Agglomerative Hierarchical Clustering

Hierarchical clustering techniques are a second important category of clustering methods. As with K-means, these approaches are relatively old compared to many clustering algorithms, but they still enjoy widespread use. There are two basic approaches for generating a hierarchical clustering:

• Agglomerative: Start with the points as individual clusters and, at each step, merge the closest pair of clusters. This requires defining a notion of cluster proximity.
• Divisive: Start with one, all-inclusive cluster and, at each step, split a cluster until only singleton clusters of individual points remain. In this case, we need to decide which cluster to split at each step and how to do the splitting.

A hierarchical clustering is often displayed graphically using a tree-like diagram called a dendrogram, which displays both the cluster-subcluster relationships and the order in which the clusters were merged (agglomerative view) or split (divisive view). For sets of two-dimensional points, such as those that we will use as examples, a hierarchical clustering can also be graphically represented using a nested cluster diagram. The following figure shows an example of these two types of figures ffor a set of four two dimensional points.

Many agglomerative hierarchical clustering techniques are variations on a single approach, starting with individual points as clusters, successively merge the two closest clusters until only one cluster remains.

Defining Proximity between Clusters

The key operation in the above algorithm is the computation of proximity between two clusters, and it is the definition of cluster proximity that differentiates the various agglomerative hierarchical techniques. Cluster proximity is typically defined with a particular type of cluster in mind. For example, many agglomerative hierarchical clustering techniques, such as MIN, MAX, and Group Average, come from a graph based clusters.

• MIN or single link: MIN defines cluster proximity as the proximity between the closest two points that are in different clusters, or using graph terms, the shortest edge between two nodes in different subsets of nodes.
• MAX or complete link: MAX takes the proximity between the farthest two points in different clusters to be the cluster proximity, or using graph terms, the longest edge between two nodes in different subsets of nodes.
• Group Average: the group average technique, defines cluster proximity to be the average pairwise proximities (average length of edges) of all pairs of points from different clusters.

If, instead, we take a prototype-based view, in which each cluster is represented by a centroid, different definitions of cluster proximity are more natural. When using centroids, the cluster proximity is commonly defined as the prox- imity between cluster centroids. An alternative technique, Ward’s method, also assumes that a cluster is represented by its centroid, but it measures the proximity between two clusters in terms of the increase in the SSE that results from merging the two clusters. Like K-means, Ward’s method attempts to minimize the sum of the squared distances of points from their cluster centroids.

Key Issues in Hierarchical Clustering

• Lack of a Global Objective Function: We previously mentioned that agglomerative hierarchical clustering cannot be viewed as globally optimizing an objective function. Instead, agglomerative hierarchical clustering techniques use various criteria to decide locally, at each step, which clusters should be merged (or split for divisive approaches). This approach yields clustering algorithms that avoid the difficulty of attempting to solve a hard combinatorial optimization problem.
• Ability to handle different cluster sizes: One aspect of agglomerative hierarchical clustering that we have not yet dis- cussed is how to treat the relative sizes of the pairs of clusters that are merged. (This discussion applies only to cluster proximity schemes that involve sums, such as centroid, Ward’s, and group average.) There are two approaches:
• weighted, which treats all clusters equally, and
• unweighted, which takes the number of points in each cluster into account.
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