nag_rngs_matrix_multi_students_t (g05lxc) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_rngs_matrix_multi_students_t (g05lxc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_rngs_matrix_multi_students_t (g05lxc) 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_rngs_matrix_multi_students_t (Nag_OrderType order, Integer mode, Integer df, Integer m, const double xmu[], const double c[], Integer pdc, Integer n, double x[], Integer pdx, Integer igen, Integer iseed[], double r[], Integer lr, 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 + vs z
where z is generated by nag_rgsn_matrix_multi_normal (g05lyc) from a Normal distribution with mean zero and covariance matrix C and s is generated by nag_rngs_chi_sq (g05lcc) from a χ2-distribution with ν degrees of freedom.
One of the initialization functions nag_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_matrix_multi_students_t (g05lxc).

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 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     modeIntegerInput
On entry: selects the operation to be performed.
Initialize and generate random numbers.
Initialize only (i.e., set up reference vector).
Generate random numbers using previously set up reference vector.
Constraint: mode=0, 1 or 2.
3:     dfIntegerInput
On entry: ν, the number of degrees of freedom of the distribution.
Constraint: df3 .
4:     mIntegerInput
On entry: m, the number of dimensions of the distribution.
Constraint: m>0.
5:     xmu[m]const doubleInput
On entry: a, the vector of means of the distribution.
6:     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.
7:     pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraint: pdcm.
8:     nIntegerInput
On entry: n, the number of random variates required.
Constraint: n1.
9:     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.
10:   pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
  • if order=Nag_ColMajor, pdxn;
  • if order=Nag_RowMajor, pdxm.
11:   igenIntegerInput
On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
12:   iseed[4]IntegerCommunication Array
On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
13:   r[lr]doubleInput/Output
On entry: if mode=2, the reference vector as set up by nag_rngs_matrix_multi_students_t (g05lxc) in a previous call with mode=0 or 1.
On exit: if mode=0 or 1, the reference vector that can be used in subsequent calls to nag_rngs_matrix_multi_students_t (g05lxc) with mode=2.
14:   lrIntegerInput
On entry: the dimension of the array r. If mode=2, it must be the same as the value of lr specified in the prior call to nag_rngs_matrix_multi_students_t (g05lxc) with mode=0 or 1.
Constraint: lr>m×m+1+1.
15:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

Dynamic memory allocation failed.
On entry, argument value had an illegal value.
On entry, df=value.
Constraint: df>2.
On entry, invalid value for igen. Ensure igen has not changed since random number generator was initialized.
On entry, m=value.
Constraint: m>0.
On entry, m=value.
Constraint: m1.
On entry, mode=value.
Constraint: mode0 and mode2.
On entry, n=value.
Constraint: n0.
On entry, n=value.
Constraint: n1.
On entry, pdc=value.
Constraint: pdc>0.
On entry, pdx=value.
Constraint: pdx>0.
m is not the same as when r was set up in a previous call. Previous value =value and m=value.
On entry, lr=value, m×m+m+1= value.
Constraint: lrm×m+m+1.
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.
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.
The covariance matrix c is not positive semidefinite to machine precision.

7  Accuracy

The maximum absolute error in LLT, and hence in the covariance matrix of the resulting vectors, is less than mε+m+3ε/2 times the maximum element of C, where ε is the machine precision. Under normal circumstances, the above will be small compared to sampling error.

8  Further Comments

The time taken by nag_rngs_matrix_multi_students_t (g05lxc) 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 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_rngs_matrix_multi_students_t (g05lxc). All ten observations are generated by a single call to nag_rngs_matrix_multi_students_t (g05lxc) with mode=0. The random number generator is initialized by nag_rngs_init_repeatable (g05kbc).

9.1  Program Text

Program Text (g05lxce.c)

9.2  Program Data


9.3  Program Results

Program Results (g05lxce.r)

nag_rngs_matrix_multi_students_t (g05lxc) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

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