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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)
f(x) = m 1/(B(α)) ∏ xi α_i − 1   and i = 1 B(α) = ( ∏ i = 1m Γ (αi) )/(
 Γ (m ) ∑ αii = 1
)
$f(x) = 1 B(α) ∏ i=1 m x i αi - 1 and B(α) = ∏ i=1 m Γ (αi) Γ ( ∑ i=1 m αi )$
where x = {x1,x2,,xm} $x=\left\{{x}_{1},{x}_{2},\dots ,{x}_{m}\right\}$ is a vector of dimension m$m$, such that xi > 0${x}_{i}>0$ for all i$i$ and i = 1m xi = 1$\sum _{\mathit{i}=1}^{m}{x}_{i}=1$.
nag_rand_dist_dirichlet (g05se) generates a draw from a Dirichlet distribution by first drawing m$m$ independent samples, yigamma(αi,1)${y}_{i}\sim \mathrm{gamma}\left({\alpha }_{i},1\right)$, i.e., independent draws from a gamma distribution with parameters αi > 0${\alpha }_{i}>0$ and one, and then setting xi = yi / j = 1m yj${x}_{i}={y}_{i}/\sum _{\mathit{j}=1}^{m}{y}_{j}$.
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:     n – int64int32nag_int scalar
n$n$, the number of pseudorandom numbers to be generated.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     a(m) – double array
m, the dimension of the array, must satisfy the constraint m > 0${\mathbf{m}}>0$.
The parameter vector for the distribution.
Constraint: a(i) > 0.0${\mathbf{a}}\left(\mathit{i}\right)>0.0$, for i = 1,2,,m$\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
3:     state( : $:$) – 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:     m – int64int32nag_int scalar
Default: The dimension of the array a.
m$m$, the number of dimensions of the distribution.
Constraint: m > 0${\mathbf{m}}>0$.

ldx

Output Parameters

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

Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
On entry, n < 0${\mathbf{n}}<0$.
ifail = 2${\mathbf{ifail}}=2$
On entry, m < 1${\mathbf{m}}<1$.
ifail = 3${\mathbf{ifail}}=3$
On entry, at least one a(i)0.0${\mathbf{a}}\left(\mathit{i}\right)\le 0.0$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, state vector was not initialized or has been corrupted.
ifail = 6${\mathbf{ifail}}=6$
On entry, ldx < n$\mathit{ldx}<{\mathbf{n}}$.

Not applicable.

None.

Example

```function nag_rand_dist_dirichlet_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
a = [2; 2; 2; 2];
% Initialize the generator to a repeatable sequence
[state, ifail] = nag_rand_init_repeat(genid, subid, seed);
[state, x, ifail] = nag_rand_dist_dirichlet(n, a, state)
```
```

state =

17
1234
1
0
4694
17841
5384
14764
17917
13895
19930
8
0
1234
1
1
1234

x =

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

ifail =

0

```
```function g05se_example
% Initialize the seed
seed = [int64(1762543)];
% genid and subid identify the base generator
genid = int64(1);
subid =  int64(1);
n = int64(5);
a = [2; 2; 2; 2];
% Initialize the generator to a repeatable sequence
[state, ifail] = g05kf(genid, subid, seed);
[state, x, ifail] = g05se(n, a, state)
```
```

state =

17
1234
1
0
4694
17841
5384
14764
17917
13895
19930
8
0
1234
1
1
1234

x =

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

ifail =

0

```