# Automated Exploratory Clustering to Democratize Clustering Analysis
## 4 Experimental Works
### Quality Functions  
In order to evaluate the clusterings, we chose six different CVIs according to the results in [67] as a Quality function. Like our chosen clustering algorithms, these CVIs encourage  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Group** & **min**\(n\) & **max**\(n\) & **min**\(d\) & **max**\(d\) & **min**\(|P|\) & **max**\(|P|\) \\ \hline Real & 101 & 14,980 & 2 & 262 & 2 & 116 \\ \hline Synthetic & 30 & 10,000 & 2 & 3 & 1 & 40 \\ \hline \end{tabular}
\end{table}
Table 2: Dataset overview.  
\begin{table}
\begin{tabular}{c c c} \hline
**Name** & **Used Parameters** & **Found Shapes** \\ \hline \(k\)Means & \(\texttt{k}\in[2,30]\) & globular clusters around a centroid \\ \hline HDBSCAN & min\_cluster\_size\(\in[2,30]\) & arbitrary shapes of similar density \\ \hline Meanshift & bandwidth\(\in[0.1,50]\) & fields of smooth density \\ \hline OPTICS & xi\(\in[0.025,0.5]\), min\_samples\(\in[3,20]\) & arbitrary shapes of similar density \\ \hline \end{tabular}
\end{table}
Table 3: Used clustering algorithms.  
different notions of clusterings in their evaluations. The utilized Quality functions are described below in Table 4.  
The _Silhouette Width Criterion_ (SWC) [52]--also known as Silhouette Coefficient--computes a silhouette for each object based on the average distance to objects in its own cluster and the average distance to objects in the second closest cluster. Non-convex clusters are valued highly using this method. We used the implementation in [65].  
_Density-Based Clustering Validation_ (DBCV) [60] is an MST-based measure using the density of an object in its cluster. We use our own implementation (available at [[https://github.com/g-schlake/ASCVI](https://github.com/g-schlake/ASCVI)]([https://github.com/g-schlake/ASCVI](https://github.com/g-schlake/ASCVI)) accessed at 19 May 2025), which matches the results of the original authors' implementation on the given test datasets.  
The _Density Separability Index_ (DSI) [68] compares the histograms of distances between a cluster and objects of all other clusters. We use the original authors' implementation (available at [[https://github.com/ShuyueG/CVI_using_DSI](https://github.com/ShuyueG/CVI_using_DSI)]([https://github.com/ShuyueG/CVI_using_DSI](https://github.com/ShuyueG/CVI_using_DSI)) accessed at 19 May 2025).  
For the _Contiguous Density Region_ (CDR) [69], the density of objects measured by the distance to its nearest neighbour is measured. Clusters should be regions where this density is constant. We used our own implementation (available at [[https://github.com/g-schlake/ASCVI](https://github.com/g-schlake/ASCVI)]([https://github.com/g-schlake/ASCVI](https://github.com/g-schlake/ASCVI)) accessed at 19 May 2025), which matches the implementation of the original authors.  
_Validity Index for Arbitrary-Shaped Clusters Based on the Kernel Density Estimation_ (VIASCKDE) [70] is an index similar to the SWC, but in this approach, objects are also weighted by the density assigned by a Kernel Density Estimation (KDE) on the dataset to value objects in dense areas higher. We used the original authors' implementation (available at [[https://github.com/senolali/VIASCKDE](https://github.com/senolali/VIASCKDE)]([https://github.com/senolali/VIASCKDE](https://github.com/senolali/VIASCKDE)) accessed at 19 May 2025). To fit the KDE, we use a Gaussian Kernel and fit the bandwidth in a random search in the space in \([0.1,100]\).  
The _Variance Ratio Criterion_ (VRC) [71]--also known as Calinski-Harabasz criterion--is a classical metric using the trace of the dispersion matrices within each cluster and between clusters. We used the implementation in [65].  
As not all CVIs can directly deal with objects assigned as outliers, we adopt the scores with a multiplication by \(\frac{n-|O|}{n}\) for maximized and by \(\frac{n}{n-|O|}\) for minimized CVIs, where \(|O|\) is the number of outliers. We did this according to [67].