Hierarchical Clustering
Description
Hierarchical cluster analysis on a set of dissimilarities and methods for analyzing it.
Usage
1 2 3 4 5 6 7 
Arguments
d 
a dissimilarity structure as produced by 
method 
the agglomeration method to be used. This should
be (an unambiguous abbreviation of) one of

members 

x 
an object of the type produced by 
hang 
The fraction of the plot height by which labels should hang below the rest of the plot. A negative value will cause the labels to hang down from 0. 
check 
logical indicating if the 
labels 
A character vector of labels for the leaves of the
tree. By default the row names or row numbers of the original data are
used. If 
axes, frame.plot, ann 
logical flags as in 
main, sub, xlab, ylab 
character strings for

... 
Further graphical arguments. E.g., 
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. Note however, that methods "median"
and
"centroid"
are not leading to a monotone distance
measure, or equivalently the resulting dendrograms can have so called
inversions or reversals which are hard to interpret,
but note the trichotomies in Legendre and Legendre (2012).
Two different algorithms are found in the literature for Ward clustering.
The one used by option "ward.D"
(equivalent to the only Ward
option "ward"
in R versions <= 3.0.3) does not implement
Ward's (1963) clustering criterion, whereas option "ward.D2"
implements
that criterion (Murtagh and Legendre 2014). With the latter, the
dissimilarities are squared before cluster updating.
Note that agnes(*, method="ward")
corresponds
to hclust(*, "ward.D2")
.
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 hclust
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 n1 merges,
there are 2^{(n1)} possible orderings for the leaves
in a cluster tree, or dendrogram.
The algorithm used 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).
Single 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 process. The object is a list with components:
merge 
an n1 by 2 matrix.
Row i of 
height 
a set of n1 real values (nondecreasing for
ultrametric trees).
The clustering height: that is, the value of
the criterion associated with the clustering

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 
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 
There are print
, plot
and identify
(see identify.hclust
) methods and the
rect.hclust()
function for hclust
objects.
Note
Method "centroid"
is typically meant to be used with
squared Euclidean distances.
Author(s)
The hclust
function is based on Fortran code
contributed to STATLIB by F. Murtagh.
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole. (S version.)
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: PhysicaVerlag (for algorithmic details of algorithms used).
McQuitty, L.L. (1966). Similarity Analysis by Reciprocal Pairs for Discrete and Continuous Data. Educational and Psychological Measurement, 26, 825–831.
Legendre, P. and L. Legendre (2012). Numerical Ecology, 3rd English ed. Amsterdam: Elsevier Science BV.
Murtagh, Fionn and Legendre, Pierre (2014). Ward's hierarchical agglomerative clustering method: which algorithms implement Ward's criterion? Journal of Classification 31 (forthcoming).
See Also
identify.hclust
, rect.hclust
,
cutree
, dendrogram
, kmeans
.
For the Lance–Williams formula and methods that apply it generally,
see agnes
from package \CRANpkgcluster.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38  require(graphics)
### Example 1: Violent crime rates by US state
hc < hclust(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 < hclust(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 < hclust(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 = "Restart from 10 clusters")
par(opar)
### Example 2: Straightline distances among 10 US cities
## Compare the results of algorithms "ward.D" and "ward.D2"
data(UScitiesD)
mds2 < cmdscale(UScitiesD)
plot(mds2, type="n", axes=FALSE, ann=FALSE)
text(mds2, labels=rownames(mds2), xpd = NA)
hcity.D < hclust(UScitiesD, "ward.D") # "wrong"
hcity.D2 < hclust(UScitiesD, "ward.D2")
opar < par(mfrow = c(1, 2))
plot(hcity.D, hang=1)
plot(hcity.D2, hang=1)
par(opar)

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