nag_rngs_copula_normal (g05rac) (PDF version)
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NAG C Library Manual

NAG Library Function Document

nag_rngs_copula_normal (g05rac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_rngs_copula_normal (g05rac) sets up a reference vector and generates an array of pseudorandom numbers from a Normal (Gaussian) copula with covariance matrix C.

2  Specification

#include <nag.h>
#include <nagg05.h>
void  nag_rngs_copula_normal (Nag_OrderType order, Integer mode, Integer m, const double c[], Integer pdc, Integer n, double x[], Integer pdx, Integer igen, Integer iseed[], double r[], Integer lr, NagError *fail)

3  Description

The Gaussian copula, c, is defined by
c u1 , u2 ,, um ; C = ΦC ϕ C11 -1 u1 , ϕ C22 -1 u2 ,, ϕ Cmm -1 um
where m is the number of dimensions, ΦC  is the multivariate Normal density function with mean zero and covariance matrix C and ϕ Cii -1  is the inverse of the univariate Normal density function with mean zero and variance Cii .
Routine nag_rgsn_matrix_multi_normal (g05lyc) is used to generate a vector from a multivariate Normal distribution and function nag_prob_normal (g01eac) is used to convert each element of that vector into a uniformly distributed value between zero and one.
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_copula_normal (g05rac).

4  References

Nelsen R B (1998) An Introduction to Copulas. Lecture Notes in Statistics 139 Springer
Sklar A (1973) Random variables: joint distribution functions and copulas Kybernetika 9 499–460

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:     mIntegerInput
On entry: m, the number of dimensions of the distribution.
Constraint: m>0.
4:     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: the covariance matrix of the distribution. Only the upper triangle need be set.
Constraint: c must be positive semidefinite to machine precision.
5:     pdcIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraint: pdcm.
6:     nIntegerInput
On entry: n, the number of random variates required.
Constraint: n1.
7:     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.
The ith element of the jth vector X is stored in
  • 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 Normal vectors.
8:     pdxIntegerInput
On entry: the stride used in the array x.
  • if order=Nag_ColMajor, pdxn;
  • if order=Nag_RowMajor, pdxm.
9:     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).
10:   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.
11:   r[lr]doubleCommunication Array
On entry: if mode=2, the reference vector as set up by nag_rngs_copula_normal (g05rac) 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_copula_normal (g05rac) with mode=2.
12:   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_copula_normal (g05rac) with mode=0 or 1.
Constraint: lr>m×m+1.
13:   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, 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+1=value.
Constraint: lrm×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

See Section 7 in nag_rgsn_matrix_multi_normal (g05lyc) for an indication of the accuracy of the underlying multivariate Normal distribution.

8  Further Comments


9  Example

This example prints ten pseudorandom observations from a Normal copula with covariance 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_copula_normal (g05rac). All ten observations are generated by a single call to nag_rngs_copula_normal (g05rac) with mode=0. The random number generator is initialized by nag_rngs_init_repeatable (g05kbc).

9.1  Program Text

Program Text (g05race.c)

9.2  Program Data


9.3  Program Results

Program Results (g05race.r)

nag_rngs_copula_normal (g05rac) (PDF version)
g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

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