% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/multivariance-functions.R
\name{cdm}
\alias{cdm}
\title{centered distance matrix}
\usage{
cdm(x, normalize = TRUE, psi = NULL, p = NULL, isotropic = FALSE,
  external.dm.fun = NULL)
}
\arguments{
\item{x}{matrix, each row of the matrix is treated as one sample}

\item{normalize}{logical, indicates if the matrix should be normalized}

\item{psi}{if it is \code{NULL}, the euclidean distance will be used. In the case of \code{isotropic = TRUE}: a real valued negative definite function of one variable (accepting vectors as arguments; returning a vector of the same length). In the case of \code{isotropic = FALSE}: a real valued function of two variables (or vectors) to compute the distance of two samples based on a continuous negative definite function.}

\item{p}{numeric, if it is a value between 1 and 2 then the Minkowski distance with parameter p is used.}

\item{isotropic}{logical, indicates if psi of the Euclidean distance matrix should be computed, i.e., if an isotropic distance should be used.}

\item{external.dm.fun}{here one can supply an external function, which computes the distance matrix given \code{x}.}
}
\description{
computes the centered distance matrix
}
\details{
The centered distance matrices are required for the computation of (total / m-) multivariance.

If \code{normalize = TRUE} then the value of multivariance is comparable and meaningful. It can be compared to the \code{\link{rejection.level}} or its p-value \code{\link{multivariance.pvalue}} can be computed.

More details: If \code{normalize = TRUE} the matrix is scaled such that the multivariance based on it, times the sample size, has in the limit - in the case of independence - the distribution of an L^2 norm of a Gaussian process with known expectation.

As default the Euclidean distance is used. The parameters \code{psi}, \code{p}, \code{isotropic} and \code{external.dm.fun} can be used to select a different distance. In particular, \code{external.dm.fun} can be used to provide any function which calculates a distance matrix for the rows of a given matrix.
}
\examples{
x = coins(100)
cdm(x) # fast euclidean distances
cdm(x,psi = function(x,y) sqrt(sum((x-y)^2))) # this is identical to the previous (but slower)

# the function cdm does the following three lines in a faster way
N = nrow(x)
C = diag(N) - matrix(1/N,nrow = N,ncol = N)
A = - C \%*\% as.matrix(stats::dist(x,method="euclidean")) \%*\% C #'
all(abs(A- cdm(x,normalize = FALSE)) < 10^(-12))

}
\references{
For the theoretic background see the references given on the main help page of this package: \link{multivariance-package}.
}