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.
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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, ...) ## S3 method for class 'monoMDS' scores(x, choices = NA, ...) ## S3 method for class 'monoMDS' plot(x, choices = c(1,2), type = "t", ...)
Starting configuration. A random configuration will be generated if this is missing.
Number of dimensions. NB., the number of points n should be n > 2*k + 1, and preferably higher in non-metric MDS.
Dissimilarity below which linear regression is used alternately with monotone regression.
Maximum number of iterations.
Use primary or weak tie treatment, where equal
observed dissimilarities are allowed to have different fitted
Use stress type 1 or 2 (see Details).
Scale final scores to unit root mean squares.
Rotate final scores to principal components.
Convergence criteria: iterations stop
when stress drops below
Dimensions returned or plotted. The default
The type of the plot:
Other parameters to the functions (ignored in
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
(smacof package) also has 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).
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
stress^2 = sum (d-dhat)^2/ sum (d-dnull)^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,
whereR2 is squared correlation between fitted values and
ordination distances, and so related to the “linear fit” of
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
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 (
that performs an eigenvector decomposition of dissimilarities (Gower
model = "hybrid" implements hybrid MDS
that uses monotone regression for all points and linear regression
for dissimilarities below or at a
in alternating steps (Faith et al. 1987). Function
stressplot can be used to display the kind of
regression in each
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
Returns an object of class
"monoMDS". The final scores
are returned in item
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).
This is the default NMDS function used in
metaMDS adds support
functions so that NMDS can be run like recommended by Minchin
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 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.
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Loading required package: permute Loading required package: lattice This is vegan 2.4-3 Call: monoMDS(dist = dis, model = "loc") Local non-metric Multidimensional Scaling 20 points, dissimilarity 'bray', call 'vegdist(x = dune)' Dimensions: 2 Stress: 0.1134962 Stress type 1, weak ties Scores scaled to unit root mean square, rotated to principal components Stopped after 77 iterations: Stress nearly unchanged (ratio > sratmax)
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