1% File src/library/stats/man/Multinomal.Rd 2% Part of the R package, https://www.R-project.org 3% Copyright 1995-2014 R Core Team 4% Distributed under GPL 2 or later 5 6\name{Multinom} 7\alias{Multinomial} 8\alias{rmultinom} 9\alias{dmultinom} 10\title{The Multinomial Distribution} 11\description{ 12 Generate multinomially distributed random number vectors and 13 compute multinomial probabilities. 14} 15\usage{ 16rmultinom(n, size, prob) 17dmultinom(x, size = NULL, prob, log = FALSE) 18} 19\arguments{ 20 \item{x}{vector of length \eqn{K} of integers in \code{0:size}.} 21 %%FUTURE: matrix of \eqn{K} rows or ... 22 \item{n}{number of random vectors to draw.} 23 \item{size}{integer, say \eqn{N}, specifying the total number 24 of objects that are put into \eqn{K} boxes in the typical multinomial 25 experiment. For \code{dmultinom}, it defaults to \code{sum(x)}.} 26 \item{prob}{numeric non-negative vector of length \eqn{K}, specifying 27 the probability for the \eqn{K} classes; is internally normalized to 28 sum 1. Infinite and missing values are not allowed.} 29 \item{log}{logical; if TRUE, log probabilities are computed.} 30} 31\note{\code{dmultinom} is currently \emph{not vectorized} at all and has 32 no C interface (API); this may be amended in the future.% yes, DO THIS! 33} 34\details{ 35 If \code{x} is a \eqn{K}-component vector, \code{dmultinom(x, prob)} 36 is the probability 37 \deqn{P(X_1=x_1,\ldots,X_K=x_k) = C \times \prod_{j=1}^K 38 \pi_j^{x_j}}{P(X[1]=x[1], \dots , X[K]=x[k]) = C * prod(j=1 , \dots, K) p[j]^x[j]} 39 where \eqn{C} is the \sQuote{multinomial coefficient} 40 \eqn{C = N! / (x_1! \cdots x_K!)}{C = N! / (x[1]! * \dots * x[K]!)} 41 and \eqn{N = \sum_{j=1}^K x_j}{N = sum(j=1, \dots, K) x[j]}. 42 \cr 43 By definition, each component \eqn{X_j}{X[j]} is binomially distributed as 44 \code{Bin(size, prob[j])} for \eqn{j = 1, \ldots, K}. 45 46 The \code{rmultinom()} algorithm draws binomials \eqn{X_j}{X[j]} from 47 \eqn{Bin(n_j,P_j)}{Bin(n[j], P[j])} sequentially, where 48 \eqn{n_1 = N}{n[1] = N} (N := \code{size}), 49 \eqn{P_1 = \pi_1}{P[1] = p[1]} (\eqn{\pi}{p} is \code{prob} scaled to sum 1), 50 and for \eqn{j \ge 2}, recursively, 51 \eqn{n_j = N - \sum_{k=1}^{j-1} X_k}{n[j] = N - sum(k=1, \dots, j-1) X[k]} 52 and 53 \eqn{P_j = \pi_j / (1 - \sum_{k=1}^{j-1} \pi_k)}{P[j] = p[j] / (1 - sum(p[1:(j-1)]))}. 54} 55\value{ 56 For \code{rmultinom()}, 57 an integer \eqn{K \times n}{K x n} matrix where each column is a 58 random vector generated according to the desired multinomial law, and 59 hence summing to \code{size}. Whereas the \emph{transposed} result 60 would seem more natural at first, the returned matrix is more 61 efficient because of columnwise storage. 62} 63\seealso{ 64 \link{Distributions} for standard distributions, including 65 \code{\link{dbinom}} which is a special case conceptually. 66%% but does not return 2-vectors 67} 68\examples{ 69rmultinom(10, size = 12, prob = c(0.1,0.2,0.8)) 70 71pr <- c(1,3,6,10) # normalization not necessary for generation 72rmultinom(10, 20, prob = pr) 73 74## all possible outcomes of Multinom(N = 3, K = 3) 75X <- t(as.matrix(expand.grid(0:3, 0:3))); X <- X[, colSums(X) <= 3] 76X <- rbind(X, 3:3 - colSums(X)); dimnames(X) <- list(letters[1:3], NULL) 77X 78round(apply(X, 2, function(x) dmultinom(x, prob = c(1,2,5))), 3) 79} 80\keyword{distribution} 81