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\name{kpca}
\alias{kpca}
\alias{kpca,formula-method}
\alias{kpca,matrix-method}
\alias{kpca,kernelMatrix-method}
\alias{kpca,list-method}
\alias{predict,kpca-method}
\title{Kernel Principal Components Analysis}
\description{
Kernel Principal Components Analysis is a nonlinear form of principal
component analysis.}
\usage{
\S4method{kpca}{formula}(x, data = NULL, na.action, ...)
\S4method{kpca}{matrix}(x, kernel = "rbfdot", kpar = list(sigma = 0.1),
features = 0, th = 1e-4, na.action = na.omit, ...)
\S4method{kpca}{kernelMatrix}(x, features = 0, th = 1e-4, ...)
\S4method{kpca}{list}(x, kernel = "stringdot", kpar = list(length = 4, lambda = 0.5),
features = 0, th = 1e-4, na.action = na.omit, ...)
}
\arguments{
\item{x}{the data matrix indexed by row or a formula describing the
model, 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 (when using a formula).}
\item{kernel}{the kernel function used in training and predicting.
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
}
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}{the list of hyper-parameters (kernel parameters).
This is a list which contains 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".
}
Hyper-parameters for user defined kernels can be passed through the
kpar parameter as well.}
\item{features}{Number of features (principal components) to
return. (default: 0 , all)}
\item{th}{the value of the eigenvalue under which principal
components are ignored (only valid when features = 0). (default : 0.0001) }
\item{na.action}{A function to specify the action to be taken if \code{NA}s are
found. The default action is \code{na.omit}, which leads to rejection of cases
with missing values on any required variable. An alternative
is \code{na.fail}, which causes an error if \code{NA} cases
are found. (NOTE: If given, this argument must be named.)}
\item{\dots}{ additional parameters}
}
\details{Using kernel functions one can efficiently compute
principal components in high-dimensional
feature spaces, related to input space by some non-linear map.\cr
The data can be passed to the \code{kpca} function in a \code{matrix} or a
\code{data.frame}, in addition \code{kpca} 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 containing the principal component vectors along with the
corresponding eigenvalues.
\item{pcv}{a matrix containing the principal component vectors (column
wise)}
\item{eig}{The corresponding eigenvalues}
\item{rotated}{The original data projected (rotated) on the principal components}
\item{xmatrix}{The original data matrix}
all the slots of the object can be accessed by accessor functions.
}
\note{The predict function can be used to embed new data on the new space}
\references{
Schoelkopf B., A. Smola, K.-R. Mueller :\cr
\emph{Nonlinear component analysis as a kernel eigenvalue problem}\cr
Neural Computation 10, 1299-1319\cr
\doi{10.1162/089976698300017467}.
}
\author{Alexandros Karatzoglou \cr
\email{alexandros.karatzoglou@ci.tuwien.ac.at}}
\seealso{\code{\link{kcca}}, \code{pca}}
\examples{
# another example using the iris
data(iris)
test <- sample(1:150,20)
kpc <- kpca(~.,data=iris[-test,-5],kernel="rbfdot",
kpar=list(sigma=0.2),features=2)
#print the principal component vectors
pcv(kpc)
#plot the data projection on the components
plot(rotated(kpc),col=as.integer(iris[-test,5]),
xlab="1st Principal Component",ylab="2nd Principal Component")
#embed remaining points
emb <- predict(kpc,iris[test,-5])
points(emb,col=as.integer(iris[test,5]))
}
\keyword{cluster}
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