Random Start and Axis Scaling for isoMDS

Description

Function initMDS gives a random start for multidimensional scaling, and postMDS performs some post-standardizations for multidimensional scaling, in order to make the configurations easier to interpret

Usage

initMDS(x, k=2)
postMDS(X, dist, pc=TRUE, center=TRUE, halfchange=TRUE, threshold=0.8,
        nthreshold=10, plot=FALSE)

Arguments

Details

Non-metric Multidimensional Scaling (NMDS) is commonly regarded as the most robust unconstrained ordination method in community ecology (Minchin 1987). Functions initMDS and postMDS together with some other functions are intended to help run NMDS in isoMDS like recommended by Minchin (1987) – NMDS is not a strong choice unless used correctly:

1. You should use a dissimilarity index that has a good rank order relation with ecological gradients. Function rankindex may help in choosing an adequate index. Often a good a priori choice is to use Bray–Curtis index, perhaps with wisconsin double standardization. Some recommended indices are available in function vegdist.
2. NMDS should be run with several random starts. It is dangerous to follow the common default of starting with metric scaling (cmdscale), because this may be close to a local optimum which will trap the iteration. Function initMDS provides such random starts.
3. NMDS solutions with minimum stress should be compared for consistency. You should be satisfied only when several minimum stress solutions have similar configurations. In particular in large data sets, single points may be unstable even with about equal stress. Function postMDS provides a simple solution for fixing the scaling of NMDS. Function procrustes provides Procrustes rotation for more formal inspection.

Function postMDS provides the following ways of ``fixing'' the indeterminacy of scaling and orientation of axes in NMDS: Centring moves the origin to the average of each axis. Principal components rotate the configuration so that the variance of points is maximized on first dimensions. Half-change scaling scales the configuration so that one unit means halving of community similarity from replicate similarity. Half-change scaling is based on closer dissimilarities where the relation between ordination distance and community dissimilarity is rather linear; the limit is controlled by parameter threshold. If there are enough points below this threshold (controlled by the the parameter nthreshold), dissimilarities are regressed on distances. The intercept of this regression is taken as the replicate dissimilarity, and half-change is the distance where similarity halves according to linear regression. Obviously the method is applicable only for dissimilarity indices scaled to 0 ... 1, such as Kulczynski, Bray-Curtis and Canberra indices.

Value

Function initMDS returns a random configuration which is intended to be used within isoMDS only. Function postMDS returns the result of isoMDS with updated configuration.

Author(s)

Jari Oksanen

References

Minchin, P.R. (1987) An evaluation of relative robustness of techniques for ecological ordinations. Vegetatio 71, 145-156.

See Also

isoMDS, cmdscale, procrustes.

Examples

## The recommended way of running NMDS (Minchin 1987)
##
data(varespec)
data(varechem)
library(MASS) ## isoMDS
library(mva)  ## cmdscale: default start to isoMDS
# Select a good dissimilarity index
rankindex(scale(varechem),varespec)
rankindex(scale(varechem),wisconsin(varespec))
vare.dist <- vegdist(wisconsin(varespec), "bray")
# Get the baseline solution: start with cmdscale
mds.null <- isoMDS(vare.dist, tol=1e-7)
## See if you can get any better.
repeat{
  mds.1<- isoMDS(vare.dist, initMDS(vare.dist), maxit=200, trace=FALSE, tol=1e-7)
  if(mds.1$stress < mds.null$stress) break
}
# Scale solutions ("fix translation, rotation and scale")
mds.null <- postMDS(mds.null, vare.dist)
mds.1 <- postMDS(mds.1, vare.dist)
# Compare solutions
plot(procrustes(mds.1, mds.null))