# Cluster-Centered Visualization Techniques for Fuzzy Clustering Results to Judge Single Clusters
## 5 Validation by Example  
This section is devoted to the validation of the new techniques for CCV using several examples. Altogether, five data sets are used. * An artificial data set with well-separated clusters is referred to as an artificial good data set in the following. The data set contains three clusters generated from multivariate normal distributions. * An artificial data set with three clusters from multivariate normal distributions where two clusters show a strong overlap. In addition, uniform noise was added to the data set. We refer to this data set as an ambiguous artificial data set. * The Iris data set. * A medical data set and
* A data from an industrial production line. With these data sets, the clustering results based on the fuzzy \(k\)-means clustering algorithm will be visualized, as will the problem of the correct choice for the number of clusters. For data generation and visualization, the statistical open-source program R is  
Figure 1: Example of a visual comparison between scatterplot and the new developed CPC technique. Figure 3: Artificial good data set with three clusters and the visualized results with the three CCV techniques. Figure 2: Artificial good data set with two clusters and the visualized results with the three CCV techniques. For the artificially ambiguous data set, the values for the center of the separated cluster based on the normal distribution is \({}_{1}=(0;0)\) and \({}_{2}=(11;3)\) and \({}_{3}=(14;4)\) for the two overlapping clusters. The standard deviation is also always 1. The interval of the uniform distribution is from \(-3\) to 18 to cover the three clusters with noisy data points. Then the x- and y-values were generated based on the distributions. To achieve the clustering results, the fuzzy \(k\)-means algorithm is also applied, and the CCV techniques are used for the visualization. The following figures show the results of the visualization for the artificially ambiguous data set with 2, 3, and 4 clusters. All three CCV techniques show the dependencies always in the direction of the first cluster. Figure 5 shows the artificially ambiguous data set with two clusters. The clustered data are located in the top left. The center of the separated red cluster is clearly visible, but the fuzzy \(k\)-means algorithm seems to be able to identify two clusters. With the CPC method in the top right, both clusters are also not clearly separated in the visualization. Although the black cluster center is in the middle and the red cluster center is at the edge of the circle, there is a smooth transition between the data points. Some data points of the black cluster are attracted to the red cluster center, so a separation is not clearly visible. The A2C method in the bottom left delivers a better visualization result, but there are also some data points very close to the black and red clusters, respectively, and the two generated overlapping clusters are identified as one cluster. The same applies for the AM method in the bottom right. Figure 6 shows the artificially ambiguous data set with three clusters. The clustered data is located again at the top left. The center of the separated green cluster is theoretically well separated, and the fuzzy \(k\)-means algorithm identified three clusters. With the CPC method in the top right, the three clusters are partially clearly separated in the visualization. The black cluster center is in the middle of the circle, and the red and green cluster centers are on the edge. Some data points in the red cluster are attracted to the black cluster. The A2C method in the bottom left delivers a not-clear separation of the clusters. Only the green cluster, in contrast to the black and red clusters, is generally separated. The black  
Figure 4: Artificial good data set with four clusters and the visualized results with the three CCV techniques. and red clusters are very close to each other, with no clear separation. The AM method in the bottom right leads to a clear separation of the separated green cluster.