# pseudoRNG: Toolbox for pseudo and quasi random number generation In randtoolbox: Toolbox for Pseudo and Quasi Random Number Generation and RNG Tests

## 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

 ```1 2 3 4 5 6``` ```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.

`.Random.seed` for what is done in R about random number generation and `runifInterface` for the `runif` interface.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154``` ```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) ```