# Investigation of Microseismic Monitoring of and Precursor Information on Roof Collapse
## 2 Cluster Theoretical Bases  
Cluster analysis is an unsupervised machine learning task. Different from supervised learning, clustering algorithms only interpret input data and find natural groups or clusters in the feature space. Clustering is the process of classifying data into different classes or clusters, where objects within the same cluster have significant similarities. Cluster analysis can be applied to the data preprocessing process. For multidimensional data with complex structures, clustering analysis can be used to aggregate the data and standardize data with complex structures.  
The Gaussian mixture model (GMM) is selected as the clustering model in 3D space. GMM is a probabilistic clustering model commonly used in unsupervised learning tasks. Each Gaussian distribution represents a cluster in the data, and GMM estimates the parameters of these distributions to assign data points to the cluster. GMM assigns probability or likelihood to each data point in each cluster. This probability represents the degree of correlation between data points and clusters.  
The probability distribution of the Gaussian mixture model can be expressed as follows:  
\[P(x)=\sum_{k=1}^{K}\alpha_{k}\phi(x;\mu_{k},\Sigma_{k}) \tag{1}\]where \(K\) is the number of models, \(x\) is the column vector of 3D spatial coordinates, \(\mu\) is the model expectation, \(\Sigma\) is the model variance, and \(\phi(x;\mu_{k},\Sigma_{k})\) is the \(k\)th component in the mixed model representing the probability distribution of the model.  
\(\alpha_{k}\) represents the probability of the model and meets the following two conditions:  
\[\sum_{k=1}^{K}\alpha_{k}=1 \tag{2}\]  
\[0\leq\alpha_{k}\leq 1 \tag{3}\]  
First, define the number of \(K\) values, and then randomly initialize a set of model parameters for each model \(k\), set each model as a random value, an identity matrix, and ensure that the initial probability of each model is equal. Then, calculate the posterior probability generated by each model \(K\) based on the current parameters.  
\[\gamma_{ik}=\frac{\alpha_{k}\phi(x_{i};\mu_{k},\Sigma_{k})}{\Sigma_{k=1}^{K} \alpha_{i}\phi(x_{i};\mu_{k},\Sigma_{k})} \tag{4}\]  
Then, re-estimate the parameters based on the current \(\gamma\). The iterative calculation is completed when convergence occurs.  
GMM can be used to model data as a combination of multiple Gaussian distributions [29; 30]. This proves particularly valuable when dealing with microseismic monitoring signals that may exhibit complex patterns, which may not be well-suited for other clustering methods. GMM is especially advantageous when working with uncertain or noisy microseismic data, as it allows for a more nuanced representation of cluster assignments. In the context of microseismic crack identification, where cracks may overlap, GMM excels in handling overlapping clusters, a challenge that some other methods struggle with. Furthermore, GMM's scalability to large datasets makes it well-suited for processing extensive collections of microseismic data gathered over time. While other methods may shine in terms of simplicity, noise handling, or the ability to identify clusters with non-Gaussian shapes, the strength of GMM lies in its flexibility and probabilistic modeling.