PCOPROCRUSTES procedure

Performs a multiple Procrustes analysis (P.G.N. Digby).


Options

PROTATE = strings
Printed output required from each Procrustes rotation (rotations, coordinates, residuals, sums); default * i.e. no output

PPCO = strings
Printed output required from the PCO analysis (roots, scores, centroid); default root, score, cent

SCALING = string
Whether isotropic scaling should be used for the Procrustes rotations (no, yes); default no

STANDARDIZE = strings
Whether to centre the configurations and/or normalize them to unit sums-of-squares for the Procrustes rotations (centre, normalize); default cent, norm


Parameters

DATA = pointers
Each pointer points to a set of matrices holding the original input configurations

LRV = LRVs
Stores the latent vectors (i.e. coordinates), roots and trace from the PCO analysis

CENTROID = diagonal matrices
Stores the squared distances of the points representing the input configurations from their overall centroid from the PCO analysis

DISTANCES = symmetric matrices
Stores the residual sums-of-squares from the Procrustes rotations


Description

An N × V matrix represents a configuration of points, for each of N units, in V dimensions. Given a set of M such matrices, a multiple Procrustes analysis compares them in pairs, keeping the residual sums-of-squares, and performs a principal coordinate analysis of the residual sums-of-squares to obtain an ordination representing the individual configurations. The rows of the matrices must represent the same set of units, in the same order; however there is no need for them to have the same number of columns (although generally they will do). An example of the use of multiple Procrustes analysis is given by Digby & Kempton (1987, pages 121-3).

   The configurations of points are specified using the DATA parameter. This supplies a pointer containing a matrix with the data for each configuration. The PROTATE option controls the output from the individual Procrustes rotations, and the PPCO option controls that from the principal coordinate analysis. There are M×(M-1)/2 Procrustes rotations so, by default, PROTATE=* to suppress any output. The SCALING and STANDARDIZE options control the way in which the Procrustes rotations are carried out, using the SCALING and STANDARDIZE options of ROTATE. However, the combination of SCALING=yes and STANDARDIZE=centre should not be used, because then the results will be dependent on the order of the input matrices.

   The LRV and CENTROID parameters can be used to save results from the principal coordinates analysis, and the DISTANCES parameter can be used to save the symmetric matrix of the residual sums-of-squares from the Procrustes analyses.


Options: PROTATE, PPCO, SCALING, STANDARDIZE.

Parameters: DATA, LRV, CENTROID, DISTANCES.


Method

The pairwise Procrustes rotations are performed using the ROTATE directive, and the residual sums-of-squares are stored in a symmetric matrix of order M. This matrix is then used as input to a principal coordinate analysis, performed using the PCO directive on a suitably transformed copy of the matrix.


Reference

Digby, P.G.N. & Kempton, R.A (1987). Multivariate Analysis of Ecological Communities. Chapman & Hall, London.