# VSFCM: A Novel Viewpoint-Driven Subspace Fuzzy C-Means Algorithm
## 4 Experimental Results
### Artificial Data Sets  
Table 2 gives the basic information of all the data sets used in the experiment. Let us first discuss the artificial data sets DATA1 and DATA2. Figure 4 is a data distribution diagram of DATA1.  
As shown in Figure 4, the clusters of three colors correspond to three classes of the DATA1, the red triangle \"\(\triangle\)\" represents the cluster center of each cluster, and the black solid square\"\(\blacksquare\)\" represents the reference point for inter-cluster separation. The cluster centers are \(V_{1}=[-6.1545,-3.8668]\), \(V_{2}=[5.3800,1.4988]\), and \(V_{3}=[-3.0078,-10.0121]\). After calculation, we get \(X_{0}=[-1.2809,-4.1416]\), \(V_{0}=[-1.2607,-4.1267]\). We express the weight of the full space as \(w_{1}^{{}^{\prime}}=w_{2}^{{}^{\prime}}=w_{3}^{{}^{\prime}}=[0.5,0.5]\), three subspace weights are expressed as \(w_{1}=[0.9,0.1],w_{2}=[0.4,0.6],w_{3}=[0.6,0.4]\). The corresponding separation between clusters can be expressed respectively as:  
\[J_{\text{full\_space}}=\sum\limits_{i}^{3}({(w_{i1}^{{}^{\prime}})}^{2}(V_{i1 }-V_{01})^{2}+{(w_{i2}^{{}^{\prime}})}^{2}(V_{i2}-V_{02})^{2})=34.36 \tag{51}\]  
\[J_{\text{ESSC\_subspace}}=\sum\limits_{i}^{3}({(w_{i1}^{2})}^{2}(V_{i1}-X_{01} )^{2}+(w_{i2})^{2}(V_{i2}-X_{02})^{2})=44.38 \tag{52}\]  
\[J_{\text{VSFCM\_subspace}}=\sum\limits_{i}^{3}({(w_{i1})}^{2}(V_{i1}-V_{01})^{ 2}+(w_{i2})^{2}(V_{i2}-V_{02})^{2})=44.49 \tag{53}\]  
Figure 4: DATA1 data distribution map.  
Among them, \(v_{0}=\sum\limits_{i=1}^{c}v_{i}/c\), \(x_{0}=\sum\limits_{j=1}^{n}x_{j}/n\). Here \(v_{i}\) is obtained by using our proposed CDCl method.  
In this example, the separation between clusters in the subspace is significantly greater than the separation between clusters in the full space, which means that the subspace clustering has higher inter-cluster separation and better clustering effect than the full-space clustering. In particular, the separation between full-space clusters and the separation between subspace clusters can also be used as separations under different distance metrics.  
Compared with the ESSC algorithm, which is also a subspace algorithm, our algorithm has been improved to some extent. That is, replace \(x_{0}\) with \(v_{0}\) in its formula. Moreover, the effect of subspace separation of our algorithm is slightly better than it, and ours is more stable. When faced with more complex data sets or more scattered and irregular data sets, our algorithms can still maintain a good result. In a word, our proposed algorithm can get better separation between clusters, which can get better results.  
Figure 5 is the distribution diagram of each dimension of DATA2, and Table 3 is the distribution of each algorithm on DATA2.  
Figure 5: (**a**) DATA2 data distribution map. (**b**) DATA2: x-y dimensional data distribution map. (**c**) DATA2: x-z data distribution map. (**d**) DATA2: y-z data distribution map.  
From Table 3, we can find that the weight of the ESSC and SCAD1 algorithm does not fluctuate very much around 0.3333, but our VSFCM algorithm has a large difference in the attribution of weights. Taking Cluster 1 as an example, three weights, 0.3433, 0.3321, and 0.3246 of ESSC, are very close. However, the weight values obtained by the VSFCM algorithm are 0.4394, 0.2980, and 0.2626. From the VSFCM algorithm, it can be seen that the first feature has a significant contribution to the clustering result, followed by the second feature, and the worst is the third feature.  
Table 4 shows the clustering results of several algorithms on two artificial data sets, DATA1 and DATA2. Obviously, the performance of our algorithm is better than other algorithms. Moreover, mainly due to the contribution of weight distribution, the values of the five clustering indicators of the subspace clustering algorithms are obviously higher than that of other algorithms.