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Hierarchical Clustering

Hierarchical clustering involves creating clusters that have a predetermined ordering from top to bottom. For example, all files and folders on the hard disk are organized in a hierarchy.
There are two types of hierarchical clustering, Divisive
and Agglomerative. 

Divisive method 
In this method we assign all of the observations to a single
cluster and then partition the cluster to two least similar clusters.
Finally, we proceed recursively on each cluster until there is one cluster for each observation. 

Agglomerative method

In this method we assign each observation to its own
cluster. Then, compute the similarity (e.g., distance) between each of the
clusters and join the two most similar clusters. Finally, repeat steps 2 and 3 until there is only a single cluster left.
The related algorithm is shown below. 

Before any clustering is performed, it is required to determine the proximity matrix containing the distance between each point using
a distance function. Then, the matrix is updated to display the distance between each cluster. The
following three methods differ in how the distance between each cluster is measured. 

Single Linkage 
In single linkage hierarchical clustering, the distance between two clusters is defined as the
shortest distance between two points in each cluster. For example, the distance between clusters “r” and “s” to the left is equal to the length of the arrow between their two closest points. 

Complete Linkage 
In complete linkage hierarchical clustering, the distance between two clusters is defined as the
longest distance between two points in each cluster. For example, the distance between clusters “r” and “s” to the left is equal to the length of the arrow between their two furthest points. 

Average Linkage 
In average linkage hierarchical clustering, the distance between two clusters is defined as the average distance between each point in one cluster to every point in the other cluster.
For example, the distance between clusters “r” and “s” to the left is equal to the average length each arrow between connecting the points of one cluster to the other. 





