Function metaMDS performs Nonmetric
Multidimensional Scaling (NMDS), and tries to find a stable solution
using several random starts. In addition, it standardizes the
scaling in the result, so that the configurations are easier to
interpret, and adds species scores to the site ordination. The
metaMDS function does not provide actual NMDS, but it calls
another function for the purpose. Currently monoMDS is
the default choice, and it is also possible to call the
isoMDS (MASS package).
metaMDS(comm, distance = "bray", k = 2, try = 20, trymax = 20,
engine = c("monoMDS", "isoMDS"), autotransform =TRUE,
noshare = (engine == "isoMDS"), wascores = TRUE, expand = TRUE,
trace = 1, plot = FALSE, previous.best, ...)
# S3 method for metaMDS
plot(x, display = c("sites", "species"), choices = c(1, 2),
type = "p", shrink = FALSE, ...)
# S3 method for metaMDS
points(x, display = c("sites", "species"),
choices = c(1,2), shrink = FALSE, select, ...)
# S3 method for metaMDS
text(x, display = c("sites", "species"), labels,
choices = c(1,2), shrink = FALSE, select, ...)
# S3 method for metaMDS
scores(x, display = c("sites", "species"), shrink = FALSE,
choices, tidy = FALSE, ...)
metaMDSdist(comm, distance = "bray", autotransform = TRUE,
noshare = TRUE, trace = 1, commname, zerodist = "ignore",
distfun = vegdist, ...)
metaMDSiter(dist, k = 2, try = 20, trymax = 20, trace = 1, plot = FALSE,
previous.best, engine = "monoMDS", maxit = 200,
parallel = getOption("mc.cores"), ...)
initMDS(x, k=2)
postMDS(X, dist, pc=TRUE, center=TRUE, halfchange, threshold=0.8,
nthreshold=10, plot=FALSE, ...)
metaMDSredist(object, ...)Function metaMDS returns an object of class
metaMDS. The final site ordination is stored in the item
points, and species ordination in the item species,
and the stress in item stress (NB, the scaling of the stress
depends on the engine: isoMDS uses
percents, and monoMDS proportions in the range \(0
\ldots 1\)). The other items store the information on the steps taken
and the items returned by the engine function. The object has
print, plot, points and text methods.
Functions metaMDSdist and metaMDSredist return
vegdist objects. Function initMDS returns a
random configuration which is intended to be used within
isoMDS only. Functions metaMDSiter and
postMDS returns the result of NMDS with updated
configuration.
Community data. Alternatively, dissimilarities either as
a dist structure or as a symmetric square matrix.
In the latter case all other stages are skipped except random
starts and centring and pc rotation of axes.
Dissimilarity index used in vegdist.
Number of dimensions. NB., the number of points \(n\) should be \(n > 2k + 1\), and preferably higher in global non-metric MDS, and still higher in local NMDS.
Minimum and maximum numbers of random starts in
search of stable solution. After try has been reached, the
iteration will stop when similar solutions were repeated or
trymax was reached.
The function used for MDS. The default is to use the
monoMDS function in vegan, but for backward
compatibility it is also possible to use isoMDS of
MASS.
Use simple heuristics for possible data
transformation of typical community data (see below). If you do
not have community data, you should probably set
autotransform = FALSE.
Triggering of calculation step-across or extended
dissimilarities with function stepacross. The
argument can be logical or a numerical value greater than zero
and less than one. If TRUE, extended dissimilarities are
used always when there are no shared species between some sites,
if FALSE, they are never used. If noshare is a
numerical value, stepacross is used when the
proportion of site pairs with no shared species exceeds
noshare. The number of pairs with no shared species is
found with no.shared function, and noshare
has no effect if input data were dissimilarities instead of
community data.
Calculate species scores using function
wascores.
Expand weighted averages of species in
wascores.
Trace the function; trace = 2 or higher will be
more voluminous.
Graphical tracing: plot interim results. You may want to set
par(ask = TRUE) with this option.
Start searches from a previous solution.
metaMDS result (or a dissimilarity structure for
initMDS).
Axes shown.
Plot type: "p" for points, "t" for text, and
"n" for axes only.
Display "sites" or "species".
Shrink back species scores if they were expanded originally.
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.
Optional test to be used instead of row names.
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.
Configuration from multidimensional scaling.
The name of comm: should not be given if the
function is called directly.
Handling of zero dissimilarities: either
"fail" or "add" a small positive value, or
"ignore". monoMDS accepts zero dissimilarities
and the default is zerodist = "ignore", but with
isoMDS you may need to set zerodist = "add".
Dissimilarity function. Any function returning a
dist object and accepting argument method can be used
(but some extra arguments may cause name conflicts).
Maximum number of iterations in the single NMDS run;
passed to the engine function monoMDS or
isoMDS.
