pseudo.randtoolbox {randtoolbox} | R Documentation |
General linear congruential generators such as Park Miller sequence, generalized feedback shift register such as SF-Mersenne Twister algorithm and WELL generator; and a quasi random generator (pseudo random generators) and the Torus algorithm (quasi random generation).
congruRand(n, dim = 1, mod = 2^31-1, mult = 16807, incr = 0, echo) SFMT(n, dim = 1, mexp = 19937, usepset = TRUE, withtorus = FALSE, usetime = FALSE) WELL(n, dim = 1, order = 512, temper = FALSE, version = "a") knuthTAOCP(n, dim = 1) setSeed(seed)
n |
number of observations. If length(n) > 1, the length is taken to be the required number. |
dim |
dimension of observations (must be <=100 000, default 1). |
seed |
a single value, interpreted as a positive integer for the seed. e.g. append your day, your month and your year of birth. |
mod |
an integer defining the modulus of the linear congruential generator. |
mult |
an integer defining the multiplier of the linear congruential generator. |
incr |
an integer defining the increment of the linear congruential generator. |
echo |
a logical to plot the seed while computing the sequence. |
mexp |
an integer for the mersenne exponent of SFMT algorithm. see details |
withtorus |
a numeric in ]0,1] defining the proportion of the torus sequence appended to the SFMT sequence; or a logical equals to FALSE (default). |
usepset |
a logical to use a set of 12 parameters set for SFMT. default TRUE. |
usetime |
a logical to use the machine time to start the Torus sequence, default TRUE. if FALSE, the Torus sequence start from the first term. |
order |
a positive integer for the order of the characteristic polynomial. see details |
temper |
a logical if you want to do a tempering stage. see details |
version |
a character either 'a' or 'b'. see details |
The currently available generator are given below.
[ ( a * u_{k-1} + c ) mod m ] / m
where a denotes the multiplier, c the increment and m the modulus, with the constraint 0 <= a < m and 0 <= c < m . The default setting is the Park Miller sequence with a=16807, m=2^31-1 and c=0.
SFMT
function implements the SIMD-oriented Fast Mersenne Twister algorithm
(cf. http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/SFMT/index.html).
The SFMT generator has a period of length 2^m-1 where m is a
Mersenne exponent. In the function SFMT
, m is given through
mexp
argument. By default it is 19937 like the ''old'' MT algorithm.
The possible values for the Mersenne exponent are 607, 1279, 2281, 4253,
11213, 19937, 44497, 86243, 132049, 216091.
There are numerous parameters
for the SFMT algorithm (see the article for details). By default, we use
a different set of parameters (among 32 sets)
at each call of SFMT
(usepset=TRUE
).
The user can use a fixed set of parameters with usepset=FALSE
. Let us
note there is for the moment just one set of parameters for 44497, 86243, 132049,
216091 mersenne exponent.
Sets of parameters can be found in appendix of the vignette.
The use of different parameter sets is motivated by the following citation of Matsumoto and Saito on this topic :
"Using one same pesudorandom number generator for generating multiple independent streams by changing the initial values may cause a problem (with negligibly small probability). To avoid the problem, using diffrent parameters for each generation is prefered. See Matsumoto M. and Nishimura T. (1998) for detailed information."
All the C code for SFMT algorithm used in this package is the code of M. Matsumoto and M. Saito (cf. http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html), except we add some C code to interface with R. Streaming SIMD Extensions 2 (SSE2) operations are not yet supported.
The order
argument of WELL
generator is the order of the characteristic polynomial, which is denoted by k in
Paneton F., L'Ecuyer P. and Matsumoto M. (2006). Possible values for order
are 512, 521, 607, 1024 where no tempering are needed (thus possible).
Order can also be 800, 19937, 21071, 23209, 44497 where a tempering stage
is possible through the temper
argument.
Furthermore a possible 'b' version of WELL RNGs are possible for the
following order 521, 607, 1024, 800, 19937, 23209 with the version
argument.
All the C code for WELL generator used in this package is the code of P. L'Ecuyer (cf. http://www.iro.umontreal.ca/~lecuyer/), except some C code, we add, to interface with R.
x_n = (x_{n-37} + x_{n-100}) mod 2^{30},
In R, there is the integer version of this generator.
All the C code for this generator called RAN_ARRAY by Knuth is the code of D. Knuth (cf. http://www-cs-faculty.stanford.edu/~knuth/news02.html#rng) except some C code, we add, to interface with R.
setSeed
is similar to the function set.seed
in R. It sets
the seed to the one given by the user. Do not use a seed with too few ones in its
binary representation. Generally, we append our day, our month and our year of birth or
append a day, a month and a year. We recall by default with use the machine time
to set the seed except for quasi random number generation.
See the pdf vignette for details.
SFMT
, WELL
, congruRand
and knuthTAOCP
generates random variables in ]0,1[, [0,1[ and [0,1[ respectively. It returns a nxdim matrix, when dim
>1 otherwise a vector of length n
.
setSeed
set the seed of the randtoolbox
package
(i.e. both for the knuthTAOCP
, SFMT
, WELL
and congruRand
functions).
Christophe Dutang.
Knuth D. (1997), The Art of Computer Programming V2 Seminumerical Algorithms, Third Edition, Massachusetts: Addison-Wesley.
