The following notes are taken from the R documentation help files (R version 2.9.2). The information outlines several types of random number generators and includes many references to related material.
.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.
.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)
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 6.9536e12 (= prod(p-1)/4, see Applied Statistics (1984) 33, 123 which corrects the original article).
"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 and has passed all tests (according to Marsaglia). 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 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 .Random.seed[1:12] is possible but we will not publish one, not least as this generator is not exactly the same as that in recent versions of S-PLUS.
"Mersenne-Twister":
From Matsumoto and Nishimura (1998). 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.
"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]) mod 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.
Initialization of this generator is done in interpreted R code and so takes a short but noticeable time.
"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 1.7.1 (now called "Buggy" had several approximation errors and should only be used for reproduction of older 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.
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 "Mersenne-Twister" and "Knuth-TAOCP").
The use of kind=NULL or normal.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".
.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).
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 two-element character vector of the RNG and normal 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.
RNGversion returns the same information as RNGkind about the defaults in a specific R version.
set.seed returns NULL, invisibly.
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.
.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.)
of RNGkind: Martin Maechler. Current implementation, B. D. Ripley
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.
De Matteis, A. and Pagnutti, S. (1993) Long-range Correlation Analysis of the Wichmann-Hill Random Number Generator, Statist. Comput., 3, 67–70.
Kinderman, A. J. and Ramage, J. G. (1976) Computer generation of normal random variables. Journal of the American Statistical Association 71, 893-896.
Knuth, D. E. (1997) The Art of Computer Programming. Volume 2, third edition.
Source code at http://www-cs-faculty.stanford.edu/~knuth/taocp.html.
Knuth, D. E. (2002) The Art of Computer Programming. Volume 2, third edition, ninth printing.
See http://Sunburn.Stanford.EDU/~knuth/news02.html.
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.
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 at 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) Algorithm AS 183: An Efficient and Portable Pseudo-random Number Generator, Applied Statistics, 31, 188–190; Remarks: 34, 198 and 35, 89.