R: Procrustes Rotation of Two Configurations and PROTEST

procrustes {vegan} | R Documentation

## Procrustes Rotation of Two Configurations and PROTEST

### Description

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", 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 `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.
`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 `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 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 `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.

### Note

The function `protest` follows Peres-Neto & 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.

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

`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

    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
    summary(vare.proc)
    plot(vare.proc)
    plot(vare.proc, kind=2)
    residuals(vare.proc)

[Package vegan version 2.4-3 Index]