Use Of Antithetic Variates Research Articles

On the effect of inducted negative correlation rate for beta acceptance–rejection algorithms

ABSTRACTOne of the variance reduction methods in simulation experiments is negative correlation induction, and in particular the use of the antithetic variates. The simultaneous use of antithetic variates and an acceptance–rejection method has been studied in some papers, where the inducted negative correlation has been calculated. In this study, the factors affecting the inducted negative correlation rate are addressed. To do this, the beta distribution is first selected to generate negatively correlated random variates using the acceptance–rejection method. The effects of both the efficiency of the acceptance–rejection method and the initial negative correlation rate on the inducted negative correlation are explored. Results show that both factors have significant effects; therefore, a combination of both can lead to algorithms better able to generate negative correlations.

PRM62 - Variance Reduction Through Antithetic Variates as a Means of Developing Complex VBA Models with Reasonable Computation Times

Modelling chronic conditions such as diabetes necessitates the development of complex models, raising issues of computational intensity and execution time. While the use of compiled languages such as C++ is more computationally efficient, concerns may exist regarding their transparency compared to commonly used VBA in Microsoft Excel. This study investigates the application of antithetic variates to the pseudo-random numbers of Monte Carlo simulation, to provide reductions in stochastic uncertainty, through the introduction of negatively correlated pairs of simulation replicates, as a means of developing complex VBA models with reasonable computation times. A simulation model of type 2 diabetes, based on the UKPDS 68 outcomes equations, was executed with and without the application of antithetic variates. The impact of the technique was evaluated through comparison of total cost and benefit estimates, predicted over a long-term horizon of 40 years. An approximate four-fold reduction was observed in the Monte Carlo Error (MCE) associated with estimated mean incremental costs and benefits, when antithetic variates were applied over 1,000 simulations of 1,000 individuals. For a fixed number of runs (1,000), the number of replicated individuals required to achieve 99% accuracy (MCE/mean<1%) in incremental cost and benefit estimates fell from approximately 500 and 550 respectively, to fewer than 50 with antithetic variates. Similarly, for a fixed cohort size (1,000) the same level of precision was produced with fewer than 10% of the simulation runs required otherwise. The use of antithetic variates can improve the precision of modelling output; reducing the number of simulation runs and thus computation time required to perform analyses. The use of such variance reduction techniques should be pursued in the simulation of chronic conditions, as a means of achieving manageable run times and facilitating the extensive scenario and sensitivity analyses required as part of economic evaluations.

Assessing the Relationship between Computational Speed and Precision

Simulation techniques are well suited to modelling diseases yet can be computationally intensive. This study explores the relationship between modelled effect size, statistical precision, and efficiency gains achieved using variance reduction and an executable programming language. A published simulation model designed to model a population with type 2 diabetes mellitus based on the UKPDS 68 outcomes equations was coded in both Visual Basic for Applications (VBA) and C++. Efficiency gains due to the programming language were evaluated, as was the impact of antithetic variates to reduce variance, using predicted QALYs over a 40-year time horizon. The use of C++ provided a 75- and 90-fold reduction in simulation run time when using mean and sampled input values, respectively. For a series of 50 one-way sensitivity analyses, this would yield a total run time of 2 minutes when using C++, compared with 155 minutes for VBA when using mean input values. The use of antithetic variates typically resulted in a 53% reduction in the number of simulation replications and run time required. When drawing all input values to the model from distributions, the use of C++ and variance reduction resulted in a 246-fold improvement in computation time compared with VBA - for which the evaluation of 50 scenarios would correspondingly require 3.8 hours (C++) and approximately 14.5 days (VBA). The choice of programming language used in an economic model, as well as the methods for improving precision of model output can have profound effects on computation time. When constructing complex models, more computationally efficient approaches such as C++ and variance reduction should be considered; concerns regarding model transparency using compiled languages are best addressed via thorough documentation and model validation.

A farewell to the use of antithetic variates in Monte Carlo simulation

Antithetic variates (AV) is one the oldest and most popular variance reduction techniques (VRTs), commonly using complementary random numbers. The AV variance reduction is generally justified by the negative correlation it produces in paired simulation estimates. A new and simpler interpretation of the AV role is presented, showing AV as solely a procedure for input sample means compensation, without any further contribution from the complementary idea. The proposed interpretation is based on the descriptive sampling framework, viewing input samples as composed of a set of values and their sequencing. Simulation experiments and third-party results give support to this interpretation. However, when newer simulation sampling methods, like Latin Hypercube Sampling, Descriptive Sampling, Moment Matching and Quasi-Monte Carlo are adopted, all of them based on a controlled selection of the input sample values, AV turns irrelevant. Other VRTs are also affected by the ideas presented here.

