## 4 Clustering Refinement Framework
### Cluster Member Based Stability  
As discussed in [2], the quality of clustering results can be evaluated through stability; and clustering instability comes from two kinds of random components: (1) different perturbations of data, such as data set sampling; and (2) different initialization and parameter settings of the algorithm. However, consider exactly the same \(D\) with no data perturbations (i.e., data randomness excluded), stable clustering solutions are not yet unique when: (1) multiple \(K\)s are produced when different criteria are used to determine optimal number of clusters; or (2) various measures are evaluated as indicators during clustering algorithm execution. Therefore, we define unstable members as individuals with inconsistent cluster labels across multiple clusterings. **Definition 1** (Unstable member). : _Suppose there exist two distinct clusterings \(\pi_{1},\pi_{2}\)\((\pi_{1}\
eq\pi_{2})\), individual \(x_{i}\) satisfying \(l_{1}(x_{i})\
eq l_{2}(x_{i})\) is called an unstable member._  
As discussed before, instability issue across multiple clusterings remains due to: (1) non-unique optimal \(K\), and (2) difficult-to-determine parameters. Take Figure 3 for example. Since our dataset is high-dimensional, cluster plots are drawn by first two principle components after Principal Component Analysis (PCA) for visualization purpose. For each individual optimal \(K\), in order to eliminate the unstability caused by initialization conditions, we use _set.seed()_ to ensure the repeatability of results, and _nstart=50_ to generate multiple initial configurations, and thus the best and stable results are reported by each clusterer. We can observe that cluster members in Cluster 1 remain stable in Figure 2(a)-c; however, when we increase \(K\), members in Cluster 2 in Figure 2(a) partially shift to Cluster 3 in Figure 2(b) and later Cluster 4 in Figure 2(c). Individuals with multiple cluster labels across different clusterings are unstable members, caused by non-unique optimal \(K\) in this scenario. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Category** & **Measure** & \(K=2\) & \(K=3\) & \(K=4\) \\ \hline \multirow{3}{*}{Internal} & Cluster average silhouette widths (\(\uparrow\)) & 0.3780, 0.4971 & 0.0646, 0.4154, 0.4909 & 0.4519, 0.1362, 0.1123, 0.3826 \\  & Average silhouette width (\(\uparrow\)) & 0.3962 \({}^{*}\) & 0.2604 & 0.2536 \\  & DI (\(\uparrow\)) & 0.2603 \({}^{*}\) & 0.1527 & 0.1845 \\ \hline \multirow{3}{*}{External} & ARI (\(\uparrow\)) & N/A & 0.3209 \({}^{*}\) & 0.2105 \\  & FM (\(\uparrow\)) & N/A & 0.6995 \({}^{*}\) & 0.5897 \\ \cline{1-1}  & Jaccard (\(\uparrow\)) & N/A & 0.4960 \({}^{*}\) & 0.3526 \\ \cline{1-1}  & VI (\(\downarrow\)) & N/A & 1.0440 & 0.7318 \({}^{*}\) \\ \hline \multirow{3}{*}{Stability} & APN (\(\downarrow\)) & 0.0000 \({}^{*}\) & 0.0000 \({}^{*}\) & 0.0014 \\ \cline{1-1}  & AD (\(\downarrow\)) & 11.7993 & 10.1856 & 9.9983 \({}^{*}\) \\ \cline{1-1}  & ADM (\(\downarrow\)) & 0.0000 \({}^{*}\) & 0.0000 \({}^{*}\) & 0.0149 \\ \cline{1-1}  & POM (\(\downarrow\)) & 0.1655 & 0.1451 & 0.1424 \({}^{*}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Clustering validation of K-means with \(K=2,3,4\) using SFD-out data. Another cause of instability issue is input parameters, which is hard to choose because of lack of established argumentation or too expensive computation, especially for some clustering algorithms especially high-dimensional algorithms. Take projected subspace clustering algorithm ProClus [61] for example. Here we use PCA to heuristically determine average cluster dimensionality \(L\). As in Figure 4, \(L=3\) is selected by the elbow point in the percentage of variances explained by each principal component; and \(L=7,11,22\) are considered according to \(85\%,90\%,95\%\) cumulative percentage of variances explained by principal components.