NAG Library Function Document

nag_rand_corr_matrix (g05pyc)

+− 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_rand_corr_matrix (g05pyc) generates a random correlation matrix with given eigenvalues.

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_corr_matrix (Integer n, const double d[], double eps, Integer state[], double c[], Integer pdc, NagError *fail)

3 Description

Given

n

eigenvalues,

λ_{1}, λ_{2}, \dots, λ_{n}

, such that

\sum_{i = 1}^{n} λ_{i} = n

and

λ_{i} \geq 0, i = 1, 2, \dots, n,

nag_rand_corr_matrix (g05pyc) will generate a random correlation matrix,

C

, of dimension

n

, with eigenvalues

λ_{1}, λ_{2}, \dots, λ_{n}

The method used is based on that described by Lin and Bendel (1985). Let

D

be the diagonal matrix with values

λ_{1}, λ_{2}, \dots, λ_{n}

and let

A

be a random orthogonal matrix generated by nag_rand_orthog_matrix (g05pxc) then the matrix

C_{0} = A D A^{T}

is a random covariance matrix with eigenvalues

λ_{1}, λ_{2}, \dots, λ_{n}

. The matrix

C_{0}

is transformed into a correlation matrix by means of

n - 1

elementary rotation matrices

P_{i}

such that

C = P_{n - 1} P_{n - 2} \dots P_{1} C_{0} P_{1}^{T} \dots P_{n - 2}^{T} P_{n - 1}^{T}

. The restriction on the sum of eigenvalues implies that for any diagonal element of

C_{0} > 1

, there is another diagonal element

< 1

. The

P_{i}

are constructed from such pairs, chosen at random, to produce a unit diagonal element corresponding to the first element. This is repeated until all diagonal elements are

1

to within a given tolerance

ε

The randomness of

C

should be interpreted only to the extent that

A

is a random orthogonal matrix and

C

is computed from

A

using the

P_{i}

which are chosen as arbitrarily as possible.

One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_corr_matrix (g05pyc).

4 References

Lin S P and Bendel R B (1985) Algorithm AS 213: Generation of population correlation on matrices with specified eigenvalues Appl. Statist. 34 193–198

5 Arguments

1: n – IntegerInput

On entry:

n

, the dimension of the correlation matrix to be generated.

Constraint:

n \geq 1

2: d[n] – const doubleInput

On entry: the

n

eigenvalues,

λ_{i}

, for

i = 1, 2, \dots, n

Constraints:

$d [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n$ ;
$\sum_{i = 1}^{n} d [i - 1] = n$ to within eps.

3: eps – doubleInput

On entry: the maximum acceptable error in the diagonal elements.

Suggested value:

eps = 0.00001

Constraint:

eps \geq n \times machine precision

(see Chapter x02).

4: state[ $\dim$ ] – IntegerCommunication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).

On entry: contains information on the selected base generator and its current state.

On exit: contains updated information on the state of the generator.

5: c[ $n \times pdc$ ] – doubleOutput

On exit: a random correlation matrix,

C

, of dimension

n

6: pdc – IntegerInput

On entry: the stride separating row elements of the matrix

C

in the array c.

Constraint:

pdc \geq n

7: 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_DIAG_ELEMENTS: The diagonals of the returned matrix are not unity, try increasing the value of eps, or rerun the code using a different seed.
NE_EIGVAL_SUM: On entry, the eigenvalues do not sum to n.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
On entry, $pdc = ⟨value⟩$ .
Constraint: $pdc > 0$ .
NE_INT_2: On entry, $pdc = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $pdc \geq n$ .
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_INVALID_STATE: On entry, state vector has been corrupted or not initialized.
NE_NEGATIVE_EIGVAL: On entry, an eigenvalue is negative.
NE_REAL: On entry, $eps = ⟨value⟩$ .
Constraint: $eps \geq n \times machine precision$ .

7 Accuracy

The maximum error in a diagonal element is given by eps.

8 Parallelism and Performance

nag_rand_corr_matrix (g05pyc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_rand_corr_matrix (g05pyc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by nag_rand_corr_matrix (g05pyc) is approximately proportional to

n^{2}

10 Example

Following initialization of the pseudorandom number generator by a call to nag_rand_init_repeatable (g05kfc), a

3

3

correlation matrix with eigenvalues of

0.7

0.9

and

1.4

is generated and printed.

NAG Library Function Documentnag_rand_corr_matrix (g05pyc)

+− 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_rand_corr_matrix (g05pyc)