## 4 Experiment
### Parameter Analysis  
In the proposed LFM-SPE model presented in this paper, two key parameters significantly impact the recommendation effectiveness of the model: the number of latent factors, \(\mathbf{K}\), and the recommendation threshold parameter, \(\mathbf{\alpha}\). In this section, we analyze the impact and variation trends of these two parameters on the recommendation performance individually. Throughout the analysis, while one parameter is examined, the remaining parameters are maintained at their optimal settings.  
We set the number of latent factors \(\mathbf{K}\) to \(\mathbf{K}=\{10,20,30,\ldots,70\}\), and observed the impact of its changes on the model RMSE and MAE, as shown in Figure 2.  
The number of latent factors \(\mathbf{K}\) reflects the number of potential association features between users and items. The optimal result obtained by the model depends on whether the appropriate number of hidden factors is selected. As shown in Figure 2, as the value of \(\mathbf{K}\) increases from 10 to 70, the RMSE and MAE indicators  
Figure 2: The impact of the number of hidden factors \(\mathbf{K}\) on the recommendation performance of LFM-SPE model. (**a**) Impact of \(\mathbf{K}\) on the evaluation index RMSE and (**b**) impact of \(\mathbf{K}\) on the evaluation index MAE.  
decreasing first and then increasing. This observation demonstrates that as the value of \(\mathbf{K}\) increases up to a certain threshold, the algorithm's performance reaches its optimum level. Beyond this threshold, a further increase in the number of hidden factors does not enhance the recommendation effectiveness; instead, it may result in a decline in model performance. This phenomenon can be attributed to overfitting, wherein an excessive number of parameters contribute to suboptimal generalization.  
Moreover, we set the recommended threshold parameter \(\mathbf{\alpha}\) to \(\mathbf{\alpha}=\{0.5,\ 0.6,0.7,\ldots,1.0\}\), and observed the impact of its changes on the model RMSE and MAE, as shown in Figure 3.  
The recommendation threshold parameter \(\mathbf{\alpha}\) represents the maximum value that the propensity score can reach, and \(\mathbf{\alpha}=1\) means that 100% of the items appear in the user's recommendation, which is impossible to observe in the real world. As depicted in Figure 3, when \(\mathbf{\alpha}\) is set to one, both the RMSE and MAE values are significantly higher, indicating a substantial decline in recommendation performance. It is evident that setting the recommendation threshold parameter \(\mathbf{\alpha}\) too high has a detrimental impact on the model's effectiveness. When \(\mathbf{\alpha}\) is varied from 0.5 to 1, the RMSE and MAE indicators exhibit a decreasing trend initially, followed by an increasing trend. This implies that there exists an optimal value for \(\mathbf{\alpha}\) that maximizes the recommendation effect. Specifically, when \(\mathbf{\alpha}\) is set to 0.8, the model achieves the best recommendation performance.  
Based on the experimental results presented above, it can be concluded that the optimal parameter setting for the recommendation system is a recommended threshold parameter \(\mathbf{\alpha}=0.8\) and a number of hidden factors \(\mathbf{K}=20\).