# NAG FL Interfaceg05rcf (copula_​students_​t)

## ▸▿ Contents

Settings help

FL Name Style:

FL Specification Language:

## 1Purpose

g05rcf sets up a reference vector and generates an array of pseudorandom numbers from a Student's $t$ copula with $\nu$ degrees of freedom and covariance matrix $\frac{\nu }{\nu -2}C$.

## 2Specification

Fortran Interface
 Subroutine g05rcf ( mode, n, df, m, c, ldc, r, lr, x, ldx,
 Integer, Intent (In) :: mode, n, df, m, ldc, lr, ldx Integer, Intent (Inout) :: state(*), ifail Real (Kind=nag_wp), Intent (In) :: c(ldc,m) Real (Kind=nag_wp), Intent (Inout) :: r(lr), x(ldx,*)
#include <nag.h>
 void g05rcf_ (const Integer *mode, const Integer *n, const Integer *df, const Integer *m, const double c[], const Integer *ldc, double r[], const Integer *lr, Integer state[], double x[], const Integer *ldx, Integer *ifail)
The routine may be called by the names g05rcf or nagf_rand_copula_students_t.

## 3Description

The Student's $t$ copula, $G$, is defined by
 $G (u1,u2,…,um;C) = T ν,C m ( t ν,C11 −1 (u1), t ν,C22 −1 (u2),…, t ν,Cmm −1 (um))$
where $m$ is the number of dimensions, ${T}_{\nu ,C}^{m}$ is the multivariate Student's $t$ density function with $\nu$ degrees of freedom, mean zero and covariance matrix $\frac{\nu }{\nu -2}C$ and ${t}_{\nu ,{C}_{\mathit{ii}}}^{-1}$ is the inverse of the univariate Student's $t$ density function with $\nu$ degrees of freedom, zero mean and variance $\frac{\nu }{\nu -2}{C}_{\mathit{ii}}$.
g05ryf is used to generate a vector from a multivariate Student's $t$ distribution and g01ebf 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 g05rcf.

## 4References

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

## 5Arguments

1: $\mathbf{mode}$Integer Input
On entry: a code for selecting the operation to be performed by the routine.
${\mathbf{mode}}=0$
Set up reference vector only.
${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to g05rcf.
${\mathbf{mode}}=2$
Set up reference vector and generate variates.
Constraint: ${\mathbf{mode}}=0$, $1$ or $2$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of random variates required.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{df}$Integer Input
On entry: $\nu$, the number of degrees of freedom of the distribution.
Constraint: ${\mathbf{df}}\ge 3$.
4: $\mathbf{m}$Integer Input
On entry: $m$, the number of dimensions of the distribution.
Constraint: ${\mathbf{m}}>0$.
5: $\mathbf{c}\left({\mathbf{ldc}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
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.
6: $\mathbf{ldc}$Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which g05rcf is called.
Constraint: ${\mathbf{ldc}}\ge {\mathbf{m}}$.
7: $\mathbf{r}\left({\mathbf{lr}}\right)$Real (Kind=nag_wp) array Communication Array
On entry: if ${\mathbf{mode}}=1$, the reference vector as set up by g05rcf in a previous call with ${\mathbf{mode}}=0$ or $2$.
On exit: if ${\mathbf{mode}}=0$ or $2$, the reference vector that can be used in subsequent calls to g05rcf with ${\mathbf{mode}}=1$.
8: $\mathbf{lr}$Integer Input
On entry: the dimension of the array r as declared in the (sub)program from which g05rcf is called. If ${\mathbf{mode}}=1$, it must be the same as the value of lr specified in the prior call to g05rcf with ${\mathbf{mode}}=0$ or $2$.
Constraint: ${\mathbf{lr}}\ge {\mathbf{m}}×\left({\mathbf{m}}+1\right)+2$.
9: $\mathbf{state}\left(*\right)$Integer array Communication 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.
10: $\mathbf{x}\left({\mathbf{ldx}},*\right)$Real (Kind=nag_wp) array Output
Note: the second dimension of the array x must be at least ${\mathbf{m}}$.
On exit: the array of values from a multivariate Student's $t$ copula, with ${\mathbf{x}}\left(i,j\right)$ holding the $j$th dimension for the $i$th variate.
11: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g05rcf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
12: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{mode}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{mode}}=0$, $1$ or $2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{df}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{df}}\ge 3$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}>0$.
${\mathbf{ifail}}=5$
On entry, the covariance matrix $C$ is not positive semidefinite to machine precision.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{ldc}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldc}}\ge {\mathbf{m}}$.
${\mathbf{ifail}}=7$
m is not the same as when r was set up in a previous call.
Previous value of ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=8$
On entry, lr is not large enough, ${\mathbf{lr}}=⟨\mathit{\text{value}}⟩$: minimum length required $\text{}=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=9$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=11$
On entry, ${\mathbf{ldx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

See Section 7 in g05ryf for an indication of the accuracy of the underlying multivariate Student's $t$-distribution.

## 8Parallelism and Performance

g05rcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g05rcf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by g05rcf is of order $n{m}^{3}$.
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 $C+E=L{L}^{\mathrm{T}}$, 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.

## 10Example

This example prints ten pseudorandom observations from a Student's $t$ copula with ten degrees of freedom 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 g05rcf. All ten observations are generated by a single call to g05rcf with ${\mathbf{mode}}=2$. The random number generator is initialized by g05kff.

### 10.1Program Text

Program Text (g05rcfe.f90)

### 10.2Program Data

Program Data (g05rcfe.d)

### 10.3Program Results

Program Results (g05rcfe.r)