# Boundary Matching and Interior Connectivity-Based Cluster Validity Analysis
## 3 Materials and Methods
### Clustering Evaluation Based on Boundary and Interior Points
To assess their differences, we further analyze their values of _bou_ (_c_). _bou_ (_c_) = 1 in CVIBI based on C-means and DPC when \(c\) = 1, while _bou_ (_c_) = 1 in CVIBI based on DBSCAN when \(c\) = 1 and 2. In views of the clustering process, we can conclude that any real cluster in all clusters is not separated when executing clustering algorithm if _bou_ (_c_) = 1. Let \(\Gamma\) be the set of numbers of clusters whose _bou_ (_c_) = 1. The maximum value in \(\Gamma\) indicates the optimal number of clusters in the multiple-peak dataset. So in the situation, both CVIBI and _bou_ (_c_) can indicate that the real number of clusters is 2, and the clustering algorithm suitable for this dataset is DBSCAN. Figure 11. Evaluation results of CVIBI based on DBSCAN. Figure 13 shows the evaluating process of the density-diverse dataset in Figure 2b. We can conclude from Figure 13 that the suggested numbers of clusters are different. CVIBI with DBSCAN and DPC both suggest 2 as the optimal number of clusters while CVIBI based on C-means regards 3 as the optimal number of clusters. To assess which clustering result is best among the three clustering algorithms, the three curves of _bou_ (_c_) are used to consist of \(\Gamma\). Finally, the optimal number of clusters can be determined from \(\Gamma\). It can be extracted that \(\Gamma\) set contains 1, 2, and 3 in total. So the suggested number of clusters is 3, which is the maximum value in \(\Gamma\). The optimal algorithm for the density-diverse dataset is C-means algorithm. Figure 14 shows the evaluating process of the dataset in Figure 2c. The results of the CVIBI index with three clustering algorithms are different. CVIBI based on C-means, DBSCAN, and DPC suggest 1, 2, and 3 as the real number of clusters, respectively. In order to obtain the optimal number of clusters, three curves of _bou_ (_c_) are used. From the relationship between \(\Gamma\) and the real number of clusters, _bou_ (_c_) of C-means, DBSCAN and DPC suggest 1, 2, and 3 as the optimal number of clusters, respectively. So the real number of clusters is 3, which is suggested by CVIBI with DPC. The suitable clustering algorithm of the investigate dataset is the DPC algorithm. Figure 12: Evaluating the process of CVIBI on the multi-peak dataset. Figure 13: Evaluating process of CVIBI on density-diverse dataset. #### 4.1.4 Comparison between CVIBI and Existing Validity Indices

Table 3 shows all evaluation results obtained by CVIBI, DB, DC, and GS when using the above three clustering algorithms. The four validity indices are analyzed as follows. 1. Size and distribution. Groups 2 and 4 denote clusters with different sizes and distributions, and there are no overlapped clusters. The evaluation results show that except set 10, all validity indices are capable of determining the correct number of clusters no matter which clustering algorithm is used. 2. Density and shape. Groups 1 and 3 contain clusters with different densities and shapes. When the densities among clusters are so skewed as in sets 1 and 3, the three validity indices cannot find the correct number of clusters. Compared with DC and DB, the cluster numbers calculated by CVIBI are closer to the real numbers of clusters. 3. Overlap. Group 5 contains datasets with overlapped clusters. DPC is good at clustering such datasets, and the evaluation results of CVIBI are nearest to the real number of clusters. However, DC and GS with DPC give a relatively smaller number. Note the two indices are originally designed to evaluate the results for nonoverlapping clusters. Thus any two overlapped clusters may be incorrectly regarded as one cluster. For instance, the clusters in set 14 are most overlapped, and its evaluation results are the weakest. How to find the correct number of clusters in a dataset containing overlapped clusters is a difficult task for most existing validity indices, such as DB and GS; fortunately, CVIBI with DPC is effective for dealing with such a problem.