The R Project SVN R

% File src/library/base/man/Random.Rd

% Part of the R package, https://www.R-project.org

% Copyright 1995-2019 R Core Team

% Distributed under GPL 2 or later

\name{Random}

\alias{Random}

\alias{RNG}

\alias{RNGkind}

\alias{RNGversion}

\alias{set.seed}

\alias{.Random.seed}

\title{Random Number Generation}

\description{

  \code{.Random.seed} is an integer vector, containing the random number

  generator (RNG) \bold{state} for random number generation in \R.  It

  can be saved and restored, but should not be altered by the user.

  \code{RNGkind} is a more friendly interface to query or set the kind

  of RNG in use.

  \code{RNGversion} can be used to set the random generators as they

  were in an earlier \R version (for reproducibility).

  \code{set.seed} is the recommended way to specify seeds.

}

\usage{

\special{.Random.seed <- c(rng.kind, n1, n2, \dots)}

RNGkind(kind = NULL, normal.kind = NULL, sample.kind = NULL)

RNGversion(vstr)

set.seed(seed, kind = NULL, normal.kind = NULL, sample.kind = NULL)

}

\arguments{

  \item{kind}{character or \code{NULL}.  If \code{kind} is a character

    string, set \R's RNG to the kind desired.  Use \code{"default"} to

    return to the \R default.  See \sQuote{Details} for the

    interpretation of \code{NULL}.}

  \item{normal.kind}{character string or \code{NULL}.  If it is a character

    string, set the method of Normal generation.  Use \code{"default"}

    to return to the \R default.  \code{NULL} makes no change.}

  \item{sample.kind}{character string or \code{NULL}.  If it is a character

    string, set the method of discrete uniform generation (used in 

    \code{\link{sample}}, for instance).  Use \code{"default"} to return to 

    the \R default.  \code{NULL} makes no change.}

  \item{seed}{a single value, interpreted as an integer, or \code{NULL}

    (see \sQuote{Details}).}

  \item{vstr}{a character string containing a version number,

    e.g., \code{"1.6.2"}.  The default RNG configuration of the current

    \R version is used if \code{vstr} is greater than the current version.} 

  \item{rng.kind}{integer code in \code{0:k} for the above \code{kind}.}

  \item{n1, n2, \dots}{integers.  See the details for how many are required

    (which depends on \code{rng.kind}).}

}

%% source and more detailed references:  ../../../main/RNG.c

\details{

  The currently available RNG kinds are given below.  \code{kind} is

  partially matched to this list.  The default is

  \code{"Mersenne-Twister"}.

  \describe{

    \item{\code{"Wichmann-Hill"}}{

    The seed, \code{.Random.seed[-1] == r[1:3]} is an integer vector of

    length 3, where each \code{r[i]} is in \code{1:(p[i] - 1)}, where

    \code{p} is the length 3 vector of primes, \code{p = (30269, 30307,

      30323)}.

    The Wichmann--Hill generator has a cycle length of

    \eqn{6.9536 \times 10^{12}}{6.9536e12} (=

    \code{prod(p-1)/4}, see \emph{Applied Statistics} (1984)

    \bold{33}, 123 which corrects the original article).}

    \item{\code{"Marsaglia-Multicarry"}:}{

    A \emph{multiply-with-carry} RNG is used, as recommended by George

    Marsaglia in his post to the mailing list \file{sci.stat.math}.

    It has a period of more than \eqn{2^{60}}{2^60} and has passed

    all tests (according to Marsaglia).  The seed is two integers (all

    values allowed).}

    \item{\code{"Super-Duper"}:}{

    Marsaglia's famous Super-Duper from the 70's.  This is the original

    version which does \emph{not} pass the MTUPLE test of the Diehard

    battery.  It has a period of \eqn{\approx 4.6\times 10^{18}}{about

      4.6*10^18} for most initial seeds.  The seed is two integers (all

    values allowed for the first seed: the second must be odd).

    We use the implementation by Reeds \emph{et al} (1982--84).

