Skip to main content

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)