## 4 Clustering Refinement Framework
### Cluster Member Based Stability
Figure 5 demonstrates member based stability with ProClus clustering with different \(L\) using the subspace ([[https://cran.r-project.org/web/packages/subspace](https://cran.r-project.org/web/packages/subspace)]([https://cran.r-project.org/web/packages/subspace](https://cran.r-project.org/web/packages/subspace))) package. For the sake of visualization and understanding, we define station IDs as follows: (1) non-transfer stations are composed of three digits, where the first digit denotes the line number, and the other two digits denote the sequential number of station; (2) transfer stations begin with \(t\) followed by three digits, where the first two digits denote intersected two lines, and the last digit means the sequential number of intersections between them. For example, 317 means the 17 th station of Line 3, and \(t122\) means the transfer station is the second intersection of Lines 1 and 2. As suggested in Figure 5, owing to difficult-to-determine parameters, which might be determined by some heuristic methods, member relationship between individuals (i.e., stations) and clusters could be unstable. For example, stations _317_, _318_, _319_ are within the same cluster in Figure 4(a),b,d, except that _317_ shifts to another cluster in Figure 4(c). That is, _318_ and _319_ are of the same group among four  
Figure 4: Illustration of difficult-to-determine parameters. Heuristically choose \(L\) by: (a) elbow point \(L=3\) in the percentage of variances explained by each principal component; (b) \(85\%,90\%,95\%\) cumulative percentage of variances explained by principal components, i.e., \(L=7,11,22\). Figure 3: K-means clustering on SFD-out data, when \(K\) takes different optimal values. Cluster plot is visualized using the factoextra package in R, where X-axis and Y-axis represent principal axes and their variable contributions. ProClus clusterers, while 317 is an unstable member. Note that ProClus detects outliers, which are not classified as objects in clustering results, and thus not shown in the figures accordingly. However, members with different cluster labels could be still organized within the same cluster, while \(l_{1}(x_{i})\
eq l_{2}(x_{i})\) is the result of unaligned clustering vectors generated by different clusterers. Therefore, we introduce corrected unstable member to illustrate clustering alignment issue. **Definition 2** (Corrected unstable member). : _Suppose there exist two distinct stable clusterings \(\pi_{1},\pi_{2}\) (\(\pi_{1}\
eq\pi_{2}\)). Given individual \(x_{i}\), its cluster labels in two clusterings are \(l_{1}(x_{i})\) and \(l_{2}(x_{i})\), and the corresponding cluster centroids are \(c(l_{1}(x_{i})),c(l_{2}(x_{i}))\) respectively. If \(l_{1}(c(l_{1}(x_{i})))\
eq l_{2}(c(l_{2}(x_{i})))\), then \(x_{i}\) is corrected unstable member._  
Take model-based high-dimensional clustering algorithm HDDC [62] for example. As illustrated in Figure 6, Expectation-Maximisation (EM), Classification E-M (CEM) and Stochastic E-M (SEM) algorithms for model selection with BIC (Bayesian Information Criterion) or ICL (Integrated Complete-data Likelihood) criteria are employed respectively for HDDC clustering [63] using the HDclassif ([[https://cran.r-project.org/web/packages/HDclassif](https://cran.r-project.org/web/packages/HDclassif)]([https://cran.r-project.org/web/packages/HDclassif](https://cran.r-project.org/web/packages/HDclassif))) package. All three clusterers report five clusters with various cluster labels, reflected as annotation of points in figures. However, it could be easily observed that the clusterings are identical except for notations, and therefore these three clusterings are regarded as the same solution. In fact, cluster labels are already aligned for previous visualization in Figures 3 and 5. From now on, we refer to unstable members as Definition 2.