\HeaderA{hclust}{Hierarchical Clustering}{hclust}
\aliasA{plclust}{hclust}{plclust}
\aliasA{plot.hclust}{hclust}{plot.hclust}
\aliasA{print.hclust}{hclust}{print.hclust}
\keyword{multivariate}{hclust}
\keyword{cluster}{hclust}
\begin{Description}\relax
Hierarchical cluster analysis on a set of dissimilarities and
methods for analyzing it.
\end{Description}
\begin{Usage}
\begin{verbatim}
hclust(d, method = "complete", members=NULL)

## S3 method for class 'hclust':
plot(x, labels = NULL, hang = 0.1,
     axes = TRUE, frame.plot = FALSE, ann = TRUE,
     main = "Cluster Dendrogram",
     sub = NULL, xlab = NULL, ylab = "Height", ...)

plclust(tree, hang = 0.1, unit = FALSE, level = FALSE, hmin = 0,
        square = TRUE, labels = NULL, plot. = TRUE,
        axes = TRUE, frame.plot = FALSE, ann = TRUE,
        main = "", sub = NULL, xlab = NULL, ylab = "Height")
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{d}] a dissimilarity structure as produced by \code{dist}.
\item[\code{method}] the agglomeration method to be used. This should
be (an unambiguous abbreviation of) one of
\code{"ward"}, \code{"single"}, \code{"complete"},
\code{"average"}, \code{"mcquitty"}, \code{"median"} or
\code{"centroid"}.
\item[\code{members}] \code{NULL} or a vector with length size of
\code{d}. See the Details section.
\item[\code{x,tree}] an object of the type produced by \code{hclust}.
\item[\code{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.
\item[\code{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 \code{labels=FALSE} no labels at all are plotted.
\item[\code{axes, frame.plot, ann}] logical flags as in \code{\LinkA{plot.default}{plot.default}}.
\item[\code{main, sub, xlab, ylab}] character strings for
\code{\LinkA{title}{title}}.  \code{sub} and \code{xlab} have a non-NULL
default when there's a \code{tree\$call}.
\item[\code{...}] Further graphical arguments.
\item[\code{unit}] logical.  If true, the splits are plotted at
equally-spaced heights rather than at the height in the object.
\item[\code{hmin}] numeric.  All heights less than \code{hmin} are regarded
as being \code{hmin}: this can be used to suppress detail at the
bottom of the tree.
\item[\code{level, square, plot.}] as yet unimplemented arguments of
\code{plclust} for S-PLUS compatibility.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
This function performs a hierarchical cluster analysis
using a set of dissimilarities for the \eqn{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.  \emph{Ward's}
minimum variance method aims at finding compact, spherical clusters.
The \emph{complete linkage} method finds similar clusters. The
\emph{single linkage} method (which is closely related to the minimal
spanning tree) adopts a \sQuote{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 \code{"median"} and
\code{"centroid"} are \emph{not} leading to a \emph{monotone distance}
measure, or equivalently the resulting dendrograms can have so called
\emph{inversions} (which are hard to interpret).

If \code{members!=NULL}, then \code{d} is taken to be a
dissimilarity matrix between clusters instead of dissimilarities
between singletons and \code{members} gives the number of observations
per cluster.  This way the hierarchical cluster algorithm can be
\dQuote{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 \code{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 \eqn{n}{} observations there are \eqn{n-1}{} merges,
there are \eqn{2^{(n-1)}}{} possible orderings for the leaves
in a cluster tree, or dendrogram.
The algorithm used in \code{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.
\end{Details}
\begin{Value}
An object of class \bold{hclust} which describes the
tree produced by the clustering process.
The object is a list with components:
\begin{ldescription}
\item[\code{merge}] an \eqn{n-1}{} by 2 matrix.
Row \eqn{i}{} of \code{merge} describes the merging of clusters
at step \eqn{i}{} of the clustering.
If an element \eqn{j}{} in the row is negative,
then observation \eqn{-j}{} was merged at this stage.
If \eqn{j}{} is positive then the merge
was with the cluster formed at the (earlier) stage \eqn{j}{}
of the algorithm.
Thus negative entries in \code{merge} indicate agglomerations
of singletons, and positive entries indicate agglomerations
of non-singletons.
\item[\code{height}] a set of \eqn{n-1}{} non-decreasing real values.
The clustering \emph{height}: that is, the value of
the criterion associated with the clustering
\code{method} for the particular agglomeration.
\item[\code{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 \code{merge} will not have
crossings of the branches.
\item[\code{labels}] labels for each of the objects being clustered.
\item[\code{call}] the call which produced the result.
\item[\code{method}] the cluster method that has been used.
\item[\code{dist.method}] the distance that has been used to create \code{d}
(only returned if the distance object has a \code{"method"}
attribute).
\end{ldescription}


There are \code{\LinkA{print}{print}}, \code{\LinkA{plot}{plot}} and \code{identify}
(see \code{\LinkA{identify.hclust}{identify.hclust}}) methods and the
\code{\LinkA{rect.hclust}{rect.hclust}()} function for \code{hclust} objects.
The \code{plclust()} function is basically the same as the plot method,
\code{plot.hclust}, primarily for back compatibility with S-plus. Its
extra arguments are not yet implemented.
\end{Value}
\begin{Author}\relax
The \code{hclust} function is based on Fortran code
contributed to STATLIB by F. Murtagh.
\end{Author}
\begin{References}\relax
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
\emph{The New S Language}.
Wadsworth \& Brooks/Cole. (S version.)

Everitt, B. (1974).
\emph{Cluster Analysis}.
London: Heinemann Educ. Books.

Hartigan, J. A. (1975).
\emph{Clustering  Algorithms}.
New York: Wiley.

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

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

Gordon, A. D. (1999).
\emph{Classification}. Second Edition.
London: Chapman and Hall / CRC

Murtagh, F. (1985).
\dQuote{Multidimensional Clustering Algorithms}, in
\emph{COMPSTAT Lectures 4}.
Wuerzburg: Physica-Verlag
(for algorithmic details of algorithms used).

McQuitty, L.L. (1966).
Similarity Analysis by Reciprocal Pairs for Discrete and Continuous
Data.
\emph{Educational and Psychological Measurement}, \bold{26}, 825--831.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{identify.hclust}{identify.hclust}}, \code{\LinkA{rect.hclust}{rect.hclust}},
\code{\LinkA{cutree}{cutree}}, \code{\LinkA{dendrogram}{dendrogram}}, \code{\LinkA{kmeans}{kmeans}}.

For the Lance--Williams formula and methods that apply it generally,
see \code{\LinkA{agnes}{agnes}} from package \pkg{cluster}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
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 = "Re-start from 10 clusters")
par(opar)
\end{ExampleCode}
\end{Examples}

