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\name{fanny}
\alias{fanny}
\title{Fuzzy Analysis Clustering}
\description{
Computes a fuzzy clustering of the data into \code{k} clusters.
}
\usage{
fanny(x, k, diss = inherits(x, "dist"), memb.exp = 2,
metric = c("euclidean", "manhattan", "SqEuclidean"),
stand = FALSE, iniMem.p = NULL, cluster.only = FALSE,
keep.diss = !diss && !cluster.only && n < 100,
keep.data = !diss && !cluster.only,
maxit = 500, tol = 1e-15, trace.lev = 0)
}
\arguments{
\item{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 of
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{k}{integer giving the desired number of clusters. It is
required that \eqn{0 < k < n/2} where \eqn{n} is the number of
observations.}
\item{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{memb.exp}{number \eqn{r} strictly larger than 1 specifying the
\emph{membership exponent} used in the fit criterion; see the
\sQuote{Details} below. Default: \code{2} which used to be hardwired
inside FANNY.}
\item{metric}{character string specifying the metric to be used for
calculating dissimilarities between observations. Options are
\code{"euclidean"} (default), \code{"manhattan"}, and
\code{"SqEuclidean"}. Euclidean distances are root sum-of-squares
of differences, and manhattan distances are the sum of absolute
differences, and \code{"SqEuclidean"}, the \emph{squared} euclidean
distances are sum-of-squares of differences. Using this last option is
equivalent (but somewhat slower) to computing so called \dQuote{fuzzy C-means}.
\cr
If \code{x} is already a dissimilarity matrix, then this argument will
be ignored.
}
\item{stand}{logical; if true, 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{iniMem.p}{numeric \eqn{n \times k}{n x k} matrix or \code{NULL}
(by default); can be used to specify a starting \code{membership}
matrix, i.e., a matrix of non-negative numbers, each row summing to
one.
} %% FIXME: add example
\item{cluster.only}{logical; if true, no silhouette information will be
computed and returned, see details.}%% FIXME: add example
\item{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 smaller results and hence also save
memory allocation \emph{time}.}
\item{maxit, tol}{maximal number of iterations and default tolerance
for convergence (relative convergence of the fit criterion) for the
FANNY algorithm. The defaults \code{maxit = 500} and \code{tol =
1e-15} used to be hardwired inside the algorithm.}
\item{trace.lev}{integer specifying a trace level for printing
diagnostics during the C-internal algorithm.
Default \code{0} does not print anything; higher values print
increasingly more.}
}
\value{
an object of class \code{"fanny"} representing the clustering.
See \code{\link{fanny.object}} for details.
}
\details{
In a fuzzy clustering, each observation is \dQuote{spread out} over
the various clusters. Denote by \eqn{u_{iv}}{u(i,v)} the membership
of observation \eqn{i} to cluster \eqn{v}.
The memberships are nonnegative, and for a fixed observation i they sum to 1.
The particular method \code{fanny} stems from chapter 4 of
Kaufman and Rousseeuw (1990) (see the references in
\code{\link{daisy}}) and has been extended by Martin Maechler to allow
user specified \code{memb.exp}, \code{iniMem.p}, \code{maxit},
\code{tol}, etc.
Fanny aims to minimize the objective function
\deqn{\sum_{v=1}^k
\frac{\sum_{i=1}^n\sum_{j=1}^n u_{iv}^r u_{jv}^r d(i,j)}{
2 \sum_{j=1}^n u_{jv}^r}}{%
SUM_[v=1..k] (SUM_(i,j) u(i,v)^r u(j,v)^r d(i,j)) / (2 SUM_j u(j,v)^r)}
where \eqn{n} is the number of observations, \eqn{k} is the number of
clusters, \eqn{r} is the membership exponent \code{memb.exp} and
\eqn{d(i,j)} is the dissimilarity between observations \eqn{i} and \eqn{j}.
\cr Note that \eqn{r \to 1}{r -> 1} gives increasingly crisper
clusterings whereas \eqn{r \to \infty}{r -> Inf} leads to complete
fuzzyness. K&R(1990), p.191 note that values too close to 1 can lead
to slow convergence. Further note that even the default, \eqn{r = 2}
can lead to complete fuzzyness, i.e., memberships \eqn{u_{iv} \equiv
1/k}{u(i,v) == 1/k}. In that case a warning is signalled and the
user is advised to chose a smaller \code{memb.exp} (\eqn{=r}).
Compared to other fuzzy clustering methods, \code{fanny} has the following
features: (a) it also accepts a dissimilarity matrix; (b) it is
more robust to the \code{spherical cluster} assumption; (c) it provides
a novel graphical display, the silhouette plot (see
\code{\link{plot.partition}}).
}
\seealso{
\code{\link{agnes}} for background and references;
\code{\link{fanny.object}}, \code{\link{partition.object}},
\code{\link{plot.partition}}, \code{\link{daisy}}, \code{\link{dist}}.
}
\examples{
## generate 10+15 objects in two clusters, plus 3 objects lying
## between those clusters.
x <- rbind(cbind(rnorm(10, 0, 0.5), rnorm(10, 0, 0.5)),
cbind(rnorm(15, 5, 0.5), rnorm(15, 5, 0.5)),
cbind(rnorm( 3,3.2,0.5), rnorm( 3,3.2,0.5)))
fannyx <- fanny(x, 2)
## Note that observations 26:28 are "fuzzy" (closer to # 2):
fannyx
summary(fannyx)
plot(fannyx)
(fan.x.15 <- fanny(x, 2, memb.exp = 1.5)) # 'crispier' for obs. 26:28
(fanny(x, 2, memb.exp = 3)) # more fuzzy in general
data(ruspini)
f4 <- fanny(ruspini, 4)
stopifnot(rle(f4$clustering)$lengths == c(20,23,17,15))
plot(f4, which = 1)
## Plot similar to Figure 6 in Stryuf et al (1996)
plot(fanny(ruspini, 5))
}
\keyword{cluster}
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