Complete-linkage clustering is one of several methods of agglomerative
hierarchical clustering
In data mining and statistics, hierarchical clustering (also called hierarchical cluster analysis or HCA) is a method of cluster analysis that seeks to build a hierarchy of clusters. Strategies for hierarchical clustering generally fall into ...
. At the beginning of the process, each element is in a cluster of its own. The clusters are then sequentially combined into larger clusters until all elements end up being in the same cluster. The method is also known as farthest neighbour clustering. The result of the clustering can be visualized as a
dendrogram
A dendrogram is a diagram representing a tree. This diagrammatic representation is frequently used in different contexts:
* in hierarchical clustering, it illustrates the arrangement of the clusters produced by the corresponding analyses.
...
, which shows the sequence of cluster fusion and the distance at which each fusion took place.
Clustering procedure
At each step, the two clusters separated by the shortest distance are combined. The definition of 'shortest distance' is what differentiates between the different agglomerative clustering methods. In complete-linkage clustering, the link between two clusters contains all element pairs, and the distance between clusters equals the distance between those two elements (one in each cluster) that are farthest away from each other. The shortest of these links that remains at any step causes the fusion of the two clusters whose elements are involved.
Mathematically, the complete linkage function — the distance
between clusters
and
— is described by the following expression :
where
*
is the distance between elements
and
;
*
and
are two sets of elements (clusters).
Algorithms
Naive scheme
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 complete linkage clustering algorithm consists of the following steps:
# Begin with the disjoint clustering having level
and sequence number
.
# Find the most similar pair of clusters in the current clustering, say pair
, according to