A Validity Index for Fuzzy Clustering Based on Bipartite Modularity

Table 1

The commonly used fuzzy clustering validity indices.

Indices	Definition	Description
PC (partition coefficient)		The partition coefficient (PC) [15] measures the fuzzy degree of final divided clusters by means of the fuzzy partition matrix, and the larger its value, the better the partition result
PE (partition entropy)		The partition entropy (PE) [16] measures the fuzzy degree of final divided clusters by means of the fuzzy partition matrix, and the smaller its value, the better the partition result
MPC (modified partition coefficient)		Because the value of PC merely depends on the membership degree uci, Dave [17] proposed the modified PC index MPC, and the larger its value, the better the partition result
MPE (modified partition entropy)		Because the value of PE merely depends on the membership degree uci, Dave [17] proposed the modified PE index MPE, and the smaller its value, the better the partition result
XB (Xie–Beni index)		Considering the membership degree and the structure of datasets, Xie and Beni [18] proposed the XB index to measure the overall average compactness and separateness, and the smaller its value, the better the partition result
FS (Fukuyama–Sugeno index)	, where	Fukuyama and Sugeno [19] also proposed the index FS considering the compactness and separateness, and when its value reaches the minimum, the partition result is the best
PCAES (partition coefficient and exponential separation)	, where	Wu and Yang [20] proposed the index PCAES by combining the normalized partition coefficient with the exponential separateness degree of each cluster, and the larger its value, the better the partition result
CO (compactness and overlap measures)	, where	Žalik [21] proposed the index CO based on the compactness and separateness, and the larger its value, the higher the compactness degree, and the lower the coverage degree between clusters, the better the partition result
WGLI (weighted global-local index)	, where	Zhang et al. [14], based on the membership degree obtained from the FCM algorithm, proposed the index WGLI combining the bipartite modularity; represents the bipartite modularity proposed by Murata [22], and the larger the value of WGLI, the better the partition result