    The two seeds are the Tausworthe and congruence long integers,

    respectively.  A one-to-one mapping to S's \code{.Random.seed[1:12]}

    is possible but we will not publish one, not least as this generator

    is \bold{not} exactly the same as that in recent versions of S-PLUS.}

    \item{\code{"Mersenne-Twister"}:}{

      From Matsumoto and Nishimura (1998); code updated in 2002.

      A twisted GFSR with period

      \eqn{2^{19937} - 1}{2^19937 - 1} and equidistribution in 623

      consecutive dimensions (over the whole period).  The \sQuote{seed} is a

      624-dimensional set of 32-bit integers plus a current position in

      that set.

      R uses its own initialization method due to B. D. Ripley and is

      not affected by the initialization issue in the 1998 code of

      Matsumoto and Nishimura addressed in a 2002 update.

    }

    \item{\code{"Knuth-TAOCP-2002"}:}{

      A 32-bit integer GFSR using lagged Fibonacci sequences with

      subtraction.  That is, the recurrence used is

      \deqn{X_j = (X_{j-100} - X_{j-37}) \bmod 2^{30}%

      }{X[j] = (X[j-100] - X[j-37]) mod 2^30}

      and the \sQuote{seed} is the set of the 100 last numbers (actually

      recorded as 101 numbers, the last being a cyclic shift of the

      buffer).  The period is around \eqn{2^{129}}{2^129}.

    }

    \item{\code{"Knuth-TAOCP"}:}{

      An earlier version from Knuth (1997).

      The 2002 version was not backwards compatible with the earlier

      version: the initialization of the GFSR from the seed was altered.

      \R did not allow you to choose consecutive seeds, the reported

      \sQuote{weakness}, and already scrambled the seeds.

      Initialization of this generator is done in interpreted \R code

      and so takes a short but noticeable time.

    }

    \item{\code{"L'Ecuyer-CMRG"}:}{

      A \sQuote{combined multiple-recursive generator} from L'Ecuyer

      (1999), each element of which is a feedback multiplicative

      generator with three integer elements: thus the seed is a (signed)

      integer vector of length 6. The period is around

      \eqn{2^{191}}{2^191}.

      The 6 elements of the seed are internally regarded as 32-bit

      unsigned integers.  Neither the first three nor the last three

      should be all zero, and they are limited to less than

      \code{4294967087} and \code{4294944443} respectively.

      This is not particularly interesting of itself, but provides the

      basis for the multiple streams used in package \pkg{parallel}.

      % See \code{\link{RngStream}}.

    }

    \item{\code{"user-supplied"}:}{

      Use a user-supplied generator.  See \code{\link{Random.user}} for

      details.

    }

  }

  \code{normal.kind} can be \code{"Kinderman-Ramage"},

  \code{"Buggy Kinderman-Ramage"} (not for \code{set.seed}),

  \code{"Ahrens-Dieter"}, \code{"Box-Muller"}, \code{"Inversion"} (the

  default), or \code{"user-supplied"}.  (For inversion, see the

  reference in \code{\link{qnorm}}.)  The Kinderman-Ramage generator

  used in versions prior to 1.7.0 (now called \code{"Buggy"}) had several

  approximation errors and should only be used for reproduction of old

  results.  The \code{"Box-Muller"} generator is stateful as pairs of

  normals are generated and returned sequentially.  The state is reset

  whenever it is selected (even if it is the current normal generator)

  and when \code{kind} is changed.

  \code{sample.kind} can be \code{"Rounding"} or \code{"Rejection"},

  or partial matches to these.  The former was the default in versions

  prior to 3.6.0:  it made \code{\link{sample}} noticeably non-uniform

  on large populations, and should only be used for reproduction of old

  results.  See \PR{17494} for a discussion. 