Matsumoto M. and Nishimura T. (1998), Dynamic Creation of Pseudorandom Number Generators, Monte Carlo and Quasi-Monte Carlo Methods, Springer, pp 56–69. (available online)
Matsumoto M., Saito M. (2008), SIMD-oriented Fast Mersenne Twister: a 128-bit Pseudorandom Number Generator. (available online)
Paneton F., L'Ecuyer P. and Matsumoto M. (2006), Improved Long-Period Generators Based on Linear Recurrences Modulo 2, ACM Transactions on Mathematical Software. (preprint available online)
Park S. K., Miller K. W. (1988), Random number generators: good ones are hard to find. Association for Computing Machinery, vol. 31, 10, pp 1192-2001. (available online)
Wikipedia (2008), a linear congruential generator.
.Random.seed
for what is done in R about random number generation.
# (1) the Park Miller sequence # # Park Miller sequence, i.e. mod = 2^31-1, mult = 16807, incr=0 # the first 10 seeds used in Park Miller sequence # 16807 1 # 282475249 2 # 1622650073 3 # 984943658 4 # 1144108930 5 # 470211272 6 # 101027544 7 # 1457850878 8 # 1458777923 9 # 2007237709 10 setSeed(1) congruRand(10, echo=TRUE) # the 9998+ th terms # 925166085 9998 # 1484786315 9999 # 1043618065 10000 # 1589873406 10001 # 2010798668 10002 setSeed(1614852353) #seed for the 9997th term congruRand(5, echo=TRUE) # (2) the SF Mersenne Twister algorithm SFMT(1000) #Kolmogorov Smirnov test #KS statistic should be around 0.037 ks.test(SFMT(1000), punif) #KS statistic should be around 0.0076 ks.test(SFMT(10000), punif) #different mersenne exponent with a fixed parameter set # SFMT(10, mexp = 607, usepset = FALSE) SFMT(10, mexp = 1279, usepset = FALSE) SFMT(10, mexp = 2281, usepset = FALSE) SFMT(10, mexp = 4253, usepset = FALSE) SFMT(10, mexp = 11213, usepset = FALSE) SFMT(10, mexp = 19937, usepset = FALSE) SFMT(10, mexp = 44497, usepset = FALSE) SFMT(10, mexp = 86243, usepset = FALSE) SFMT(10, mexp = 132049, usepset = FALSE) SFMT(10, mexp = 216091, usepset = FALSE) #use different sets of parameters [default when possible] # for(i in 1:7) print(SFMT(1, mexp = 607)) for(i in 1:7) print(SFMT(1, mexp = 2281)) for(i in 1:7) print(SFMT(1, mexp = 4253)) for(i in 1:7) print(SFMT(1, mexp = 11213)) for(i in 1:7) print(SFMT(1, mexp = 19937)) #use a fixed set and a fixed seed #should be the same output setSeed(08082008) SFMT(1, usepset = FALSE) setSeed(08082008) SFMT(1, usepset = FALSE) # (3) withtorus argument # # one third of outputs comes from Torus algorithm u <- SFMT(1000, with=1/3) # the third term of the following code is the first term of torus sequence print(u[666:670] ) # (4) WELL generator # # 'basic' calls # WELL512 WELL(10, order = 512) # WELL1024 WELL(10, order = 1024) # WELL19937 WELL(10, order = 19937) # WELL44497 WELL(10, order = 44497) # WELL19937 with tempering WELL(10, order = 19937, temper = TRUE) # WELL44497 with tempering WELL(10, order = 44497, temper = TRUE) # tempering vs no tempering setSeed(08082008) WELL(10, order =19937) setSeed(08082008) WELL(10, order =19937, temper=TRUE) # (5) Knuth TAOCP generator # knuthTAOCP(10) knuthTAOCP(10, 2) # (6) How to set the seed? # all example is duplicated to ensure setSeed works # congruRand setSeed(1302) congruRand(1) setSeed(1302) congruRand(1) # SFMT setSeed(1302) SFMT(1, usepset=FALSE) setSeed(1302) SFMT(1, usepset=FALSE) # BEWARE if you do not set usepset to FALSE setSeed(1302) SFMT(1) setSeed(1302) SFMT(1) # WELL setSeed(1302) WELL(1) setSeed(1302) WELL(1) # Knuth TAOCP setSeed(1302) knuthTAOCP(1) setSeed(1302) knuthTAOCP(1) # (7) computation times on my macbook, mean of 1000 runs # ## Not run: # algorithm time in seconds for n=10^6 # classical Mersenne Twister 0.066 # SF Mersenne Twister 0.044 # WELL generator 0.065 # Knuth TAOCP 0.046 # Park Miller 0.108 n <- 1e+06 mean( replicate( 1000, system.time( runif(n), gcFirst=TRUE)[3]) ) mean( replicate( 1000, system.time( SFMT(n), gcFirst=TRUE)[3]) ) mean( replicate( 1000, system.time( WELL(n), gcFirst=TRUE)[3]) ) mean( replicate( 1000, system.time( knuthTAOCP(n), gcFirst=TRUE)[3]) ) mean( replicate( 1000, system.time( congruRand(n), gcFirst=TRUE)[3]) ) ## End(Not run)