monoMDS  R Documentation 
Function implements Kruskal's (1964a,b) nonmetric 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.
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 = 1e4, sfgrmin = 1e7,
sratmax=0.999999, ...)
## S3 method for class 'monoMDS'
scores(x, choices = NA, ...)
## S3 method for class 'monoMDS'
plot(x, 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, ...)
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 
model 
MDS model: 
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 
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 
x 
A 
choices 
Dimensions returned or plotted. The default 
type 
The type of the plot: 
select 
Items to be displayed. This can either be a logical
vector which is 
labels 
Labels to be use used instead of row names. 
... 
Other parameters to the functions (ignored in

There are several versions of nonmetric 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
,
whereR^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
.
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).
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.
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).
Peter R. Michin (Fortran core) and Jari Oksanen (R interface).
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 goodnessoffit 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.
metaMDS
for the vegan way of running NMDS,
and isoMDS
and smacof for some
alternative implementations of NMDS.
data(dune)
dis < vegdist(dune)
m < monoMDS(dis, model = "loc")
m
plot(m)
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