# Classification of Tropical Cyclone Tracks in the Northwest Pacific Based on the SD-K-Means Model
## 3 Model Performance  
To evaluate the performance of the proposed clustering model and to compare it with alternative approaches, this study employs two widely used internal validation metrics: the Calinski-Harabasz (CH) index and the Davies-Bouldin index (DBI). These metrics are used to assess the clustering results of four different algorithms: the proposed SD-K-Means model, the DTW-based K-Means model, the standard K-Means algorithm, and DBSCAN. The CH index, also known as the variance ratio criterion, evaluates clustering quality based on the ratio of between-cluster dispersion to within-cluster compactness. Its formulation is given in Equation (14):  
\[CH=\frac{SS_{B}}{K-1}/\frac{SS_{W}}{n-K}=\frac{tr(B_{K})}{tr(W_{K})}\times \frac{n-K}{K-1} \tag{14}\]  
where \(n\) represents the number of samples, \(K\) represents the number of clusters, \(SS_{B}\) represents the variance between clusters, \(SS_{W}\) represents the variance within clusters, \(B_{K}\) represents the distance between clusters, and \(W_{K}\) represents the distance within clusters. In essence, a higher CH index implies that the clusters are more distinct (greater inter-cluster separation) and internally more cohesive (smaller intra-cluster variance). Thus, larger CH index values indicate better clustering performance. The specific CH index values of the SD-K-Means clustering model, the DTW-based K-Means clustering model, and the K-Means clustering model are shown in Table 2.  
As shown in Table 2, the SD-K-Means model achieves the highest CH index (230.3705), indicating the best clustering performance among the four models. In contrast, the K-Means model shows the lowest CH index (122.7050), suggesting the poorest performance in separating and compacting clusters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Clustering Model** & **K-Means** & **DBSCAN** & **DTW-Based K-Means** & **SD-K-Means** \\ \hline CH Index & 122.7050 & 173.1103 & 221.8391 & 230.3705 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of CH index values of multi-clustering models. Figure 2: Average SSE value under different numbers of clusters. The blue line shows the different average SSE values as the number of clusters changes. DBI, also known as the classification accuracy index, is also an evaluation index for measuring the quality of clustering effects. It comprehensively considers the similarity of samples within the cluster and the difference in samples between clusters. A lower DBI value indicates that clusters are compact and well separated. The DBI is defined as follows:  
\[DBI=\frac{1}{n}\underset{i=1}{\overset{n}{\sum}}max\frac{\overline{S_{i}}+ \overline{S_{j}}}{\left\|\overline{u_{i}}-\overline{w_{j}}\right\|_{2}},\ i\
eq j \tag{15}\]  
where, \(\overline{S_{i}}\) represents the average distance from the \(i\)-th cluster sample to its cluster center, and \(\left\|w_{i}-w_{j}\right\|_{2}\) represents the Euclidean distance between the \(i\)-th cluster and the \(j\)-th cluster center. The specific DBI values of the SD-K-Means clustering model, the DTW-based K-Means clustering model, and the K-Means clustering model are shown in Table 3.  
As observed in Table 3, the K-Means model again performs the worst, yielding the highest DBI value (1.6221), indicating poor intra-cluster cohesion and inter-cluster separation. The SD-K-Means model achieves a relatively low DBI of 1.1422, outperforming the DTW-based K-Means model (1.2229), although slightly underperforming compared with DBSCAN (0.9268). However, when considering both metrics in combination, the SD-K-Means model demonstrates a more balanced and superior performance. The high CH index coupled with a relatively low DBI suggests that the SD-K-Means algorithm produces clusters that are both well separated and internally cohesive, making it more robust than the other models in handling the nonlinear and variable-length nature of TC tracks. Notably, the DBSCAN model exhibits the lowest DBI, indicating its tendency to form highly compact clusters.