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).
- 1:
order – 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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
mode – 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:
n – IntegerInput
On entry: , the number of pseudorandom numbers to be generated.
Constraint:
.
- 4:
m – IntegerInput
On entry: , the number of trials of the multinomial distribution.
Constraint:
.
- 5:
k – IntegerInput
On entry: , the number of possible outcomes of the multinomial distribution.
Constraint:
.
- 6:
p[k] – const doubleInput
On entry: contains the probabilities
, for , of the possible outcomes of the multinomial distribution.
Constraint:
and .
- 7:
r[lr] – 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:
lr – 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:
state[] – 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:
x[] – 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:
pdx – IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 12:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
Not applicable.
Not applicable.
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.
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).
None.