Antithetic Variates Research Articles

Combining antithetic variates and control variates in simulation experiments

Antithetic variates and control variates are two well-known variance reduction techniques. We consider combining antithetic variates and control variates to estimate the mean response in a stochastic simulation experiment. When applying antithetic variates to generate control variates across paired replications, we show that the integrated control-variate estimator is unbiased and yields, under the assumption of common correlations induced for all control variates, a smaller variance than the conventional control-variate estimator without using antithetic variates. We examine the proposed estimator and two alternative integrated control-variate estimators when applying antithetic variates on control variates and show that the proposed estimator is the optimal integrated control-variate estimator We implement these three integrated control-variate estimators and the conventional control-variate estimator in a simulation model of a stochastic network to evaluate the performance of each control-variate estimator Empirical results show that the proposed estimator outperforms the other control-variate estimators.

Integrated Variance Reduction Strategies for Simulation

We develop strategies for integrated use of certain well-known variance reduction techniques to estimate a mean response in a finite-horizon simulation experiment. The building blocks for these integrated variance reduction strategies are the techniques of conditional expectation, correlation induction (including antithetic variates and Latin hypercube sampling), and control variates; all pairings of these techniques are examined. For each integrated strategy, we establish sufficient conditions under which that strategy will yield a smaller response variance than its constituent variance reduction techniques will yield individually. We also provide asymptotic variance comparisons between many of the methods discussed, with emphasis on integrated strategies that incorporate Latin hypercube sampling. An experimental performance evaluation reveals that in the simulation of stochastic activity networks, substantial variance reductions can be achieved with these integrated strategies. Both the theoretical and experimental results indicate that superior performance is obtained via joint application of the techniques of conditional expectation and Latin hypercube sampling.

Variance reduction in Ueno's method and cylinder sampling for forest volume estimation

Three basic techniques are proposed for reducing the variance of the stand volume estimate provided by cylinder sampling and Ueno's method. Ueno's method is based on critical height sampling but does not require measurement of critical heights. Instead, a count of trees whose critical heights are less than randomly generated heights is used to estimate stand volume. Cylinder sampling selects sample trees for which randomly generated heights fall within cylinders formed by tree heights and point sampling plot sizes. The methods proposed here for variance reduction in cylinder sampling and Ueno's method are antithetic variates, importance sampling, and control variates. Cylinder sampling without variance reduction was the most efficient of 12 methods compared in computer simulation that used estimated measurement times. However, cylinder sampling requires knowledge of a combined variable individual tree volume equation. Of the three variance reduction techniques applied to Ueno's method, antithetic variates performed best in computer simulation.

Faster Valuation of Financial Derivatives

High-dimensional integrals are usually solved with Monte Carlo algorithms although theory suggests that low-discrepancy algorithms are sometimes superior. We report on numerical testing which compares low-discrepancy and Monte Carlo algorithms on the evaluation of financial derivatives. The testing is performed on a Collateralized Mortgage Obligation (CMO) which is formulated as the computation of ten integrals of dimension up to 360. We tested two low-discrepancy algorithms (Sobol and Halton) and two randomized algorithms (classical Monte Carlo and Monte Carlo combined with antithetic variables). We conclude that for this CMO the Sobol algorithm is always superior to the other algorithms. We believe that it will be advantageous to use the Sobol algorithm for many other types of financial derivatives. Our conclusion regarding the superiority of the Sobol algorithm also holds when a rather small number of sample points are used, an important case in practice. We have built a software system called FINDER for computing high-dimensional integrals. FINDER runs on a heterogeneous network of workstations under PVM 3.2 (Parallel Virtual Machine). Since workstations are ubiquitous, this is a cost-effect way to do large computations fast. The measured speedup is at least .9N for $N$ workstations, $N$ less than or equal to 25. The software can also be used to compute high-dimensional integrals on a single workstation. A routine for generating Sobol points may be found, for example, in Numerical Recipes in C by Press et al. However, we incorporated major improvements in FINDER and we stress that the results reported in this paper were obtained using FINDER. One of the improvements was developing the table of primitive polynomials and initial direction numbers for dimensions up to 360.

