 Singlelinkage clustering

In cluster analysis, single linkage, nearest neighbour or shortest distance is a method of calculating distances between clusters in hierarchical clustering. In single linkage, the distance between two clusters is computed as the distance between the two closest elements in the two clusters.
Mathematically, the linkage function – the distance D(X,Y) between clusters X and Y – is described by the expression
where X and Y are any two sets of elements considered as clusters, and d(x,y) denotes the distance between the two elements x and y.
A drawback of this method is the socalled chaining phenomenon: clusters 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.
Contents
Naive 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 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:
 Begin with the disjoint clustering having level L(0) = 0 and sequence number m = 0.
 Find the most similar 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 as d[(k), (r,s)] = min d[(k),(r)], d[(k),(s)].
 If all objects are in one cluster, stop. Else, go to step 2.
Optimally efficient algorithm
The algorithm explained above is easy to understand but of complexity . In 1973, R. Sibson proposed an optimally efficient algorithm of only complexity known as SLINK.^{[1]}
Other linkages
This is essentially the same as Kruskal's algorithm for minimum spanning trees. However, in single linkage clustering, the order in which clusters are formed is important, while for minimum spanning trees what matters is the set of pairs of points that form distances chosen by the algorithm.
Alternative linkage schemes include complete linkage and Average linkage clustering  implementing a different linkage in the naive algorithm is simply a matter of using a different formula to calculate intercluster distances in the initial computation of the proximity matrix and in step 4 of the above algorithm. An optimally efficient algorithm is however not available for arbitrary linkages. The formula that should be adjusted has been highlighted using bold text.
External links
References
 ^ R. Sibson (1973). "SLINK: an optimally efficient algorithm for the singlelink cluster method". The Computer Journal (British Computer Society) 16 (1): 30–34. http://www.cs.gsu.edu/~wkim/index_files/papers/sibson.pdf.
Categories: Data clustering algorithms
Wikimedia Foundation. 2010.