Complete-linkage clustering

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.

Mathematically, the complete linkage function — the distance ${\displaystyle D(X,Y)}$ between clusters ${\displaystyle X}$ and ${\displaystyle Y}$ — is described by the following expression : ${\displaystyle D(X,Y)=\max _{x\in X,y\in Y}d(x,y)}$

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