NAG Library Function Document

nag_mv_promax (g03bdc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

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

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

X_{i j}

and

Y_{i j}

denote the

(i, j)

th element of matrices

X

and

Y

. Given a value greater than one for the exponent

p

Y_{i j} = δ_{i j} {‖X_{i j}‖}^{p},

for

$i = 1, 2, \dots, n$ ;
$j = 1, 2, \dots, m$ ;
$δ_{i j} = \{\begin{matrix} - 1, if X_{i j} < 0; \\ 1, otherwise. \end{matrix}$

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:

X W = Y, X, Y \in ℝ^{n \times m}, ​ W \in ℝ^{m \times 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 = O W \tilde{W}, O, ​ \tilde{W} \in ℝ^{m \times m},

where

O

is the rotation matrix from an orthogonal transformation, and

\tilde{W}

is a matrix with the square root of diagonal elements of

{(W^{T} W)}^{- 1}

on its diagonal and zeros elsewhere.

The ProMax factor pattern matrix

P

is given by

P = X W \tilde{W}, P \in ℝ^{n \times m};

the inter-factor correlations

Φ

are given by

Φ = {(Q^{T} Q)}^{- 1}, Φ \in ℝ^{m \times m};

where

Q = W \tilde{W}

; and the factor structure

S

is given by

S = P Φ, S \in ℝ^{n \times 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

Nag_RoLoadStand

2: n – IntegerInput

On entry:

n

the number of points.

Constraint:

n \geq m

3: m – IntegerInput

On entry:

m

, the number of dimensions.

Constraint:

m \geq 1

4: x[ $n \times pdx$ ] – const doubleInput

Note: the

(i, j)

th element of the matrix

X

is stored in

x [(i - 1) \times 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:

pdx \geq m

6: ro[ $m \times pdro$ ] – const doubleInput

Note: the

(i, j)

th element of the matrix is stored in

ro [(i - 1) \times 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:

pdro \geq m

8: power – doubleInput

On entry:

p

, the value of exponent.

Constraint:

power > 1.0

9: fp[ $n \times pdfp$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

fp [(i - 1) \times 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:

pdfp \geq m

11: r[ $m \times pdr$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix

R

is stored in

r [(i - 1) \times 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:

pdr \geq m

13: phi[ $m \times pdphi$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

phi [(i - 1) \times 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:

pdphi \geq m

15: fs[ $n \times pdfs$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix is stored in

fs [(i - 1) \times 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:

pdfs \geq m

17: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .; 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: $n \geq m$ .; On entry, $pdfp = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdfp \geq m$ .; On entry, $pdfs = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdfs \geq m$ .; On entry, $pdphi = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdphi \geq m$ .; On entry, $pdr = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdr \geq m$ .; On entry, $pdro = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdro \geq m$ .; On entry, $pdx = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdx \geq m$ .
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 Further Comments

None.

9 Example

This example reads a loadings matrix and calculates a varimax transformation before calculating

P

R

and

σ

for a ProMax rotation.

NAG Library Function Documentnag_mv_promax (g03bdc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Function Document

nag_mv_promax (g03bdc)