Complete-linkage clustering

In cluster analysis, complete linkage or farthest neighbour is a method of calculating distances between clusters in hierarchical clustering. In complete linkage, the distance between two clusters is computed as the distance between the two farthest elements in the two clusters.

Mathematically, the linkage function — the distance $D(X,Y)$ between clusters $X$ and $Y$ — is described by the following expression : $D(X,Y)=\max(d(x,y))$

where

• $d(x,y)$ is the distance between elements $x\in X$ and $y\in Y$ ;
• $X$ and $Y$ are two sets of elements (clusters)

Complete linkage clustering avoids a drawback of the alternative single linkage method - the so-called chaining phenomenon, where clusters formed via single linkage clustering may be forced together due to single elements being close to each other, even though many of the elements in each cluster may be very distant to each other.

Algorithm

The following algorithm is an agglomerative scheme that erases rows and columns in a proximity matrix as old clusters are merged into new ones. The $N\times N$ proximity matrix D contains all distances d(i,j). The clusterings are assigned sequence numbers 0,1,......, (n − 1) and L(k) is the level of the kth clustering. A cluster with sequence number m is denoted (m) and the proximity between clusters (r) and (s) is denoted d[(r),(s)].

The algorithm is composed of the following steps:

1. Begin with the disjoint clustering having level L(0) = 0 and sequence number m = 0.
2. Find the most similar pair of clusters in the current clustering, say pair (r), (s), according to d[(r),(s)] = max d[(i),(j)] where the maximum is over all pairs of clusters in the current clustering.
3. Increment the sequence number: m = m + 1. Merge clusters (r) and (s) into a single cluster to form the next clustering m. Set the level of this clustering to L(m) = d[(r),(s)]
4. Update the proximity matrix, D, by deleting the rows and columns corresponding to clusters (r) and (s) and adding a row and column corresponding to the newly formed cluster. The proximity between the new cluster, denoted (r,s) and old cluster (k) is defined as d[(k), (r,s)] = max d[(k),(r)], d[(k),(s)].
5. If all objects are in one cluster, stop. Else, go to step 2.

Other linkages

Alternative linkage schemes include single linkage and average linkage clustering - implementing a different linkage is simply a matter of using a different formula to calculate inter-cluster distances in the initial computation of the proximity matrix and in step 4 of the above algorithm. The formula that should be adjusted has been highlighted using bold text.

External links

• Statistical Glossary - Complete linkage clustering