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=0.99999, ...)
"scores"(x, choices = NA, ...)
"plot"(x, choices = c(1,2), type = "t", ...)"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.FALSE, then secondary (strong) tie treatment is
used, and tied values are not broken.smin, scale factor of the gradient
drops below sfgrmin, or stress ratio between two iterations
goes over sratmax (but is still $< 1$).monoMDS result.NA
returns all dimensions. "t" for text, "p"
for points, and "n" for none.monoMDS, passed to graphical functions in plot.)."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). monoMDS via
metaMDS) to assess if the solutions is likely a global
optimum. The stopping criteria are:
monoMDS offers the following unique
combination of features:
smacofSym
(smacof package) also has adequate tie treatment.
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,
$dhat$ are the fitted ordination distances, and
$dnull$ are the ordination distances under null model. For
“stress 1” $dnull = 0$, and for “stress 2”
$dnull = dbar$ or mean distances. “Stress 2”
can be expressed as $stress^2 = 1 - R2$,
where$R2$ 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)
or smacofSym (smacof). 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 ot 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.
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.
metaMDS for the vegan way of
running NMDS, and isoMDS and
smacofSym for some alternative implementations
of NMDS. data(dune)
dis <- vegdist(dune)
m <- monoMDS(dis, model = "loc")
m
plot(m)
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