quasiRNG: Toolbox for quasi random number generation

Description

the Torus algorithm, the Sobol and Halton sequences.

Usage

torus(n, dim = 1, prime, init = TRUE, mixed = FALSE, usetime = FALSE, 
                  normal = FALSE, mexp = 19937, start = 1)
sobol(n, dim = 1, init = TRUE, scrambling = 0, seed = NULL, normal = FALSE,
                   mixed = FALSE, method = "C", mexp = 19937, start = 1,
                   maxit = 10)
halton(n, dim = 1, init = TRUE, normal = FALSE, usetime = FALSE, 
                    mixed = FALSE, method = "C", mexp = 19937, start = 1)

Value

torus, halton and sobol generates random variables in [0,1). It returns a $n$x$dim$ matrix, when dim>1

otherwise a vector of length n.

Arguments

n: number of observations. If length(n) > 1, the length is taken to be the required number.
dim: dimension of observations default 1.
init: a logical, if TRUE the sequence is initialized and restarts to the start value, otherwise not. By default TRUE.
normal: a logical if normal deviates are needed, default FALSE.
scrambling: an integer value, if 1, 2 or 3 the sequence is scrambled otherwise not. If scrambling=1, Owen type type of scrambling is applied, if scrambling=2, Faure-Tezuka type of scrambling, is applied, and if scrambling=3, both Owen+Faure-Tezuka type of scrambling is applied. By default 0.
seed: an integer value, the random seed for initialization of the scrambling process (only for sobol with scrambling>0). If NULL, set to 4711.
prime: a single prime number or a vector of prime numbers to be used in the Torus sequence. (optional argument).
mixed: a logical to combine the QMC algorithm with the SFMT algorithm, default FALSE.
usetime: a logical to use the machine time to start the Torus sequence, default FALSE, i.e. when usetime=FALSE the Torus sequence start from the first term. usetime is only used when mixed=FALSE.
method: a character string either "C". Note that mixed=TRUE is only available when method="C".
mexp: an integer for the Mersenne exponent of SFMT algorithm, only used when mixed=TRUE.
start: an integer 0 or 1 to initiliaze the sequence, default to 1, only used when init=TRUE.
maxit: a positive integer used to control inner loops both for generating randomized seed and for controlling outputs (when needed).

Author

Christophe Dutang and Diethelm Wuertz

Details

Scrambling is temporarily disabled and will be reintroduced in a future release.

The currently available generator are given below. Whatever the sequence, when normal=TRUE, outputs are transformed with the quantile of the standard normal distribution qnorm. If init=TRUE, the default, unscrambled and unmixed-SFMT quasi-random sequences start from start. If start != 0 and normal=FALSE, we suggest to use 0 as recommended by Owen (2020). One must handle the starting value (0) correctly if a quantile function of a non-lower-bounded distribution is used.

Torus algorithm:

The $k$th term of the Torus algorithm in d dimension is given by $$u_k = \left(frac(k \sqrt{p_1}), ..., frac(k \sqrt{p_d}) \right)$$ where $p_i$ denotes the ith prime number, $frac$ the fractional part (i.e. $frac(x) = x-floor(x)$). We use the 100 000 first prime numbers from https://primes.utm.edu/, thus the dimension is limited to 100 000. If the user supplys prime numbers through the argument prime, we do NOT check for primality and we cast numerics to integers, (i.e. prime=7.1234 will be cast to prime=7 before computing Torus sequence). The Torus sequence starts from $k=1$ when initialized with init = TRUE and so not depending on machine time usetime = FALSE. This is the default. When init = FALSE, the sequence is not initialized (to 1) and starts from the last term. We can also use the machine time to start the sequence with usetime = TRUE, which overrides init or a randomized when mixed = TRUE.

(scrambled) Sobol sequences

Computes uniform Sobol low discrepancy numbers. The sequence starts from $k=1$ when initialized with init = TRUE (default). When scrambling > 0, a scrambling is performed or when mixed = TRUE, a randomized seed is performed. If some number of Sobol sequences are generated outside [0,1) with scrambling, the seed is randomized until we obtain all numbers in [0,1). One version of Sobol sequences is available the current version in Fortran (method = "Fortran") since method = "C" is under development.

