## Hierarchical Clustering

## Usage

    hclust(d, method="complete")

    plot.hclust(hclust.obj, labels, hang=0.1, ...)

## Arguments

`d` | a dissimilarity structure as produced by `dist`.
`method` | the agglomeration method to be used. This should be (an unambiguous abbreviation of) one of `"ward"`, `"single"`, `"complete"`, `"average"`, `"mcquitty"`, `"median"` or `"centroid"`.
`hclust.obj` | an object of the type produced by `hclust`.
`hang` | The fraction of the plot height which labels should hang below the rest of the plot. A negative value will cause the labels to hang down from 0.
`labels` | A character vector of of labels for the leaves of the tree. By default the row names or row numbers of the original data are used. If `labels=FALSE` no labels at all are plotted.

## Description

This function performs a hierarchical cluster analysis using a set of dissimilarities for the n objects being clustered. Initially, each object is assigned to its own cluster and then the algorithm proceeds iteratively, at each stage joining the two most similar clusters, continuing until there is just a single cluster. At each stage distances between clusters are recomputed by the Lance-Williams dissimilarity update formula according to the particular clustering method being used.

An number of different clustering methods are provided. Ward's minimum variance method aims at finding compact, spherical clusters. The complete linkage method finds similar clusters. The single linkage method (which is closely related to the minimal spanning tree) adopts a `friends of friends' clustering strategy. The other methods can be regarded as aiming for clusters with characteristics somewhere between the single and complete link methods.

In hierarchical cluster displays, a decision is needed at each merge to specify which subtree should go on the left and which on the right. Since, for n observations there are n-1 merges, there are 2^(n-1) possible orderings for the leaves in a cluster tree, or dendrogram. The algorithm in `hclust` is to order the subtree so that the tighter cluster is on the left (the last, i.e. most recent, merge of the left subtree is at a lower value than the last merge of the right subtree). Observations are the tightest clusters possible, and merges involving two observations place them in order by their observation sequence number.

## Value

An object of class hclust which describes the tree produced by the clustering processs. The object is a list with components:

`merge` | an n-1 by 2 matrix. Row i of `merge` describes the merging of clusters at step i of the clustering. If an element j in the row is negative, then observation -j was merged at this stage. If j is positive then the merge was with the cluster formed at the (earlier) stage j of the algorithm. Thus negative entries in `merge` indicate agglomerations of singletons, and positive entries indicate agglomerations of non-singletons.
`height` | a set of n-1 non-decreasing real values. The clustering height: that is, the value of the criterion associated with the clustering `method` for the particular agglomeration.
`order` | a vector giving the permutation of the original observations suitable for plotting, in the sense that a cluster plot using this ordering and matrix `merge` will not have crossings of the branches.
`labels` | labels for each of the objects being clustered.

## Author(s)

The `hclust` function is based on Fortran code contributed to STATLIB by F. Murtagh.

## References

Everitt, B. (1974). Cluster Analysis. London: Heinemann Educ. Books.

Hartigan, J. A. (1975). Clustering Algorithms. New York: Wiley.

Sneath, P. H. A. and R. R. Sokal (1973). Numerical Taxonomy. San Francisco: Freeman.

Anderberg, M. R. (1973). Cluster Analysis for Applications. Academic Press: New York.

Gordon, A. D. (1981). Classification. London: Chapman and Hall.

Murtagh, F. (1985). ``Multidimensional Clustering Algorithms'', in COMPSTAT Lectures 4. Wuerzburg: Physica-Verlag (for algorithmic details of algorithms used).

## See Also

`kmeans`.

## Examples

    library(mva)
    data(crimes)
    hc <- hclust(dist(crimes), "ave")
    plot(hc, hang=-1)
    plot(hc)

[Package Contents]