On the use of antithetic variates in particle transport problems

On the use of antithetic variates in particle transport problems

On the use of antithetic variates in particle transport problems

The possible use of antithetic variates as a method of variance reduction in particle transport problems is investigated, by performing some numerical experiments. It is found that if variance reduction is not very carefully defined, it is possible, with antithetic variates, to spuriously detect reduction, or not detect true reduction. Once such subtleties are overcome, it is shown that antithetic variates can reduce variance in multidimensional integration up to a point. The phenomenon of spontaneous correlation is defined and identified as the cause of failure. The surprising result that it sometimes pays to track non-contributing particle histories is demonstrated by means of a zero variance integration analogue. The principles developed in the investigation of multi-variable integration are then employed in a simple calculation of energy deposition using the EGS4 computer code. Promising results are obtained for the total energy deposition problem, but the depth/dose problem remains unsolved. Possible means of overcoming the difficulties are suggested.

Variance reduction in Monte-Carlo radiation transport using antithetic variates

The use of antithetic variates in conjunction with geometrical surface splitting/Russian roulette is numerically studied for one-group Monte-Carlo radiation transport problems in slab geometry. Numerical solutions of the coupled set of integral equations for the first and second moments around the scores, and thereby estimates of variances are obtained. Derivations of the equations for the score accumulation probability and the moments around the score are presented. Two types of antithetic transformations are considered. The results indicate that antithetic transformations always lead to the reduction of variance. A substantial reduction in variance is obtained when antithetic transformations are used with the rule of thumb splitting. The reduction in variance is attractively large for thick slabs (∼20 m.f.p.) and low scattering probability.

Estimation of individual tree volume by importance sampling and antithetic variates from the cylindrical shells integral

A method of estimating individual tree volume by importance sampling is developed that is based on the cylindrical shells volume integral. The cylindrical shells volume integral conceptualizes a tree stem as being a solid of revolution comprising a collection of thin cylinders. Previous applications of importance sampling to individual tree volume estimation have been based on the method of slices integral, in which the tree stem is viewed as a solid of revolution comprising a collection of thin disks. Both approaches provide unbiased estimates of individual tree volume for any monotonic tree taper, if it can be assumed that the tree is circular in cross section. An advantage of the cylindrical shells approach is that total tree height need not be measured to accomplish the estimation process, as would be required when using importance sampling estimators based on the method of slices integral. Tree volume estimation via importance sampling based on the cylindrical shells integral requires measurement of stump cross-sectional area, which is usually more easily obtained than total or merchantable height measurements. Both approaches require one or more upper stem diameter measurements. Computer simulation shows that use of antithetic variates with importance sampling reduces the variance of the volume estimate significantly when a paraboloid is used as a proxy taper function.

ANTITHETIC VARIATES FOR MONTE CARLO ESTIMATION OF PROBABILITIES

AbstractThis paper explores some possibilities for variance reduction by the use of antithetic variates when estimating probabilities.

The Use of Antithetic Variates in Computer Simulations

The method of antithetic variates is a well known technique for reducing the variability of estimators in computer simulation experiments. It is shown that in many, if not most simulations, the usually suggested way of applying the method suffers from two weaknesses. One is so serious that it can lead to estimators which are worse than if the technique is not used at all. Simple alternatives are suggested which do not suffer either weakness. A numerical example is given illustrating the advantages of these alternative procedures.

The Use of Antithetic Variates in Computer Simulations

The Use of Antithetic Variates in Computer Simulations

Some properties of variance reduction techniques where hydrological extremes are estimated by Monte Carlo Simulation

An estimate of a water resource system's performance, when it is derived by simulation using synthetic streamflow sequences, is subject to at least three errors: first, model errors, arising from the approximation to the ‘true’ streamflow mechanism which the model represents; second, sampling errors in the model parameters θ when they are calculated from the historic records; and third, errors introduced by the Monte Carlo calculation from which is derived. This paper presents some observations on the effects of the first two types of error on the estimate but concentrates on the application of variance reduction techniques to the derivation of Monte Carlo estimates for given θ. These techniques are, first, the use of control variates and, second, the use of antithetic variates, and their application is illustrated by using some hypothetical examples of the calculation of probabilities of extreme hydrological events and of the calculation of reliability measures for a much oversimplified storage system. Considerable reduction in the variance of resulted from the application of the control variate method; the reduction in Var resulting from the use of antithetic variates was much less but still probably worth the small additional programing effort required. It is concluded that the promise of the control variate method suggests that it should be applied to assist in the efficient simulation of more realistic water resource systems than the trivial ones considered in this paper.

Variance Reduction by Antithetic Variates in GI/G/1 Queuing Simulations

This paper considers the use of antithetic variates to reduce the variance of estimates obtained in the simulation of a GI/G/1 queue. Two experimental configurations are considered: in the first, 2n observations are taken in a single run; in the second, n observations are taken in each of two runs. If the sequences of uniform random variables that generate the realizations of the queuing system in the two runs are antithetic, we show that the variance of estimates of the mean and distribution of stationary waiting time and number in the queue is less in the second configuration than in the first. We also obtain sufficient conditions for the covariance of functions of a vector of uniform random variables to be nonnegative. Experimental results are given for M/M/1 queuing simulations to illustrate the magnitude of the variance reduction.