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 2.7 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: $m \geq 1$ .

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 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.

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 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_promax (g03bdc)