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\name{kmmd}
\alias{kmmd}
\alias{kmmd,matrix-method}
\alias{kmmd,list-method}
\alias{kmmd,kernelMatrix-method}
\alias{show,kmmd-method}
\alias{H0}
\alias{Asymbound}
\alias{Radbound}
\alias{mmdstats}
\alias{AsympH0}
\title{Kernel Maximum Mean Discrepancy.}
\description{The Kernel Maximum Mean Discrepancy \code{kmmd} performs
a non-parametric distribution test.}
\usage{
\S4method{kmmd}{matrix}(x, y, kernel="rbfdot",kpar="automatic", alpha = 0.05,
asymptotic = FALSE, replace = TRUE, ntimes = 150, frac = 1, ...)
\S4method{kmmd}{kernelMatrix}(x, y, Kxy, alpha = 0.05,
asymptotic = FALSE, replace = TRUE, ntimes = 100, frac = 1, ...)
\S4method{kmmd}{list}(x, y, kernel="stringdot",
kpar = list(type = "spectrum", length = 4), alpha = 0.05,
asymptotic = FALSE, replace = TRUE, ntimes = 150, frac = 1, ...)
}
\arguments{
\item{x}{data values, in a \code{matrix},
\code{list}, or \code{kernelMatrix}}
\item{y}{data values, in a \code{matrix},
\code{list}, or \code{kernelMatrix}}
\item{Kxy}{\code{kernlMatrix} between \eqn{x} and \eqn{y} values (only for the
kernelMatrix interface)}
\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. \code{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}{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".
\item \code{lenght, 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 \code{kpar} parameter as well. In the case of a Radial
Basis kernel function (Gaussian) kpar can also be set to the
string "automatic" which uses the heuristics in 'sigest' to
calculate a good 'sigma' value for the Gaussian RBF or
Laplace kernel, from the data. (default = "automatic").
}
\item{alpha}{the confidence level of the test (default: 0.05)}
\item{asymptotic}{calculate the bounds asymptotically (suitable for
smaller datasets) (default: FALSE)}
\item{replace}{use replace when sampling for computing the asymptotic
bounds (default : TRUE)}
\item{ntimes}{number of times repeating the sampling procedure (default
: 150)}
\item{frac}{fraction of points to sample (frac : 1) }
\item{\dots}{additional parameters.}
}
\details{\code{kmmd} calculates the kernel maximum mean discrepancy for
samples from two distributions and conducts a test as to whether the samples are
from different distributions with level \code{alpha}.
}
\value{
An S4 object of class \code{kmmd} containing the
results of whether the H0 hypothesis is rejected or not. H0 being
that the samples \eqn{x} and \eqn{y} come from the same distribution.
The object contains the following slots :
\item{\code{H0}}{is H0 rejected (logical)}
\item{\code{AsympH0}}{is H0 rejected according to the asymptotic bound (logical)}
\item{\code{kernelf}}{the kernel function used.}
\item{\code{mmdstats}}{the test statistics (vector of two)}
\item{\code{Radbound}}{the Rademacher bound}
\item{\code{Asymbound}}{the asymptotic bound}
see \code{kmmd-class} for more details.
}
\references{Gretton, A., K. Borgwardt, M. Rasch, B. Schoelkopf and A. Smola\cr
\emph{A Kernel Method for the Two-Sample-Problem}\cr
Neural Information Processing Systems 2006, Vancouver \cr
\url{https://papers.neurips.cc/paper/3110-a-kernel-method-for-the-two-sample-problem.pdf}
}
\author{Alexandros Karatzoglou \cr \email{alexandros.karatzoglou@ci.tuwien.ac.at}}
\seealso{\code{ksvm}}
\examples{
# create data
x <- matrix(runif(300),100)
y <- matrix(runif(300)+1,100)
mmdo <- kmmd(x, y)
mmdo
}
\keyword{htest}
\keyword{nonlinear}
\keyword{nonparametric}
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