G05RDF (PDF version)
G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

NAG Library Routine Document


Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

G05RDF sets up a reference vector and generates an array of pseudorandom numbers from a Normal (Gaussian) copula with covariance matrix C.

2  Specification

REAL (KIND=nag_wp)  C(LDC,M), R(LR), X(LDX,M)

3  Description

The Gaussian copula, G, is defined by
G 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 .
G05RZF is used to generate a vector from a multivariate Normal distribution and G01EAF is used to convert each element of that vector into a uniformly distributed value between zero and one.
One of the initialization routines G05KFF (for a repeatable sequence if computed sequentially) or G05KGF (for a non-repeatable sequence) must be called prior to the first call to G05RDF.

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  Parameters

1:     MODE – INTEGERInput
On entry: a code for selecting the operation to be performed by the routine.
Set up reference vector only.
Generate variates using reference vector set up in a prior call to G05RDF.
Set up reference vector and generate variates.
Constraint: MODE=0, 1 or 2.
2:     N – INTEGERInput
On entry: n, the number of random variates required.
Constraint: N0.
3:     M – INTEGERInput
On entry: m, the number of dimensions of the distribution.
Constraint: M>0.
4:     C(LDC,M) – REAL (KIND=nag_wp) arrayInput
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:     LDC – INTEGERInput
On entry: the first dimension of the array C as declared in the (sub)program from which G05RDF is called.
Constraint: LDCM.
6:     R(LR) – REAL (KIND=nag_wp) arrayCommunication Array
On entry: if MODE=1, the reference vector as set up by G05RDF in a previous call with MODE=0 or 2.
On exit: if MODE=0 or 2, the reference vector that can be used in subsequent calls to G05RDF with MODE=1.
7:     LR – INTEGERInput
On entry: the dimension of the array R as declared in the (sub)program from which G05RDF is called. If MODE=1, it must be the same as the value of LR specified in the prior call to G05RDF with MODE=0 or 2.
Constraint: LRM×M+1+1.
8:     STATE(*) – INTEGER arrayCommunication Array
Note: the actual argument supplied must be the array STATE supplied to the initialization routines G05KFF or G05KGF.
On entry: contains information on the selected base generator and its current state.
On exit: contains updated information on the state of the generator.
9:     X(LDX,M) – REAL (KIND=nag_wp) arrayOutput
On exit: the array of values from a multivariate Gaussian copula, with Xij holding the jth dimension for the ith variate.
10:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which G05RDF is called.
Constraint: LDXN.
11:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
On entry, MODE0, 1 or 2.
On entry, N<0.
On entry, M<1.
The covariance matrix C is not positive semidefinite to machine precision.
On entry, LDC<M.
The reference vector R has been corrupted or M has changed since R was set up in a previous call to G05RDF with MODE=0 or 2.
On entry, LRM×M+1.
On entry,STATE vector was not initialized or has been corrupted.
On entry, LDX<N.

7  Accuracy

See Section 7 in G05RZF for an indication of the accuracy of the underlying multivariate Normal distribution.

8  Further Comments

The time taken by G05RDF 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 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 G05RDF. All ten observations are generated by a single call to G05RDF with MODE=2. The random number generator is initialized by G05KFF.

9.1  Program Text

Program Text (g05rdfe.f90)

9.2  Program Data

Program Data (g05rdfe.d)

9.3  Program Results

Program Results (g05rdfe.r)

G05RDF (PDF version)
G05 Chapter Contents
G05 Chapter Introduction
NAG Library Manual

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