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\name{kernelMatrix}
\alias{kernelMatrix}
\alias{kernelMult}
\alias{kernelPol}
\alias{kernelFast}
\alias{kernelPol,kernel-method}
\alias{kernelMatrix,kernel-method}
\alias{kernelMult,kernel-method}
\alias{kernelFast,kernel-method}
\alias{kernelMatrix,rbfkernel-method}
\alias{kernelMatrix,polykernel-method}
\alias{kernelMatrix,vanillakernel-method}
\alias{kernelMatrix,tanhkernel-method}
\alias{kernelMatrix,laplacekernel-method}
\alias{kernelMatrix,anovakernel-method}
\alias{kernelMatrix,splinekernel-method}
\alias{kernelMatrix,besselkernel-method}
\alias{kernelMatrix,stringkernel-method}
\alias{kernelMult,rbfkernel,ANY-method}
\alias{kernelMult,splinekernel,ANY-method}
\alias{kernelMult,polykernel,ANY-method}
\alias{kernelMult,tanhkernel,ANY-method}
\alias{kernelMult,laplacekernel,ANY-method}
\alias{kernelMult,besselkernel,ANY-method}
\alias{kernelMult,anovakernel,ANY-method}
\alias{kernelMult,vanillakernel,ANY-method}
\alias{kernelMult,character,kernelMatrix-method}
\alias{kernelMult,stringkernel,ANY-method}
\alias{kernelPol,rbfkernel-method}
\alias{kernelPol,splinekernel-method}
\alias{kernelPol,polykernel-method}
\alias{kernelPol,tanhkernel-method}
\alias{kernelPol,vanillakernel-method}
\alias{kernelPol,anovakernel-method}
\alias{kernelPol,besselkernel-method}
\alias{kernelPol,laplacekernel-method}
\alias{kernelPol,stringkernel-method}
\alias{kernelFast,rbfkernel-method}
\alias{kernelFast,splinekernel-method}
\alias{kernelFast,polykernel-method}
\alias{kernelFast,tanhkernel-method}
\alias{kernelFast,vanillakernel-method}
\alias{kernelFast,anovakernel-method}
\alias{kernelFast,besselkernel-method}
\alias{kernelFast,laplacekernel-method}
\alias{kernelFast,stringkernel-method}
\alias{kernelFast,splinekernel-method}
\title{Kernel Matrix functions}
\description{
\code{kernelMatrix} calculates the kernel matrix \eqn{K_{ij} = k(x_i,x_j)} or \eqn{K_{ij} =
k(x_i,y_j)}.\cr
\code{kernelPol} computes the quadratic kernel expression \eqn{H = z_i z_j
k(x_i,x_j)}, \eqn{H = z_i k_j k(x_i,y_j)}.\cr
\code{kernelMult} calculates the kernel expansion \eqn{f(x_i) =
\sum_{i=1}^m z_i k(x_i,x_j)}\cr
\code{kernelFast} computes the kernel matrix, identical
to \code{kernelMatrix}, except that it also requires the squared
norm of the first argument as additional input, useful in iterative
kernel matrix calculations.
}
\usage{
\S4method{kernelMatrix}{kernel}(kernel, x, y = NULL)
\S4method{kernelPol}{kernel}(kernel, x, y = NULL, z, k = NULL)
\S4method{kernelMult}{kernel}(kernel, x, y = NULL, z, blocksize = 256)
\S4method{kernelFast}{kernel}(kernel, x, y, a)
}
\arguments{
\item{kernel}{the kernel function to be used to calculate the kernel
matrix.
This has to be a function of class \code{kernel}, i.e. which can be
generated either one of the build in
kernel generating functions (e.g., \code{rbfdot} etc.) or a user defined
function of class \code{kernel} taking two vector arguments and returning a scalar.}
\item{x}{a data matrix to be used to calculate the kernel matrix, or a
list of vector when a \code{stringkernel} is used}
\item{y}{second data matrix to calculate the kernel matrix, or a
list of vector when a \code{stringkernel} is used}
\item{z}{a suitable vector or matrix}
\item{k}{a suitable vector or matrix}
\item{a}{the squared norm of \code{x}, e.g., \code{rowSums(x^2)}}
\item{blocksize}{the kernel expansion computations are done block wise
to avoid storing the kernel matrix into memory. \code{blocksize}
defines the size of the computational blocks.}
}
\details{
Common functions used during kernel based computations.\cr
The \code{kernel} parameter can be set to any function, of class
kernel, which computes the inner product in feature space between two
vector arguments. \pkg{kernlab} provides the most popular kernel functions
which can be initialized by using the following
functions:
\itemize{
\item \code{rbfdot} Radial Basis kernel function
\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} the Spline kernel
} (see example.)
\code{kernelFast} is mainly used in situations where columns of the
kernel matrix are computed per invocation. In these cases,
evaluating the norm of each row-entry over and over again would
cause significant computational overhead.
}
\value{
\code{kernelMatrix} returns a symmetric diagonal semi-definite matrix.\cr
\code{kernelPol} returns a matrix.\cr
\code{kernelMult} usually returns a one-column matrix.
}
\author{Alexandros Karatzoglou \cr
\email{alexandros.karatzoglou@ci.tuwien.ac.at}}
\seealso{\code{\link{rbfdot}}, \code{\link{polydot}},
\code{\link{tanhdot}}, \code{\link{vanilladot}}}
\examples{
## use the spam data
data(spam)
dt <- as.matrix(spam[c(10:20,3000:3010),-58])
## initialize kernel function
rbf <- rbfdot(sigma = 0.05)
rbf
## calculate kernel matrix
kernelMatrix(rbf, dt)
yt <- as.matrix(as.integer(spam[c(10:20,3000:3010),58]))
yt[yt==2] <- -1
## calculate the quadratic kernel expression
kernelPol(rbf, dt, ,yt)
## calculate the kernel expansion
kernelMult(rbf, dt, ,yt)
}
\keyword{algebra}
\keyword{array}
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