Random Number Generation

Description

.Random.seed is an integer vector, containing the random number generator (RNG) state for random number generation in R. It can be saved and restored, but should not be altered by the user.

RNGkind is a more friendly interface to query or set the kind of RNG in use.

RNGversion can be used to set the random generators as they were in an earlier R version (for reproducibility).

set.seed is the recommended way to specify seeds.

Usage

.Random.seed <- c(rng.kind, n1, n2, ...)

RNGkind(kind = NULL, normal.kind = NULL, sample.kind = NULL)
RNGversion(vstr)
set.seed(seed, kind = NULL, normal.kind = NULL, sample.kind = NULL)

Arguments

Details

The currently available RNG kinds are given below. kind is partially matched to this list. The default is "Mersenne-Twister".

"Wichmann-Hill"

The seed, .Random.seed[-1] == r[1:3] is an integer vector of length 3, where each r[i] is in 1:(p[i] - 1), where p is the length 3 vector of primes, p = (30269, 30307, 30323). The Wichmann–Hill generator has a cycle length of 6.9536 \times 10^{12} (= prod(p-1)/4, see Applied Statistics (1984) 33, 123 which corrects the original article). It exhibits 12 clear failures in the TestU01 Crush suite and 22 in the BigCrush suite (L'Ecuyer, 2007).

"Marsaglia-Multicarry":

A multiply-with-carry RNG is used, as recommended by George Marsaglia in his post to the mailing list ‘sci.stat.math’. It has a period of more than 2^{60}.

It exhibits 40 clear failures in L'Ecuyer's TestU01 Crush suite. Combined with Ahrens-Dieter or Kinderman-Ramage it exhibits deviations from normality even for univariate distribution generation. See PR#18168 for a discussion.

The seed is two integers (all values allowed).

"Super-Duper":

Marsaglia's famous Super-Duper from the 70's. This is the original version which does not pass the MTUPLE test of the Diehard battery. It has a period of \approx 4.6\times 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.

It exhibits 25 clear failures in the TestU01 Crush suite (L'Ecuyer, 2007).

"Mersenne-Twister":

From Matsumoto and Nishimura (1998); code updated in 2002. A twisted GFSR with period 2^{19937} - 1 and equidistribution in 623 consecutive dimensions (over the whole period). The ‘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.

It exhibits 2 clear failures in each of the TestU01 Crush and the BigCrush suite (L'Ecuyer, 2007).

"Knuth-TAOCP-2002":

A 32-bit integer GFSR using lagged Fibonacci sequences with subtraction. That is, the recurrence used is

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

and the ‘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 2^{129}.

"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 ‘weakness’, and already scrambled the seeds. Otherwise, the algorithm is identical to Knuth-TAOCP-2002, with the same lagged Fibonacci recurrence formula.

Initialization of this generator is done in interpreted R code and so takes a short but noticeable time.

It exhibits 3 clear failure in the TestU01 Crush suite and 4 clear failures in the BigCrush suite (L'Ecuyer, 2007).

"L'Ecuyer-CMRG":

A ‘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 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 4294967087 and 4294944443 respectively.

This is not particularly interesting of itself, but provides the basis for the multiple streams used in package parallel.

It exhibits 6 clear failures in each of the TestU01 Crush and the BigCrush suite (L'Ecuyer, 2007).

"user-supplied":

Use a user-supplied generator. See Random.user for details.

normal.kind can be "Kinderman-Ramage", "Buggy Kinderman-Ramage" (not for set.seed), "Ahrens-Dieter", "Box-Muller", "Inversion" (the default), or "user-supplied". (For inversion, see the reference in qnorm.) The Kinderman-Ramage generator used in versions prior to 1.7.0 (now called "Buggy") had several approximation errors and should only be used for reproduction of old results. The "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 kind is changed.

sample.kind can be "Rounding" or "Rejection", or partial matches to these. The former was the default in versions prior to 3.6.0: it made sample noticeably non-uniform on large populations, and should only be used for reproduction of old results. See PR#17494 for a discussion.

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 "Mersenne-Twister" and "Knuth-TAOCP"). There is no guarantee that different values of seed will seed the RNG differently, although any exceptions would be extremely rare. If called with seed = NULL it re-initializes (see ‘Note’) as if no seed had yet been set.

The use of kind = NULL, normal.kind = NULL or sample.kind = NULL in RNGkind or 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 "default".

Value

.Random.seed is an integer vector whose first element codes the kind of RNG and normal generator. The lowest two decimal digits are in 0:(k-1) where k is the number of available RNGs. The hundreds represent the type of normal generator (starting at 0), and the ten thousands represent the type of discrete uniform sampler.

In the underlying C, .Random.seed[-1] is unsigned; therefore in R .Random.seed[-1] can be negative, due to the representation of an unsigned integer by a signed integer.

RNGkind returns a three-element character vector of the RNG, normal and sample kinds selected before the call, invisibly if either argument is not NULL. A type starts a session as the default, and is selected either by a call to RNGkind or by setting .Random.seed in the workspace. (NB: prior to R 3.6.0 the first two kinds were returned in a two-element character vector.)

RNGversion returns the same information as RNGkind about the defaults in a specific R version.

set.seed returns NULL, invisibly.

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.

.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 set.seed (preferably with explicit values for kind and normal.kind) rather than set .Random.seed.

The object .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 2^{32} distinct values and long runs will return duplicated values (Wichmann-Hill is the exception, and all give at least 30 varying bits.)

Author(s)

of RNGkind: Martin Maechler. Current implementation, B. D. Ripley with modifications by Duncan Murdoch.

References

Ahrens, J. H. and Dieter, U. (1973). Extensions of Forsythe's method for random sampling from the normal distribution. Mathematics of Computation, 27, 927–937.

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988). The New S Language. Wadsworth & Brooks/Cole. (set.seed, storing in .Random.seed.)

Box, G. E. P. and Muller, M. E. (1958). A note on the generation of normal random deviates. Annals of Mathematical Statistics, 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. Statistics and Computing, 3, 67–70. doi:10.1007/BF00153065.

Kinderman, A. J. and Ramage, J. G. (1976). Computer generation of normal random variables. Journal of the American Statistical Association, 71, 893–896. doi:10.2307/2286857.

Knuth, D. E. (1997). The Art of Computer Programming. Volume 2, third edition.
Source code at https://www-cs-faculty.stanford.edu/~knuth/taocp.html.

Knuth, D. E. (2002). 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. Operations Research, 47, 159–164. doi:10.1287/opre.47.1.159.

L'Ecuyer, P. and Simard, R. (2007). TestU01: A C Library for Empirical Testing of Random Number Generators ACM Transactions on Mathematical Software, 33, Article 22. doi:10.1145/1268776.1268777.
The TestU01 C library is available from https://simul.iro.umontreal.ca/testu01/tu01.html or also https://github.com/umontreal-simul/TestU01-2009.

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

Marsaglia, G. and Zaman, A. (1994). Some portable very-long-period random number generators. Computers in Physics, 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, ACM Transactions on Modeling and Computer Simulation, 8, 3–30.
Source code formerly at http://www.math.keio.ac.jp/~matumoto/emt.html.
Now see 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. Applied Statistics, 31, 188–190; Remarks: 34, 198 and 35, 89. doi:10.2307/2347988.

See Also

sample for random sampling with and without replacement.

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))

## 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