ROTATE directive

Does a Procrustes rotation of one configuration of points to fit another.


Options

PRINT = strings
Printed output required (rotations, coordinates, residuals, sums); default * i.e. no printing

SCALING = string
Whether or not isotropic scaling is allowed (yes, no); default no

STANDARDIZE = strings
Whether to centre the configurations (at the origin), and/or to normalize them (to unit sum of squares) prior to rotation (centre, normalize); default cent,norm

SUPPRESS = string
Whether to suppress reflection (yes, no); default no


Parameters

XINPUT = matrices
Inputs the fixed configuration

YINPUT = matrices
Inputs the configuration to be fitted

XOUTPUT = matrices
To store the (standardized) fixed configuration

YOUTPUT = matrices
To store the fitted configuration

ROTATION = matrices
To store the rotation matrix

RESIDUALS = matrices or variates
To store distances between the (standardized) fixed and fitted configurations

RSS = scalars
To store the residual sum of squares


Description

The ROTATE directive provides orthogonal Procrustes rotation. You must set the parameters XINPUT and YINPUT, which specify respectively the fixed configuration and the configuration that you want to be translated and rotated; these are called X and Y above. The other parameters are used for saving results from the analysis. For X and Y to refer to the same set of objects they must have the same number of rows, and each object must be represented by the same row in both X and Y. If the XINPUT matrix is n×p and the YINPUT matrix is n×q, GenStat does the analysis using matrices that are n×r, where r is max(p, q). The smaller matrix is expanded with columns of zeros, as explained above.

   The PRINT option specifies which results you want to print; the settings are as follows.

    coordinates
specifies that the fixed and fitted configurations are to be printed; note that the fixed configuration is printed after any standardization (see below), and the fitted configuration is printed after standardization and rotation.

    residuals
prints the residual distances of the points in the fixed configuration from the fitted points; this is after any standardization and rotation.

    rotations
prints the orthogonal rotation matrix.

    sums
prints an analysis of variance giving the sums of squares of each configuration, and the residual sum of squares; if scaling is used, the scaling factor is also printed.

The three other options of the ROTATE directive control the form of analysis. The SCALING option specifies whether you want least-squares scaling to be applied to the standardized YINPUT matrix when finding the best fit to the fixed configuration. You should set SCALING=yes if you want scaling; GenStat will then print the least-squares scaling factor with the analysis of variance. By default there is no scaling.

   The STANDARDIZE option specifies what preliminary standardization is to be applied to the XINPUT and YINPUT matrices. It has settings:

    centre
centre the matrices to have zero column means;

    normalize
normalize the matrices to unit sums of squares.

The default is STANDARDIZE=centre,normalize. The initial centring ensures that the configurations are translated to have a common centroid, and thus automatically provides the best translation of Y to match X. The normalization arranges that the residual sum of squares from rotating X to Y is the same as that for rotating Y to X. Switching off both centring and standardization is rarely advisable, but can be requested by putting STANDARDIZE=*.

   With some methods of multivariate analysis, for example the analysis of skew-symmetry, the direction of travel about the origin is important. It is then undesirable to perform a reflection as part of the rotation: the SUPPRESS option can be used to prevent this. The default setting is no, which allows reflection to take place.

 

Options: PRINT, SCALING, STANDARDIZE, SUPPRESS.

Parameters: XINPUT, YINPUT, XOUTPUT, YOUTPUT, ROTATION, RESIDUALS, RSS.