NAG Library Function Document

nag_rand_dirichlet (g05sec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_rand_dirichlet (g05sec) generates a vector of pseudorandom numbers taken from a Dirichlet distribution.

2 Specification

#include <nag.h>

#include <nagg05.h>

void	nag_rand_dirichlet (Nag_OrderType order, Integer n, Integer m, const double a[], Integer state[], double x[], Integer pdx, NagError *fail)

3 Description

The distribution has PDF (probability density function)

\begin{array}{l} f (x) & = & \frac{1}{B (α)} \prod_{i = 1}^{m} x_{i}^{α_{i} - 1} and \\ B (α) & = & \frac{\prod_{i = 1}^{m} Γ (α_{i})}{Γ (\sum_{i = 1}^{m} α_{i})} \end{array}

where

x = \{x_{1}, x_{2}, \dots, x_{m}\}

is a vector of dimension

m

, such that

x_{i} > 0

for all

i

and

\sum_{i = 1}^{m} x_{i} = 1

nag_rand_dirichlet (g05sec) generates a draw from a Dirichlet distribution by first drawing

m

independent samples,

y_{i} \sim gamma (α_{i}, 1)

, i.e., independent draws from a gamma distribution with parameters

α_{i} > 0

and one, and then setting

x_{i} = y_{i} / \sum_{j = 1}^{m} y_{j}

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_dirichlet (g05sec).

4 References

Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

5 Arguments

1: order – Nag_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

Nag_ColMajor

2: n – IntegerInput

On entry:

n

, the number of pseudorandom numbers to be generated.

Constraint:

n \geq 0

3: m – IntegerInput

On entry:

m

, the number of dimensions of the distribution.

Constraint:

m > 0

4: a[m] – const doubleInput

On entry: the parameter vector for the distribution.

Constraint:

a [i - 1] > 0.0

, for

i = 1, 2, \dots, m

5: state[ $\dim$ ] – IntegerCommunication Array

Note: the dimension,

\dim

, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument state in the previous call to 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.

6: x[ $\dim$ ] – doubleOutput

Note: the dimension, dim, of the array x must be at least

$\max (1, pdx \times m)$ when $order = Nag_ColMajor$ ;
$\max (1, n \times pdx)$ when $order = Nag_RowMajor$ .

Where

X (i, j)

appears in this document, it refers to the array element

$x [(j - 1) \times pdx + i - 1]$ when $order = Nag_ColMajor$ ;
$x [(i - 1) \times pdx + j - 1]$ when $order = Nag_RowMajor$ .

On exit: the

n

pseudorandom numbers from the specified Dirichlet distribution, with

X (i, j)

holding the

j

th dimension for the

i

th variate.

7: pdx – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraints:

if $order = Nag_ColMajor$ , $pdx \geq n$ ;
if $order = Nag_RowMajor$ , $pdx \geq m$ .

8: fail – NagError *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, $m = ⟨value⟩$ .
Constraint: $m > 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $pdx = ⟨value⟩$ and $m = ⟨value⟩$ .
Constraint: $pdx \geq m$ .
On entry, $pdx = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $order = Nag_ColMajor$ or $pdx \geq n$ .
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_REAL_ARRAY: On entry, at least one $a [i] \leq 0$ .