Multinom
The Multinomial Distribution
Generate multinomially distributed random number vectors and compute multinomial probabilities.
- Keywords
- distribution
Usage
rmultinom(n, size, prob)
dmultinom(x, size = NULL, prob, log = FALSE)Arguments
- x
- vector of length $K$ of integers in
0:size. - n
- number of random vectors to draw.
- size
- integer, say $N$, specifying the total number
of objects that are put into $K$ boxes in the typical multinomial
experiment. For
dmultinom, it defaults tosum(x). - prob
- numeric non-negative vector of length $K$, specifying the probability for the $K$ classes; is internally normalized to sum 1. Infinite and missing values are not allowed.
- log
- logical; if TRUE, log probabilities are computed.
Details
If x is a $K$-component vector, dmultinom(x, prob)
is the probability
$$P(X_1=x_1,\ldots,X_K=x_k) = C \times \prod_{j=1}^K
\pi_j^{x_j}$$
where $C$ is the ‘multinomial coefficient’
$C = N! / (x[1]! * \dots * x[K]!)$
and $N = sum(j=1, \dots, K) x[j]$.
By definition, each component $X[j]$ is binomially distributed as
Bin(size, prob[j]) for $j = 1, \ldots, K$.
The rmultinom() algorithm draws binomials $X[j]$ from
$Bin(n[j], P[j])$ sequentially, where
$n[1] = N$ (N := size),
$P[1] = p[1]$ ($p$ is prob scaled to sum 1),
and for $j \ge 2$, recursively,
$n[j] = N - sum(k=1, \dots, j-1) X[k]$
and
$P[j] = p[j] / (1 - sum(p[1:(j-1)]))$.
Value
-
For
rmultinom(),
an integer $K x n$ matrix where each column is a
random vector generated according to the desired multinomial law, and
hence summing to size. Whereas the transposed result
would seem more natural at first, the returned matrix is more
efficient because of columnwise storage.
Note
dmultinom is currently not vectorized at all and has
no C interface (API); this may be amended in the future.
See Also
Distributions for standard distributions, including
dbinom which is a special case conceptually.
Examples
library(stats)
rmultinom(10, size = 12, prob = c(0.1,0.2,0.8))
pr <- c(1,3,6,10) # normalization not necessary for generation
rmultinom(10, 20, prob = pr)
## all possible outcomes of Multinom(N = 3, K = 3)
X <- t(as.matrix(expand.grid(0:3, 0:3))); X <- X[, colSums(X) <= 3]
X <- rbind(X, 3:3 - colSums(X)); dimnames(X) <- list(letters[1:3], NULL)
X
round(apply(X, 2, function(x) dmultinom(x, prob = c(1,2,5))), 3)