\HeaderA{cmdscale}{Classical (Metric) Multidimensional Scaling}{cmdscale}
\keyword{multivariate}{cmdscale}
\begin{Description}\relax
Classical multidimensional scaling of a data matrix.
Also known as \emph{principal coordinates analysis} (Gower, 1966).
\end{Description}
\begin{Usage}
\begin{verbatim}
cmdscale(d, k = 2, eig = FALSE, add = FALSE, x.ret = FALSE)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{d}] a distance structure such as that returned by \code{dist}
or a full symmetric matrix containing the dissimilarities.
\item[\code{k}] the dimension of the space which the data are to be
represented in; must be in \eqn{\{1,2,\ldots,n-1\}}{}.
\item[\code{eig}] indicates whether eigenvalues should be returned.
\item[\code{add}] logical indicating if an additive constant \eqn{c*}{} should
be computed, and added to the non-diagonal dissimilarities such that
all \eqn{n-1}{} eigenvalues are non-negative.
\item[\code{x.ret}] indicates whether the doubly centred symmetric distance
matrix should be returned.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
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.

The functions \code{isoMDS} and \code{sammon} in package \pkg{MASS}
provide alternative ordination techniques.

When \code{add = TRUE}, an additive constant \eqn{c*}{} is computed, and
the dissimilarities \eqn{d_{ij} + c*}{d[i,j] + c*} are used instead of
the original \eqn{d_{ij}}{d[i,j]}'s.

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 (1994).
\end{Details}
\begin{Value}
If \code{eig = FALSE} and \code{x.ret = FALSE} (default), a matrix
with \code{k} columns whose rows give the coordinates of the points
chosen to represent the dissimilarities.

Otherwise, a list containing the following components.
\begin{ldescription}
\item[\code{points}] a matrix with \code{k} columns whose rows give the
coordinates of the points chosen to represent the dissimilarities.
\item[\code{eig}] the \eqn{n-1}{} eigenvalues computed during the scaling process if
\code{eig} is true.
\item[\code{x}] the doubly centered distance matrix if \code{x.ret} is true.
\item[\code{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
decreasingly),
\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)}.

\end{ldescription}
\end{Value}
\begin{References}\relax
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.

Cox, T. F. and Cox, M. A. A. (1994)
\emph{Multidimensional Scaling}.
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.

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.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{dist}{dist}}.
Also \code{\LinkA{isoMDS}{isoMDS}} and \code{\LinkA{sammon}{sammon}}
in package \pkg{MASS}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
loc <- cmdscale(eurodist)
x <- loc[,1]
y <- -loc[,2]
plot(x, y, type="n", xlab="", ylab="", main="cmdscale(eurodist)")
text(x, y, rownames(loc), cex=0.8)

cmdsE <- cmdscale(eurodist, k=20, add = TRUE, eig = TRUE, x.ret = TRUE)
str(cmdsE)
\end{ExampleCode}
\end{Examples}

