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.

Usage

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", length=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 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.
to.target: Draw arrows to point to target.
type: The type of plot drawn. In plot, the type can be "points" or "text" to select the marker for the tail of the arrow, or "none" for drawing an empty plot. In lines the type selects either arrows or line segments to connect target and rotated configuration.
truemean: Use the original range of target matrix instead of centring the fitted values. Function plot.procrustes needs truemean = FALSE, and adding graphical items to the plots from the original results may need truemean = TRUE.
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 how, or the number of permutations required, or a permutation matrix where each row gives the permuted indices.
ar.col: Arrow colour.
length: 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 procrustes and protest parameters are passed to scores, in graphical functions to underlying graphical functions.

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 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 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 predict can be used to add new rotated coordinates to the target. The predict function will always translate coordinates to the original non-centred 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 correlation-like statistic derived from the symmetric Procrustes sum of squares \(ss\) as \(r =\sqrt{1-ss}\), and also prints the sum of squares of the symmetric analysis, sometimes called \(m_{12}^2\). 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 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.
xmean: The centroid of the target.
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. The distribution of these correlations can be inspected with permustats function.
signif: Significance of t
permutations: Number of permutations.
control: A list of control values for the permutations as returned by the function how.
control: the list passed to argument control describing the permutation design.

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.

Author

Jari Oksanen

Note

The function protest follows Peres-Neto & Jackson (2001), but the implementation is still after Mardia et al. (1979).

Examples

## IGNORE_RDIFF_BEGIN
data(varespec)
vare.dist <- vegdist(wisconsin(varespec))
mds.null <- monoMDS(vare.dist, y = cmdscale(vare.dist))
mds.alt <- monoMDS(vare.dist)
vare.proc <- procrustes(mds.alt, mds.null)
vare.proc
#> 
#> Call:
#> procrustes(X = mds.alt, Y = mds.null) 
#> 
#> Procrustes sum of squares:
#> 12.17 
#> 
summary(vare.proc)
#> 
#> Call:
#> procrustes(X = mds.alt, Y = mds.null) 
#> 
#> Number of objects: 24    Number of dimensions: 2 
#> 
#> Procrustes sum of squares:  
#>  12.17414 
#> Procrustes root mean squared error: 
#>  0.7122189 
#> Quantiles of Procrustes errors:
#>        Min         1Q     Median         3Q        Max 
#> 0.08310146 0.23201442 0.39936990 0.69726217 2.46689284 
#> 
#> Rotation matrix:
#>            [,1]        [,2]
#> [1,] 0.99973618  0.02296875
#> [2,] 0.02296875 -0.99973618
#> 
#> Translation of averages:
#>               [,1]         [,2]
#> [1,] -2.369052e-17 1.053365e-17
#> 
#> Scaling of target:
#> [1] 0.7019575
#> 
plot(vare.proc)

plot(vare.proc, kind=2)

residuals(vare.proc)
#>         18         15         24         27         23         19         22 
#> 0.08310146 0.41041125 0.45194386 0.71307369 0.30096268 0.20156824 0.22919248 
#>         16         28         13         14         20         25          7 
#> 0.42200459 0.80784709 0.69199166 1.05489472 0.19554776 0.23226762 0.23125479 
#>          5          6          3          4          2          9         12 
#> 0.88831939 0.18779158 0.38832855 0.57436957 2.46689284 0.30775037 0.28591142 
#>         10         11         21 
#> 0.29093369 0.51065673 0.78854129 
## IGNORE_RDIFF_END