cmdscale
Classical (Metric) Multidimensional Scaling
Classical multidimensional scaling of a data matrix. Also known as principal coordinates analysis (Gower, 1966).
 Keywords
 multivariate
Usage
cmdscale(d, k = 2, eig = FALSE, add = FALSE, x.ret = FALSE)
Arguments
 d
 a distance structure such as that returned by
dist
or a full symmetric matrix containing the dissimilarities.  k
 the maximum dimension of the space which the data are to be represented in; must be in ${1, 2, \dots, n1}$.
 eig
 indicates whether eigenvalues should be returned.
 add
 logical indicating if an additive constant $c*$ should be computed, and added to the nondiagonal dissimilarities such that the modified dissimilarities are Euclidean.
 x.ret
 indicates whether the doubly centred symmetric distance matrix should be returned.
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 ‘ordination’.)
A set of Euclidean distances on $n$ points can be represented
exactly in at most $n  1$ dimensions. cmdscale
follows
the analysis of Mardia (1978), and returns the bestfitting
$k$dimensional representation, where $k$ may be less than the
argument k
.
The representation is only determined up to location (cmdscale
takes the column means of the configuration to be at the origin),
rotations and reflections. The configuration returned is given in
principalcomponent axes, so the reflection chosen may differ between
R platforms (see prcomp
).
When add = TRUE
, a minimal additive constant $c*$ is
computed such that the the dissimilarities $d[i,j] +
c*$ are Euclidean and hence can be represented in n  1
dimensions. Whereas S (Becker 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 nonnegative, and even theoretically the representation
could be in fewer than n  1
dimensions.
Value

If
 points
 a matrix with up to
k
columns whose rows give the coordinates of the points chosen to represent the dissimilarities.  eig
 the $n$ eigenvalues computed during the scaling process if
eig
is true. NB: versions of R before 2.12.1 returned onlyk
but were documented to return $n  1$.  x
 the doubly centered distance matrix if
x.ret
is true.  ac
 the additive constant $c*$,
0
ifadd = FALSE
.  GOF
 a numeric vector of length 2, equal to say $(g.1,g.2)$, where $g.i = (sum{j=1..k} \lambda[j]) / (sum{j=1..n} T.i(\lambda[j]))$, where $\lambda[j]$ are the eigenvalues (sorted in decreasing order), $T.1(v) = abs(v)$, and $T.2(v) = max(v, 0)$.
eig = FALSE
, add = FALSE
and x.ret = FALSE
(default), a matrix with k
columns whose rows give the
coordinates of the points chosen to represent the dissimilarities.Otherwise, a list containing the following components.
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.
Cailliez, F. (1983) The analytical solution of the additive constant problem. Psychometrika 48, 343349.
Cox, T. F. and Cox, M. A. A. (2001) Multidimensional Scaling. Second edition. Chapman and Hall.
Gower, J. C. (1966) Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53, 325328.
Krzanowski, W. J. and Marriott, F. H. C. (1994) Multivariate Analysis. Part I. Distributions, Ordination and Inference. London: Edward Arnold. (Especially pp.\ifelse{latex}{\out{~}}{ } 108111.)
Mardia, K.V. (1978) Some properties of classical multidimensional scaling. Communications on Statistics  Theory and Methods, A7, 123341.
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Chapter 14 of Multivariate Analysis, London: Academic Press.
Seber, G. A. F. (1984). Multivariate Observations. New York: Wiley.
Torgerson, W. S. (1958). Theory and Methods of Scaling. New York: Wiley.
See Also
dist
.
isoMDS
and sammon
in package \href{https://CRAN.Rproject.org/package=#1}{\pkg{#1}}MASSMASS provide alternative methods of
multidimensional scaling.
Examples
library(stats)
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)