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Table 1 Cluster validity indexes used as objective functions

From: A multi-objective gene clustering algorithm guided by apriori biological knowledge with intensification and diversification strategies

Validity index Equation Type
Xie-Beni index (XB) [47] measures \({XB=\frac {\sum \limits _{k=1}^{K}\sum \limits _{i=1}^{n} D^{2}(z_{k}, x_{i})}{ n \times \min _{k\neq l}\{D^{2}(z_{k}, z_{l})\}}} \) Minimisation
the quotient between the total  
variance and the minimum  
separation of the elements  
in the clusters.  
Overall cluster deviation (Dev) [48] \(Dev= \sum \limits _{k=1}^{K} \sum \limits _{x_{i} \in C_{k}} D(z_{k}, x_{i})\) Minimisation
is defined as the overall summed  
distances between genes and their  
corresponding cluster medoid.  
Cluster separation (Sep) [49] is \( Sep= \frac {2}{K(K-1)}\sum \limits _{k=1}^{K} \sum \limits _{j=1, j\not =i}^{K} D^{2}(z_{i}, z_{j})\) Maximization
defined as inter-cluster distances  
between cluster medoids.  
  1. The distance D in each formula is measured using both expression profiles-based distance (DEB) and biological-based distance (DBB)