# Research on Slope Stability Based on Bayesian Gaussian Mixture Model and Random Reduction Method
## 3 Comprehensive Safety Factor
### Performance Evaluation and Metric Analysis of the Bayesian Gaussian Mixture Model  
To comprehensively evaluate the classification performance of the Bayesian Gaussian Mixture Model (BGMM) for slope energy dissipation patterns, this study uses the comparison of data distributions and four categories of complementary indicators to comprehensively verify the model's effectiveness. By programming in Python (v.3.11.) to retrieve the model output results (see Figure 10 and Table 5), the evaluation covers the following: internal compactness indicators (silhouette score, quantifying intra-cluster aggregation, and inter-cluster separation); statistical fitting indicators (log-likelihood, testing the probability fitting goodness of the model to energy dissipation data distributions); clustering validity indices (Calinski-Harabasz Index and Davies-Bouldin Index, providing standardized assessments of clustering distinguishability and compactness); and stability indicators (Weight Variability and Mean Separation, verifying the model's anti-interference ability and robustness in identifying slope critical states). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{c}\)-Reduction} & \multirow{2}{*}{\(\mathbf{\varphi}\)-Reduction} & \multirow{2}{*}{**Dissipated Energy/J**} & \multicolumn{2}{c}{**Membership Probability**} & \multirow{2}{*}{**Cluster Label**} \\ \cline{3-3} \cline{5-6}  & & & **Cluster 1** & & **Cluster 2** \\ \hline \(1\) & \(1\) & 241.4 & 0.980 & 0.020 & Stable \\ \(1\) & \(1.1\) & 241.4 & 0.980 & 0.020 & Stable \\ \(1\) & \(1.2\) & 241.5 & 0.980 & 0.020 & Stable \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(1.368\) & \(1.366\) & 3522 & 0.743 & 0.257 & Stable \\ \(1.368\) & \(1.368\) & 3801.5 & 0.564 & 0.436 & Stable \\ \(1.368\) & \(1.37\) & 4107 & 0.327 & 0.673 & Unstable \\ \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) & \(\ldots\) \\ \(1.5\) & \(1\) & 408 & 0.983 & 0.017 & Stable \\ \(1.5\) & \(1.1\) & 674.4 & 0.986 & 0.014 & Stable \\ \(1.5\) & \(1.2\) & 1272.2 & 0.987 & 0.013 & Stable \\ \hline \hline \end{tabular}
\end{table} 
Table 4: BGMM clustering outcomes. Figure 9: Visualization of training results for the Bayesian Gaussian Mixture Model. These four categories of indicators complement each other, constructing a cross-validation system from the following four dimensions: clustering structure rationality, probability distribution matching degree, statistical validity, and engineering application stability. This ensures that the model maintains both theoretical rigor and engineering practicality in capturing energy mutation characteristics. This figure illustrates the comparison between synthetic data (generated by sampling from the trained BGMM) and original data. The peaks of both the original and synthetic data align in the same position, indicating that the model successfully captured the primary modal characteristics of slope dissipated energy. The left-skewed extensions of both datasets exhibit a steep decline, reflecting the rapid convergence of low energy dissipation. The right tails of the original and synthetic data show an extremely high degree of overlap, demonstrating that the synthetic data can accurately replicate the features of the original data. To quantitatively evaluate the similarity between the distributions of the original and synthetic data, the following three statistical parameters are presented in Figure 10: the Overlap Coefficient (0.8180), Bhattacharyya Coefficient (0.9386), and Jensen-Shannon Divergence (0.0490). Specifically, the Overlap Coefficient (0.8180) indicates a high spatial consistency in energy dissipation ranges. The Bhattacharyya Coefficient (0.9386), approaching 1, demonstrates an accurate replication of the distribution morphology of the original data. The Jensen-Shannon Divergence is below 0.05 (0.0490), implying that the information loss during distribution reconstruction is less than 5%.