## 3 Methodology
### Model Evaluation System  
(1) Clustering evaluation index  
To evaluate the performance of dimensionality reduction and clustering, the Davies-Bouldin Index (DBI) is adopted as a quantitative evaluation metric [54]. DBI is widely used to assess clustering quality by simultaneously considering intra-cluster compactness and inter-cluster separation. Specifically, it quantifies the average similarity between each cluster and its most similar counterpart, where similarity is a function of the ratio between within-cluster scatter and between-cluster distance. A lower DBI value indicates more compact clusters with better separation, suggesting improved clustering performance and more effective feature space transformation following dimensionality reduction. The compactness \(C_{k}\) of cluster \(k\) is defined as the average Euclidean distance between all points within the cluster and the cluster centroid, and is expressed as  
\[C_{k}=\frac{1}{n_{k}}\sum_{i=1}^{n_{k}}\parallel x_{i}-\mu_{k}\parallel \tag{8}\]  
Here, \(x_{i}\) represents the \(i\)-th sample within cluster \(k\), \(\mu_{k}\) denotes the centroid of cluster \(k\), and \(n_{k}\) is the number of samples in cluster \(k\). The separation \(S_{kj}\) between cluster \(k\) and cluster \(j\) is defined as the ratio of the distance between their centroids to the sum of their within-cluster compactness, as expressed by  
\[S_{kj}=\frac{\parallel\mu_{k}-\mu_{j}\parallel}{C_{k}+C_{j}} \tag{9}\]  
where \(\mu_{k}\) and \(\mu_{j}\) are the centroid vectors of clusters \(k\) and \(j\), respectively, and \(C_{k}\) and \(C_{j}\) represent the compactness of clusters \(k\) and \(j\). The compactness \(C_{k}\) is computed as the average Euclidean distance between all samples in cluster \(k\) and the cluster centroid \(\mu_{k}\). Based on the separation metric \(S_{kj}\), the Davies-Bouldin Index (DBI) is calculated as the average of the maximum similarity between each cluster and all other clusters, as follows:  
\[DBI=\frac{1}{N}\sum_{k=1}^{N}\underset{j\
eq k}{max}S_{kj} \tag{10}\]  
where \(N\) is the total number of clusters, and \(S_{kj}\) denotes the similarity (inverse separability) between clusters \(k\) and \(j\). A lower DBI value indicates a better clustering result, characterized by high intra-cluster compactness and strong inter-cluster separation. Conversely, a higher DBI suggests poor clustering performance due to overlapping or poorly separated clusters. By leveraging the DBI metric, the quality of clustering outcomes under different dimensionality reduction or clustering algorithms can be quantitatively assessed, thereby guiding the selection of appropriate methods and parameter configurations. (2) Recognition model evaluation index  
To evaluate the performance of lithology classification models, it is essential to select appropriate evaluation metrics. Balanced Accuracy and F1-score are widely used indicators, particularly effective for datasets with class imbalance. Figure 7: Schematic diagram of the CatBoost algorithm. Balanced Accuracy accounts for imbalanced class distributions by computing the sensitivity (recall) for each class and then averaging the results. It is defined as  
\[\text{Balanced Accuracy}=\frac{1}{N}\sum_{k=1}^{N}\frac{TP_{k}}{TP_{k}+FN_{k}} \tag{11}\]  
where \(TP_{k}\) denotes the number of correctly classified samples of class \(k\), \(FN_{k}\) is the number of samples of class \(k\) that were incorrectly classified, and \(N\) is the total number of classes. A higher Balanced Accuracy indicates that the model performs well across all classes, regardless of sample distribution. The F1-score is the harmonic mean of precision and recall, and is particularly effective for evaluating models under imbalanced class conditions.