Recall in the previous section that we spoke about Monothetic and Polythetic methods. Monothetic methods only looks at a single variable at a time while Polythetic looks at multiple variables simultaneously. In this section, we will speak more about polythetic divisive methods.

## Polythetic Divisive Methods

Polythetic methods operate via a distance matrix.

This procedure avoids considering all possible splits by

- Finding the object that is furthest away from the others within a group and using that as a seed for a splinter group.
- Each object is then considered for entry to that separate splinter group: any that are closer to the splinter group than the main group is moved to the splinter one.
- The step is then repeated.

This process has been developed into a program named `DIANA`

(DIvisive ANAlysis Clustering) which is implemented in `R`

.

### Similarities to Politics

This somewhat resembles a way a political party might split due to inner conflicts.

Firstly, the most discontented member leaves the party and starts a new one, and then some others follow him until a kind of equilibrium is attained.

## Methods for Large Data Sets

There are two common hierarchical methods used for large data sets `BIRCH`

and `CURE`

. Both of these algorithms employ a pre-clustering phase in where dense regions are summarized, the summaries being then clustered using a hierarchical method based on centroids.

### CURE

`CURE`

starts with a random sample of points and represents clusters by a smaller number of points that capture the shape of the cluster- Which are then shrunk towards the centroid as to dampen the effect of the outliers
- Hierarchical clustering then operates on the representative points

`CURE`

has been shown to be able to cope with arbitrary-shaped clusters and in that respect may be superior to `BIRCH`

, although it does require judgment as to the number of clusters and also a parameter which favors either more or less compact clusters.

## Revisiting Topics: Cluster Dissimilarity

In order to decide where clusters should be combined (for agglomerative), or where a cluster should be split (for divisive), a measure of dissimilarity between sets of observations is required.

In most methods of hierarchical clustering this is achieved by a use of an appropriate

- Metric (a measure of distance between pairs of observations)
- Linkage Criterion (which specifies the dissimilarities of sets as functions of pairwise distances observations in the sets)

## Advantages of Hierarchical Clustering

- Any valid measure of distance measure can be used
- In most cases, the observations themselves are not required, just hte matrix of distances
- This can have the advantage of only having to store a distance matrix in memory as opposed to a n-dimensional matrix.