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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{\I{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 (\abbr{MDS}) of a data matrix.
  Also known as \emph{principal coordinates analysis} (\bibcite{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 \bibcite{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 \abbr{et al.}, 1988) computes this
  constant using an approximation suggested by \I{Torgerson}, \R uses the
  analytical solution of \bibcite{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}