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)
# S3 method for class 'wcmdscale'
plot(x, choices = c(1, 2), display = "sites", type = "t", ...)
# S3 method for class 'wcmdscale'
scores(x, choices = NA, display = "sites", tidy = FALSE, ...)

Arguments

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 \(\{1,2,\ldots,n-1\}\). If missing, all dimensions with above zero eigenvalue.
eig: indicates whether eigenvalues should be returned.
add: an additive constant \(c\) is added to the non-diagonal dissimilarities such that all \(n-1\) eigenvalues are non-negative. Alternatives are "lingoes" (default, also used with TRUE) and "cailliez" (which is the only alternative in cmdscale). See Legendre & Anderson (1999).
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.
display: Kind of scores. Normally only "sites" are available, but "species" can be supplemented with sppscores.
type: Type of graph which may be "t"ext, "p"oints or "n"one.
tidy: Return scores that are compatible with ggplot2: scores are in a data.frame, score type is in the variable score labelled as "sites", weights in variable weigth, and names in variable label.
...: Other arguments passed to graphical functions.

Details

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.

The method is Euclidean, and with non-Euclidean dissimilarities some eigenvalues can be negative. If this disturbs you, this can be avoided by adding a constant to non-diagonal dissimilarities making all eigenvalues non-negative. The function implements methods discussed by Legendre & Anderson (1999): The method of Lingoes (add="lingoes") adds the constant \(c\) to squared dissimilarities \(d\) using \(\sqrt{d^2 + 2 c}\) and the method of Cailliez (add="cailliez") to dissimilarities using \(d + c\). Legendre & Anderson (1999) recommend the method of Lingoes, and base R function cmdscale implements the method of Cailliez.

Value

If eig = FALSE and x.ret = FALSE (default), a matrix with k columns whose rows give the coordinates of points corresponding to positive eigenvalues. 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.
ac, add: additive constant and adjustment method used to avoid negative eigenvalues. These are NA and FALSE if no adjustment was done.
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.

References

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

Legendre, P. & Anderson, M. J. (1999). Distance-based redundancy analysis: testing multispecies responses in multifactorial ecological experiments. Ecology 69, 1–24.

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

Examples

## 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)

## IGNORE_RDIFF_BEGIN
## Eigevalues are numerically similar
ca$CA$eig - ord$eig
#>           CA1           CA2           CA3           CA4           CA5 
#> -5.551115e-16 -6.661338e-16 -8.881784e-16  1.942890e-16 -1.110223e-16 
#>           CA6           CA7           CA8           CA9          CA10 
#>  4.302114e-16 -1.942890e-16 -1.526557e-16  4.163336e-17  4.163336e-17 
#>          CA11          CA12          CA13          CA14          CA15 
#>  3.469447e-17  7.632783e-17  3.469447e-17  6.938894e-18  1.387779e-17 
#>          CA16          CA17          CA18          CA19 
#> -2.081668e-17 -1.040834e-17 -1.821460e-17  5.074066e-17 
## Configurations are similar when site scores are scaled by
## eigenvalues in CA
procrustes(ord, ca, choices=1:19, scaling = "sites")
#> 
#> Call:
#> procrustes(X = ord, Y = ca, choices = 1:19, scaling = "sites") 
#> 
#> Procrustes sum of squares:
#> -5.684e-14 
#> 
## IGNORE_RDIFF_END

plot(procrustes(ord, ca, choices=1:2, scaling="sites"))

## Reconstruction of non-Euclidean distances with negative eigenvalues
d <- vegdist(dune)
ord <- wcmdscale(d, eig = TRUE)
## Only positive eigenvalues:
cor(d, dist(ord$points))
#> [1] 0.9975185
## Correction with negative eigenvalues:
cor(d, sqrt(dist(ord$points)^2 - dist(ord$negaxes)^2))
#> [1] 1