  \code{set.seed} uses a single integer argument to set as many seeds

  as are required.  It is intended as a simple way to get quite different

  seeds by specifying small integer arguments, and also as a way to get

  valid seed sets for the more complicated methods (especially

  \code{"Mersenne-Twister"} and \code{"Knuth-TAOCP"}).  There is no

  guarantee that different values of \code{seed} will seed the RNG

  differently, although any exceptions would be extremely rare.  If

  called with \code{seed = NULL} it re-initializes (see \sQuote{Note})

  as if no seed had yet been set.

  The use of \code{kind = NULL}, \code{normal.kind = NULL} or 

  \code{sample.kind = NULL} in

  \code{RNGkind} or \code{set.seed} selects the currently-used

  generator (including that used in the previous session if the

  workspace has been restored): if no generator has been used it selects

  \code{"default"}.

}

\note{

  Initially, there is no seed; a new one is created from the current

  time and the process ID when one is required.  Hence different

  sessions will give different simulation results, by default.  However,

  the seed might be restored from a previous session if a previously

  saved workspace is restored.

  \code{.Random.seed} saves the seed set for the uniform random-number

  generator, at least for the system generators.  It does not

  necessarily save the state of other generators, and in particular does

  not save the state of the Box--Muller normal generator.  If you want

  to reproduce work later, call \code{set.seed} (preferably with

  explicit values for \code{kind} and \code{normal.kind}) rather than

  set \code{.Random.seed}.

  The object \code{.Random.seed} is only looked for in the user's

  workspace.

  Do not rely on randomness of low-order bits from RNGs.  Most of the

  supplied uniform generators return 32-bit integer values that are

  converted to doubles, so they take at most \eqn{2^{32}}{2^32} distinct

  values and long runs will return duplicated values (Wichmann-Hill is

  the exception, and all give at least 30 varying bits.)

}

\value{

  \code{.Random.seed} is an \code{\link{integer}} vector whose first

  element \emph{codes} the kind of RNG and normal generator. The lowest

  two decimal digits are in \code{0:(k-1)}

  where \code{k} is the number of available RNGs.  The hundreds

  represent the type of normal generator (starting at \code{0}), and

  the ten thousands represent the type of discrete uniform sampler.

  In the underlying C, \code{.Random.seed[-1]} is \code{unsigned};

  therefore in \R \code{.Random.seed[-1]} can be negative, due to

  the representation of an unsigned integer by a signed integer.

  \code{RNGkind} returns a three-element character vector of the RNG,

  normal and sample kinds selected \emph{before} the call, invisibly if 

  either argument is not \code{NULL}.  A type starts a session as the 

  default, and is selected either by a call to \code{RNGkind} or by setting

  \code{.Random.seed} in the workspace. (NB: prior to \R 3.6.0 the first

  two kinds were returned in a two-element character vector.)

  \code{RNGversion} returns the same information as \code{RNGkind} about

  the defaults in a specific \R version.

  \code{set.seed} returns \code{NULL}, invisibly.

}

\references{

  Ahrens, J. H. and Dieter, U. (1973).

  Extensions of Forsythe's method for random sampling from the normal

  distribution.

  \emph{Mathematics of Computation}, \bold{27}, 927--937.

  Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988).

  \emph{The New S Language}.

  Wadsworth & Brooks/Cole.

  (\code{set.seed}, storing in \code{.Random.seed}.)

  Box, G. E. P. and Muller, M. E. (1958).

  A note on the generation of normal random deviates.

  \emph{Annals of Mathematical Statistics}, \bold{29}, 610--611.

  \doi{10.1214/aoms/1177706645}.

  De Matteis, A. and Pagnutti, S. (1993).

  Long-range Correlation Analysis of the Wichmann-Hill Random Number

  Generator.

  \emph{Statistics and Computing}, \bold{3}, 67--70.

  \doi{10.1007/BF00153065}.

  Kinderman, A. J. and Ramage, J. G. (1976).

  Computer generation of normal random variables.

  \emph{Journal of the American Statistical Association}, \bold{71},

  893--896.

  \doi{10.2307/2286857}.

  Knuth, D. E. (1997).

  \emph{The Art of Computer Programming}.

