\Header{agnes}{Agglomerative Nesting}
\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{\Link{daisy}} or \code{\Link{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" (group average method), "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{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{\Link{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{\Link{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{\Link{diana}}, and \code{\Link{mona}}
construct a hierarchy of clusterings, with the number of clusters
ranging from one to the number of observations.
\item[Partitioning methods] like
\code{\Link{pam}}, \code{\Link{clara}}, and \code{\Link{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{\Link{agnes.object}}, \code{\Link{daisy}}, \code{\Link{diana}},
\code{\Link{dist}}, \code{\Link{hclust}}, \code{\Link{plot.agnes}},
\code{\Link{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}

