\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)
\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. If it is \code{NULL},
return the currently used RNG. Use \code{"default"} to return to the
\R{} default.
\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.
\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 \dQuote{seed} is a
624-dimensional set of 32-bit integers plus a current position in
that set.


\item[\code{"Knuth-TAOCP":}] From Knuth (1997).  A 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 \dQuote{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-2002":}] The 2002 version which 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.


\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"},
\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 had several approximation errors and should only be used for
reproduction of older results.

\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"}).
\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 in use \emph{before} the call, invisibly if either
argument is not \code{NULL}. \code{RNGversion} returns the same
information.

\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 will give
different simulation results, by default.

\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} rather than set
\code{.Random.seed}.

As from \R{} 1.8.0, \code{.Random.seed} is only looked for in the user's
workspace.

All 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.
\end{Note}
\begin{Author}\relax
of RNGkind: Martin Maechler. Current implementation, B. D. Ripley
\end{Author}
\begin{References}\relax
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}.)

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.

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.

Marsaglia, G. (1997) \emph{A random number generator for C.} Discussion
paper, posting on Usenet newsgroup \code{sci.stat.math} on
September 29, 1997.

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.)

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}.

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}.


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.

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.

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.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{runif}{runif}}, \code{\LinkA{rnorm}{rnorm}}, \ldots.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
## 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]

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
ok <- RNGkind()
RNGkind("Super")#matches  "Super-Duper"
RNGkind()
.Random.seed # new, corresponding to  Super-Duper

## Reset:
RNGkind(ok[1])

## ----
sum(duplicated(runif(1e6))) # around 110
## and we would expect about almost sure duplicates beyond about
qbirthday(1-1e-6, classes=2e9) # 235,000
\end{ExampleCode}
\end{Examples}

