\HeaderA{Random}{Random Number Generation}{Random}
\aliasA{.Random.seed}{Random}{.Random.seed}
\aliasA{RNG}{Random}{RNG}
\aliasA{RNGkind}{Random}{RNGkind}
\aliasA{RNGversion}{Random}{RNGversion}
\aliasA{set.seed}{Random}{set.seed}
\keyword{distribution}{Random}
\keyword{sysdata}{Random}
\begin{Description}\relax
\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.
\end{Description}
\begin{Usage}
\begin{verbatim}
.Random.seed <- c(rng.kind, n1, n2, ...)
save.seed <- .Random.seed

RNGkind(kind = NULL, normal.kind = NULL)
RNGversion(vstr)
set.seed(seed, kind = NULL, normal.kind = NULL)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{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[\code{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[\code{seed}] a single value, interpreted as an integer.
\item[\code{vstr}] a character string containing a version number,
e.g., \code{"1.6.2"}
\item[\code{rng.kind}] integer code in \code{0:k} for the above \code{kind}.
\item[\code{n1, n2, ...}] integers.  See the details for how many are required
(which depends on \code{rng.kind}).
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
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 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). 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.


\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{"user-supplied":}] Use a user-supplied generator. See \code{\LinkA{Random.user}{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{\LinkA{qnorm}{qnorm}}.)  The Kinderman-Ramage generator used in versions
prior to 1.7.1 (now called \code{"Buggy"} had several approximation
errors and should only be used for reproduction of older 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{set.seed} uses its 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"}).

The use of \code{kind=NULL} or \code{normal.kind=NULL} in
\code{RNGkind} or \code{set.seed} selects the currently-used
generator (including as from \R{} 2.8.0 that used in the previous
session if the workspace has been restored): if no generator has been
used it selects \code{"default"}.
\end{Details}
\begin{Value}
\code{.Random.seed} is an \code{\LinkA{integer}{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}).

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 two-element character vector of the RNG and
normal 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.

\code{RNGversion} returns the same information as \code{RNGkind} about
the defaults in a specific \R{} version.

\code{set.seed} returns \code{NULL}, invisibly.
\end{Value}
\begin{Note}\relax
Initially, there is no seed;  a new one is created from the current
time when one is required.  Hence, different sessions started at
(sufficiently) different times 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.)
\end{Note}
\begin{Author}\relax
of RNGkind: Martin Maechler. Current implementation, B. D. Ripley
\end{Author}
\begin{References}\relax
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 Mathmatical Statistics} \bold{29},
610--611.

De Matteis, A. and Pagnutti, S. (1993)
\emph{Long-range Correlation Analysis of the Wichmann-Hill Random
Number Generator}, Statist. Comput., \bold{3}, 67--70.

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.

Knuth, D. E. (1997)
\emph{The Art of Computer Programming.} Volume 2, third edition.\\
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.\\
See \url{http://Sunburn.Stanford.EDU/~knuth/news02.html}.

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.

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.\\
Source code at \url{http://www.math.keio.ac.jp/~matumoto/emt.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)
\emph{Algorithm AS 183: An Efficient and Portable Pseudo-random Number
Generator}, Applied Statistics, \bold{31}, 188--190; Remarks:
\bold{34}, 198 and \bold{35}, 89.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{runif}{runif}}, \code{\LinkA{rnorm}{rnorm}}, \ldots.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
require(stats)

## 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 }
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])

## ----
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
\end{ExampleCode}
\end{Examples}

