Agglomerative hierarchical procedure

One of the most intuitive ways to describe how cluster analysis works in practice is by referring to the agglomerative hierarchical cluster analysis (HCA) method. Beside the common preliminary steps already discussed, that is definition of the metric (Euclidean, Mahalanobis, Manhattan distance, etc.) and calculation of the distance matrix and the corresponding similarity matrix, the analysis continues according to a recursive procedure such as... 

There are two main types of clustering techniques hierarchical and nonhierarchical. Hierarchical cluster analysis may follow either an agglomerative or a divisive scheme agglomerative techniques start with as many clusters as objects and, by means of repeated similarity-based fusion steps, they reach a final situation with a unique cluster containing all of the objects. Divisive methods follow exactly the opposite procedure they start from an all-inclusive cluster and then perform a number of consecutive partitions until there is a bijective correspondence between clusters and objects (see Fig. 2.12). In both cases, the number of clusters is defined by the similarity level selected. 

The most commonly used procedures in environmental studies are clustering performed using hierarchical agglomerative procedures because of the comprehensible graphical output and clear hierarchical relations between clusters.2,3... 

Agglomerative methods, such as single link and complete link, are stepwise procedures. The formulation in (5)-(7) allows us to define the hierarchical clustering problem in terms of combinatorial optimization. To do this, however, we need an appropriate internal clustering criterion. The most obvious is squared error. 

In fact, agglomerative procedures differ only by the way of updating the new distance matrix in each step. As an essentia drawback of hierarchic clustering, mistakes of fusion at a low level of aggregation are retained, due to hierarchy, in the whole sequence of partitions. 

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