g05 Chapter Contents
g05 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_rngs_gen_multinomial (g05mrc)

## 1  Purpose

nag_rngs_gen_multinomial (g05mrc) generates a sequence of $n$ variates, each consisting of $k$ pseudorandom integers, from the discrete multinomial distribution with $k$ outcomes and $m$ trials, where the outcomes have probabilities ${p}_{1},{p}_{2},\dots ,{p}_{k}$ respectively.

## 2  Specification

 #include #include
 void nag_rngs_gen_multinomial (Nag_OrderType order, Integer mode, Integer m, Integer k, const double p[], Integer n, Integer x[], Integer pdx, Integer igen, Integer iseed[], double r[], NagError *fail)

## 3  Description

nag_rngs_gen_multinomial (g05mrc) generates a sequence of $n$ groups of $k$ integers ${x}_{\mathit{i},\mathit{j}}$, for $\mathit{j}=1,2,\dots ,k$ and $\mathit{i}=1,2,\dots ,n$, from a multinomial distribution with $m$ trials and $k$ outcomes, where the probability of ${x}_{i,j}={I}_{j}$ for each $j=1,2,\dots ,k$ is
 $Pi1=I1,…,ik=Ik= m! ∏j=1k Ij! ∏j=1k pjIj= m! I1!I2!⋯Ik! p1I1p2I2⋯pkIk,$
where
 $∑j= 1k pj= 1 and ∑j= 1k Ij=m.$
A single trial can have several outcomes ($k$, say) and the probability of achieving each outcome is known (${p}_{j}$, say). After $m$ trials each outcome will have occurred a certain number of times. The $k$ numbers representing the numbers of occurrences for each outcome after $m$ trials is then a single sample from the multinomial distribution defined by the parameters $k$, $m$ and ${p}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,k$. This function returns $n$ such samples with each sample being stored as a row in a two-dimensional array of integers.
When $k=2$ this distribution is equivalent to the binomial distribution with parameters $m$ and $p={p}_{1}$ (nag_rngs_binomial (g05mjc)).
The variates can be generated with or without using a search table and index. If a search table is used then it is stored with the index in a reference vector and subsequent calls to nag_rngs_gen_multinomial (g05mrc) with the same parameter values can then use this reference vector to generate further variates. The reference array is only generated for the outcome with greatest probability. The number of successes for the outcome with greatest probability is calculated first as for the binomial distribution (nag_rngs_binomial (g05mjc)); the number of successes for other outcomes are calculated in turn for the remaining reduced multinomial distribution; the number of successes for the final outcome is simply calculated to ensure that the total number of successes is $m$.
One of the initialization functions nag_rngs_init_repeatable (g05kbc) (for a repeatable sequence if computed sequentially) or nag_rngs_init_nonrepeatable (g05kcc) (for a non-repeatable sequence) must be called prior to the first call to nag_rngs_gen_multinomial (g05mrc).

## 4  References

Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley

## 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:     modeIntegerInput
On entry: a code for selecting the operation to be performed by the function.
${\mathbf{mode}}=0$
Set up reference vector only.
${\mathbf{mode}}=1$
Generate variates using reference vector set up in a prior call to nag_rngs_gen_multinomial (g05mrc).
${\mathbf{mode}}=2$
Set up reference vector and generate variates.
${\mathbf{mode}}=3$
Generate variates without using the reference vector.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
3:     mIntegerInput
On entry: $m$, the number of trials of the multinomial distribution.
Constraint: ${\mathbf{m}}\ge 0$.
4:     kIntegerInput
On entry: $k$, the number of possible outcomes of the multinomial distribution.
Constraint: ${\mathbf{k}}\ge 2$.
5:     p[k]const doubleInput
On entry: contains the probabilities ${p}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,k$, of the $k$ possible outcomes of the multinomial distribution.
Constraint: $0.0\le {\mathbf{p}}\left[j-1\right]\le 1.0$ and $\sum _{j=1}^{k}{\mathbf{p}}\left[j-1\right]=1.0$.
6:     nIntegerInput
On entry: $n$, the number of pseudorandom numbers to be generated.
Constraint: ${\mathbf{n}}\ge 1$.
7:     x[$\mathit{dim}$]IntegerOutput
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{k}}\right)$ 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 first $n$ rows of ${\mathbf{X}}\left(i,j\right)$ each contain $k$ pseudorandom numbers representing a $k$-dimensional variate from the specified multinomial distribution.
8:     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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{k}}$.
9:     igenIntegerInput
On entry: must contain the identification number for the generator to be used to return a pseudorandom number and should remain unchanged following initialization by a prior call to nag_rngs_init_repeatable (g05kbc) or nag_rngs_init_nonrepeatable (g05kcc).
10:   iseed[$4$]IntegerCommunication Array
On entry: contains values which define the current state of the selected generator.
On exit: contains updated values defining the new state of the selected generator.
11:   r[$\mathit{dim}$]doubleCommunication Array
Note: the dimension, dim, of the array r must be at least
• $22+20\sqrt{{\mathbf{m}}×\mathit{p_max}\left(1-\mathit{p_max}\right)}$ when ${\mathbf{mode}}\ne 3$;
• $1$ otherwise.
On entry: if ${\mathbf{mode}}=1$, the reference vector from the previous call to nag_rngs_gen_multinomial (g05mrc).
On exit: the reference vector.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
On entry, ${\mathbf{p}}\left[i-1\right]<0.0$ or ${\mathbf{p}}\left[i-1\right]>1.0$ where: $i=〈\mathit{\text{value}}〉$ and ${\mathbf{p}}\left[i-1\right]=〈\mathit{\text{value}}〉$.
NE_INT
On entry, ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{k}}\ge 2$.
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 0$.
On entry, ${\mathbf{mode}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mode}}=0$, $1$, $2$ or $3$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}>0$.
NE_INT_2
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{k}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{k}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
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_PREV_CALL
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{p}}\left[i-1\right]\right)$ or m is not the same as when r was set up in a previous call. Previous value of $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{p}}\left[i-1\right]\right)=〈\mathit{\text{value}}〉$ and $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{p}}\left[i-1\right]\right)=〈\mathit{\text{value}}〉$. Previous value of ${\mathbf{m}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{mode}}=1$, but either r has not been set up in prior call or r has become corrupted.
NE_REAL
On entry, the sum of ${\mathbf{p}}\left[\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{k}}$, is not unity. The difference from unity in the summation is: $〈\mathit{\text{value}}〉$.

## 7  Accuracy

Not applicable.

Only the reference vector for one outcome can be set up because the conditional distributions cannot be known in advance of the generation of variates. The outcome with greatest probability of success is chosen for the reference vector because it will have the greatest spread of likely values.

## 9  Example

This example prints $20$ pseudorandom $k$-dimensional variates from a multinomial distribution with $k=4$, $m=6000$, ${p}_{1}=0.08$, ${p}_{2}=0.1$, ${p}_{3}=0.8$ and ${p}_{4}=0.02$, generated by a single call to nag_rngs_gen_multinomial (g05mrc), after initialization by nag_rngs_init_repeatable (g05kbc).

### 9.1  Program Text

Program Text (g05mrce.c)

None.

### 9.3  Program Results

Program Results (g05mrce.r)