NAG Library Function Document
nag_mv_promax (g03bdc)
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
and
denote
by
matrices each representing a set of
points in an
-dimensional space. The
matrix is a matrix of loadings as returned by
nag_mv_orthomax (g03bac), that is following an orthogonal rotation of a loadings matrix
. The target matrix
is calculated as a power transformation of
that preserves the sign of the loadings. Let
and
denote the
th element of matrices
and
. Given a value greater than one for the exponent
:
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:
is found for
in the least squares sense by use of singular value decomposition of the orthogonal loadings
. The ProMax rotation matrix
is then given by
where
is the rotation matrix from an orthogonal transformation, and
is a matrix with the square root of diagonal elements of
on its diagonal and zeros elsewhere.
The ProMax factor pattern matrix
is given by
the inter-factor correlations
are given by
where
; and the factor structure
is given by
Optionally, the rows of target matrix can be scaled by the communalities of loadings.
4
References
None.
5
Arguments
- 1:
– Nag_RotationLoadingInput
-
On entry: indicates how loadings are normalized.
- Rows of are (Kaiser) normalized by the communalities of the loadings.
- Rows are not normalized.
Constraint:
or .
- 2:
– IntegerInput
-
On entry: , the number of points.
Constraint:
.
- 3:
– IntegerInput
-
On entry: , the number of dimensions.
Constraint:
.
- 4:
– const doubleInput
-
Note: the th element of the matrix is stored in .
On entry: the loadings matrix following an orthogonal rotation, .
- 5:
– IntegerInput
-
On entry: the stride separating matrix column elements in the array
x.
Constraint:
.
- 6:
– const doubleInput
-
Note: the th element of the matrix is stored in .
On entry: the orthogonal rotation matrix, .
- 7:
– IntegerInput
-
On entry: the stride separating matrix column elements in the array
ro.
Constraint:
.
- 8:
– doubleInput
-
On entry: , the value of exponent.
Constraint:
.
- 9:
– doubleOutput
-
Note: the th element of the matrix is stored in .
On exit: the factor pattern matrix, .
- 10:
– IntegerInput
-
On entry: the stride separating matrix column elements in the array
fp.
Constraint:
.
- 11:
– doubleOutput
-
Note: the th element of the matrix is stored in .
On exit: the ProMax rotation matrix, .
- 12:
– IntegerInput
-
On entry: the stride separating matrix column elements in the array
r.
Constraint:
.
- 13:
– doubleOutput
-
Note: the th element of the matrix is stored in .
On exit: the matrix of inter-factor correlations, .
- 14:
– IntegerInput
-
On entry: the stride separating matrix column elements in the array
phi.
Constraint:
.
- 15:
– doubleOutput
-
Note: the th element of the matrix is stored in .
On exit: the factor structure matrix, .
- 16:
– IntegerInput
-
On entry: the stride separating matrix column elements in the array
fs.
Constraint:
.
- 17:
– 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, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- 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, .
Constraint: .
- 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.
None.
10
Example
This example reads a loadings matrix and calculates a varimax transformation before calculating , and 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)