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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_dist_dirichlet (g05se)

## Purpose

nag_rand_dist_dirichlet (g05se) generates a vector of pseudorandom numbers taken from a Dirichlet distribution.

## Syntax

[state, x, ifail] = g05se(n, a, state, 'm', m)
[state, x, ifail] = nag_rand_dist_dirichlet(n, a, state, 'm', m)

## Description

The distribution has PDF (probability density function)
 $fx = 1 Bα ∏ i=1 m x i αi - 1 and Bα = ∏ i=1 m Γ αi Γ ∑ i=1 m αi$
where $x=\left\{{x}_{1},{x}_{2},\dots ,{x}_{m}\right\}$ is a vector of dimension $m$, such that ${x}_{i}>0$ for all $i$ and $\sum _{\mathit{i}=1}^{m}{x}_{i}=1$.
nag_rand_dist_dirichlet (g05se) generates a draw from a Dirichlet distribution by first drawing $m$ independent samples, ${y}_{i}\sim \mathrm{gamma}\left({\alpha }_{i},1\right)$, i.e., independent draws from a gamma distribution with parameters ${\alpha }_{i}>0$ and one, and then setting ${x}_{i}={y}_{i}/\sum _{\mathit{j}=1}^{m}{y}_{j}$.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_dist_dirichlet (g05se).

## References

Dagpunar J (1988) Principles of Random Variate Generation Oxford University Press
Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
$n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{a}\left({\mathbf{m}}\right)$ – double array
The parameter vector for the distribution.
Constraint: ${\mathbf{a}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the array a.
$m$, the number of dimensions of the distribution.
Constraint: ${\mathbf{m}}>0$.

### Output Parameters

1:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Contains updated information on the state of the generator.
2:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{m}}\right)$ – double array
The $n$ pseudorandom numbers from the specified Dirichlet distribution, with ${\mathbf{x}}\left(i,j\right)$ holding the $j$th dimension for the $i$th variate.
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{m}}>0$.
${\mathbf{ifail}}=3$
On entry, at least one ${\mathbf{a}}\left(i\right)\le 0$.
${\mathbf{ifail}}=4$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=6$
Constraint: $\mathit{ldx}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

Not applicable.

None.

## Example

This example prints a set of five pseudorandom numbers from a Dirichlet distribution with parameters $m=4$ and $\alpha =\left\{2.0,2.0,2.0,2.0\right\}$, generated by a single call to nag_rand_dist_dirichlet (g05se), after initialization by nag_rand_init_repeat (g05kf).
```function g05se_example

fprintf('g05se example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(1762543)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
genid, subid, seed);

% Number of variates
n = int64(5);
% Parameters
a = [2; 2; 2; 2];

% Generate variates from Dirichlet distribution
[state, x, ifail] = g05se( ...
n, a, state);

disp('Variates');
disp(x);

```
```g05se example results

Variates
0.3600    0.3138    0.0837    0.2426
0.2874    0.5121    0.1497    0.0509
0.2286    0.2190    0.3959    0.1566
0.1744    0.3961    0.2764    0.1530
0.1522    0.2845    0.2074    0.3559

```