Global and Local Non-metric Multidimensional Scaling and Linear and Hybrid Scaling

Function implements Kruskal's (1964a,b) non-metric multidimensional scaling (NMDS) using monotone regression and primary (“weak”) treatment of ties. In addition to traditional global NMDS, the function implements local NMDS, linear and hybrid multidimensional scaling.

Usage

monoMDS(dist, y, k = 2, model = c("global", "local", "linear", "hybrid"),
    threshold = 0.8, maxit = 200, weakties = TRUE, stress = 1,
    scaling = TRUE, pc = TRUE, smin = 1e-4, sfgrmin = 1e-7,
    sratmax=0.999999, ...)
# S3 method for class 'monoMDS'
scores(x, display = "sites", shrink = FALSE, choices,
    tidy = FALSE, ...)
# S3 method for class 'monoMDS'
plot(x, display = "sites", choices = c(1,2), type = "t", ...)
# S3 method for class 'monoMDS'
points(x, choices = c(1,2), select, ...)
# S3 method for class 'monoMDS'
text(x, labels, choices = c(1,2), select, ...)

Arguments

dist: Input dissimilarities.
y: Starting configuration. A random configuration will be generated if this is missing.
k: Number of dimensions. NB., the number of points $n$ should be $n > 2k + 1$, and preferably higher in non-metric MDS.
model: MDS model: "global" is normal non-metric MDS with a monotone regression, "local" is non-metric MDS with separate regressions for each point, "linear" uses linear regression, and "hybrid" uses linear regression for dissimilarities below a threshold in addition to monotone regression. See Details.
threshold: Dissimilarity below which linear regression is used alternately with monotone regression.
maxit: Maximum number of iterations.
weakties: Use primary or weak tie treatment, where equal observed dissimilarities are allowed to have different fitted values. if FALSE, then secondary (strong) tie treatment is used, and tied values are not broken.
stress: Use stress type 1 or 2 (see Details).
scaling: Scale final scores to unit root mean squares.
pc: Rotate final scores to principal components.
smin, sfgrmin, sratmax: Convergence criteria: iterations stop when stress drops below smin, scale factor of the gradient drops below sfgrmin, or stress ratio between two iterations goes over sratmax (but is still $< 1$).
x: A monoMDS result.
display: Kind of scores. Normally there are only scores for "sites", but "species" scores can be added with sppscores.
shrink: Shrink back species scores if they were expanded in wascores.
tidy: Return scores that are compatible with ggplot2: all scores are in a single data.frame, score type is identified by factor variable code ("sites" or "species"), the names by variable label. These scores are incompatible with conventional plot functions, but they can be used in ggplot2.
choices: Dimensions returned or plotted. The default NA returns all dimensions.
type: The type of the plot: "t" for text, "p" for points, and "n" for none.
select: Items to be displayed. This can either be a logical vector which is TRUE for displayed items or a vector of indices of displayed items.
labels: Labels to be use used instead of row names.
...: Other parameters to the functions (ignored in monoMDS, passed to graphical functions in plot.).

Details

There are several versions of non-metric multidimensional scaling in R, but monoMDS offers the following unique combination of features:

“Weak” treatment of ties (Kruskal 1964a,b), where tied dissimilarities can be broken in monotone regression. This is especially important for cases where compared sites share no species and dissimilarities are tied to their maximum value of one. Breaking ties allows these points to be at different distances and can help in recovering very long coenoclines (gradients). Functions in the smacof package also hav adequate tie treatment.
Handles missing values in a meaningful way.
Offers “local” and “hybrid” scaling in addition to usual “global” NMDS (see below).
Uses fast compiled code (isoMDS of the MASS package also uses compiled code).

Function monoMDS uses Kruskal's (1964b) original monotone regression to minimize the stress. There are two alternatives of stress: Kruskal's (1964a,b) original or “stress 1” and an alternative version or “stress 2” (Sibson 1972). Both of these stresses can be expressed with a general formula

$$s^2 = \frac{\sum (d - \hat d)^2}{\sum(d - d_0)^2}$$

where $d$ are distances among points in ordination configuration, $\hat d$ are the fitted ordination distances, and $d_0$ are the ordination distances under null model. For “stress 1” $d_0 = 0$, and for “stress 2” $d_0 = \bar{d}$ or mean distances. “Stress 2” can be expressed as $s^2 = 1 - R^2$, where$R^2$ is squared correlation between fitted values and ordination distances, and so related to the “linear fit” of stressplot.

