# Research on Slope Stability Based on Bayesian Gaussian Mixture Model and Random Reduction Method
## 2 Construction of Slope Instability Criteria Using Bayesian Gaussian Mixture Model
### Gaussian Mixture Model (GMM)  
The Gaussian Mixture Model (GMM) is a probabilistic model and an unsupervised clustering machine learning algorithm [22]. It assumes that all data points \(X=\{x_{1},x_{2},\ldots,x_{n}\}\) are independently sampled from the same probability distribution. Its probability density function can be expressed as a linear combination of multiple Gaussian functions, which are as follows:  
\[p(x)=\sum\limits_{k=1}^{K}\pi_{k}\mathcal{N}\bigg{(}x;\mu_{k},\sum\limits_{k} \bigg{)} \tag{8}\]  
\[s.t.\sum\limits_{k=1}^{K}\pi_{k}=1,0\leq\pi_{k}\leq 1,\forall k=1,\ldots,K \tag{9}\]  
where \(K\) represents the number of Gaussian distributions, i.e., the number of mixture components; \(\pi_{k}\) is the prior probability, i.e., the mixing coefficient; \(\mathcal{N}(x;\mu_{k},\sum_{k})\) is the probability density function of the k-th Gaussian distribution, where \(\mu_{k}\) is the mean vector of the k-th Gaussian distribution; and \(\sum_{k}\) is the covariance matrix of the k-th Gaussian distribution. The specific form of the probability density function \(\mathcal{N}(x;\mu_{k},\sum_{k})\) for the Gaussian distribution is as follows:  
\[\mathcal{N}\Big{(}x;\mu_{k},\sum_{k}\Big{)}=\frac{1}{\frac{d}{ (2\pi)^{\frac{1}{2}}\frac{1}{|\sum_{k}|^{2}}}}e^{-\frac{1}{2}(x-\mu_{k})^{T} \frac{1}{\sum}(x-\mu_{k})} \tag{10}\]  
where \(d\) is the dimensionality of the data point x; \(|\sum_{k}|\) is the determinant of the covariance matrix \(\sum_{k}\); and \(\sum\limits_{k}^{-1}\) is the inverse matrix of the covariance matrix \(\sum_{k}\). The core idea of the Gaussian Mixture Model lies in the assumption that the data is generated from a mixture of multiple Gaussian distributions, with each Gaussian distribution corresponding to a cluster. The model fits the probability distribution of the data by estimating the mean, covariance matrix, and mixing coefficient for each cluster. Its operational mechanism involves first identifying the Gaussian function center corresponding to each sample point, then calculating the posterior probability of the sample point belonging to each cluster, and finally assigning the sample point to the appropriate cluster based on these probabilities [23]. In practical implementation, within the Gaussian Mixture Model the mean, covariance matrix, and mixing coefficient for each Gaussian distribution can be estimated using either the maximum likelihood estimation method or the Expectation Maximization (EM) algorithm. Once these parameters have been estimated, the sample points can be assigned to the cluster corresponding to the Gaussian function with the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Feature** & **BGMM** & **K-Means** & **Hierarchical** & **DBSCAN** \\ \hline Core principle & Probabilistic density modeling & Distance minimization & Tree-based merging & Density expansion \\ Critical state & Handles overlapping states & Ignores transition & Depend on subjective thresholds & Density-sensitive \\ Computation & High (Bayesian iteration) & Low (linear complexity) & Medium (matrix calculation) & Medium (Neighborhood search) \\ Slope data fit & Optimal for energy mutation & Limited for critical zones & Moderate for hierarchies & Moderate for uniform density \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparative analysis of clustering algorithms for slope energy data. highest probability density by comparing the probability density of each sample point across the Gaussian distributions. The likelihood estimation method is a commonly used statistical inference approach. Its fundamental idea is that, given a probability model and a set of observed data, the goal of likelihood estimation is to find the model parameter values that are most likely to have generated the observed data. The likelihood function is the core tool of likelihood estimation, representing the probability of observing the data given the parameter values.