nag_rand_quasi_normal (g05yj) generates a quasi-random sequence from a Normal (Gaussian) distribution. It must be preceded by a call to one of the initialization functions nag_rand_quasi_init (g05yl) or nag_rand_quasi_init_scrambled (g05yn).

Syntax

[quas, iref, ifail] = g05yj(xmean, std, n, iref)

[quas, iref, ifail] = nag_rand_quasi_normal(xmean, std, n, iref)

Description

nag_rand_quasi_normal (g05yj) generates a quasi-random sequence from a Normal distribution by first generating a uniform quasi-random sequence which is then transformed into a Normal sequence using the inverse of the Normal CDF. The type of uniform sequence used depends on the initialization function called and can include the low-discrepancy sequences proposed by Sobol, Faure or Niederreiter. If the initialization function nag_rand_quasi_init_scrambled (g05yn) was used then the underlying uniform sequence is first scrambled prior to being transformed (see Description in nag_rand_quasi_init_scrambled (g05yn) for details).

References

Bratley P and Fox B L (1988) Algorithm 659: implementing Sobol's quasirandom sequence generator ACM Trans. Math. Software 14(1) 88–100

Fox B L (1986) Algorithm 647: implementation and relative efficiency of quasirandom sequence generators ACM Trans. Math. Software 12(4) 362–376

Wichura (1988) Algorithm AS 241: the percentage points of the Normal distribution Appl. Statist. 37 477–484

Parameters

Note: the following variables are used in the parameter descriptions:

$idim = idim$ , the number of dimensions required, see nag_rand_quasi_init (g05yl) or nag_rand_quasi_init_scrambled (g05yn);
$liref = liref$ , the length of iref as supplied to the initialization functions nag_rand_quasi_init (g05yl) or nag_rand_quasi_init_scrambled (g05yn).

Compulsory Input Parameters

1: $xmean (idim)$ – double array: Specifies, for each dimension, the mean of the Normal distribution.
2: $std (idim)$ – double array: Specifies, for each dimension, the standard deviation of the Normal distribution.

Constraint: $std (i) \geq 0.0$ , for $i = 1, 2, \dots, idim$ .
3: $n$ – int64int32nag_int scalar: The number of quasi-random numbers required.

Constraint: $n \geq 0$ and $n + previous number of generated values \leq 2^{31} - 1$ .
4: $iref (liref)$ – int64int32nag_int array: Contains information on the current state of the sequence.

Optional Input Parameters

None.

Output Parameters

1: $quas (n, idim)$ – double array: Contains the n quasi-random numbers of dimension idim
2: $iref (liref)$ – int64int32nag_int array: Contains updated information on the state of the sequence.
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:

$ifail = 1$: On entry, incorrect initialization has been detected.

$ifail = 2$

On entry,

n < 1

$ifail = 3$: A standard deviation is negative.

$ifail = 4$: There have been too many calls to the generator.

$ifail = - 99$: 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

None.

Example

This example calls nag_rand_quasi_init (g05yl) to initialize the generator and then nag_rand_quasi_normal (g05yj) to generate a sequence of five four-dimensional variates.

Open in the MATLAB editor: g05yj_example

function g05yj_example


fprintf('g05yj example results\n\n');

% Initialize the Sobol generator, skipping some variates
iskip = int64(1000);
idim = int64(4);
genid = int64(1);
% Initialize the Sobol generator
[iref, ifail] = g05yl( ...
                       genid,idim,iskip);

% Number of variates
n = int64(5);

% Parameters
xmean = [1; 2; 3; 4];
std   = [1; 1; 1; 1];

% Generate N values for the normal distribution
[quasi, iref, ifail] = g05yj( ...
                              xmean, std, n, iref);

disp('Variates');
disp(quasi);

g05yj example results

Variates
    1.5820    2.2448    0.9154    3.0722
    2.8768    1.6057    3.7341    5.4521
    0.9240    3.0223    2.3828    3.8154
    0.6004    1.9290    1.9355    3.4806
    2.0141    3.9061    3.3680    4.8479

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

Chapter Contents

Chapter Introduction

NAG Toolbox