NAG Library Function Document
nag_rand_gen_multinomial (g05tgc)
1
Purpose
nag_rand_gen_multinomial (g05tgc) generates a sequence of variates, each consisting of pseudorandom integers, from the discrete multinomial distribution with outcomes and trials, where the outcomes have probabilities respectively.
2
Specification
#include <nag.h> |
#include <nagg05.h> |
void |
nag_rand_gen_multinomial (Nag_OrderType order,
Nag_ModeRNG mode,
Integer n,
Integer m,
Integer k,
const double p[],
double r[],
Integer lr,
Integer state[],
Integer x[],
Integer pdx,
NagError *fail) |
|
3
Description
nag_rand_gen_multinomial (g05tgc) generates a sequence of
groups of
integers
, for
and
, from a multinomial distribution with
trials and
outcomes, where the probability of
for each
is
where
A single trial can have several outcomes (
) and the probability of achieving each outcome is known (
). After
trials each outcome will have occurred a certain number of times. The
numbers representing the numbers of occurrences for each outcome after
trials is then a single sample from the multinomial distribution defined by the parameters
,
and
, for
. This function returns
such samples.
When
this distribution is equivalent to the binomial distribution with parameters
and
(see
nag_rand_binomial (g05tac)).
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_rand_gen_multinomial (g05tgc) with the same parameter values can then use this reference vector to generate further variates. The reference array is generated only for the outcome with greatest probability. The number of successes for the outcome with greatest probability is calculated first as for the binomial distribution (see
nag_rand_binomial (g05tac)); 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
.
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_gen_multinomial (g05tgc).
4
References
Knuth D E (1981) The Art of Computer Programming (Volume 2) (2nd Edition) Addison–Wesley
5
Arguments
- 1:
– 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
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_ModeRNGInput
-
On entry: a code for selecting the operation to be performed by the function.
- Set up reference vector only.
- Generate variates using reference vector set up in a prior call to nag_rand_gen_multinomial (g05tgc).
- Set up reference vector and generate variates.
- Generate variates without using the reference vector.
Constraint:
, , or .
- 3:
– IntegerInput
-
On entry: , the number of pseudorandom numbers to be generated.
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of trials of the multinomial distribution.
Constraint:
.
- 5:
– IntegerInput
-
On entry: , the number of possible outcomes of the multinomial distribution.
Constraint:
.
- 6:
– const doubleInput
-
On entry: contains the probabilities
, for , of the possible outcomes of the multinomial distribution.
Constraint:
and .
- 7:
– doubleCommunication Array
-
On entry: if
, the reference vector from the previous call to
nag_rand_gen_multinomial (g05tgc).
If
,
r is not referenced and may be
NULL.
On exit: if , the reference vector.
- 8:
– IntegerInput
-
Note: for convenience p_max will be used here to denote the expression .
On entry: the dimension of the array
r.
Suggested values:
- if , ;
- otherwise .
Constraints:
- if or ,
;
- if , lr must remain unchanged from the previous call to nag_rand_gen_multinomial (g05tgc).
- 9:
– IntegerCommunication Array
-
Note: the dimension,
, 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.
- 10:
– IntegerOutput
-
Note: the dimension,
dim, of the array
x
must be at least
- when ;
- when .
Where
appears in this document, it refers to the array element
- when ;
- when .
On exit: the first rows of each contain pseudorandom numbers representing a -dimensional variate from the specified multinomial distribution.
- 11:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 12:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry,
lr is too small when
or
:
, minimum length required
.
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- 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.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_INVALID_STATE
-
On entry,
state vector has been corrupted or not initialized.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_PREV_CALL
-
The value of
m or
k is not the same as when
r was set up in a previous call.
Previous value of
and
.
Previous value of
and
.
- NE_REAL_ARRAY
-
On entry, at least one element of the vector
p is less than
or greater than
.
On entry, the sum of the elements of
p do not equal one.
- NE_REF_VEC
-
On entry, some of the elements of the array
r have been corrupted or have not been initialized.
7
Accuracy
Not applicable.
8
Parallelism and Performance
nag_rand_gen_multinomial (g05tgc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The reference vector for only 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.
10
Example
This example prints
pseudorandom
-dimensional variates from a multinomial distribution with
,
,
,
,
and
, generated by a single call to
nag_rand_gen_multinomial (g05tgc), after initialization by
nag_rand_init_repeatable (g05kfc).
10.1
Program Text
Program Text (g05tgce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (g05tgce.r)