## 4 Clustering Refinement Framework
### Clustering Refinement with Member Based Stability  
As discussed before, isomorphic or heterogeneous clusterers could generate multiple stable clusterings, and it is difficult to confirm the best clustering due to various validation measures. In addition, some individuals keep the same membership among multiple clusterings, while unstable members shift across clusters during multiple clusterers. To this end, we propose a clustering refinement procedure based on unstable members to integrate multiple stable clusterings produced by isomorphic or heterogeneous clusterers. The proposed clustering refinement is illustrated in Algorithm 1. As mentioned before, input clusterer \(M\) is supposed to be relatively stable in its own nature. That is, input parameters are carefully selected using existing effective or heuristic methods. The intuition is that in most circumstances there are multiple good clustering solutions available, and each of them could be outstanding from different perspectives. Thus, we ensemble multiple sufficiently good clusterings for refinement. Suppose we have a set of models with proper parameter settings to ensemble, and a candidate list of potential optimal \(K\) determined by various methods (Figure 2). Basically, there are three steps in clustering refinement framework. **Step 1. ** Initialization clusters by running qualified clusterer with \(\min\{K^{(0)}\}\). Since we would align cluster labels as \(1,2,\ldots\) sequentially, minimum \(K\) clusterer is first executed to produce minimum number of clusters for the purpose of clustering vector alignment later. **Step 2. ** Run other clusterers with remaining candidate \(K\), and relabel members by clustering vector alignment. Parameter settings should be carefully chosen to reduce running time; however, details about how to choose parameters for specific clustering algorithm are not considered in the framework. Moreover, clustering vector alignment guarantees cluster labels produced by all clusterers are consistent. Figure 7 illustrates the principle of clustering vector alignment. Suppose initially we have two clusters as in Figure 7a, centroids are \(c_{1}\) and \(c_{2}\) respectively, and object \(p_{1}\) belongs to Cluster 1, while \(p_{2}\) belongs to Cluster 2. Later, clusterings Figure 7b,c are produced when using alternative \(K\), and all subsequent clusterings are relabeled according to Figure 7a by nearest centroid. For example, suppose originally \(p_{1}\) belongs to Cluster 2 in upper Figure 7b. Among three centroids, \(p_{1}\) is closest to \(c_{1}\), so the cluster in which \(p_{1}\) is located (Cluster 2) is relabeled as 1 (Cluster 1) and thus \(p_{1}\) has label 1. Similarly, if originally \(p_{2}\) belongs to Cluster 1 in upper Figure 7b, compared to existing \(c_{1},c_{2}\) in Figure 7a, \(p_{2}\) is more close to \(c_{3}\), and thus a new label (Cluster 3) is assigned to the cluster in which \(p_{2}\) is located (Cluster 1), as illustrated in lower Figure 7b. Later on, when a larger \(K\) is applied, members closest to newly generated centroid are assigned to greater label. For instance, in clustering Figure 7c, the cluster in which \(p_{2}\) is located is relabeled as Cluster 4 because \(c_{4}\) is the nearest centroid. In this way, we reconstruct clustering vectors by consistently labeling \(1,2,\ldots,\max\{K^{(0)}\}\) without changing clustering partitions. Figure 6: Illustration of clustering alignment issue: HDDC clusterings on SFD-out data with EM, CEM or SEM models using either ICL or BIC report identical \(K=5\) clusters. Cluster plot is visualized using the factoextra package in R.  
**Step 3. ** Make decisions about final cluster labels for all members. We use voting strategies for clustering integration. * Majority rule: if there is a single label with more than a third of the votes, regard it as the final label directly. Here we use \(1/3\) instead of \(1/2\) to eliminate the randomness error in clustering vector alignment. For instance, some clusterers assign NA for detected outliers.