## 3 Materials and Methods
### Cluster Analysis
#### 3.3.1 Cluster Analysis Using Statistical Models  
For the clustering analysis, we used three statistical models, including the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) [48], the Gaussian Mixture Model (GMM) [49], and the Agglomerative Clustering Model (ACM) [49]. These models were selected to capture a range of clustering behaviors--DBSCAN for its ability to detect arbitrary-shaped clusters and outliers, GMM for its probabilistic approach to overlapping clusters, and ACM for its hierarchical structure that does not require a predefined number of clusters. DBSCAN is a density-based clustering algorithm that does not require the user to predefine the number of clusters. Unlike partitioning algorithms such as k-means, DBSCAN can identify clusters of arbitrary shape and size, which is particularly advantageous when working with complex geospatial datasets or data with non-uniform density distributions. The algorithm groups together points that are closely packed, i.e., those in high-density regions, while labeling points in low-density areas as noise or outliers. This ability to differentiate between dense clusters and sparse regions makes DBSCAN well-suited for identifying natural groupings in geophysical and geological data, where meaningful structures often do not conform to simple geometric boundaries. GMM adopts a probabilistic framework that allows data points to belong to multiple clusters with varying degrees of membership or probabilities, unlike k-means, which assigns each piece of data exclusively to a single cluster. This soft clustering approach provides greater flexibility, making GMM particularly effective for identifying complex, overlapping clusters with different shapes, sizes, and densities. GMM assumes that the data is generated from a mixture of several Gaussian (normal) distributions, each representing a cluster. The model implements the Expectation-Maximization algorithm to iteratively estimate the parameters, including means, covariances, and mixing coefficients, of the underlying Gaussian distributions, enabling more nuanced clustering compared to hard assignment methods like k-means. This probabilistic nature also allows GMM to model uncertainty in cluster assignments, making it especially useful in applications where data points do not clearly belong to one cluster or where cluster boundaries are ambiguous. As a result, GMM offers a more nuanced and flexible clustering solution compared to hard-assignment methods like k-means. ACM is a hierarchical clustering technique that builds a nested cluster structure by iteratively merging the closest pairs of data points or existing clusters based on a defined linkage criterion (e.g., single, complete, or average linkage). Unlike partitioning methods such as k-means, ACM does not require the number of clusters to be specified in advance. One of the key strengths of agglomerative clustering is its flexibility in identifying clusters with complex shapes, sizes, and densities, as it makes no assumptions about the underlying cluster geometry. This makes it particularly suitable for datasets where cluster boundaries are non-linear or irregular. For these statistical models, we applied Principal Component Analysis (PCA) to reduce the dimensionality of the feature set while preserving as much variance as possible. PCA achieves this by projecting the data onto a new set of orthogonal axes--principal components--that represent the directions of greatest variance in the data, providing a simplified yet informative representation of the dataset. In our case, the feature space was reduced to two principal components. All statistical models were subsequently applied to the PCA-transformed variables to enhance interpretability and computational efficiency. We then applied the Elbow method to determine the optimal number of clusters, ensuring a consistent cluster count was used across all model results. The elbow point, identified in the inertia plot, marks where the rate of decrease in inertia begins to level off.