# CDSTAEP: Cross-Domain Spatial-Temporal Association Learning for Abnormal Events Prediction
## 5 Experiments and Evaluation
### Experimental Results
#### 5.2.2 Effect of Different Representation Spaces on Prediction of Abnormal Events  
This experiment aims to compare and analyze the impact of different curvature spaces on anomaly prediction. To evaluate the proposed model's ability to predict abnormal events better, we separate the representation learning part into single curvature space and mixed-curvature space.  
**CDSTAEP-H:** Representation of event association graphs using hyperbolic spaces only.  
**CDSTAEP-S:** Representation of event association graphs using only spherical spaces.  
**CDSTAEP-E** Representation of event association graphs using only Euclidean spaces.  
**CDSTAEP:** Representation of event association graphs using mixed-curvature spaces.  
After resampling the original data, we set the ratios of the training set to 0.2, 0.4, 0.6, and 0.8, respectively, and the rest as the test set. We recorded the results under different ratios of the training set. As shown in Figure 7 and Table 6, based on the same data set, the CDSTAEP-H, CDSTAEP-S, CDSTAEP-E, and CDSTAEP models all achieved good robustness in predicting abnormal events. The AUC score is better as the proportion of the training set increases, reflecting the fact that Riemannian spaces with different curvatures can effectively represent complex network data.  
Specifically, the data representation space of the CDSTAEP model is a mixed-curvature space. The result after classification by the abnormal behavior prediction model is better than the other models mentioned above. When the training ratio is 0.2, the difference in event prediction effect is the largest. The prediction effect of the CDSTAEP model is 0.4981, which is 1.80%, 2.19%, and 1.90% higher than that of CDSTAEP-H, CDSTAEP-S, and CDSTAEP-E, respectively. When the training ratio is 0.8, the AUC score of the CDSTAEP model is 0.7440, and the prediction effect is the best, which is 0.51%, 0.83%, and 0.67% higher than the prediction effect of CDSTAEP-H, CDSTAEP-S, and CDSTAEP-E respectively. There is less variance in the model's predictive performance. The above experimental results show that the event association graph represented by the mixed-curvature space can retain more relationship features between events. The prediction effect is better than that represented by a single curvature space and has the strongest correlation with the prediction of abnormal events.  
This experiment demonstrates that mixed-curvature spaces are suitable for representing complex network data and can capture rich event relationships. The comprehensive prediction effect of the CDSTAEP model is better than the performance of single-curvature space representation, but the difference is small. The reason is the sparsity of abnormal events in the real data, which leads to the relatively uniform distribution of abnormal events in the tree structure, ring structure, and network structure. Spaces with different  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{**AUC**} \\ \cline{2-5}  & **Training Ratio** & **Training Ratio** & **Training Ratio** & **Training Ratio** \\ \hline  & **0.20** & **0.40** & **0.60** & **0.80** \\ CDSTAEP & 0.4981 & 0.5430 & 0.6685 & 0.7440 \\ CDSTAEP-H & 0.4801 & 0.5389 & 0.6575 & 0.7389 \\ CDSTAEP-S & 0.4762 & 0.5285 & 0.6510 & 0.7357 \\ CDSTAEP-E & 0.4791 & 0.5290 & 0.6517 & 0.7373 \\ \hline \hline \end{tabular}
\end{table}
Table 6: AUC for prediction by different representation spaces on the dataset.  
Figure 7: Comparison of the effects of different representation spaces on event prediction: the horizontal axis is the ratio of the training set, and the vertical axis is the AUC score.  
curvatures are suitable for representing various types of data. Specifically, refer to the last column of Table 3.