wcmdscale: Weighted Classical (Metric) Multidimensional Scaling
In vegan: Community Ecology Package

Description Usage Arguments Details Value References See Also Examples

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

wcmdscale(d, k, eig = FALSE, add = FALSE, x.ret = FALSE, w)
## S3 method for class 'wcmdscale'
plot(x, choices = c(1, 2), type = "t", ...)
## S3 method for class 'wcmdscale'
scores(x, choices = NA, ...)

`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.
`x`	The `wcmdscale` result object when the function was called with options `eig = TRUE` or `x.ret = TRUE` (See Details).
`choices`	Axes to be returned; `NA` returns all real axes.
`type`	Type of graph which may be `"t"`ext, `"p"`oints or `"n"`one.
`...`	Other arguments passed to graphical functions.

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.

With default options, the function returns only a matrix of scores scaled by eigenvalues for all real axes. If the function is called with eig = TRUE or x.ret = TRUE, the function returns an object of class "wcmdscale" with print, plot, scores, eigenvals and stressplot methods, and described in section Value.

If eig = FALSE and x.ret = FALSE (default), a matrix with k columns whose rows give the coordinates of points corresponding to positive eignenvalues. Otherwise, an object of class wcmdscale containing the components that are mostly similar as in cmdscale:

`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.
`GOF`	Goodness of fit statistics for `k` axes. The first value is based on the sum of absolute values of all eigenvalues, and the second value is based on the sum of positive eigenvalues
`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.
`call`	Function call.

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.

The function is modelled after cmdscale, but adds weights (hence name) and handles negative eigenvalues differently. eigenvals.wcmdscale and stressplot.wcmdscale are some specific methods. Other multidimensional scaling methods are monoMDS, and isoMDS and sammon in package MASS.

## Correspondence analysis as a weighted principal coordinates
## analysis of Euclidean distances of Chi-square transformed data
data(dune)
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))

Loading required package: permute
Loading required package: lattice
This is vegan 2.4-3
          CA1           CA2           CA3           CA4           CA5 
 1.110223e-16 -6.661338e-16 -6.661338e-16  8.326673e-17  2.498002e-16 
          CA6           CA7           CA8           CA9          CA10 
 1.249001e-16 -1.249001e-16 -1.249001e-16  6.938894e-17  4.857226e-17 
         CA11          CA12          CA13          CA14          CA15 
 1.179612e-16 -2.081668e-17 -1.387779e-17  5.898060e-17 -1.040834e-17 
         CA16          CA17          CA18          CA19 
 1.734723e-18  1.908196e-17 -4.683753e-17  2.385245e-17 

Call:
procrustes(X = ord, Y = ca, choices = 1:19, scaling = 1) 

Procrustes sum of squares:
-7.105e-14 

[1] 0.9975185
[1] 1