nag_rand_matrix_multi_students_t (g05ryc) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_rand_matrix_multi_students_t (g05ryc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_rand_matrix_multi_students_t (g05ryc) 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 .

2  Specification

#include <nag.h>
#include <nagg05.h>
void  nag_rand_matrix_multi_students_t (Nag_OrderType order, Nag_ModeRNG mode, Integer n, Integer df, Integer m, const double xmu[], const double c[], Integer pdc, double r[], Integer lr, Integer state[], double x[], Integer pdx, NagError *fail)

3  Description

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_normal (g05skc) from a Normal distribution with mean zero and covariance matrix C and s is generated by nag_rand_chi_sq (g05sdc) from a χ2-distribution with ν degrees of freedom.
One of the initialization functions nag_rand_init_repeatable (g05kfc) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeatable (g05kgc) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_matrix_multi_students_t (g05ryc).

4  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

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     modeNag_ModeRNGInput
On entry: a code for selecting the operation to be performed by the function.
mode=Nag_InitializeReference
Set up reference vector only.
mode=Nag_GenerateFromReference
Generate variates using reference vector set up in a prior call to nag_rand_matrix_multi_students_t (g05ryc).
mode=Nag_InitializeAndGenerate
Set up reference vector and generate variates.
Constraint: mode=Nag_InitializeReference, Nag_GenerateFromReference or Nag_InitializeAndGenerate.
3:     nIntegerInput
On entry: n, the number of random variates required.
Constraint: n0.
4:     dfIntegerInput
On entry: ν, the number of degrees of freedom of the distribution.
Constraint: df3 .
5:     mIntegerInput
On entry: m, the number of dimensions of the distribution.
Constraint: m>0.
6:     xmu[m]const doubleInput
On entry: a, the vector of means of the distribution.
7:     c[dim]const doubleInput
Note: the dimension, dim, of the array c must be at least pdc×m.
The i,jth element of the matrix C is stored in
  • c[j-1×pdc+i-1] when order=Nag_ColMajor;
  • c[i-1×pdc+j-1] when order=Nag_RowMajor.
On entry: 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.
8:     pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraint: pdcm.
9:     r[lr]doubleInput/Output
On entry: if mode=Nag_GenerateFromReference, the reference vector as set up by nag_rand_matrix_multi_students_t (g05ryc) in a previous call with mode=Nag_InitializeReference or Nag_InitializeAndGenerate.
On exit: if mode=Nag_InitializeReference or Nag_InitializeAndGenerate, the reference vector that can be used in subsequent calls to nag_rand_matrix_multi_students_t (g05ryc) with mode=Nag_GenerateFromReference.
10:   lrIntegerInput
On entry: the dimension of the array r. If mode=Nag_GenerateFromReference, it must be the same as the value of lr specified in the prior call to nag_rand_matrix_multi_students_t (g05ryc) with mode=Nag_InitializeReference or Nag_InitializeAndGenerate.
Constraint: lrm×m+1+2.
11:   state[dim]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions nag_rand_init_repeatable (g05kfc) or nag_rand_init_nonrepeatable (g05kgc).
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
12:   x[dim]doubleOutput
Note: the dimension, dim, of the array x must be at least
  • max1,pdx×m when order=Nag_ColMajor;
  • max1,n×pdx when order=Nag_RowMajor.
Where Xi,j appears in this document, it refers to the array element
  • x[j-1×pdx+i-1] when order=Nag_ColMajor;
  • x[i-1×pdx+j-1] when order=Nag_RowMajor.
On exit: the array of pseudorandom multivariate Student's t vectors generated by the function, with Xi,j holding the jth dimension for the ith variate.
13:   pdxIntegerInput
On entry: the stride used in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxn;
  • if order=Nag_RowMajor, pdxm.
14:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, df=value.
Constraint: df3.
On entry, lr is not large enough, lr=value: minimum length required =value.
On entry, m=value.
Constraint: m>0.
On entry, n=value.
Constraint: n0.
NE_INT_2
On entry, pdc=value and m=value.
Constraint: pdcm.
On entry, pdx=value and m=value.
Constraint: pdxm.
On entry, pdx=value and n=value.
Constraint: pdxn.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_INVALID_STATE
On entry, state vector has been corrupted or not initialized.
NE_POS_DEF
On entry, the covariance matrix C is not positive semidefinite to machine precision.
NE_PREV_CALL
m is not the same as when r was set up in a previous call.
Previous value of m=value and m=value.

7  Accuracy

Not applicable.

8  Further Comments

The time taken by nag_rand_matrix_multi_students_t (g05ryc) 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.

9  Example

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_matrix_multi_students_t (g05ryc). All ten observations are generated by a single call to nag_rand_matrix_multi_students_t (g05ryc) with mode=Nag_InitializeAndGenerate. The random number generator is initialized by nag_rand_init_repeatable (g05kfc).

9.1  Program Text

Program Text (g05ryce.c)

9.2  Program Data

None.

9.3  Program Results

Program Results (g05ryce.r)


nag_rand_matrix_multi_students_t (g05ryc) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012