wcmdscale {vegan}R Documentation

Weighted Classical (Metric) Multidimensional Scaling


Weighted classical multidimensional scaling, also known as weighted principal coordinates analysis.


wcmdscale(d, k, eig = FALSE, add = FALSE, x.ret = FALSE, w)


d a distance structure such as that returned by dist or a full symmetric matrix containing the dissimilarities.
k the dimension of the space which the data are to be represented in; must be in {1,2,...,n-1}. If missing, all dimensions with above zero eigenvalue.
eig indicates whether eigenvalues should be returned.
add logical indicating if an additive constant c* should be computed, and added to the non-diagonal dissimilarities such that all n-1 eigenvalues are non-negative. Not implemented.
x.ret indicates whether the doubly centred symmetric distance matrix should be returned.
w Weights of points.


Function wcmdscale is based on function cmdscale (package stats of base R), but it uses point weights. Points with high weights will have a stronger influence on the result than those with low weights. Setting equal weights w = 1 will give ordinary multidimensional scaling.


If eig = 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, an object of class wcmdscale list containing the following components.

points a matrix with k columns whose rows give the coordinates of the points chosen to represent the dissimilarities.
eig the n-1 eigenvalues computed during the scaling process if eig is true.
x the doubly centred and weighted distance matrix if x.ret is true.
weights Weights.
negaxes A matrix of scores for axes with negative eigenvalues scaled by the absolute eigenvalues similarly as points. This is NULL if there are no negative eigenvalues or k was specified, and would not include negative eigenvalues.


Gower, J. C. (1966) Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika 53, 325–328.

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

See Also

cmdscale. Also isoMDS and sammon in package MASS.


## Correspondence analysis as a weighted principal coordinates
## analysis of Euclidean distances of Chi-square transformed data
rs <- rowSums(dune)/sum(dune)
d <- dist(decostand(dune, "chi"))
ord <- wcmdscale(d, w = rs, eig = TRUE)
## Ordinary CA
ca <- cca(dune)
## Eigevalues are numerically similar
ca$CA$eig - ord$eig
## Configurations are similar when site scores are scaled by
## eigenvalues in CA
procrustes(ord, ca, choices=1:19, scaling = 1)
plot(procrustes(ord, ca, choices=1:2, scaling=1))
## Reconstruction of non-Euclidean distances with negative eigenvalues
d <- vegdist(dune)
ord <- wcmdscale(d, eig = TRUE)
## Only positive eigenvalues:
cor(d, dist(ord$points))
## Correction with negative eigenvalues:
cor(d, sqrt(dist(ord$points)^2 - dist(ord$negaxes)^2))

[Package vegan version 1.16-32 Index]