hclustglad: Hierarchical Clustering

Description

Hierarchical cluster analysis on a set of dissimilarities and methods for analyzing it.

Usage

hclustglad(d, method = "complete", members=NULL)

Arguments

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".

members

NULL or a vector with length size of d.

Value

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.
call: the call which produced the result.
method: the cluster method that has been used.
dist.method: the distance that has been used to create d (only returned if the distance object has a "method" attribute).

Details

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.

A 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.

If members!=NULL, then d is taken to be a dissimilarity matrix between clusters instead of dissimilarities between singletons and members gives the number of observations per cluster. This way the hierarchical cluster algorithm can be ``started in the middle of the dendrogram'', e.g., in order to reconstruct the part of the tree above a cut (see examples). Dissimilarities between clusters can be efficiently computed (i.e., without hclustglad itself) only for a limited number of distance/linkage combinations, the simplest one being squared Euclidean distance and centroid linkage. In this case the dissimilarities between the clusters are the squared Euclidean distances between cluster means.

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 used in hclustglad 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). Single observations are the tightest clusters possible, and merges involving two observations place them in order by their observation sequence number.

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. (1999). Classification. Second Edition. London: Chapman and Hall / CRC

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

Examples

Run this code

data(USArrests)
hc <- hclustglad(dist(USArrests), "ave")
plot(hc)
plot(hc, hang = -1)

## Do the same with centroid clustering and squared Euclidean distance,
## cut the tree into ten clusters and reconstruct the upper part of the
## tree from the cluster centers.
hc <- hclustglad(dist(USArrests)^2, "cen")
memb <- cutree(hc, k = 10)
cent <- NULL
for(k in 1:10){
  cent <- rbind(cent, colMeans(USArrests[memb == k, , drop = FALSE]))
}
hc1 <- hclustglad(dist(cent)^2, method = "cen", members = table(memb))
opar <- par(mfrow = c(1, 2))
plot(hc,  labels = FALSE, hang = -1, main = "Original Tree")
plot(hc1, labels = FALSE, hang = -1, main = "Re-start from 10 clusters")
par(opar)

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