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_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_matrix_corr (g05py).

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

Parameters

Compulsory Input Parameters

1: $d (n)$ – double array

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.

2: $state (:)$ – int64int32nag_int array

Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).

Contains information on the selected base generator and its current state.

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array d.
$n$ , the dimension of the correlation matrix to be generated.

Constraint: $n \geq 1$ .
2: $eps$ – double scalar: Default: $0.00001$
The maximum acceptable error in the diagonal elements.

Constraint: $eps \geq n \times machine precision$ (see Chapter X02).

Output Parameters

1: $state (:)$ – int64int32nag_int array: Contains updated information on the state of the generator.
2: $c (ldc, n)$ – double array: A random correlation matrix, $C$ , of dimension $n$ .
3: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$: Constraint: $n \geq 1$ .

$ifail = 2$: On entry, an eigenvalue is negative.

On entry, the eigenvalues do not sum to n.

$ifail = 3$: Constraint: $eps \geq n \times machine precision$ .

$ifail = 4$: On entry, state vector has been corrupted or not initialized.

$ifail = 5$: The diagonals of the returned matrix are not unity, try increasing the value of eps, or rerun the code using a different seed.

$ifail = 6$: Constraint: $ldc \geq n$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

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

Further Comments

The time taken by nag_rand_matrix_corr (g05py) is approximately proportional to

n^{2}

Example

Following initialization of the pseudorandom number generator by a call to nag_rand_init_repeat (g05kf), a

3

3

correlation matrix with eigenvalues of

0.7

0.9

and

1.4

is generated and printed.

Open in the MATLAB editor: g05py_example

function g05py_example


fprintf('g05py example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
                        genid, subid, seed);

% Eigenvalues
d = [0.7; 0.9; 1.4];

% Generate the correlation matrix with eigenvalues d
[state, c, ifail] = g05py( ...
                           d, state);

disp('Correlation Matrix');
disp(c);

g05py example results

Correlation Matrix
    1.0000   -0.2549   -0.1004
   -0.2549    1.0000    0.2343
   -0.1004    0.2343    1.0000

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox