\HeaderA{agnes}{Agglomerative Nesting (Hierarchical Clustering)}{agnes}
\keyword{cluster}{agnes}
\begin{Description}\relax
Computes agglomerative hierarchical clustering of the dataset.
\end{Description}
\begin{Usage}
\begin{verbatim}
agnes(x, diss = inherits(x, "dist"), metric = "euclidean",
      stand = FALSE, method = "average", par.method,
      keep.diss = n < 100, keep.data = !diss)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] data matrix or data frame, or dissimilarity matrix, depending on the
value of the \code{diss} argument.

In case of a matrix or data frame, each row corresponds to an observation,
and each column corresponds to a variable. All variables must be numeric.
Missing values (NAs) are allowed.

In case of a dissimilarity matrix, \code{x} is typically the output of
\code{\LinkA{daisy}{daisy}} or \code{\LinkA{dist}{dist}}.
Also a vector with length n*(n-1)/2 is allowed (where n is the number
of observations), and will be interpreted in the same way as the
output of the above-mentioned functions. Missing values (NAs) are not
allowed.

\item[\code{diss}] logical flag: if TRUE (default for \code{dist} or
\code{dissimilarity} objects), then \code{x} is assumed to be a
dissimilarity matrix.  If FALSE, then \code{x} is treated as
a matrix of observations by variables.

\item[\code{metric}] character string specifying the metric to be used for calculating
dissimilarities between observations.
The currently available options are "euclidean" and "manhattan".
Euclidean distances are root sum-of-squares of differences, and
manhattan distances are the sum of absolute differences.
If \code{x} is already a dissimilarity matrix, then this argument will
be ignored.

\item[\code{stand}] logical flag: if TRUE, then the measurements in \code{x} are
standardized before calculating the dissimilarities. Measurements
are standardized for each variable (column), by subtracting the
variable's mean value and dividing by the variable's mean absolute
deviation.  If \code{x} is already a dissimilarity matrix, then this
argument will be ignored.

\item[\code{method}] character string defining the clustering method.  The six methods
implemented are "average" ([unweighted pair-]group average method, UPGMA),
"single" (single linkage), "complete" (complete linkage),
"ward" (Ward's method), "weighted" (weighted average linkage) and
its generalization \code{"flexible"} which uses (a constant version of)
the Lance-Williams formula and the \code{par.method} argument.  Default
is "average".

\item[\code{par.method}] if \code{method == "flexible"}, numeric vector of
length 1, 3, or 4, see in the details section.

\item[\code{keep.diss, keep.data}] logicals indicating if the dissimilarities
and/or input data \code{x} should be kept in the result.  Setting
these to \code{FALSE} can give much smaller results and hence even save
memory allocation \emph{time}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
\code{agnes} is fully described in chapter 5 of Kaufman and Rousseeuw (1990).
Compared to other agglomerative clustering methods such as \code{hclust},
\code{agnes} has the following features: (a) it yields the
agglomerative coefficient (see \code{\LinkA{agnes.object}{agnes.object}})
which measures the amount of clustering structure found; and (b)
apart from the usual tree it also provides the banner, a novel
graphical display (see \code{\LinkA{plot.agnes}{plot.agnes}}).

The \code{agnes}-algorithm constructs a hierarchy of clusterings.\\
At first, each observation is a small cluster by itself.  Clusters are
merged until only one large cluster remains which contains all the
observations.  At each stage the two \emph{nearest} clusters are combined
to form one larger cluster.

For \code{method="average"}, the distance between two clusters is the
average of the dissimilarities between the points in one cluster and the
points in the other cluster.
\\
In \code{method="single"}, we use the smallest dissimilarity between a
point in the first cluster and a point in the second cluster (nearest
neighbor method).
\\
When \code{method="complete"}, we use the largest dissimilarity
between a point in the first cluster and a point in the second cluster
(furthest neighbor method).

The \code{method = "flexible"} allows (and requires) more details:
The Lance-Williams formula specifies how dissimilarities are
computed when clusters are agglomerated (equation (32) in K.\&R.,
p.237).  If clusters \eqn{C_1}{} and \eqn{C_2}{} are agglomerated into a
new cluster, the dissimilarity between their union and another
cluster \eqn{Q}{} is given by
\deqn{
D(C_1 \cup C_2, Q) = \alpha_1 * D(C_1, Q) + \alpha_2 * D(C_2, Q) +
\beta * D(C_1,C_2) + \gamma * |D(C_1, Q) - D(C_2, Q)|,
}{}
where the four coefficients \eqn{(\alpha_1, \alpha_2, \beta, \gamma)}{}
are specified by the vector \code{par.method}:

If \code{par.method} is of length 1,
say \eqn{= \alpha}{}, \code{par.method} is extended to
give the \dQuote{Flexible Strategy} (K. \& R., p.236 f) with
Lance-Williams coefficients \eqn{(\alpha_1 = \alpha_2 = \alpha, \beta =
1 - 2\alpha, \gamma=0)}{}.\\
If of length 3, \eqn{\gamma = 0}{} is used.

\bold{Care} and expertise is probably needed when using \code{method
    = "flexible"} particularly for the case when \code{par.method} is
specified of longer length than one.
The \emph{weighted average} (\code{method="weighted"}) is the same as
\code{method="flexible", par.method = 0.5}.
\end{Details}
\begin{Value}
an object of class \code{"agnes"} representing the clustering.
See \code{\LinkA{agnes.object}{agnes.object}} for details.
\end{Value}
\begin{Section}{BACKGROUND}
Cluster analysis divides a dataset into groups (clusters) of
observations that are similar to each other.
\describe{
\item[Hierarchical methods] like
\code{agnes}, \code{\LinkA{diana}{diana}}, and \code{\LinkA{mona}{mona}}
construct a hierarchy of clusterings, with the number of clusters
ranging from one to the number of observations.
\item[Partitioning methods] like
\code{\LinkA{pam}{pam}}, \code{\LinkA{clara}{clara}}, and \code{\LinkA{fanny}{fanny}}
require that the number of clusters be given by the user.
}
\end{Section}
\begin{References}\relax
Kaufman, L. and Rousseeuw, P.J. (1990).
\emph{Finding Groups in Data: An Introduction to Cluster Analysis}.
Wiley, New York.

Anja Struyf, Mia Hubert \& Peter J. Rousseeuw (1996):
Clustering in an Object-Oriented Environment.
\emph{Journal of Statistical Software}, \bold{1}.
\url{http://www.stat.ucla.edu/journals/jss/}

Struyf, A., Hubert, M. and Rousseeuw, P.J. (1997). Integrating
Robust Clustering Techniques in S-PLUS,
\emph{Computational Statistics and Data Analysis}, \bold{26}, 17--37.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{agnes.object}{agnes.object}}, \code{\LinkA{daisy}{daisy}}, \code{\LinkA{diana}{diana}},
\code{\LinkA{dist}{dist}}, \code{\LinkA{hclust}{hclust}}, \code{\LinkA{plot.agnes}{plot.agnes}},
\code{\LinkA{twins.object}{twins.object}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
data(votes.repub)
agn1 <- agnes(votes.repub, metric = "manhattan", stand = TRUE)
agn1
plot(agn1)

op <- par(mfrow=c(2,2))
agn2 <- agnes(daisy(votes.repub), diss = TRUE, method = "complete")
plot(agn2)
agnS <- agnes(votes.repub, method = "flexible", par.meth = 0.6)
plot(agnS)
par(op)

data(agriculture)
## Plot similar to Figure 7 in ref
## Not run:  plot(agnes(agriculture), ask = TRUE)

\end{ExampleCode}
\end{Examples}

