vegan (version 1.4-2)

procrustes: Procrustes Rotation of Two Configurations

Description

Rotates a configuration to maximum similarity with another configuration.

Usage

procrustes(X, Y, scale = TRUE, symmetric = FALSE)
## S3 method for class 'procrustes':
summary(object, ...)
## S3 method for class 'procrustes':
plot(x, kind=1, choices=c(1,2), xlab, ylab, main, ...)
## S3 method for class 'procrustes':
residuals(object, ...)
## S3 method for class 'procrustes':
fitted(object, truemean = TRUE, ...)
protest(X, Y, permutations = 1000, strata)

Arguments

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).

x

An object of class procrustes.

object

An object of class procrustes.

kind

The kind of plot produced: kind=1 plots shifts in two configurations 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.

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.

...

Other parameters passed to functions.

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 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.

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 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 form which scores can extract results. Function plot plots a procrustes object and 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.''

References

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.

Examples

data(varespec)
vare.dist <- vegdist(wisconsin(varespec))
library(MASS)  ## isoMDS
library(mva)   ## cmdscale to start isoMDS
mds.null <- isoMDS(vare.dist, tol=1e-7)
## This was a good seed for me: your rng may vary.
set.seed(237)
mds.alt <- isoMDS(vare.dist, initMDS(vare.dist), maxit=200, tol=1e-7)
vare.proc <- procrustes(mds.alt$points, mds.null$points)
vare.proc
summary(vare.proc)
plot(vare.proc)
plot(vare.proc, kind=2)
residuals(vare.proc)
## Reset rng:
rm(.Random.seed)