# Cluster-Centered Visualization Techniques for Fuzzy Clustering Results to Judge Single Clusters
## 5 Validation by Example
### Iris Data Set  
In addition to the two artificial data sets, the CCV techniques are also applied to a real data set. The data set is the simple Iris data set [32]. This data set was created as part of the investigation of Iris flowers in 1935. The data set contains measured data from 150 iris  
Figure 7: Artificial ambiguous data set with four clusters and the visualized results with the three CCV techniques.  
plants. These are divided into 50 Virginia irises (Iris Virginica), bristly irises (Iris setosa), and irises of different colors (iris versicolor) each. The length and width of the sepal leaves are included, as well as the petal leaves of each individual plant. The categorical attribute species will not be considered here. The fuzzy \(k\)-means clustering algorithm is also applied to determine the clustering results. For the visualization of the clustering results, CCV techniques are used. The following figures show the results of the visualization for the Iris data set with 2, 3, and 4 clusters. The CPC method always puts the first cluster center in the middle, all other centers on the edge, and all data points inside the circle. The A2C method always uses the first and second cluster centers for the angle-based visualization. The AM method also always uses the first cluster center for the angle-based visualization. The colors of the clusters are red for the first, black for the second, and green for the third, and blue for the fourth cluster.  
Figure 8 shows the Iris data set with two clusters. With the CPC method (left), two clusters are separated, but some data points from the red cluster are attracted to the black cluster. Therefore, a clear separation is not visible. The A2C method (middle) leads to a better visualization because the two clusters are partially separated, but some data points of the red cluster are also attracted to the black cluster. The AM method (right) shows the same result.  
Figure 9 shows the Iris data set with three clusters. With the CPC method (left), three clusters are clearly separated, and the black cluster is very well distinguished from the other two clusters. The A2C method (middle) also shows that the red cluster is clearly separated. But the black and green clusters are slightly mixed. With the AM method (right), the same results are indicated.  
Figure 8: Iris data set with two clusters and visualized results with the CPC (**left**), A2C (**middle**) and AM (**right**) method.  
Figure 9: Iris data set with three clusters and visualized results with the CPC (**left**), A2C (**middle**), and AM (**right**) methods.  
Figure 10 shows the Iris data set in four clusters. The CPC method (left) leads to a clear separation of the different clusters. But the green and red clusters are attracted to each other. With the A2C method (middle), the black and blue cluster are slightly mixed, and the other two clusters are mixed with no distinction. The same applies to the AM method (right).  
The use of the three newly developed CCV techniques has indicated that the results based on the \(k\)-means clustering algorithm deliver more information about the dependencies between the individual clusters when focusing on a single cluster. This was also demonstrated with the artificial data sets. In general, two clusters could be too few for the Iris dataset because there are three iris plants. With three clusters, one cluster is clearly separated but not the other two clusters. With four clusters, a large part of the data points are mixed, and only one cluster differs from the others. With the red cluster as the center, all three CCV techniques achieved a good separation of the three other clusters.