Toolbox for pseudo and quasi random number generation

Description

General linear congruential generators such as Park Miller sequence, generalized feedback shift register such as SF-Mersenne Twister algorithm and WELL generator.

The list of supported generators consists of generators available via direct functions and generators available via set.generator() and runif() interface. Most of the generators belong to both these groups, but some generators are only available directly (SFMT) and some only via runif() interface (Mersenne Twister 2002). This help page describes the list of all the supported generators and the functions for the direct access to those, which are available in this way. See set.generator() for the generators available via runif() interface.

Usage

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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)

Arguments

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

Details

The currently available generator are given below.

Linear congruential generators:

The kth term of a linear congruential generator is defined as

[ ( 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.

Knuth TAOCP 2002 (double version):

The Knuth-TACOP-2002 is a Fibonnaci-lagged generator invented by Knuth(2002), based on the following recurrence.

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/~uno/programs.html) except some C code, we add, to interface with R.

Mersenne Twister 2002 generator:

The generator suggested by Makoto Matsumoto and Takuji Nishimura with the improved initialization from 2002. See http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html for more information on the generator itself. This generator is available only via set.generator() and runif() interface. Mersenne Twister generator used in base R is the same generator (the recurrence), but with a different initialization and the output transformation. The implementation included in randtoolbox allows to generate the same random numbers as in Matlab, see examples in set.generator().

SF Mersenne-Twister algorithm:

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.

WELL generator:

The WELL (which stands for Well Equidistributed Long-period Linear) is in a sentence a generator with better equidistribution than Mersenne Twister algorithm but this gain of quality has to be paid by a slight higher cost of time. See Panneton et al. (2006) for details.

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.

Set the seed:

The function 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.

Set the generator:

Some of the generators are available using runif() interface. See set.generator() for more information.

See the pdf vignette for details.

Value

SFMT, WELL, congruRand and knuthTAOCP generate 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 sets the seed of the randtoolbox package (i.e. both for the knuthTAOCP, SFMT, WELL and congruRand functions).

Author(s)

Christophe Dutang and Petr Savicky

References

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.

See Also

.Random.seed for what is done in R about random number generation and runifInterface for the runif interface.

Examples

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require(rngWELL)

# (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
setSeed4WELL(08082008)
WELL(10, order =19937)
setSeed4WELL(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)