Function monoMDS can fit several alternative NMDS variants that can be selected with argument model. The default model = "global" fits global NMDS, or Kruskal's (1964a,b) original NMDS similar to isoMDS (MASS). Alternative model = "local" fits local NMDS where independent monotone regression is used for each point (Sibson 1972). Alternative model = "linear" fits a linear MDS. This fits a linear regression instead of monotone, and is not identical to metric scaling or principal coordinates analysis (cmdscale) that performs an eigenvector decomposition of dissimilarities (Gower 1966). Alternative model = "hybrid" implements hybrid MDS that uses monotone regression for all points and linear regression for dissimilarities below or at a threshold dissimilarity in alternating steps (Faith et al. 1987). Function stressplot can be used to display the kind of regression in each model.

Scaling, orientation and direction of the axes is arbitrary. However, the function always centres the axes, and the default scaling is to scale the configuration of unit root mean square and to rotate the axes (argument pc) to principal components so that the first dimension shows the major variation. It is possible to rotate the solution so that the first axis is parallel to a given environmental variable using function MDSrotate.

Convergence Criteria

NMDS is iterative, and the function stops when any of its convergence criteria is met. There is actually no criterion of assured convergence, and any solution can be a local optimum. You should compare several random starts (or use monoMDS via metaMDS) to assess if the solutions is likely a global optimum.

The stopping criteria are:

maxit:: Maximum number of iterations. Reaching this criterion means that solutions was almost certainly not found, and maxit should be increased.
smin:: Minimum stress. If stress is nearly zero, the fit is almost perfect. Usually this means that data set is too small for the requested analysis, and there may be several different solutions that are almost as perfect. You should reduce the number of dimensions (k), get more data (more observations) or use some other method, such as metric scaling (cmdscale, wcmdscale).
sratmax:: Change in stress. Values close to one mean almost unchanged stress. This may mean a solution, but it can also signal stranding on suboptimal solution with flat stress surface.
sfgrmin:: Minimum scale factor. Values close to zero mean almost unchanged configuration. This may mean a solution, but will also happen in local optima.

Value

Returns an object of class "monoMDS". The final scores are returned in item points (function scores extracts these results), and the stress in item stress. In addition, there is a large number of other items (but these may change without notice in the future releases). There are no species scores, but these can be added with sppscores function.

References

Faith, D.P., Minchin, P.R and Belbin, L. 1987. Compositional dissimilarity as a robust measure of ecological distance. Vegetatio 69, 57–68.

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

Kruskal, J.B. 1964a. Multidimensional scaling by optimizing goodness-of-fit to a nonmetric hypothesis. Psychometrika 29, 1–28.

Kruskal, J.B. 1964b. Nonmetric multidimensional scaling: a numerical method. Psychometrika 29, 115–129.

Minchin, P.R. 1987. An evaluation of relative robustness of techniques for ecological ordinations. Vegetatio 69, 89–107.

Sibson, R. 1972. Order invariant methods for data analysis. Journal of the Royal Statistical Society B 34, 311–349.

Author

Peter R. Michin (Fortran core) and Jari Oksanen (R interface).

Note

This is the default NMDS function used in metaMDS. Function metaMDS adds support functions so that NMDS can be run like recommended by Minchin (1987).

Examples

data(dune)
dis <- vegdist(dune)
m <- monoMDS(dis, model = "loc")
m
#> 
#> Call:
#> monoMDS(dist = dis, model = "loc") 
#> 
#> Local non-metric Multidimensional Scaling
#> 
#> 20 points, dissimilarity ‘bray’, call ‘vegdist(x = dune)’
#> 
#> Dimensions: 2 
#> Stress:     0.07596745 
#> Stress type 1, weak ties
#> Scores scaled to unit root mean square, rotated to principal components
#> Stopped after 62 iterations: Stress nearly unchanged (ratio > sratmax)
plot(m)