NAG Library Function Document

nag_mv_procustes (g03bcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

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

2 Specification

#include <nag.h>

#include <nagg03.h>

void	nag_mv_procustes (Nag_TransNorm stand, Nag_RotationScale pscale, Integer n, Integer m, double x[], Integer tdx, double y[], Integer tdy, double yhat[], double r[], Integer tdr, double alpha, double rss, double res[], NagError *fail)

3 Description

Let

X

and

Y

n

m

matrices. They can be considered as representing sets of

n

points in an

m

-dimensional space. The

X

matrix may be a matrix of loadings from say factor or canonical variate analysis, and the

Y

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

Y

matrix is considered as the target matrix and the

X

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

X_{c}

and

Y_{c}

, i.e., the matrices will have zero column means. Then the rotation of the translated

X_{c}

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:

X_{c}^{T} Y_{c} = {U D V}^{T},

where

U

and

V

are orthogonal matrices and

D

is a diagonal matrix. The matrix of rotations,

R

, is computed as:

R = {U V}^{T} .

After rotation, a scaling or dilation factor,

α

, may be estimated by least squares. Thus, the final set of points that best match

Y_{c}

is given by:

{\hat{Y}}_{c} = α X_{c} R .

Before rotation, both sets of points may be normalized to have unit sums of squares or the

X

matrix may be normalized to have the same sum of squares as the

Y

matrix. After rotation, the results may be translated to the original

Y

centroid.

The

i

th residual,

r_{i}

, is given by the distance between the point given in the

i

th row of

Y

and the point given in the

i

th row of

\hat{Y}

. The residual sum of squares is also computed.

4 References

Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

Lawley D N and Maxwell A E (1971) Factor Analysis as a Statistical Method (2nd Edition) Butterworths

5 Arguments

1: stand – Nag_TransNormInput

On entry: indicates if translation/normalization is required.

$stand = Nag_NoTransNorm$: No translation or normalization.
$stand = Nag_Orig$: Translation to the origin.
$stand = Nag_OrigCentroid$: Translation to the origin and then to the $Y$ centroid after rotation.
$stand = Nag_Norm$: Unit normalization.
$stand = Nag_OrigNorm$: Translation and normalization.
$stand = Nag_OrigNormCentroid$: Translation and normalization to $Y$ scale, then translation to the $Y$ centroid after rotation.

Constraint:

stand = Nag_NoTransNorm

Nag_Orig

Nag_OrigCentroid

Nag_Norm

Nag_OrigNorm

Nag_OrigNormCentroid

2: pscale – Nag_RotationScaleInput

On entry: indicates if least squares scaling is applied after rotation.

$pscale = Nag_LsqScale$: Scaling is to be applied.
$pscale = Nag_NotLsqScale$: No scaling is applied.

Constraint:

pscale = Nag_LsqScale

Nag_NotLsqScale

3: n – IntegerInput

On entry: the number of points,

n

Constraint:

n \geq 1

4: m – IntegerInput

On entry: the number of dimensions,

m

Constraints:

$m \geq 1$ ;
$m \leq n$ .

5: x[ $n \times tdx$ ] – doubleInput/Output

Note: the

(i, j)

th element of the matrix

X

is stored in

x [(i - 1) \times tdx + j - 1]

On entry: the matrix to be rotated,

X

On exit: if

stand = Nag_NoTransNorm

, x will be unchanged.

stand = Nag_Orig

Nag_OrigCentroid

Nag_OrigNorm

Nag_OrigNormCentroid

, x will be translated to have zero column means.

stand = Nag_Norm

Nag_OrigNorm

, x will be scaled to have unit sum of squares.

stand = Nag_OrigNormCentroid

, x will be scaled to have the same sum of squares as y.

6: tdx – IntegerInput

On entry: the stride separating matrix column elements in the array x.

Constraint:

tdx \geq m

7: y[ $n \times tdy$ ] – doubleInput/Output

Note: the

(i, j)

th element of the matrix

Y

is stored in

y [(i - 1) \times tdy + j - 1]

On entry: the target matrix,

Y

On exit: if

stand = Nag_NoTransNorm

, then y will be unchanged.

stand = Nag_Orig

Nag_OrigNorm

, then y will be translated to have zero column means.

stand = Nag_Norm

Nag_OrigNorm

, then y will be scaled to have unit sum of squares.

stand = Nag_OrigCentroid

Nag_OrigNormCentroid

, then y will be translated and then after rotation, translated back. The output y should be the same as the input y except for rounding errors.

8: tdy – IntegerInput

On entry: the stride separating matrix column elements in the arrays y, yhat.

Constraint:

tdy \geq m

9: yhat[ $n \times tdy$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

yhat [(i - 1) \times tdy + j - 1]

On exit: the fitted matrix,

\hat{Y}

10: r[ $m \times tdr$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix

R

is stored in

r [(i - 1) \times tdr + j - 1]

On exit: the matrix of rotations,

R

, see Section 9.

11: tdr – IntegerInput

On entry: the stride separating matrix column elements in the array r.

Constraint:

tdr \geq m

12: alpha – double *Output

On exit: if

pscale = Nag_LsqScale

the scaling factor,

α

; otherwise alpha is not set.

13: rss – double *Output

On exit: the residual sum of squares.

14: res[n] – doubleOutput

On exit: the residuals,

r_{i}

, for

i = 1, 2, \dots, n

15: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_GT: On entry, $m = ⟨value⟩$ while $n = ⟨value⟩$ . These arguments must satisfy $m \leq n$ .
NE_2_INT_ARG_LT: On entry, $tdr = ⟨value⟩$ while $m = ⟨value⟩$ . These arguments must satisfy $tdr \geq m$ .
On entry, $tdx = ⟨value⟩$ while $m = ⟨value⟩$ . These arguments must satisfy $tdx \geq m$ .
On entry, $tdy = ⟨value⟩$ while $m = ⟨value⟩$ . These arguments must satisfy $tdy \geq m$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument pscale had an illegal value.
On entry, argument stand had an illegal value.
NE_INT_ARG_LT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 1$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
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 contact NAG for assistance.
NE_LSQ_SCAL_ZERO_PTS: The fitted matrix $\hat{Y},$ 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.

7 Accuracy

The accuracy of the calculation of the rotation matrix largely depends upon the singular value decomposition. See the f08 Chapter Introduction for further details.

8 Parallelism and Performance

Not applicable.

9 Further Comments

Note that if the matrix

X_{c}^{T} Y

is not of full rank, then the matrix of rotations,

R

, may not be unique even if there is a unique solution in terms of the rotated matrix,

{\hat{Y}}_{c}

. The matrix

R

may also include reflections as well as pure rotations, see Krzanowski (1990).

If the column dimensions of the

X

and

Y

matrices are not equal, the smaller of the two should be supplemented by columns of zeros. Adding a column of zeros to both

X

and

Y

will have the effect of allowing reflections as well as rotations.

10 Example

Three points representing the vertices of a triangle in two dimensions are input. The points are translated and rotated to match the triangle given by (0,0),(1,0),(0,2) and scaling is applied after rotation. The target matrix and fitted matrix are printed along with additional information.

NAG Library Function Documentnag_mv_procustes (g03bcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_mv_procustes (g03bcc)