NAG Library Function Document

nag_mv_promax (g03bdc)

 Contents

    1  Purpose
    7  Accuracy

1
Purpose

nag_mv_promax (g03bdc) calculates a ProMax rotation, given information following an orthogonal rotation.

2
Specification

#include <nag.h>
#include <nagg03.h>
void  nag_mv_promax (Nag_RotationLoading stand, Integer n, Integer m, const double x[], Integer pdx, const double ro[], Integer pdro, double power, double fp[], Integer pdfp, double r[], Integer pdr, double phi[], Integer pdphi, double fs[], Integer pdfs, NagError *fail)

3
Description

Let X and Y denote n by m matrices each representing a set of n points in an m-dimensional space. The X matrix is a matrix of loadings as returned by nag_mv_orthomax (g03bac), that is following an orthogonal rotation of a loadings matrix Z. The target matrix Y is calculated as a power transformation of X that preserves the sign of the loadings. Let Xij and Yij denote the i,jth element of matrices X and Y. Given a value greater than one for the exponent p:
Yij = δij Xij p ,  
for
The above power transformation tends to increase the difference between high and low values of loadings and is intended to increase the interpretability of a solution.
In the second step a solution of:
XW=Y , X,Y n×m , ​ W m×m ,  
is found for W in the least squares sense by use of singular value decomposition of the orthogonal loadings X. The ProMax rotation matrix R is then given by
R=OW W~ , O, ​ W~ m×m ,  
where O is the rotation matrix from an orthogonal transformation, and W~ is a matrix with the square root of diagonal elements of WTW-1 on its diagonal and zeros elsewhere.
The ProMax factor pattern matrix P is given by
P = X W W~ , P n×m ;  
the inter-factor correlations Φ are given by
Φ= QT Q-1 , Φ m×m ;  
where Q=WW~; and the factor structure S is given by
S=PΦ , S n×m .  
Optionally, the rows of target matrix Y can be scaled by the communalities of loadings.

4
References

None.

5
Arguments

1:     stand Nag_RotationLoadingInput
On entry: indicates how loadings are normalized.
stand=Nag_RoLoadStand
Rows of Y are (Kaiser) normalized by the communalities of the loadings.
stand=Nag_RoLoadNotStand
Rows are not normalized.
Constraint: stand=Nag_RoLoadNotStand or Nag_RoLoadStand.
2:     n IntegerInput
On entry: n, the number of points.
Constraint: nm.
3:     m IntegerInput
On entry: m, the number of dimensions.
Constraint: m1.
4:     x[n×pdx] const doubleInput
Note: the i,jth element of the matrix X is stored in x[i-1×pdx+j-1].
On entry: the loadings matrix following an orthogonal rotation, X.
5:     pdx IntegerInput
On entry: the stride separating matrix column elements in the array x.
Constraint: pdxm.
6:     ro[m×pdro] const doubleInput
Note: the i,jth element of the matrix is stored in ro[i-1×pdro+j-1].
On entry: the orthogonal rotation matrix, O.
7:     pdro IntegerInput
On entry: the stride separating matrix column elements in the array ro.
Constraint: pdrom.
8:     power doubleInput
On entry: p, the value of exponent.
Constraint: power>1.0.
9:     fp[n×pdfp] doubleOutput
Note: the i,jth element of the matrix is stored in fp[i-1×pdfp+j-1].
On exit: the factor pattern matrix, P.
10:   pdfp IntegerInput
On entry: the stride separating matrix column elements in the array fp.
Constraint: pdfpm.
11:   r[m×pdr] doubleOutput
Note: the i,jth element of the matrix R is stored in r[i-1×pdr+j-1].
On exit: the ProMax rotation matrix, R.
12:   pdr IntegerInput
On entry: the stride separating matrix column elements in the array r.
Constraint: pdrm.
13:   phi[m×pdphi] doubleOutput
Note: the i,jth element of the matrix is stored in phi[i-1×pdphi+j-1].
On exit: the matrix of inter-factor correlations, Φ.
14:   pdphi IntegerInput
On entry: the stride separating matrix column elements in the array phi.
Constraint: pdphim.
15:   fs[n×pdfs] doubleOutput
Note: the i,jth element of the matrix is stored in fs[i-1×pdfs+j-1].
On exit: the factor structure matrix, S.
16:   pdfs IntegerInput
On entry: the stride separating matrix column elements in the array fs.
Constraint: pdfsm.
17:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_INT
On entry, m=value.
Constraint: m1.
On entry, pdfp=value.
Constraint: pdfp>0.
On entry, pdfs=value.
Constraint: pdfs>0.
On entry, pdphi=value.
Constraint: pdphi>0.
On entry, pdr=value.
Constraint: pdr>0.
On entry, pdro=value.
Constraint: pdro>0.
On entry, pdx=value.
Constraint: pdx>0.
NE_INT_2
On entry, n=value and m=value.
Constraint: nm.
On entry, pdfp=value and m=value.
Constraint: pdfpm.
On entry, pdfs=value and m=value.
Constraint: pdfsm.
On entry, pdphi=value and m=value.
Constraint: pdphim.
On entry, pdr=value and m=value.
Constraint: pdrm.
On entry, pdro=value and m=value.
Constraint: pdrom.
On entry, pdx=value and m=value.
Constraint: pdxm.
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_REAL_ARG_LE
On entry, power=value.
Constraint: power>1.0.
NE_SVD_FAIL
SVD failed to converge.

7
Accuracy

The calculations are believed to be stable.

8
Parallelism and Performance

nag_mv_promax (g03bdc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_mv_promax (g03bdc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9
Further Comments

None.

10
Example

This example reads a loadings matrix and calculates a varimax transformation before calculating P, R and σ for a ProMax rotation.

10.1
Program Text

Program Text (g03bdce.c)

10.2
Program Data

Program Data (g03bdce.d)

10.3
Program Results

Program Results (g03bdce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017