Runuran (version 0.40)

use.aux.urng-method: Use auxiliary random number generator for Runuran objects

Description

Some UNU.RAN methods that are based on the rejection method can make use of a second auxiliary uniform random number generator. It is only used when a rejection has occurred, see below for details. This allows to keep two streams of random variates (almost) synchronized. This is in particular necessary for variance reduction methods like common or antithetic random variates.

[Advanced] -- Use auxiliary URNG for rejection method.

Usage

# S4 method for unuran
use.aux.urng(unr)
use.aux.urng(unr) <- value
set.aux.seed(seed)

Value

use.aux.urng returns

TRUE, if using the auxiliary generator is enabled,

FALSE, it is disabled, and

NA, if this feature does not exist at all.

set.aux.seed returns NULL, invisibly.

Arguments

unr

a unuran generator object.

value

TRUE when an auxiliary URNG is used, FALSE when no auxiliary URNG is used (the default).

seed

seed for the auxiliary URNG.

Methods

Currently the following UNU.RAN methods support this feature. (Currently the last four methods are only available via unuran.new, see the UNU.RAN manual for more details.)

method name	Runuran call	URN per variate
“tdr” (ps)	tdr.new	2
“arou” (ia)	--	1
“tabl” (ia=false)	--	2
“tdr” (gw)	--	2
“tdr” (ia)	--	1
	inversion

Details

Variance reduction techniques like common or antithetic random variates require correlated streams of non-uniform random variates. Such streams can be easily created by means of the inversion method using the same source of uniform random numbers (URNs). However, the quantile function (inverse CDF) or even the CDF often is not available in closed form and thus numerical method are required that are expensive or only return approximate random numbers or both.

On the other hand two streams of non-uniformly distributed random variates are completely uncorrelated when the acceptance-rejection method is used. Then the two streams run “out-of-sync” when the first rejection occurs.

Schmeiser and Kachitvichyanukul, however, have shown that this problem can be overcome when the proposal point is generated by inversion and a fixed number \(k\) of URNs is used for generating one non-uniform random variate. This can be accomplished by means of a second auxiliary stream of URNs which is used when the required number of URNs exceeds \(k\).

By this approach two streams of non-uniform random variates run synchronized except when a rejection occurs in one of the two streams. Therefore the two generated streams are respected mixtures of highly correlated streams and independent streams. The induced correlation thus decreases when the rejection constants of the acceptance-rejection algorithms used for generating the two streams increases.

In UNU.RAN some of the acceptance-rejection algorithms make use of a second auxiliary stream of URNs. It is implemented in one of the following ways:

• The primary uniform random number generator is used for the first loop of the acceptance-rejection step. When a rejection occurs the algorithms switches to auxiliary generator until the proposal point is accepted. Thus exactly two URNs from the primary generator are used to generate one non-uniform random variate.

• The primary uniform random number generator is used just for the first proposal point and then switches to the auxiliary generator until the proposal point is accepted. Thus exactly one URN from the primary generator is used to generate one non-uniform random variate.

The call use.aux.urng(unr) returns FALSE if this feature is disabled for Runuran generator object unr (the default) and TRUE if this feature is enabled. It auxiliary URNs are not supported at all for object unr then use.aux.urng(unr) returns NA.

The replacement method use.aux.urng(unr) <- TRUE enables this feature for generator unr. It can be disabled by means of use.aux.urng(unr) <- FALSE. (One gets an error if this feature is not supported at all.)

The seed of the auxiliary uniform random number generator can be set by means of set.aux.seed(seed). The auxiliary generator is a combined multiple recursive generator (MRG31k3p) by L'Ecuyer and Touzin and is built into package Runuran. Currently it cannot be replaced by some other generator.

References

W. H\"ormann, J. Leydold, and G. Derflinger (2004): Automatic Nonuniform Random Variate Generation, Sect. 8.4.2. Springer-Verlag, Berlin Heidelberg

B. W. Schmeiser and V. Kachitvichyanukul (1986): Correlation induction without the inverse transformation. In: Proc. 1986 Winter Simulation Conf., J. Wilson, J. Henriksen, S. Roberts (eds.), 266-274.

B. W. Schmeiser and V. Kachitvichyanukul (1990): Non-inverse correlation induction: guidelines for algorithm development. J. Comput. Appl. Math. 31, 173-180.

W. H\"ormann and G. Derflinger (1994): Universal generators for correlation induction. In: Compstat, Proceedings in Computational Statistics, R. Dutter and W. Grossmann (eds.), 52-57.

J. Leydold, E. Janka, and W. H\"ormann (2002): Variants of Transformed Density Rejection and Correlation Induction. In: Monte Carlo and Quasi-Monte Carlo Methods 2000, K.-T. Fang, F. Hickernell, and H. Niederreiter (eds.), 345--356.

P. L'Ecuyer and R. Touzin (2000): Fast combined multiple recursive generators with multipliers of the form a = +/- 2^q +/- 2^r. In: J.A. Jones, R.R. Barton, K. Kang, and P.A. Fishwick (eds.), Proc. 2000 Winter Simulation Conference, 683-689.

See Also

tdr.new.

Examples

## Create respective generators for normal and exponential distribution.
## Use method TDR
gen1 <- tdrd.new(udnorm())
gen2 <- tdrd.new(udexp())

## The two streams are independent even we use the same seed
set.seed(123); x1 <- ur(gen1,1e5)
set.seed(123); x2 <- ur(gen2,1e5)
cor(x1,x2)

## We can enable the auxiliary URNG and get correlated streams
use.aux.urng(gen1) <- TRUE
use.aux.urng(gen2) <- TRUE
set.seed(123); x1 <- ur(gen1,1e5)
set.seed(123); x2 <- ur(gen2,1e5)
cor(x1,x2)

## This feature can be disabled again
use.aux.urng(gen1)
use.aux.urng(gen1) <- FALSE
use.aux.urng(gen2) <- FALSE

## Notice that TDR cannot simply mixed with an inversion method
## as the number of URNG per random point differs
gen3 <- pinvd.new(udexp())
set.seed(123); x3 <- ur(gen3,1e5)
cor(x1,x3)

## But a trick would do this
set.seed(123); x3 <- ur(gen3,2*1e5)
x3 <- x3[seq(1,2*1e5,2)]
cor(x1,x3)
## or ...
set.seed(123); u3 <- runif(2*1e5); u3 <- u3[seq(1,2*1e5,2)]
x3 <- uq(gen3,u3)
cor(x1,x3)

## Maybe method AROU is more appropriate
gen4 <- unuran.new(udnorm(), "arou")
use.aux.urng(gen4) <- TRUE
set.seed(123); x3 <- ur(gen3,1e5)
set.seed(123); x4 <- ur(gen4,1e5)
cor(x3,x4)