## 4 Experiments
### Numeric Association Rules Mining
#### 4.2.3 Results and Evaluation of Invariant Rule Generation  
To comprehensively assess the performance of different swarm intelligence algorithms in the task of numerical association rule mining, we evaluated them using metrics such as fitness, support, interestingness, comprehensibility, and the number of rules generated. Table 4 presents the results of these evaluation metrics for each swarm intelligence algorithm under their optimal hyperparameter configurations, providing quantitative evidence for comparative analysis.  
As shown in Table 4, PSO outperformed BA and DE in most metrics. Specifically, PSO achieved the highest average fitness (0.8573), support (0.7102), and interestingness (0.7468). These superior results can be attributed to PSO's global search capability and rapid convergence, which enable it to explore the solution space effectively and avoid local optima. The high support and interestingness indicate that the rules generated by PSO are both frequent and novel, reflecting the true behavior of the system.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Factor** & **Level 1** & **Level 2** & **Level 3** \\ \hline \multirow{8}{*}{PSO} & \(w\) & 0.4 & 0.8 & – \\  & \(C_{1}\) & 0.5 & 1 & 2 \\  & \(C_{2}\) & 0.5 & 1 & 2 \\  & \(C_{2}\) & 20 & 30 & 50 \\  & Population & 50 & 100 & 500 \\  & Iteration & 50 & 100 & 500 \\  & \(C_{2}\) & 0.5 & 100 & 500 \\ \hline \multirow{8}{*}{DE} & \(F\) & 0.5 & 0.8 & 1 \\  & \(CR\) & 0.7 & 0.8 & 0.9 \\ \cline{1-1}  & Population & 20 & 30 & 50 \\ \cline{1-1}  & Iteration & 50 & 100 & 500 \\ \hline \multirow{8}{*}{BA} & Loudness & 1 & 0.9 & 0.8 \\ \cline{1-1}  & Pulse Rate & 1 & 0.9 & 0.8 \\ \cline{1-1}  & \(\alpha\) & 0.7 & 0.8 & 0.9 \\ \cline{1-1}  & \(\gamma\) & 0.1 & 0.2 & 0.3 \\ \cline{1-1}  & Population & 20 & 30 & 50 \\ \cline{1-1}  & & 50 & 100 & 500 \\ \cline{1-1}  & Iteration & 50 & 100 & 500 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameter settings for different optimization methods.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & **Factor** & **Value** \\ \hline \multirow{4}{*}{PSO} & \(w\) & 0.8 \\  & \(C_{1}\) & 2 \\  & \(C_{2}\) & 2 \\ \hline \multirow{2}{*}{DE} & \(F\) & 1 \\  & \(CR\) & 0.8 \\ \hline \multirow{4}{*}{BA} & Loudness & 1 \\  & Pulse rate & 1 \\ \cline{1-1}  & \(\alpha\) & 0.9 \\ \cline{1-1}  & \(\gamma\) & 0.1 \\ \hline \hline \multirow{2}{*}{PSO, DE, BA} & population & 50 \\ \cline{1-1}  & Iteration & 50 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Best parameter setting.  
While PSO's comprehensibility (0.5866) was slightly lower than BA's (0.5417) but higher than DE's (0.5410), it still generated a reasonable number of rules (21,075), balancing quality and quantity. This suggests that PSO can produce high-quality, interpretable rules without generating excessive redundancy.  
In contrast, BA performed well in fitness and support but was less effective in interestingness and comprehensibility. DE had relatively lower performance in all metrics except comprehensibility. These differences can be attributed to the distinct search mechanisms and optimization strategies of BA and DE.  
Overall, PSO demonstrated superior performance in numerical association rule mining, particularly in terms of fitness, support, and interestingness. Its global search capability and rapid convergence make it a preferred choice for this task.