procrustes {vegan}R Documentation

Procrustes Rotation of Two Configurations and PROTEST


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':
text(x, display = c("target", "rotated"), labels, ...)
## 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 = 999, 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.
labels Character vector of text labels. Rownames of the result object are used as default.
... 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.

See Also

isoMDS, initMDS for obtaining objects for procrustes, and mantel for an alternative to protest without need of dimension reduction.


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)
plot(vare.proc, kind=2)

[Package vegan version 1.16-32 Index]