Six intercluster distances and three intracluster diameters can be used to
calculate such validity indices as Dunn and Davies-Bouldin like.
Let d(x,y) be a distance function between two objects comming from our data set.
Intracluster diameters The complete diameter represents the distance between two the most remote objects belonging
to the same cluster.
diam1(C) = max{ d(x,y): x,y belongs to cluster C }
The average diameter distance defines the average distance between all of the
samples belonging to the same cluster.
diam2(C) = 1/|C|(|C|-1) * sum{ forall x,y belongs to cluster C and x != y } d(x,y)
The centroid diameter distance reflects the double average distance between all of the
samples and the cluster's center (v(C) - cluster center).
diam3(C) = 1/|C| * sum{ forall x belonging to cluster C} d(x,v(C))
Intercluster distances
The single linkage distance defines the closest distance between two samples
belonging to two different clusters.
dist1(Ci,Cj) = min{ d(x,y): x belongs to Ci and y to Cj cluster }
The complete linkage distance represents the distance between the most remote samples
belonging to two different clusters.
dist2(Ci,Cj) = max{ d(x,y): x belongs to Ci and y to Cj cluster }
The average linkage distance defines the average distance between all of the samples
belonging to two different clusters.
dist3(Ci,Cj) = 1/(|Ci|*|Cj|) * sum{ forall x belongs Ci and y to Cj } d(x,y)
The centroid linkage distance reflects the distance between the centres of two clusters
(v(i), v(j) - clusters' centers).
dist4(Ci,Cj) = d(v(i), V(j))
The average of centroids linkage represents the distance between the centre of a cluster
and all of samples belonging to a different cluster.
dist5(Ci,Cj) = 1/(|Ci|+|Cj|) *
( sum{ forall x belongs Ci } d(x,v(j)) + sum{ forall y belongs Cj } d(y,v(i)) )
Hausdorff metrics are based on the discovery of a maximal distance from samples of one
cluster to the nearest sample of another cluster.
dist6(Ci,Cj) = max{ distH(Ci,Cj), distH(Cj,Ci) }
where: distH(A,B) = max{ min{ d(x,y): y belongs to B}: x belongs to A }