## 5 Experiment
### Evaluation Indicator  
When evaluating the performance of unsupervised clustering algorithms, key indicators include the Silhouette Coefficient, the Rand Index, the Adjusted Rand Index, and the Normalized Mutual Information. The Silhouette Coefficient (SC) [46] can be used to assess the compactness of clustering results. Its value range is between \(-1\) and \(1\), and a larger value indicates better clustering effects. The calculation formula of the Silhouette Coefficient is as follows:  
\[S(i)=\frac{b(i)-a(i)}{\max\{a(i),b(i)\}} \tag{6}\]  
Here, \(a(i)\) represents the average distance of other samples within the cluster to which \(i\) belongs, and \(b(i)\) represents the minimum average distance of \(i\) to the samples of other clusters. The Rand Index (RI) provides a method to measure the similarity of clustering results [47] and is used to evaluate the accuracy of clustering algorithms in data classification tasks. This index is calculated based on the comparison of the number of similar and dissimilar sample pairs within two clustering results. Its value ranges from 0 to 1. A value close to 1 indicates a high degree of consistency in the clustering results, while a value close to 0 indicates significant differences between the two results. When using the Rand Index to evaluate the performance of clustering algorithms in unsupervised learning, it should be noted that the Rand Index may be biased in situations with a large number of categories. Therefore, a comprehensive evaluation should be conducted in combination with other evaluation indicators. In addition, when the Rand Index is applied to cluster analysis with a known label set, it can be used as an indicator to measure the difference between the clustering results and the true labels. The calculation formula of the Rand Index is as follows:  
\[RI=\frac{TP+TN}{TP+FP+TN+FN} \tag{7}\]  
Here, \(TP\) represents the number of samples that are of the same category and are classified in the same cluster, \(TN\) refers to the number of samples of different categories and distributed in different clusters, \(FP\) involves the number of samples of different categories but classified in the same cluster, and \(FN\) refers to the number of samples of the same category but distributed in different clusters. The Adjusted Rand Index (ARI) optimizes the original Rand Index. The Rand Index does not always remain stable close to zero when dealing with random classification results. Therefore, ARI is introduced to reflect the clustering effect more accurately [48]. ARI is similar to the Rand Index and is used to assess the consistency of two clustering results. It estimates the similarity by counting the similar or dissimilar sample pairs in the two clustering results and incorporates the consideration of the influence of randomness on clustering. The value of ARI ranges from \(-1\) to \(1\). The closer the value is to \(1\), the higher the similarity of the two clustering results; the closer to \(-1\), the greater the difference. When ARI is 0, it means that the clustering results are no different from random assignment. The feature of ARI is that it can eliminate the interference of randomness on clustering judgment and is usually used to evaluate the performance of the clustering algorithm on the actual dataset. The calculation formula of the Adjusted Rand Index is as follows:  
\[ARI=\frac{RI-E[RI]}{\max(RI)-E[RI]} \tag{8}\]  
Here, \(RI\) is the Rand Index. The Normalized Mutual Information (NMI) is a measure of the similarity of two clustering results. After applying different clustering algorithms, in order to evaluate theeffect of the algorithm or the superiority of parameter configuration, it is often necessary to compare the clustering results [49]. At this time, the NMI metric can be used to estimate the similarity of the two clustering results. NMI normalizes the mutual information between the two clustering results to generate a value between 0 and 1, which reflects the similarity degree of the two clustering results.