The Generalized Dunn’s index da2020incrementalgeocmeans is a
ratio of the worst pair-wise separation of clusters and the worst compactness
of clusters. A higher value indicates a better clustering. The formula
is:
$$GD_{r s}=\frac{\min_{i \neq j}\left[\delta_{r}\left(\omega_{i}, \omega_{j}\right)\right]}{\max_{k}\left[\Delta_{s}\left(\omega_{k}\right)\right]}$$
The numerator is a measure of the minimal separation between all the clusters
i and j given by the formula:
$$\delta_{r}\left(\omega_{i}, \omega_{j}\right)=\frac{\sum_{l=1}^{n}\left\|\boldsymbol{x_{l}}-\boldsymbol{c_{i}}\right\|^{\frac{1}{2}} . u_{il}+\sum_{l=1}^{n}\left\|\boldsymbol{x_{l}}-\boldsymbol{c_{j}}\right\|^{\frac{1}{2}} . u_{jl}}{\sum{u_{i}} + \sum{u_{j}}}$$
where u is the membership matrix and \(u_{i}\) is the column of
u describing the membership of the n observations to cluster
i. \(c_{i}\) is the center of the cluster i.
The denominator is a measure of the maximal dispersion of all clusters, given
by the formula:
$$\frac{2*\sum_{l=1}^{n}\left\|\boldsymbol{x}_{l}-\boldsymbol{c_{i}}\right\|^{\frac{1}{2}}}{\sum{u_{i}}}$$