Single-Linkage Clustering: The Algorithm
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The algorithm is an agglomerative scheme that erases rows and columns in the proximity matrix as old clusters are merged into new ones.
The N*N proximity matrix is D = [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:
- Begin with the disjoint clustering having level L(0) = 0 and sequence number m = 0.
- Find the least dissimilar pair of clusters in the current clustering, say pair (r), (s), according to
d[(r),(s)] = min d[(i),(j)]
where the minimum is over all pairs of clusters in the current clustering. - 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)] - 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 in this way:
d[(k), (r,s)] = min d[(k),(r)], d[(k),(s)] - If all objects are in one cluster, stop. Else, go to step 2.
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