nag_mv_procustes (g03bcc) computes Procrustes rotations in which an orthogonal rotation is found so that a transformed matrix best matches a target matrix.

Let  and  be  by  matrices. They can be considered as representing sets of  points in an -dimensional space. The  matrix may be a matrix of loadings from say factor or canonical variate analysis, and the  matrix may be a postulated pattern matrix or the loadings from a different sample. The problem is to relate the two sets of points without disturbing the relationships between the points in each set. This can be achieved by translating, rotating and scaling the sets of points. The  matrix is considered as the target matrix and the  matrix is rotated to match that matrix.

First the two sets of points are translated so that their centroids are at the origin to give  and , i.e., the matrices will have zero column means. Then the rotation of the translated  matrix which minimizes the sum of squared distances between corresponding points in the two sets is found. This is computed from the singular value decomposition of the matrix:

where  and  are orthogonal matrices and  is a diagonal matrix. The matrix of rotations, , is computed as:

After rotation, a scaling or dilation factor, , may be estimated by least-squares. Thus, the final set of points that best match  is given by:

Before rotation, both sets of points may be normalized to have unit sums of squares or the  matrix may be normalized to have the same sum of squares as the  matrix. After rotation, the results may be translated to the original  centroid.

The th residual, , is given by the distance between the point given in the th row of  and the point given in the th row of . The residual sum of squares is also computed.

"NE_2_INT_ARG_GT"

On entry,  while . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument stand had an illegal value.

"NE_INT_ARG_LT"

On entry, n must not be less than 1: .

"NE_INTERNAL_ERROR"

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.

"NE_LSQ_SCAL_ZERO_PTS"

The fitted matrix  contains only zero-points when least-squares scaling is applied.

"NE_NORM_ZERO_PTS"

On entry, either x or y contains only zero-points (possibly after translation) when normalization is to be applied.

"NE_SVD_NOT_CONV"

The singular value decomposition has failed to converge. This is an unlikely error exit.

The accuracy of the calculation of the rotation matrix largely depends upon the singular value decomposition. See f02wec (nag_real_svd) for further details.

Note that if the matrix  is not of full rank, then the matrix of rotations, , may not be unique even if there is a unique solution in terms of the rotated matrix, . The matrix  may also include reflections as well as pure rotations, see Krzanowski (1990).

If the column dimensions of the  and  matrices are not equal, the smaller of the two should be supplemented by columns of zeros. Adding a column of zeros to both  and  will have the effect of allowing reflections as well as rotations.