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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_rand_multivar_students_t (g05ry)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_rand_multivar_students_t (g05ry) sets up a reference vector and generates an array of pseudorandom numbers from a multivariate Student's t distribution with ν degrees of freedom, mean vector a and covariance matrix ν ν-2 C .


[r, state, x, ifail] = g05ry(mode, n, df, xmu, c, r, state, 'm', m, 'lr', lr)
[r, state, x, ifail] = nag_rand_multivar_students_t(mode, n, df, xmu, c, r, state, 'm', m, 'lr', lr)


When the covariance matrix is nonsingular (i.e., strictly positive definite), the distribution has probability density function
fx = Γ ν+m 2 πv m/2 Γ ν/2 C 12 1 + x-aT C-1 x-a ν -ν+m 2  
where m is the number of dimensions, ν is the degrees of freedom, a is the vector of means, x is the vector of positions and ν ν-2 C  is the covariance matrix.
The function returns the value
x = a + νs z  
where z is generated by nag_rand_dist_normal (g05sk) from a Normal distribution with mean zero and covariance matrix C and s is generated by nag_rand_dist_chisq (g05sd) from a χ2-distribution with ν degrees of freedom.
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).


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


Compulsory Input Parameters

1:     mode int64int32nag_int scalar
A code for selecting the operation to be performed by the function.
Set up reference vector only.
Generate variates using reference vector set up in a prior call to nag_rand_multivar_students_t (g05ry).
Set up reference vector and generate variates.
Constraint: mode=0, 1 or 2.
2:     n int64int32nag_int scalar
n, the number of random variates required.
Constraint: n0.
3:     df int64int32nag_int scalar
ν, the number of degrees of freedom of the distribution.
Constraint: df3 .
4:     xmum – double array
a, the vector of means of the distribution.
5:     cldcm – double array
ldc, the first dimension of the array, must satisfy the constraint ldcm.
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:     rlr – double array
If mode=1, the reference vector as set up by nag_rand_multivar_students_t (g05ry) in a previous call with mode=0 or 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: lrm×m+1+2.

Output Parameters

1:     rlr – 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:     xldxm – double array
The array of pseudorandom multivariate Student's t vectors generated by the function, with xij holding the jth dimension for the ith 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:
Constraint: mode=0, 1 or 2.
Constraint: n0.
Constraint: df3.
Constraint: m > 0.
On entry, the covariance matrix C is not positive semidefinite to machine precision.
Constraint: ldcm.
m is not the same as when r was set up in a previous call.
On entry, lr is not large enough, lr=_: minimum length required .
On entry, state vector has been corrupted or not initialized.
Constraint: ldxn.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Not applicable.

Further Comments

The time taken by nag_rand_multivar_students_t (g05ry) is of order nm3.
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=LLT, 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.


This example prints ten pseudorandom observations from a multivariate Student's t-distribution with ten degrees of freedom, means vector
1.0 2.0 -3.0 0.0  
and c 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 ,  
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).
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');

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

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