The partition results are dependent on the choice of C. There exist validity indices to evaluate the goodness of clustering according to a given number of clusters; therefore, these validity indices

can be used to acquire the optimal value of C [27]. The XB index presents a fuzzy-validity criterion based on a validity function which identifies overall compact kinase inhibitors and separate fuzzy c-partitions. This function depends upon the data set, geometric distance measure, and distance between cluster centroids and fuzzy partition, irrespective of any fuzzy algorithm used. For evaluating the goodness of the data partition, both cluster compactness and intercluster separation should be taken into account. For the FCM algorithm with m = 2.0, the Xie-Beni index can be shown to be SXB=JFCMNdmin⁡2, (13) where dmin = mini,j‖βi − βj‖ is the minimum distance between cluster centroids. The more separate the clusters, the larger the dmin and the smaller the SXB. 3. Shadowed Sets-Based PSO-Fuzzy Clustering: SP-FCM FCM strives to find C compact clusters in X where C is one of the specified parameters. But the process of selecting and adjusting C manually to obtain desirable cluster partitions in a given data set is very subjective and somewhat arbitrary. To seek the optimal cluster structure, C is always

allowed to be overestimated [28], such that the distances between some clusters are not big enough or the membership values of some objects with different clusters are adjacent and ambiguous in a given data set. And, in this case, the modification of prototypes through long time iteration

is meaningless. The main subject of cluster validation is the evaluation of clustering results to find the partitioning that best fits the data set. Based on the foregoing algorithms, we wish to find cluster partitions that contain compact and well-separated clusters. In our algorithm C is also overestimated and the clusters compete for data membership. We can set [Cmin , Cmax ] as the reasonable range of cluster number based on the knowledge of Batimastat the data. This provides a more transparent and tractable process of cluster number reduction. Considering the fuzzy partition matrix U = [uij]N×C, each column is comprised of the membership values of all feature vectors xi with a single cluster center. Thus, an optimal threshold αj (j = 1,2,…C) for each column should be found to create a harder partition by (12). The amount of data which are assigned membership value equal to 1 is identified as the cardinality of corresponding cluster. According to αj, the cardinality of the jth column is Mj=carduij ∣ uij≥ujmax⁡−αj. (14) Here, the threshold is not subjectively user-defined but it is established on the balance of uncertainty and can be adjusted automatically in the clustering process. This property of shadowed sets can be used to reduce the cluster number.

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