|
Complete-linkage clustering is one of several methods of agglomerative hierarchical clustering. 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. 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. The method is also known as farthest neighbour clustering. The result of the clustering can be visualized as a dendrogram, which shows the sequence of cluster fusion and the distance at which each fusion took place.〔Legendre, P. & Legendre, L. 1998. Numerical Ecology. Second English Edition. 853 pages.〕 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) 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. Complete linkage tends to find compact clusters of approximately equal diameters.〔Everitt, Landau and Leese (2001), pp. 62-64.〕 == 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''(). 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''() = max ''d''() where the maximum 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''() # 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''((''r'',''s'') ) = max ''d''(), ''d''(). # If all objects are in one cluster, stop. Else, go to step 2. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Complete-linkage clustering」の詳細全文を読む スポンサード リンク
|