Table 3.

Some internal and external validity indexes.

INTERNAL CLUSTERING MEASURES
Measures	Formular
The partition entropy159(PE) (Bezdek 1973)	$V_{PE}=-\frac{1}{n}{\displaystyle \sum _{j=1}^c{\displaystyle \sum _{j=1}^n{u}_{ij}log{u}_{ij}}}$
Dunn index141	$Dunn=\min \left(D\left(c{l}_{i}c{l}_j\right)\right)/{\max }_{1\le i\le k}\left(D\left(c{l}_i\right)\right)$
The partition coefficient160(Pc)	$V_{PC}=\frac{1}{n}{\displaystyle \sum _{j=1}^c{\displaystyle \sum _{j=1}^n{u}_{ij}{ }^2}}$
Calinski-Harabasz161	$\frac{traceB/\left(k-1\right)}{traceW\left(n-k\right)}$
C-Index162	$\frac{S-S_{mm}}{S_{\max }-S_{mm}}$
Davies-Bouldin163	$\frac{1}{k_{i=I}}{\displaystyle \sum _{j=i\dots k:j\ne i}^k\mathit{max}\left(d_{ij}\right)}$
Krzanowski-Lai index164	$\text{KL}=\left|diff(k)/diff(k+1)\right|$
VFS165	$V_{FS}=J_m(U,V)-K_m(U,V)={\displaystyle \sum _{i=1}^c{u}_{ij}^m}{\displaystyle \sum _{j=1}^n{u}_{ij}^m}{\Vert x_j-v_i\Vert }^2-{\displaystyle \sum _{i=1}^c{u}_{ij}^m}{\displaystyle \sum _{j=1}^n{u}_{ij}^m}{\Vert v_i-\overline{i}\Vert }^2$
VXB38	$V_{FS}=\frac{J_m(U,V)/n}{Sep(V)}=\frac{{\displaystyle {\sum }_{i=1}^c{u}_{ij}^m}{\displaystyle {\sum }_{j=1}^n{u}_{ij}^m{\Vert x_j-v_i\Vert }^2}}{n\min {\Vert x_j-v_i\Vert }^2}$
EXTERNAL CLUSTERING MEASURES
Measures	Formular
The fuzzy hypervolume166 (FHV) validity	$FHV={\displaystyle \sum _{i=I}^c{V}_i,V_i}={\left|\sum _i\right|}^{1/2},{\displaystyle \sum _i=\frac{{\displaystyle {\sum }_{j=1}^n{u}_{ij}^m\left(x_j-c_i\right){\left(x_j-c_i\right)}^T}}{{\displaystyle {\sum }_{j=1}^n{u}_{ij}^m}}}$
Modification of the MPE index146	$MPE=\frac{n\times I_{PE}}{n-C}$
Modification of the VPC index146	$V_{MPC}=1-\frac{c}{c-1}\left(1-V_{PC}\right)$
Kwon index167 (KI)	$K=\frac{{\displaystyle {\sum }_{i=1}^c{u}_{ij}^m}{\displaystyle {\sum }_{j=1}^n{u}_{ij}^m}{\Vert x_j-v_i\Vert }^2-{\displaystyle {\sum }_{i=1}^c{u}_{ij}^m}{\displaystyle {\sum }_{j=1}^n{u}_{ij}^m}{\Vert v_i-\overline{v}\Vert }^2}{n\min {\Vert x_j-v_i\Vert }^2}$
The PBM index168	$PBM=(1/k\times E_r/E_w\times D_E)$
PBM-index for Fuzzy c-means168	$PBMF={\left(\frac{1}{C}\times \frac{E_1}{J_m}\times D_c\right)}^2$
VPCAES169	$V_{PCAES}={\displaystyle \sum _{i=1}^{c}PCAE{S}_i}={\displaystyle \sum _{i=1}^c{\displaystyle \sum _{j=1}^n{u}_{ij}^2/u_M}}-{\displaystyle \sum _{j=1}^n\exp }\left(-\min \left\{\raisebox{1ex}{${\Vert x_j-v_i\Vert }^2$}\!\left/ \!\raisebox{-1ex}{${\beta }_{\mathrm{r}}$}\right.\right\}\right)$
Mutual information and variation of information170	$MI={\displaystyle \sum _i{\displaystyle \sum _j{P}_{ij}\log \frac{P_{ij}}{P_i{P}_j}}}$
Zhang index139 (ZI)	$Z_c(V,U)=\frac{Va{r}_{N,C}(V,U)}{Se{p}_N(c,U)}$
VSC(c,U)171	$VSC(c,U)=Se{p}^N(c,U)+Com{p}^N(c,U)$
Rezaee compactness and separation171 (RSC)	$RSC=Se{p}^N(c,U)+Com{p}^N(C,X,U,V)$
WGLI39	$WGLI=(2MMD+Q_{/B})/3.$
Graded Distance Index172 (GD_index)	$G{D}_{index,c}=\frac{{\displaystyle {\sum }_{i=1}^N(u_{i,1st\max }-u_{i,2nd\max })}}{N}-\left(\frac{c}{N}\right)$