g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

NAG Library Function Documentnag_rand_dirichlet (g05sec)

1  Purpose

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

2  Specification

 #include #include
 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)
 $fx = 1 Bα ∏ i=1 m x i αi - 1 and Bα = ∏ i=1 m Γ αi Γ ∑ i=1 m αi$
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_dirichlet (g05sec) 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_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:     orderNag_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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 0$.
3:     mIntegerInput
On entry: $m$, the number of dimensions of the distribution.
Constraint: ${\mathbf{m}}>0$.
4:     a[m]const doubleInput
On entry: the parameter vector for the distribution.
Constraint: ${\mathbf{a}}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{m}}$.
5:     state[$\mathit{dim}$]IntegerCommunication Array
Note: the actual argument supplied must be the array state supplied to the initialization functions 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[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{m}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdx}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Where ${\mathbf{X}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: 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.
7:     pdxIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{m}}$.
8:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}>0$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INT_2
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{order}}\ne \mathrm{Nag_ColMajor}$ or ${\mathbf{pdx}}\ge {\mathbf{n}}$.
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 ${\mathbf{a}}\left[i\right]\le 0$.

Not applicable.

None.

9  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_dirichlet (g05sec), after initialization by nag_rand_init_repeatable (g05kfc).

9.1  Program Text

Program Text (g05sece.c)

None.

9.3  Program Results

Program Results (g05sece.r)