# Boundary Matching and Interior Connectivity-Based Cluster Validity Analysis
## 3 Materials and Methods
### Clustering Evaluation Based on Boundary and Interior Points
CVIBI can point out these optimal clustering results on the above six datasets no matter what clustering algorithms are used. In terms of various shapes in _Group_ 3, CVIBI works well based on DPC and DBSCAN, but C-means cannot. Because C-means originally is designed to partition spherical clusters rather than arbitrary shapes, and then incorrect partitions of boundary points and incorrect values of _con_ (\(c\)) are caused. When the clusters in datasets contain different distributions (e.g., group 4), CVIBI based on DPC shows better performance than C-means and DBSCAN according to accuracy. Because C-means and DBSCAN cannot give correct clustering results, DPC can obtain relatively correct results. When the dataset has clusters that partially overlap with each other (e.g., group 5), DPC can obtain relatively correct clustering results than C-means and DBSCAN, as shown by the corresponding curves. The merit of CVIBI is to select the best one from any candidates of clustering results, no matter what clustering algorithm is used. However, if all available clustering results cannot contain the real number of clusters, CVIBI can find it as well. Figure 9: The solved range of the number of neighbors in 15 datasets. #### 4.1.2 Relationship between CVIBI and \(\varepsilon\) in DBSCAN

When using DBSCAN, any certain number of clusters responds to a continuous interval of _Eps_ (i.e., \(\varepsilon\)). The clustering results of DBSCAN highly depend on the value of _Eps_. In order to solve the optimal values of _Eps_, we determine the possible range of _Eps_ first. Then, a group of _Eps_ values is evenly sampled to approximate the _Eps_ variable itself. Then, we cluster the dataset using DBSCAN with the group of _Eps_ values. Finally, the optimal values of _Eps_ can be determined by CVIBI. Figure 11 shows the evaluation results of CVIBI with respect to _Eps_, where piecewise numbers chain in these CVIBI curves is the obtained number of clusters. The abscissa and vertical axes denote the value of _Eps_ and CVIBI, respectively. The number along with each red point denotes the number of clusters computed by DBSCAN. If CVIBI can compute the real number of clusters, then a green box will be drawn and point out the optimal value of _Eps_. The maximum value of each curve in Figure 11 indicates the suggested optimal number of clusters. The corresponding values of _Eps_ are the suggested optimal parameters. It also can be seen that the optimal _Eps_ is not one certain number but a range of values. CVIBI with DBSCAN can point out the correct clustering results except _Set_ 10 and group 5. Set 10 has three clusters and one cluster is separated from the other two clusters. The other two clusters are close to each other, which are easy to be treated as one cluster. CVIBI with DBSCAN algorithm regards these two clusters as one cluster, and the maximum value occurs when \(c=2\). The clusters in group 5 are

Figure 10: Evaluation results of CVIBI obtained using DPC, C-means, and DBSCAN. significantly overlapped, so CVIBI cannot recognize the real numbers of clusters. Consequently, CVIBI can point out the optimal clustering results by finding the optimal parameter of _Eps_. #### 4.1.3. CVIBI Evaluation under Various Clustering Algorithms

Once a dataset is clustered by several clustering algorithms that lead to different clustering results, how one can select the optimal clustering result remain a challenging task for the existing validity indices. Nevertheless, CVIBI can realize this purpose. For example, CVIBI obtains three curves based on C-means, DBSCAN, and DPC algorithms. Each curve points out a suggested number of clusters, which may be different from the others. The optimal number of clusters can be selected by comparing the values of the boundary matching degree of _bou_ (_c_). Figure 12 shows the evaluating process of the multiple-peak dataset that is shown in Figure 2a. It shows that CVIBI based on DBSCAN suggests 2 as the optimal number of clusters, but CVIBI based on C-means and DPC both do 1.