Single-linkage 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

$D(X,Y)=\min_{x\in X, y\in Y} d(x,y),$

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 so-called 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.

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 $N \times N$ 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)] = min d[(i),(j)] where the minimum 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)] = min d[(k),(r)], d[(k),(s)].
5. 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 $\mathcal{O}(n^3)$. In 1973, R. Sibson proposed an optimally efficient algorithm of only complexity $\mathcal{O}(n^2)$ 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 inter-cluster 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

• Single linkage clustering algorithm implementation in Ruby (AI4R)
• Linkages used in Matlab

References

1. ^ R. Sibson (1973). "SLINK: an optimally efficient algorithm for the single-link cluster method". The Computer Journal (British Computer Society) 16 (1): 30–34. http://www.cs.gsu.edu/~wkim/index_files/papers/sibson.pdf.