Halton sequences

Calculates a matrix of uniform or normal deviated halton low discrepancy numbers. Let us note that Halton sequence in dimension is the Van Der Corput sequence. The Halton sequence starts from $k=1$ when initialized with init = TRUE (default) and no not depending on machine time usetime = FALSE. When init = FALSE, the sequence is not initialized (to 1) and starts from the last term. We can also use the machine time to start the sequence with usetime = TRUE, which overrides init. Two versions of Halton sequences are available the historical version in Fortran (method = "Fortran") and the new version in C (method = "C"). If method = "C", mixed argument can be used to randomized the Halton sequences.

See the pdf vignette for details.

References

Bratley P., Fox B.L. (1988), Algorithm 659: Implementing Sobol's Quasirandom Sequence Generator, ACM Transactions on Mathematical Software 14, 88--100.

Joe S., Kuo F.Y. (2003), Remark on Algorithm 659: Implementing Sobol's Quaisrandom Seqence Generator, ACM Transactions on Mathematical Software 29, 49--57.

Joe S., Kuo F.Y. (2008), Constructing Sobol sequences with better two-dimensional projections, SIAM J. Sci. Comput. 30, 2635--2654, https://web.maths.unsw.edu.au/~fkuo/sobol/index.html.

Owen A.B. (2020), On dropping the first Sobol' point, https://arxiv.org/abs/2008.08051.

Planchet F., Jacquemin J. (2003), L'utilisation de methodes de simulation en assurance. Bulletin Francais d'Actuariat, vol. 6, 11, 3-69. (available online)

Examples

Run this code

# (1) the Torus algorithm
#
torus(100)

# example of setting the seed
setSeed(1)
torus(5)
setSeed(6)
torus(5)
#the same
setSeed(1)
torus(10)

#no use of the machine time
torus(10, use=FALSE)

#Kolmogorov Smirnov test
#KS statistic should be around 0.0019
ks.test(torus(1000), punif) 
	
#KS statistic should be around 0.0003
ks.test(torus(10000), punif) 

#the mixed Torus sequence
torus(10, mixed=TRUE)
if (FALSE) {
  par(mfrow = c(1,2))
  acf(torus(10^6))
  acf(torus(10^6, mixed=TRUE))
}

#usage of the init argument
torus(5)
torus(5, init=FALSE)

#should be equal to the combination of the two
#previous call
torus(10)

# (2) Halton sequences
#

# uniform variate
halton(n = 10, dim = 5)

# normal variate
halton(n = 10, dim = 5, normal = TRUE)

#usage of the init argument
halton(5)
halton(5, init=FALSE)

#should be equal to the combination of the two
#previous call
halton(10)

# some plots
par(mfrow = c(2, 2), cex = 0.75)
hist(halton(n = 500, dim = 1), main = "Uniform Halton", 
  xlab = "x", col = "steelblue3", border = "white")

hist(halton(n = 500, dim = 1, norm = TRUE), main = "Normal Halton", 
  xlab = "x", col = "steelblue3", border = "white")
   
# (3) Sobol sequences
#

# uniform variate
sobol(n = 10, dim = 5)

# normal variate
sobol(n = 10, dim = 5, normal = TRUE)

# some plots
hist(sobol(500, 1), main = "Uniform Sobol", 
  xlab = "x", col = "steelblue3", border = "white")

hist(sobol(500, 1, normal = TRUE), main = "Normal Sobol", 
  xlab = "x", col = "steelblue3", border = "white")

#usage of the init argument
sobol(5)
sobol(5, init=FALSE)

#should be equal to the combination of the two
#previous call
sobol(10)

# (4) computation times on my 2017 macbook (on my 2007 macbook), mean of 1000 runs
#

if (FALSE) {
# algorithm			time in seconds for n=10^6
# Torus algo					  0.012 (0.058)
# mixed Torus algo 	    0.018 (0.087)
# Halton sequence				0.180 (0.878)
# Sobol sequence				0.027 (0.214)
n <- 1e+06
mean( replicate( 1000, system.time( torus(n), gcFirst=TRUE)[3]) )
mean( replicate( 1000, system.time( torus(n, mixed=TRUE), gcFirst=TRUE)[3]) )
mean( replicate( 1000, system.time( halton(n), gcFirst=TRUE)[3]) )
mean( replicate( 1000, system.time( sobol(n), gcFirst=TRUE)[3]) )
	}

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