NAG Library Routine Document

G05PYF

Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

G05PYF generates a random correlation matrix with given eigenvalues.

2 Specification

SUBROUTINE G05PYF ( N, D, EPS, STATE, C, LDC, IFAIL)

INTEGER N, STATE(*), LDC, IFAIL

REAL (KIND=nag_wp) D(N), EPS, C(LDC,N)

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,

G05PYF 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 G05PXF 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 routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05PYF.

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 Parameters

1: N – INTEGERInput

On entry:

n

, the dimension of the correlation matrix to be generated.

Constraint:

N \geq 1

2: D(N) – REAL (KIND=nag_wp) arrayInput

On entry: the

n

eigenvalues,

λ_{i}

, for

i = 1, 2, \dots, n

Constraints:

$D (i) \geq 0.0$ , for $i = 1, 2, \dots, n$ ;
$\sum_{i = 1}^{n} D (i) = n$ to within EPS.

3: EPS – REAL (KIND=nag_wp)Input

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( $*$ ) – INTEGER arrayCommunication Array

Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.

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(LDC,N) – REAL (KIND=nag_wp) arrayOutput

On exit: a random correlation matrix,

C

, of dimension

n

6: LDC – INTEGERInput

On entry: the first dimension of the array C as declared in the (sub)program from which G05PYF is called.

Constraint:

LDC \geq N

7: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: On entry, $N < 1$ .

$IFAIL = 2$: On entry, $D (i) < 0.0$ for some $i$ ,
or $\sum_{i = 1}^{n} D (i) \neq n$ .

$IFAIL = 3$: On entry, $EPS < N \times machine precision$ .

$IFAIL = 4$: On entry, STATE vector was not initialized or has been corrupted.

$IFAIL = 5$: The error in a diagonal element is greater than EPS. The value of EPS should be increased. Otherwise the program could be rerun with a different value used for the seed of the random number generator, see G05KFF or G05KGF.

$IFAIL = 6$: On entry, $LDC < N$ .

7 Accuracy

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

8 Further Comments

The time taken by G05PYF is approximately proportional to

n^{2}

9 Example

Following initialization of the pseudorandom number generator by a call to G05KFF, a

3

3

correlation matrix with eigenvalues of

0.7

0.9

and

1.4

is generated and printed.

NAG Library Routine DocumentG05PYF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Routine Document

G05PYF