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% File src/library/stats/man/cmdscale.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2018 R Core Team
% Distributed under GPL 2 or later

\name{cmdscale}
\alias{cmdscale}
\concept{ordination}
\concept{MDS}
\title{Classical (Metric) Multidimensional Scaling}
\usage{
cmdscale(d, k = 2, eig = FALSE, add = FALSE, x.ret = FALSE,
         list. = eig || add || x.ret)
}
\description{
  Classical multidimensional scaling (MDS) of a data matrix.
  Also known as \emph{principal coordinates analysis} (Gower, 1966).
}
\arguments{
  \item{d}{a distance structure such as that returned by \code{dist}
    or a full symmetric matrix containing the dissimilarities.}
  \item{k}{the maximum dimension of the space which the data are to be
    represented in; must be in \eqn{\{1, 2, \ldots, n-1\}}{{1, 2, \dots, n-1}}.}
  \item{eig}{indicates whether eigenvalues should be returned.}
  \item{add}{logical indicating if an additive constant \eqn{c*} should
    be computed, and added to the non-diagonal dissimilarities such that
    the modified dissimilarities are Euclidean.}
  \item{x.ret}{indicates whether the doubly centred symmetric distance
    matrix should be returned.}
  \item{list.}{logical indicating if a \code{\link{list}} should be
    returned or just the \eqn{n \times k}{n * k} matrix, see \sQuote{Value:}.}
}
\details{
  Multidimensional scaling takes a set of dissimilarities and returns a
  set of points such that the distances between the points are
  approximately equal to the dissimilarities.  (It is a major part of
  what ecologists call \sQuote{ordination}.)

  A set of Euclidean distances on \eqn{n} points can be represented
  exactly in at most \eqn{n - 1} dimensions.  \code{cmdscale} follows
  the analysis of Mardia (1978), and returns the best-fitting
  \eqn{k}-dimensional representation, where \eqn{k} may be less than the
  argument \code{k}.

  The representation is only determined up to location (\code{cmdscale}
  takes the column means of the configuration to be at the origin),
  rotations and reflections.  The configuration returned is given in
  principal-component axes, so the reflection chosen may differ between
  \R platforms (see \code{\link{prcomp}}).

  When \code{add = TRUE}, a minimal additive constant \eqn{c*} is
  computed such that the dissimilarities \eqn{d_{ij} + c*}{d[i,j] +
  c*} are Euclidean and hence can be represented in \code{n - 1}
  dimensions.  Whereas S (Becker \emph{et al}, 1988) computes this
  constant using an approximation suggested by Torgerson, \R uses the
  analytical solution of Cailliez (1983), see also Cox and Cox (2001).
  Note that because of numerical errors the computed eigenvalues need
  not all be non-negative, and even theoretically the representation
  could be in fewer than \code{n - 1} dimensions.
}
\value{
  If \code{.list} is false (as per default), a matrix with \code{k}
  columns whose rows give the coordinates of the points chosen to
  represent the dissimilarities.

  Otherwise, a \code{\link{list}} containing the following components.
  \item{points}{a matrix with up to \code{k} columns whose rows give the
    coordinates of the points chosen to represent the dissimilarities.}
  \item{eig}{the \eqn{n} eigenvalues computed during the scaling process if
    \code{eig} is true.  \strong{NB}: versions of \R before 2.12.1
    returned only \code{k} but were documented to return \eqn{n - 1}.}
  \item{x}{the doubly centered distance matrix if \code{x.ret} is true.}
  \item{ac}{the additive constant \eqn{c*}, \code{0} if \code{add = FALSE}.}
  \item{GOF}{a numeric vector of length 2, equal to say
    \eqn{(g_1,g_2)}{(g.1,g.2)}, where
    \eqn{g_i = (\sum_{j=1}^k \lambda_j)/ (\sum_{j=1}^n T_i(\lambda_j))}{g.i = (sum{j=1..k} \lambda[j]) / (sum{j=1..n} T.i(\lambda[j]))},
    where \eqn{\lambda_j}{\lambda[j]} are the eigenvalues (sorted in
    decreasing order),
    \eqn{T_1(v) = \left| v \right|}{T.1(v) = abs(v)}, and
    \eqn{T_2(v) = max( v, 0 )}{T.2(v) = max(v, 0)}.
  }
}
\references{
  Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988).
  \emph{The New S Language}.
  Wadsworth & Brooks/Cole.

  Cailliez, F. (1983).
  The analytical solution of the additive constant problem.
  \emph{Psychometrika}, \bold{48}, 343--349.
  \doi{10.1007/BF02294026}.

  Cox, T. F. and Cox, M. A. A. (2001).
  \emph{Multidimensional Scaling}.  Second edition.
  Chapman and Hall.

  Gower, J. C. (1966).
  Some distance properties of latent root and vector
  methods used in multivariate analysis.
  \emph{Biometrika}, \bold{53}, 325--328.
  \doi{10.2307/2333639}.

  Krzanowski, W. J. and Marriott, F. H. C. (1994).
  \emph{Multivariate Analysis. Part I. Distributions, Ordination and
    Inference.}
  London: Edward Arnold.
  (Especially pp.\sspace{}108--111.)

  Mardia, K.V. (1978).
  Some properties of classical multidimensional scaling.
  \emph{Communications on Statistics -- Theory and Methods}, \bold{A7},
  1233--41.
  \doi{10.1080/03610927808827707}

  Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979).
  Chapter 14 of \emph{Multivariate Analysis}, London: Academic Press.

  Seber, G. A. F. (1984).
  \emph{Multivariate Observations}.
  New York: Wiley.

  Torgerson, W. S. (1958).
  \emph{Theory and Methods of Scaling}.
  New York: Wiley.
}
\seealso{
  \code{\link{dist}}.

  \code{\link[MASS]{isoMDS}} and \code{\link[MASS]{sammon}}
  in package \CRANpkg{MASS} provide alternative methods of
  multidimensional scaling.
}
\examples{
require(graphics)

loc <- cmdscale(eurodist)
x <- loc[, 1]
y <- -loc[, 2] # reflect so North is at the top
## note asp = 1, to ensure Euclidean distances are represented correctly
plot(x, y, type = "n", xlab = "", ylab = "", asp = 1, axes = FALSE,
     main = "cmdscale(eurodist)")
text(x, y, rownames(loc), cex = 0.6)
}
\keyword{multivariate}