Procrustes Rotation of Two Configurations and PROTEST
Description
Function procrustes
rotates a configuration to maximum similarity
with another configuration. Function protest
tests the
nonrandomness (‘significance’) between two configurations.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23  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), to.target = TRUE,
type = "p", xlab, ylab, main, ar.col = "blue", len=0.05,
cex = 0.7, ...)
## S3 method for class 'procrustes'
points(x, display = c("target", "rotated"),
choices = c(1,2), truemean = FALSE, ...)
## S3 method for class 'procrustes'
text(x, display = c("target", "rotated"),
choices = c(1,2), labels, truemean = FALSE, ...)
## S3 method for class 'procrustes'
lines(x, type = c("segments", "arrows"),
choices = c(1, 2), truemean = FALSE, ...)
## S3 method for class 'procrustes'
residuals(object, ...)
## S3 method for class 'procrustes'
fitted(object, truemean = TRUE, ...)
## S3 method for class 'procrustes'
predict(object, newdata, truemean = TRUE, ...)
protest(X, Y, scores = "sites", permutations = how(nperm = 999), ...)

Arguments
X 
Target matrix 
Y 
Matrix to be rotated. 
scale 
Allow scaling of axes of 
symmetric 
Use symmetric Procrustes statistic (the rotation will still be nonsymmetric). 
scores 
Kind of scores used. This is the 
x, object 
An object of class 
digits 
Number of digits in the output. 
kind 
For 
choices 
Axes (dimensions) plotted. 
xlab, ylab 
Axis labels, if defaults unacceptable. 
main 
Plot title, if default unacceptable. 
display 
Show only the 
to.target 
Draw arrows to point to target. 
type 
The type of plot drawn. In 
truemean 
Use the original range of target matrix instead of
centring the fitted values. Function 
newdata 
Matrix of coordinates to be rotated and translated to the target. 
permutations 
a list of control values for the permutations
as returned by the function 
ar.col 
Arrow colour. 
len 
Width of the arrow head. 
labels 
Character vector of text labels. Rownames of the result object are used as default. 
cex 
Character expansion for points or text. 
... 
Other parameters passed to functions. In 
Details
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 nonsymmetric. 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 if to.target = TRUE
, and to rotated
configuration if to.target = FALSE
. Target and original
rotated axes are shown as cross hairs in twodimensional 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 predict
can be used to add new rotated
coordinates to the target. The predict
function will always
translate coordinates to the original noncentred matrix. The
function cannot be used with newdata
for symmetric
analysis.
Function protest
performs symmetric Procrustes analysis
repeatedly to estimate the ‘significance’ of the Procrustes
statistic. Function protest
uses a correlationlike statistic
derived from the symmetric Procrustes sum of squares ss as
r = sqrt(1ss), and also prints the sum of
squares of the symmetric analysis, sometimes called
squared m12. Function protest
has own
print
method, but otherwise uses procrustes
methods. Thus plot
with a protest
object yields a
“Procrustean superimposition plot.”
Value
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 
X 
Target matrix. 
ss 
Sum of squared differences between 
rotation 
Orthogonal rotation matrix. 
translation 
Translation of the origin. 
scale 
Scaling factor. 
xmean 
The centroid of the target. 
symmetric 
Type of 
call 
Function call. 
t0 
This and the following items are only in class

t 
Procrustes correlations from permutations. The distribution
of these correlations can be inspected with 
signif 
‘Significance’ of 
permutations 
Number of permutations. 
control 
A list of control values for the permutations
as returned by the function 
control 
the list passed to argument 
Note
The function protest
follows PeresNeto & Jackson (2001),
but the implementation is still after Mardia et al.
(1979).
Author(s)
Jari Oksanen
References
Mardia, K.V., Kent, J.T. and Bibby, J.M. (1979). Multivariate Analysis. Academic Press.
PeresNeto, 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: 169178.
See Also
monoMDS
, for obtaining
objects for procrustes
, and mantel
for an
alternative to protest
without need of dimension reduction. See
how
for details on specifying the type of
permutation required.
Examples
1 2 3 4 5 6 7 8 9 10 