Ward’s linkage is a method for hierarchical cluster analysis. The idea has much in common with analysis of variance (ANOVA). The linkage function specifying the distance between two clusters is computed as the increase in the “error sum of squares” (ESS) after fusing two clusters into a single cluster. Ward’s Method seeks to choose the successive clustering steps so as to minimize the increase in ESS at each step.
The ESS of a set of values is the sum of squares of the deviations from the mean value or the mean vector ( centroid). For a set the ESS is described by the following expression:
where
is the absolute value of a scalar value or the norm (the “length”) of a vector.
Mathematically the linkage function – the distance between clusters and – is described by the following expression
where
is the combined cluster resulting from fusion clusters and ;
is the error sum of squares describe above.