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_multivar_students_t (g05ry).

References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

Parameters

Compulsory Input Parameters

1: $mode$ – int64int32nag_int scalar

A code for selecting the operation to be performed by the function.

$mode = 0$: Set up reference vector only.
$mode = 1$: Generate variates using reference vector set up in a prior call to nag_rand_multivar_students_t (g05ry).
$mode = 2$: Set up reference vector and generate variates.

Constraint:

mode = 0

1

2

2: $n$ – int64int32nag_int scalar

n

, the number of random variates required.

Constraint:

n \geq 0

3: $df$ – int64int32nag_int scalar

ν

, the number of degrees of freedom of the distribution.

Constraint:

df \geq 3

4: $xmu (m)$ – double array

a

, the vector of means of the distribution.

5: $c (ldc, m)$ – double array

ldc, the first dimension of the array, must satisfy the constraint

ldc \geq m

Matrix which, along with df, defines the covariance of the distribution. Only the upper triangle need be set.

Constraint: c must be positive semidefinite to machine precision.

6: $r (lr)$ – double array

mode = 1

, the reference vector as set up by nag_rand_multivar_students_t (g05ry) in a previous call with

mode = 0

2

7: $state (:)$ – 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: $m$ – int64int32nag_int scalar: Default: the dimension of the array xmu and the first dimension of the array c and the second dimension of the array c. (An error is raised if these dimensions are not equal.)
$m$ , the number of dimensions of the distribution.

Constraint: $m > 0$ .
2: $lr$ – int64int32nag_int scalar: Default: the dimension of the array r.
The dimension of the array r. if $mode = 1$ , it must be the same as the value of lr specified in the prior call to nag_rand_multivar_students_t (g05ry) with $mode = 0$ or $2$ .

Constraint: $lr \geq m \times (m + 1) + 2$ .

Output Parameters

1: $r (lr)$ – double array: If $mode = 0$ or $2$ , the reference vector that can be used in subsequent calls to nag_rand_multivar_students_t (g05ry) with $mode = 1$ .
2: $state (:)$ – int64int32nag_int array: Contains updated information on the state of the generator.
3: $x (ldx, m)$ – double array: The array of pseudorandom multivariate Student's $t$ vectors generated by the function, with $x (i, j)$ holding the $j$ th dimension for the $i$ th variate.
4: $ifail$ – 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: $mode = 0$ , $1$ or $2$ .

$ifail = 2$: Constraint: $n \geq 0$ .

$ifail = 3$: Constraint: $df \geq 3$ .

$ifail = 4$: Constraint: $m > 0$ .

$ifail = 6$: On entry, the covariance matrix $C$ is not positive semidefinite to machine precision.

$ifail = 7$: Constraint: $ldc \geq m$ .

$ifail = 8$: m is not the same as when r was set up in a previous call.

$ifail = 9$: On entry, lr is not large enough, $lr =_$ : minimum length required .

$ifail = 10$: On entry, state vector has been corrupted or not initialized.

$ifail = 12$: Constraint: $ldx \geq n$ .

$ifail = - 99$: 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

Not applicable.

Further Comments

The time taken by nag_rand_multivar_students_t (g05ry) is of order

n m^{3}

It is recommended that the diagonal elements of

C

should not differ too widely in order of magnitude. This may be achieved by scaling the variables if necessary. The actual matrix decomposed is

C + E = L L^{T}

, where

E

is a diagonal matrix with small positive diagonal elements. This ensures that, even when

C

is singular, or nearly singular, the Cholesky factor

L

corresponds to a positive definite covariance matrix that agrees with

C

within machine precision.

Example

This example prints ten pseudorandom observations from a multivariate Student's

t

-distribution with ten degrees of freedom, means vector

[\begin{matrix} 1.0 \\ 2.0 \\ - 3.0 \\ 0.0 \end{matrix}]

and c matrix

[\begin{matrix} 1.69 & 0.39 & - 1.86 & 0.07 \\ 0.39 & 98.01 & - 7.07 & - 0.71 \\ - 1.86 & - 7.07 & 11.56 & 0.03 \\ 0.07 & - 0.71 & 0.03 & 0.01 \end{matrix}],

generated by nag_rand_multivar_students_t (g05ry). All ten observations are generated by a single call to nag_rand_multivar_students_t (g05ry) with

mode = 2

. The random number generator is initialized by nag_rand_init_repeat (g05kf).

Open in the MATLAB editor: g05ry_example

function g05ry_example


fprintf('g05ry 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);

% Number of variates and degrees of freedom
n  = int64(10);
df = int64(10);

% Distribution means
xmu = [1; 2; -3; 0];

% Upper triangular part of covariance matrix
c = [ 1.69,  0.39, -1.86,  0.07;
      0,    98.01, -7.07, -0.71;
      0,     0,    11.56,  0.03;
      0,     0,     0,     0.01];
m = size(c,1);

% Setup and generate in one go
mode = int64(2);

% Generate variates from a multivariate Student t distrobution
lr = m*(m+1) + 2;
r = zeros(lr, 1);
[r, state, x, ifail] = g05ry( ...
                              mode, n, df, xmu, c, r, state);

disp('Variates from multivariate Student t distribution');
disp(x);

g05ry example results

Variates from multivariate Student t distribution
    1.4957  -15.6226   -3.8101    0.1294
   -1.0827   -6.7473    0.6696   -0.0391
    2.1369    6.3861   -5.7413    0.0140
    2.2481  -16.0417   -1.0982    0.1641
   -0.2550    3.5166   -0.2541   -0.0592
    0.9731   -4.3553   -4.4181    0.0043
    0.7098   -3.4281    1.1741    0.0586
    1.8827   23.2619    1.5140   -0.0704
    0.9904   22.7479    0.1811   -0.0893
    1.5026    2.7753   -2.2805   -0.0112

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

Chapter Contents

Chapter Introduction

NAG Toolbox