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\name{specc}
\alias{specc}
\alias{specc,matrix-method}
\alias{specc,formula-method}
\alias{specc,list-method}
\alias{specc,kernelMatrix-method}
\alias{show,specc-method}
\title{Spectral Clustering}
\description{
A spectral clustering algorithm. Clustering is performed by
embedding the data into the subspace of the eigenvectors
of an affinity matrix.
}
\usage{
\S4method{specc}{formula}(x, data = NULL, na.action = na.omit, ...)
\S4method{specc}{matrix}(x, centers,
kernel = "rbfdot", kpar = "automatic",
nystrom.red = FALSE, nystrom.sample = dim(x)[1]/6,
iterations = 200, mod.sample = 0.75, na.action = na.omit, ...)
\S4method{specc}{kernelMatrix}(x, centers, nystrom.red = FALSE, iterations = 200, ...)
\S4method{specc}{list}(x, centers,
kernel = "stringdot", kpar = list(length=4, lambda=0.5),
nystrom.red = FALSE, nystrom.sample = length(x)/6,
iterations = 200, mod.sample = 0.75, na.action = na.omit, ...)
}
\arguments{
\item{x}{the matrix of data to be clustered, or a symbolic
description of the model to be fit, or a kernel Matrix of class
\code{kernelMatrix}, or a list of character vectors.}
\item{data}{an optional data frame containing the variables in the model.
By default the variables are taken from the environment which
`specc' is called from.}
\item{centers}{Either the number of clusters or a set of initial cluster
centers. If the first, a random set of rows in the eigenvectors
matrix are chosen as the initial centers.}
\item{kernel}{the kernel function used in computing the affinity matrix.
This parameter can be set to any function, of class kernel, which computes a dot product between two
vector arguments. kernlab provides the most popular kernel functions
which can be used by setting the kernel parameter to the following
strings:
\itemize{
\item \code{rbfdot} Radial Basis kernel function "Gaussian"
\item \code{polydot} Polynomial kernel function
\item \code{vanilladot} Linear kernel function
\item \code{tanhdot} Hyperbolic tangent kernel function
\item \code{laplacedot} Laplacian kernel function
\item \code{besseldot} Bessel kernel function
\item \code{anovadot} ANOVA RBF kernel function
\item \code{splinedot} Spline kernel
\item \code{stringdot} String kernel
}
The kernel parameter can also be set to a user defined function of
class kernel by passing the function name as an argument.
}
\item{kpar}{a character string or the list of hyper-parameters (kernel parameters).
The default character string \code{"automatic"} uses a heuristic to determine a
suitable value for the width parameter of the RBF kernel.
The second option \code{"local"} (local scaling) uses a more advanced heuristic
and sets a width parameter for every point in the data set. This is
particularly useful when the data incorporates multiple scales.
A list can also be used containing the parameters to be used with the
kernel function. Valid parameters for existing kernels are :
\itemize{
\item \code{sigma} inverse kernel width for the Radial Basis
kernel function "rbfdot" and the Laplacian kernel "laplacedot".
\item \code{degree, scale, offset} for the Polynomial kernel "polydot"
\item \code{scale, offset} for the Hyperbolic tangent kernel
function "tanhdot"
\item \code{sigma, order, degree} for the Bessel kernel "besseldot".
\item \code{sigma, degree} for the ANOVA kernel "anovadot".
\item \code{length, lambda, normalized} for the "stringdot" kernel
where length is the length of the strings considered, lambda the
decay factor and normalized a logical parameter determining if the
kernel evaluations should be normalized.
}
Hyper-parameters for user defined kernels can be passed through the
kpar parameter as well.}
\item{nystrom.red}{use nystrom method to calculate eigenvectors. When
\code{TRUE} a sample of the dataset is used to calculate the
eigenvalues, thus only a \eqn{n x m} matrix where \eqn{n} the sample size
is stored in memory (default: \code{FALSE}}
\item{nystrom.sample}{number of data points to use for estimating the
eigenvalues when using the nystrom method. (default : dim(x)[1]/6)}
\item{mod.sample}{proportion of data to use when estimating sigma (default: 0.75)}
\item{iterations}{the maximum number of iterations allowed. }
\item{na.action}{the action to perform on NA}
\item{\dots}{additional parameters}
}
\details{
Spectral clustering works by embedding the data points of the
partitioning problem into the
subspace of the \eqn{k} largest eigenvectors of a normalized affinity/kernel matrix.
Using a simple clustering method like \code{kmeans} on the embedded points usually
leads to good performance. It can be shown that spectral clustering methods boil down to
graph partitioning.\cr
The data can be passed to the \code{specc} function in a \code{matrix} or a
\code{data.frame}, in addition \code{specc} also supports input in the form of a
kernel matrix of class \code{kernelMatrix} or as a list of character
vectors where a string kernel has to be used.}
\value{
An S4 object of class \code{specc} which extends the class \code{vector}
containing integers indicating the cluster to which
each point is allocated. The following slots contain useful information
\item{centers}{A matrix of cluster centers.}
\item{size}{The number of point in each cluster}
\item{withinss}{The within-cluster sum of squares for each cluster}
\item{kernelf}{The kernel function used}
}
\references{
Andrew Y. Ng, Michael I. Jordan, Yair Weiss\cr
\emph{On Spectral Clustering: Analysis and an Algorithm}\cr
Neural Information Processing Symposium 2001\cr
\url{https://papers.neurips.cc/paper/2092-on-spectral-clustering-analysis-and-an-algorithm.pdf}
}
\author{Alexandros Karatzoglou \cr \email{alexandros.karatzoglou@ci.tuwien.ac.at}
}
\seealso{\code{\link{kkmeans}}, \code{\link{kpca}}, \code{\link{kcca}} }
\examples{
## Cluster the spirals data set.
data(spirals)
sc <- specc(spirals, centers=2)
sc
centers(sc)
size(sc)
withinss(sc)
plot(spirals, col=sc)
}
\keyword{cluster}
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