# 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
Collectively, these metrics verify that the BGMM accurately captures the statistical patterns of dissipated energy data. This enables the trained model to replicate the transition of slope energy dissipation from stable to unstable states, thereby achieving the reliable classification of slope models into two distinct categories: stable and unstable. Figure 10: Comparison of data distributions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Log-Likelihood** & **Silhouette Score** & \begin{tabular}{c} **Calinski–** \\ **Harabasz Index** \\ \end{tabular} & \begin{tabular}{c} **Davies–Bouldin** \\ **Index** \\ \end{tabular} & \
\begin{tabular}{c} **Weight** \\ **Variability** \\ \end{tabular} &
\begin{tabular}{c} **Mean** \\ **Separation** \\ \end{tabular} \\ \hline \(-\)0.69 & 0.88 & 916.52 & 0.64 & 0.36 & 2.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Model evaluation metrics. (1) Log-likelihood value: This value measures the model's fit to the data, representing the logarithm of the probability of observing the data given the model parameters. A positive value indicates the model's explanatory power, with higher values signifying better performance [26]. A negative value reflects model bias, with smaller absolute values being preferable. The log-likelihood value of the model in this study is \(-0.69\), with an extremely small absolute value, indicating that the model effectively captures the distribution characteristics of the data and can accurately identify the critical state of slopes. (2) Silhouette coefficient: This metric evaluates the clarity of sample assignments in clustering results, reflecting the balance between intra-cluster cohesion and inter-cluster separation. Values closer to 1 indicate higher clustering quality [27]. The silhouette coefficient of the model in this study is 0.88, approaching the theoretical optimum (1.0), demonstrating significant separability between the energy dissipation patterns of stable and unstable slopes and low error rates in critical state identification. (3) Calinski-Harabasz index: This index assesses clustering effectiveness by the ratio of inter-cluster variance to intra-cluster variance, with higher values indicating greater inter-cluster differences [28]. The model in this study achieves an exceptionally high value of 916.52 (typical model range: 200-500), revealing substantial differences in energy distribution between stable and unstable states, which aligns well with the nonlinear instability mechanisms of slopes. (4) Davies-Bouldin index: This metric quantifies inter-cluster similarity, with smaller values indicating lower risk of cluster overlap [29]. The index for the model in this study is 0.64 (significantly better than the threshold of 1.0), proving that the overlap region between stable and unstable modes accounts for less than 5%, with minimal misjudgment risk and robust performance even in noisy monitoring data. (5) Weight volatility: This parameter characterizes the stability of the model's component weights, with smaller values indicating stronger consistency in identifying primary and secondary states. The volatility of the model in this study is 0.36, demonstrating highly stable weight assignments for distinguishing between stable and unstable slope states, making it widely applicable in engineering contexts. (6) Mean difference degree: This value quantifies the degree of difference between two probability distributions, model components, or system states. Its core objective is to transform abstract differences into computable numerical values to support decision-making. The mean difference degree in this study is 2.02 (in normalized space), reflecting the significance of changes in dissipated energy, which aligns with the abrupt energy dissipation observed during slope instability.