Table 1
Selected fuzzy clustering validation indices for optimizing the number of clusters (c)
| Index | Source | Equation | Parameters |
|---|---|---|---|
| Partition coefficient | Bezdek (1974a, 1974b) |  | VPC: Partition coefficient |
| Partition entropy | Bezdek (1974a, 1974b) |  | VPE: Partition entropy |
| Fukuyama–Sugeno | Fukuyama & Sugeno (1989) |  | VFS: Fukuyama and Sugeno validity measure |
 | VEXB: The extended FCM | ||
| Extended Xie–Beni | Xie & Beni (1991) and Pal & Bezdek (1995) |  | Xie–Beni index VK: Kwon index N: Number of cells in each layer |
| Kwon's index | Kwon (1998) |  | c: Number of clusters |
 | uij: Degree of membership xj: Rescaled feature value for each cell ci and cj : The fuzzy centroids of the ith cluster |
| Index | Source | Equation | Parameters |
|---|---|---|---|
| Partition coefficient | Bezdek (1974a, 1974b) | | VPC: Partition coefficient |
| Partition entropy | Bezdek (1974a, 1974b) | | VPE: Partition entropy |
| Fukuyama–Sugeno | Fukuyama & Sugeno (1989) | | VFS: Fukuyama and Sugeno validity measure |
| | VEXB: The extended FCM | ||
| Extended Xie–Beni | Xie & Beni (1991) and Pal & Bezdek (1995) | | Xie–Beni index VK: Kwon index N: Number of cells in each layer |
| Kwon's index | Kwon (1998) | | c: Number of clusters |
| | uij: Degree of membership xj: Rescaled feature value for each cell ci and cj : The fuzzy centroids of the ith cluster |