Number of parallel processes or a predefined socket
cluster. If you use pre-defined socket clusters (say,
clus), you must issue clusterEvalQ(clus,
library(vegan)) to make available internal vegan
functions. With parallel = 1 uses ordinary, non-parallel
processing. The parallel processing is done with parallel
package.
Dissimilarity matrix used in multidimensional scaling.
Rotate to principal components.
Centre the configuration.
Scale axes to half-change units. This defaults
TRUE when dissimilarities are known to have a theoretical
maximum value (ceiling). Function vegdist will have that
information in attribute maxdist, and for other distfun
this is interpreted in a simple test (that can fail), and the
information may not available when input data are distances. If
FALSE, the ordination dissimilarities are scaled to the same
range as the input dissimilarities.
Largest dissimilarity used in half-change scaling. If
dissimilarities have a known (or inferred) ceiling, threshold
is relative to that ceiling (see halfchange).
Minimum number of points in half-change scaling.
A result object from metaMDS.
Other parameters passed to functions. Function
metaMDS passes all arguments to its component functions
metaMDSdist, metaMDSiter, postMDS, and to
distfun and engine.
Non-linear optimization is a hard task, and the best possible solution
(“global optimum”) may not be found from a random starting
configuration. Most software solve this by starting from the result of
metric scaling (cmdscale). This will probably give a
good result, but not necessarily the “global
optimum”. Vegan does the same, but metaMDS tries to
verify or improve this first solution (“try 0”) using several
random starts and seeing if the result can be repeated or improved and
the improved solution repeated. If this does not succeed, you get a
message that the result could not be repeated. However, the result
will be at least as good as the usual standard strategy of starting
from metric scaling or it may be improved. You may not need to do
anything after such a message, but you can be satisfied with the
result. If you want to be sure that you probably have a “global
optimum” you may try the following instructions.
With default engine = "monoMDS" the function will
tabulate the stopping criteria used, so that you can see which
criterion should be made more stringent. The criteria can be given
as arguments to metaMDS and their current values are
described in monoMDS. In particular, if you reach
the maximum number of iterations, you should increase the value of
maxit. You may ask for a larger number of random starts
without losing the old ones giving the previous solution in
argument previous.best.
In addition to slack convergence criteria and too low number
of random starts, wrong number of dimensions (argument k)
is the most common reason for not being able to repeat similar
solutions. NMDS is usually run with a low number dimensions
(k=2 or k=3), and for complex data increasing
k by one may help. If you run NMDS with much higher number
of dimensions (say, k=10 or more), you should reconsider
what you are doing and drastically reduce k. For very
heterogeneous data sets with partial disjunctions, it may help to
set stepacross, but for most data sets the default
weakties = TRUE is sufficient.
Please note that you can give all arguments of other
metaMDS* functions and NMDS engine (default
monoMDS) in your metaMDS command,and you
should check documentation of these functions for details.
NMDS is often misunderstood and wrong claims of its properties are common on the Web and even in publications. It is often claimed that the NMDS configuration is non-metric which means that you cannot fit environmental variables or species onto that space. This is a false statement. In fact, the result configuration of NMDS is metric, and it can be used like any other ordination result. In NMDS the rank orders of Euclidean distances among points in ordination have a non-metric monotone relationship to any observed dissimilarities. The transfer function from observed dissimilarities to ordination distances is non-metric (Kruskal 1964a, 1964b), but the ordination result configuration is metric and observed dissimilarities can be of any kind (metric or non-metric).
The ordination configuration is usually rotated to principal
components in metaMDS. The rotation is performed after
finding the result, and it only changes the direction of the
reference axes. The only important feature in the NMDS solution are
the ordination distances, and these do not change in
rotation. Similarly, the rank order of distances does not change in
uniform scaling or centring of configuration of points. You can also
rotate the NMDS solution to external environmental variables with
MDSrotate. This rotation will also only change the
orientation of axes, but will not change the configuration of points
or distances between points in ordination space.
Function stressplot displays the method graphically:
it plots the observed dissimilarities against distances in
ordination space, and also shows the non-metric monotone
regression.
Jari Oksanen
metaMDS uses monoMDS as its
NMDS engine from vegan version 2.0-0, when it replaced
the isoMDS function. You can set argument
engine to select the old engine.
Non-metric Multidimensional Scaling (NMDS) is commonly
regarded as the most robust unconstrained ordination method in
community ecology (Minchin 1987). Function metaMDS is a
wrapper function that calls several other functions to combine
Minchin's (1987) recommendations into one command. The complete
steps in metaMDS are:
Transformation: If the data values are larger than common
abundance class scales, the function performs a Wisconsin double
standardization (wisconsin). If the values look
very large, the function also performs sqrt
transformation. Both of these standardizations are generally found
to improve the results. However, the limits are completely
arbitrary (at present, data maximum 50 triggers sqrt
and \(>9\) triggers wisconsin). If you want to
have a full control of the analysis, you should set
autotransform = FALSE and standardize and transform data
independently. The autotransform is intended for community
data, and for other data types, you should set
autotransform = FALSE. This step is perfomed using
metaMDSdist, and the step is skipped if input were
dissimilarities.
