or small eigenvalues. Min-max normalization was employed to linearly transform the original data, mapping the results to the range [0, \(-\)1]. Data normalization accelerates the convergence of weight parameters. The processed data were split into a training set and a test set, with cross-validation selected as the verification method. Figure 11: Elman neural network topology. The neural network was tasked with predicting the \(HIC_{36}\) value of the driver's head, the first-order modes of the body-in-white, and the body mass during a crash. Figure 11 presents a comparison between the prediction results of the driver's head \(HIC_{36}\) value and the simulation results. To assess the prediction accuracy in comparison with other neural network algorithms, the radial basis function (RBF) neural network, the generalized regression neural network (GRNN), and the back propagation (BP) neural network were also used for the prediction. Evaluation metrics such as the mean squared error, root mean squared error, mean absolute error, and mean absolute percentage error were employed to assess the prediction accuracy, as displayed in Table 5.  
From the table, it is evident that the predictions made by the Elman neural network exhibit a high degree of agreement with the actual values and have lower prediction errors compared to the other neural networks. The ordering of the prediction accuracy, from highest to lowest, is as follows: Elman, GRNN, REF, BP. The mean squared error (\(MSE\)) quantifies the sum of the squares of the differences between each predicted value and the actual value. Smaller values indicate lower prediction errors and a higher accuracy. \[MSE=\frac{1}{n}\sum_{i=1}^{n}{(a_{o}-y_{o})^{2}} \tag{5}\]  
The root mean squared error (\(RMSE\)) is the square root of the MSE and measures the deviation of observations from the true values. \[RMSE=\sqrt{MSE} \tag{6}\]  
The mean absolute error (\(MAE\)) is not affected by the sign of the deviation due to its use of absolute values, providing a better reflection of the actual prediction error. \[MAE=\frac{1}{n}\sum_{i=1}^{n}{\left|a_{o}-y_{o}\right|} \tag{7}\]  
The mean absolute percentage error (\(MAPE\)) offers a more intuitive representation of forecast accuracy, resembling the concept of the relative error. It compares the difference between the predicted value and the actual value to the actual value itself to determine the percentage error. \[MAPE=\frac{1}{n}\sum_{i=1}^{n}{\left|\frac{y_{i}-y_{i}}{y_{i}}\right|} \tag{8}\]  
The Elman neural network algorithm also predicts the first-order modes of the body-in-white, and the associated prediction errors for the body mass are detailed in Table 6.  
Tables 5 and 6 clearly demonstrate that the Elman neural network outperforms the other three neural network algorithms, obtaining the smallest prediction error for the driver's head injury value across the four metrics of \(MSE\), \(RMSE\), \(MAE\), and \(MAPE\). The Elman neural network also exhibits a low prediction error when predicting body mass and the first-order modal state of the body-in-white, thus meeting the prediction requirements. To ensure processing accuracy, the step size for each input variable in the solution space was set to 0.1, resulting in 244,140,625 feasible solutions. The Elman neural network was \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Neural Network** & \(MSE\) & \(RMSE\) & \(MAE\) & \(MAPE\) \\ \hline Elman & 1847.729 & 42.985 & 35.131 & 3.835 \\ RBF & 10,995.161 & 104.858 & 91.378 & 11.345 \\ GRNN & 9030.242 & 95.028 & 90.139 & 10.936 \\ BP & 24,737.367 & 157.21 & 135.874 & 15.975 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of errors of algorithms in predicting head injury values.