# Cluster-Centered Visualization Techniques for Fuzzy Clustering Results to Judge Single Clusters
## 5 Validation by Example
### Sugar Production Data Set  
The CCV techniques are also applied to a third real data set. The data was collected in a sugar factory of Nordzucker AG, where the sugar production process consists of several stages that are monitored automatically. The sugar production data set contains 43 parameters that influence a target parameter during sugar production. The parameters represent the sugar production over a period of four weeks. The collected data was completely anonymized and additionally manipulated with normally distributed noise. 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 sugar production 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, green for the third, and blue for the fourth cluster. Figure 14 shows the sugar production data set with two clusters. With the CPC method (left), two clusters are separated, but the points are placed in the middle of both clusters and are very close to each other. Therefore, a clear separation is not visible. The A2C method (middle) leads to a better visualization of the fact that the two clusters are partially separated. The AM method (right) indicates that the points of the black cluster are distributed in a circle around the center, even if some points deviate from this. The points of the red cluster are distributed around the black cluster. Figure 15 shows the sugar production data set in three clusters. With the CPC method (left), three clusters are separated, but the data points of the green cluster are near the center between the green and red clusters. The data points of the black cluster are in the center between the red and the black cluster but are more attracted to the black cluster. The A2C method (middle) shows a better separation where the points of the black and green clusters are close to each other. The red cluster is more separated. The AM method (right) shows that the data points of the black cluster are also distributed in a circle around the center with no deviations. The data points of the red and green clusters are distributed around the black data points. Figure 14: Sugar production data set with two clusters and visualized results with the CPC (**left**), A2C (**middle**), and AM (**right**) methods. Figure 16 shows the sugar production data set with four clusters. The CPC method (left) leads to a clear separation of the different clusters, with green data points in the center of the green and red clusters. With the A2C method (middle), a clear separation is also visible even if the red and blue clusters are slightly mixed. The AM method (right) shows that the points of the blue cluster are distributed in a circle around the center with slight deviations from some data points. The data points of the red, green, and black clusters are distributed around the blue cluster. 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. Only two clusters could be too few for the sugar production dataset because the whole process varies and the influence of the parameters on the target variables cannot be divided into only two classes. With three clusters, the clusters are close together but more or less clearly separated.