| procrustes {vegan} | R Documentation |
Function procrustes rotates a configuration to maximum similarity
with another configuration. Function protest tests the
non-randomness (`significance') between two configurations.
procrustes(X, Y, scale = TRUE, symmetric = FALSE, scores = "sites", ...)
## S3 method for class 'procrustes':
summary(object, digits = getOption("digits"), ...)
## S3 method for class 'procrustes':
plot(x, kind=1, choices=c(1,2), xlab, ylab, main,
ar.col = "blue", len=0.05, ...)
## S3 method for class 'procrustes':
points(x, display = c("target", "rotated"), ...)
## S3 method for class 'procrustes':
lines(x, type = c("segments", "arrows"), choices = c(1, 2), ...)
## S3 method for class 'procrustes':
residuals(object, ...)
## S3 method for class 'procrustes':
fitted(object, truemean = TRUE, ...)
protest(X, Y, scores = "sites", permutations = 1000, strata, ...)
X |
Target matrix |
Y |
Matrix to be rotated. |
scale |
Allow scaling of axes of Y. |
symmetric |
Use symmetric Procrustes statistic (the rotation will still be non-symmetric). |
scores |
Kind of scores used. This is the display argument
used with the corresponding scores function: see
scores, scores.cca and
scores.cca for alternatives. |
x, object |
An object of class procrustes. |
digits |
Number of digits in the output. |
kind |
For plot function, the kind of plot produced:
kind = 1 plots shifts in two configurations, kind = 0
draws a corresponding empty plot, and kind = 2
plots an impulse diagram of residuals. |
choices |
Axes (dimensions) plotted. |
xlab, ylab |
Axis labels, if defaults unacceptable. |
main |
Plot title, if default unacceptable. |
display |
Show only the "target" or "rotated"
matrix as points. |
type |
Combine target and rotated points with line
segments or arrows. |
truemean |
Use the original range of target matrix instead of centring the fitted values. |
permutations |
Number of permutation to assess the significance of the symmetric Procrustes statistic. |
strata |
An integer vector or factor specifying the strata for permutation. If supplied, observations are permuted only within the specified strata. |
ar.col |
Arrow colour. |
len |
Width of the arrow head. |
... |
Other parameters passed to functions. In procrustes
and protest parameters are passed to scores, in
graphical functions to underlying graphical functions. |
Procrustes rotation rotates a matrix to maximum similarity with a
target matrix minimizing sum of squared differences. Procrustes
rotation is typically used in comparison of ordination results. It is
particularly useful in comparing alternative solutions in
multidimensional scaling. If scale=FALSE, the function only
rotates matrix Y. If scale=TRUE, it scales linearly
configuration Y for maximum similarity. Since Y is scaled
to fit X, the scaling is non-symmetric. However, with
symmetric=TRUE, the configurations are scaled to equal
dispersions and a symmetric version of the Procrustes statistic
is computed.
Instead of matrix, X and Y can be results from an
ordination from which scores can extract results.
Function procrustes passes extra arguments to
scores, scores.cca etc. so that you can
specify arguments such as scaling.
Function plot plots a procrustes
object and returns invisibly an ordiplot object so that
function identify.ordiplot can be used for identifying
points. The items in the ordiplot object are called
heads and points with kind=1 (ordination diagram)
and sites with kind=2 (residuals). In ordination
diagrams, the arrow heads point to the target configuration, which may
be either logical or illogical. Target and original rotated axes are
shown as cross hairs in two-dimensional Procrustes analysis, and with
a higher number of dimensions, the rotated axes are projected onto
plot with their scaled and centred range. Function
plot passes
parameters to underlying plotting functions. For full control of
plots, you can draw the axes using plot with kind = 0,
and then add items with points or lines. These
functions pass all parameters to the underlying functions so that you
can select the plotting characters, their size, colours etc., or you
can select the width, colour and type of line segments or
arrows, or you can select the orientation and head width of
arrows.
Function residuals returns the pointwise
residuals, and fitted the fitted values, either centred to zero
mean (if truemean=FALSE) or with the original scale (these
hardly make sense if symmetric = TRUE). In
addition, there are summary and print methods.
If matrix X has a lower number of columns than matrix
Y, then matrix X will be filled with zero columns to
match dimensions. This means that the function can be used to rotate
an ordination configuration to an environmental variable (most
practically extracting the result with the fitted function).
Function protest calls procrustes(..., symmetric = TRUE)
repeatedly to estimate the `significance' of the Procrustes
statistic. Function protest uses a correlation-like statistic
derived from the symmetric Procrustes sum of squares ss as
r =sqrt{(1-ss)}, and sometimes called m_{12}. Function
protest has own print method, but otherwise uses
procrustes methods. Thus plot with a protest object
yields a ``Procrustean superimposition plot.''
Function procrustes returns an object of class
procrustes with items. Function protest inherits from
procrustes, but amends that with some new items:
Yrot |
Rotated matrix Y. |
X |
Target matrix. |
ss |
Sum of squared differences between X and Yrot. |
rotation |
Orthogonal rotation matrix. |
translation |
Translation of the origin. |
scale |
Scaling factor. |
symmetric |
Type of ss statistic. |
call |
Function call. |
t0 |
This and the following items are only in class
protest: Procrustes correlation from non-permuted solution. |
t |
Procrustes correlations from permutations. |
signif |
`Significance' of t |
permutations |
Number of permutations. |
strata |
The name of the stratifying variable. |
stratum.values |
Values of the stratifying variable. |
The function protest follows Peres-Neto & Jackson (2001),
but the implementation is still after Mardia et al.
(1979).
Jari Oksanen
Mardia, K.V., Kent, J.T. and Bibby, J.M. (1979). Multivariate Analysis. Academic Press.
Peres-Neto, P.R. and Jackson, D.A. (2001). How well do multivariate data sets match? The advantages of a Procrustean superimposition approach over the Mantel test. Oecologia 129: 169-178.
isoMDS, initMDS for obtaining
objects for procrustes, and mantel for an
alternative to protest without need of dimension reduction.
data(varespec) vare.dist <- vegdist(wisconsin(varespec)) library(MASS) ## isoMDS mds.null <- isoMDS(vare.dist, tol=1e-7) mds.alt <- isoMDS(vare.dist, initMDS(vare.dist), maxit=200, tol=1e-7) vare.proc <- procrustes(mds.alt, mds.null) vare.proc summary(vare.proc) plot(vare.proc) plot(vare.proc, kind=2) residuals(vare.proc)