Choice of dissimilarity: For a good result, you should use
dissimilarity indices that have a good rank order relation to
ordering sites along gradients (Faith et al. 1987). The default
is Bray-Curtis dissimilarity, because it often is the test
winner. However, any other dissimilarity index in
vegdist can be used. Function
rankindex can be used for finding the test winner
for you data and gradients. The default choice may be bad if you
analyse other than community data, and you should probably select
an appropriate index using argument distance. This step is
performed using metaMDSdist, and the step is skipped if
input were dissimilarities.
Step-across dissimilarities: Ordination may be very difficult
if a large proportion of sites have no shared species. In this
case, the results may be improved with stepacross
dissimilarities, or flexible shortest paths among all sites. The
default NMDS engine is monoMDS which is able
to break tied values at the maximum dissimilarity, and this often
is sufficient to handle cases with no shared species, and
therefore the default is not to use stepacross with
monoMDS. Function isoMDS does
not handle tied values adequately, and therefore the default is to
use stepacross always when there are sites with no
shared species with engine = "isoMDS". The
stepacross is triggered by option noshare. If
you do not like manipulation of original distances, you should set
noshare = FALSE. This step is skipped if input data were
dissimilarities instead of community data. This step is performed
using metaMDSdist, and the step is skipped always when
input were dissimilarities.
NMDS with random starts: NMDS easily gets trapped into local
optima, and you must start NMDS several times from random starts
to be confident that you have found the global solution. The
strategy in metaMDS is to first run NMDS starting with the
metric scaling (cmdscale which usually finds a good
solution but often close to a local optimum), or use the
previous.best solution if supplied, and take its solution
as the standard (Run 0). Then metaMDS starts NMDS
from several random starts (minimum number is given by try
and maximum number by trymax). These random starts are
generated by initMDS. If a solution is better (has a lower
stress) than the previous standard, it is taken as the new
standard. If the solution is better or close to a standard,
metaMDS compares two solutions using Procrustes analysis
(function procrustes with option
symmetric = TRUE). If the solutions are very similar in their
Procrustes rmse and the largest residual is very small, the
solutions are regarded as repeated and the better one is taken
as the new standard. The conditions are stringent, and you may
have found good and relatively similar solutions although the
function is not yet satisfied. Setting trace = TRUE will
monitor the final stresses, and plot = TRUE will display
Procrustes overlay plots from each comparison. This step is
performed using metaMDSiter. This is the first step
performed if input data (comm) were dissimilarities. Random
starts can be run with parallel processing (argument
parallel).
Scaling of the results: metaMDS will run postMDS
for the final result. 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 the axes; Principal components rotate the configuration
so that the variance of points is maximized on first dimension
(with function MDSrotate you can alternatively
rotate the configuration so that the first axis is parallel to an
environmental variable); 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 set by argument threshold). If there are
enough points below this threshold (controlled by 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 \(0 \ldots
1\), such as Kulczynski, Bray-Curtis and Canberra indices. If
half-change scaling is not used, the ordination is scaled to the
same range as the original dissimilarities. Half-change scaling is
skipped by default if input were dissimilarities, but can be
turned on with argument halfchange = TRUE. NB., The PC
rotation only changes the directions of reference axes, and it
does not influence the configuration or solution in general.
Species scores: Function adds the species scores to the final
solution as weighted averages using function
wascores with given value of parameter
expand. The expansion of weighted averages can be undone
with shrink = TRUE in plot or scores
functions, and the calculation of species scores can be suppressed
with wascores = FALSE. This step is skipped if input were
dissimilarities and community data were unavailable. However, the
species scores can be added or replaced with
sppscores.
Faith, D. P, Minchin, P. R. and Belbin, L. (1987). Compositional dissimilarity as a robust measure of ecological distance. Vegetatio 69, 57--68.
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.
monoMDS (and isoMDS),
decostand, wisconsin,
vegdist, rankindex, stepacross,
procrustes, wascores, sppscores,
MDSrotate, ordiplot, stressplot.
## The recommended way of running NMDS (Minchin 1987)
##
data(dune)
## IGNORE_RDIFF_BEGIN
## Global NMDS using monoMDS
sol <- metaMDS(dune)
sol
plot(sol, type="t")
## Start from previous best solution
sol <- metaMDS(dune, previous.best = sol)
## Local NMDS and stress 2 of monoMDS
sol2 <- metaMDS(dune, model = "local", stress=2)
sol2
## Use Arrhenius exponent 'z' as a binary dissimilarity measure
sol <- metaMDS(dune, distfun = betadiver, distance = "z")
sol
## IGNORE_RDIFF_END
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