distance labels figure lefta cluster hierarchy of clusters rightthe corresponding dendrogram linkage these two clusters by (ijby specifying the function dwe indicate how the clusters are linked for this reason it is also referred to as the linkage criterion we give number of examplessingle linkage the closest distance between the clusters dmin (ij:min dist(xi iijj complete linkage the furthest distance between the clusters dmax (ij:max dist(xi iijj group average the mean distance between the clusters note that this depends on the cluster sizes xx dist(xi davg (ij:| |jii jj ward' linkage for these linkage criteriax is usually assumed to be rd with the euclidean distance another notable measure for the distance between clusters is ward' minimum variance linkage criterion herethe distance between clusters is expressed as the additional amount of "variance(expressed in terms of the sum of squaresthat would be introduced if the two clusters were merged more preciselyfor any set of indices (labelslet xk kk xk /|kdenote its corresponding cluster mean then kx xj ( dward (ij:kxk xij kxi xi kij ii jj it can be shown (see exercise that the ward linkage depends only on the cluster means and the cluster sizes for and jdward (ij| |jkxi xj | |