## 6 Case Validation
### Prediction of Accident Losses from Equipment Safety Investment in Coal Mines  
(1) Dataset Partitioning  
In this study, the original dataset Table 2 is divided into training and testing sets. Due to the small dataset, the \"self-help method\" (bootstrapping) is used. The total dataset is set as N, the training set is set as M, and the test set is set as C. The specific steps are as follows:  
Training set (M): a sample is randomly selected from the total dataset (N) and copied into the training set, the sample is put back after each selection and this operation is repeated 13 times. The resulting training set (M) contains 13 samples. Test set (C): unselected samples from the total dataset (N) are used as the test set. The detailed dataset division results are shown in Table 4.  
(2) Method Comparison  
Currently, the commonly used prediction methods in accidental loss prediction mainly include BP [48; 49], SVR [50; 51], gray model [52; 53], etc. In order to further compare the advantages and disadvantages of the three different methods in accident loss prediction of coal mine equipment safety resource investment, this paper takes the original dataset \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**Features**} & \multirow{2}{*}{**Name**} & \multirow{2}{*}{**Total (N)**} & \multirow{2}{*}{**Training Set (M)**} & \multirow{2}{*}{**Test Set (C)**} \\ \cline{2-2} \cline{5-6}  & **Input** & & & & & \\ \hline bootstrapping & \(X_{1}\), \(X_{2}\),\(X_{3}\),\(X_{4}\),\(X_{5}\) & \(Y\) & sample size & 13 & 13 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Dataset division. Figure 8: Scatter plot of preprocessed data. between 1 and 10. For \(m=5\) and \(n=1\), the range for \(L\) is set between 4 and 13. Specific parameter settings are provided in Table 6.  
(4)Results Analysis  
Figure 11 demonstrates the change curve of the fitness value, where the horizontal axis indicates the number of evolutions and the vertical axis indicates the fitness value. As can be seen from the figure, the fitness value of the particle gradually decreases during the evolution process, indicating that the error is continuously decreasing. The whole running process takes 44.17 s. In the PSO algorithm, when the number of iterations reaches 44, the fitness value is \(8.58\times 10^{-5}\) and the running time is 16.23 s, which successfully reaches the target value. In comparison, the IPSO algorithm achieved a fitness value of \(6.7\times 10^{-5}\) and a running time of 10.82 s in only 22 iterations, also reaching the target value. Figure 12 shows the comparison between the predicted and true values of the IPSO-BP training samples, and the results show that the predicted values of the IPSO-BP model are highly consistent with the true values, and the fitting accuracy is significantly better than that of the BP and PSO-BP models. This further confirms the effectiveness of the IPSO algorithm in improving the prediction performance. In order to further evaluate and compare the model prediction performance objectively, the calculation results of relevant indexes are shown in Table 7. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \multirow{5}{*}{BP, PSO, IPSO} & training epochs & 1000 \\  & target error & \(10^{-6}\) \\  & learning rate & 0.001 \\  & population update iterations & 100 \\  & population size & 20 \\  & \(L\) & 5 \\  & \(\omega\) & 0.9 \\ PSO & \(c_{1}\) & 2 \\  & \(c_{2}\) & 2 \\  & \(\omega_{max}\) & 0.9 \\  & \(\omega_{min}\) & 0.4 \\ IPSO & \(c_{start}\) & 4 \\  & \(c_{end}\) & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: IPSO-BP parameter settings. Figure 11: Fitness value change curve. From Table 7, it can be observed that the IPSO-BP algorithm achieves the best performance on the training set, followed by the PSO-BP algorithm, while the BP algorithm exhibits the poorest performance. Figure 13 presents a comparison of the predicted values and actual values for the IPSO-BP test samples.