  Volume 2, third edition.\cr

  Source code at \url{http://www-cs-faculty.stanford.edu/~knuth/taocp.html}.

  Knuth, D. E. (2002).

  \emph{The Art of Computer Programming}.

  Volume 2, third edition, ninth printing.

  L'Ecuyer, P. (1999).

  Good parameters and implementations for combined multiple recursive

  random number generators.

  \emph{Operations Research}, \bold{47}, 159--164.

  \doi{10.1287/opre.47.1.159}.

  Marsaglia, G. (1997).

  \emph{A random number generator for C}.

  Discussion paper, posting on Usenet newsgroup \code{sci.stat.math} on

  September 29, 1997.

  Marsaglia, G. and Zaman, A. (1994).

  Some portable very-long-period random number generators.

  \emph{Computers in Physics}, \bold{8}, 117--121.

  \doi{10.1063/1.168514}.

  Matsumoto, M. and Nishimura, T. (1998).

  Mersenne Twister: A 623-dimensionally equidistributed uniform

  pseudo-random number generator,

  \emph{ACM Transactions on Modeling and Computer Simulation},

  \bold{8}, 3--30.\cr

  Source code formerly at \code{http://www.math.keio.ac.jp/~matumoto/emt.html}.\cr

  Now see \url{http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/VERSIONS/C-LANG/c-lang.html}.

  Reeds, J., Hubert, S. and Abrahams, M. (1982--4).

  C implementation of SuperDuper, University of California at Berkeley.

  (Personal communication from Jim Reeds to Ross Ihaka.)

  Wichmann, B. A.  and Hill, I. D. (1982).

  Algorithm AS 183: An Efficient and Portable Pseudo-random Number

  Generator.

  \emph{Applied Statistics}, \bold{31}, 188--190; Remarks:

  \bold{34}, 198 and \bold{35}, 89.

  \doi{10.2307/2347988}.

}

\author{of RNGkind: Martin Maechler. Current implementation, B. D. Ripley

  with modifications by Duncan Murdoch.}

\seealso{

  \code{\link{sample}} for random sampling with and without replacement.

  \link{Distributions} for functions for random-variate generation from

  standard distributions.

}

\examples{

require(stats)

## Seed the current RNG, i.e., set the RNG status

set.seed(42); u1 <- runif(30)

set.seed(42); u2 <- runif(30) # the same because of identical RNG status:

stopifnot(identical(u1, u2))

\donttest{## the default random seed is 626 integers, so only print a few

 runif(1); .Random.seed[1:6]; runif(1); .Random.seed[1:6]

 ## If there is no seed, a "random" new one is created:

 rm(.Random.seed); runif(1); .Random.seed[1:6]

}

ok <- RNGkind()

RNGkind("Wich")  # (partial string matching on 'kind')

## This shows how 'runif(.)' works for Wichmann-Hill,

## using only R functions:

p.WH <- c(30269, 30307, 30323)

a.WH <- c(  171,   172,   170)

next.WHseed <- function(i.seed = .Random.seed[-1])

  { (a.WH * i.seed) \%\% p.WH }

my.runif1 <- function(i.seed = .Random.seed)

  { ns <- next.WHseed(i.seed[-1]); sum(ns / p.WH) \%\% 1 }

set.seed(1998-12-04)# (when the next lines were added to the souRce)

rs <- .Random.seed

(WHs <- next.WHseed(rs[-1]))

u <- runif(1)

stopifnot(

 next.WHseed(rs[-1]) == .Random.seed[-1],

 all.equal(u, my.runif1(rs))

)

## ----

.Random.seed

RNGkind("Super") # matches  "Super-Duper"

RNGkind()

.Random.seed # new, corresponding to  Super-Duper

## Reset:

RNGkind(ok[1])

RNGversion(getRversion()) # the default version for this R version

## ----

sum(duplicated(runif(1e6))) # around 110 for default generator

## and we would expect about almost sure duplicates beyond about

qbirthday(1 - 1e-6, classes = 2e9) # 235,000

}

\keyword{distribution}

\keyword{sysdata}