wcmdscale {vegan} R Documentation

## Weighted Classical (Metric) Multidimensional Scaling

### Description

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

### Usage

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

### 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,...,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.

### 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.

### Value

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.

### References

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.

`cmdscale`. Also `isoMDS` and `sammon` in package MASS.

### 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)
## 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]