Generating Antithetic Random Variates in Simulation of a Replacement Process by Rejection Method

Analysis of a replacement process provides the needed insight into prediction of the replacement requirements of a system. Accurate prediction of such requirements can lead to cost savings as well as improved system performance. When the component lives are not exponentially distributed, simulation becomes a potentially suitable method of analysis. This paper deals with Gamma distributed component lives whose deviates are generated through the so-called Rejection Method. It will be shown that antithetic random deviates can be generated through the rejection method to estimate the renewal function in a single component replacement process. A theorem proved by George is modified and extended to the case at hand. Finally, an algorithm is developed, and along with the experimental results, will be presented. A replacement process is chosen as an example through which to verify the merits of the proposed algorithm.

Sequential Monte Carlo simulation for composite power system reliability analysis with time varying loads

The paper illustrates the development and utilization of an annual chronological load curve for each load bus in a composite generation and transmission system and a sequential Monte Carlo simulation approach for composite system reliability assessment. Antithetic variates as a variance reduction technique has been applied to the simulation model to increase the efficiency of the simulation. An approximate method using a load duration curve of the system load and an enumeration process have been applied to the developed load model and the results are compared in this paper. >

On the Forecasting Performance of Estimators for a Structural Equation in a Large System: Some Monte Carlo Results

This paper examines the relative forecasting performance of published estimators proposed for a structural equation in a large system using Monte Carlo experiments with antithetic variates. The performances of the estimators are compared in terms of the accuracy of the within-sample as well as post-sample predictions for 10 structural equations by using the mean absolute percentage error of forecasts. It is concluded that the ridge-type estimator developed by Kadiyala and Nunns and the estimator proposed by Swamy and Holmes performed consistently better than other estimators in both the within-sample predictions and ex post forecasts.

The pricing of forward-starting asian options

Asian options are path-dependent options whose payoff is based on an average. In some cases, the underlying asset of the option is an average; in others, the strike price itself is computed as an average of the underlying asset recent prices. Asian currency options, as an example, allow corporate users to cover seasonal and so-called strategic foreign exchange positions. Averages are also found in some recent warrant issues, where they are mostly used as poison pills to preclude hostile takeovers. This paper derives a closed-form solution for the valuation of European asian options whose strike price is an average. Both “plain vanilla” average options—i.e. those for which the time interval taken into account for the strike average calculation is the life of the option—and forward-starting average options are considered. The valuation formula is obtained by relying upon a slight linear approximation. Although some previous contributions in the literature already use approximation techniques, our approach contrary to others allows the derivation of a formal upper bound to the approximation error. The numerical values given by this formula are then compared to those generated by an antithetic variate Monte-Carlo method. Based on this comparison and the computation of an upper bound to the approximation error, it is shown that the closed-form solution performs quite well and is obviously computationally efficient.

Strategies for Combining Antithetic Variates and Control Variates in Designed Simulation Experiments

In this paper we examine three methods for combining the variance reduction techniques of antithetic variates and control variates to estimate the mean response in a designed simulation experiment. In Combined Method I, we perform h independent pairs of simulation runs as follows—on the second run of each such pair, we use random number streams that are antithetic (complementary) to the streams used on the first run of the pair to drive the non-control-variate components of the simulation model; and we use independent random number streams to drive the control-variate components of the simulation model. In Combined Method II, we also perform h independent pairs of runs; but on each pair of runs we use independent random number streams to drive the non-control-variate model components, and we use antithetic random number streams to drive the control-variate components. In Combined Method III, all of the random number streams driving the second run of each pair of runs are antithetic to the streams driving the first run of the pair. For each of these three methods we drive the variance of the resulting estimator of the mean response to make a theoretical comparison of the efficiency of each method. We implemented these three methods, along with the classical method of control variates, in a simulation model of a resource-constrained activity network to show how each combined method is implemented in practice and to evaluate the performance of each combined method experimentally. The results indicate that: (a) Combined Method III outperformed all other methods, and (b) the effectiveness of Combined Method III as well as the choice of whether to use Combined Method I or Combined Method II depends on the degree of correlation between the control variates and the response.

ESTIMATING SIMULATION METAMODELS USING COMBINED CORRELATION-BASED VARIANCE REDUCTION TECHNIQUES

This paper develops a procedure for jointly applying all of the correlation-based variance reduction techniques (namely, die methods of antithetic variates, common random numbers, and control variates) in a simulation experiment that is designed to estimate a linear metamodel (that is, a linear regression model) for a single response variable expressed in terms of an input vector of design variables for the target system. This procedure combines (a) the Schruben-Margolin strategy for metamodel estimation based on joint application of the methods of common random numbers and antithetic variates, and (b) a metamodel estimation scheme based on the method of control variates. Under specified conditions on the dependency structure of die simulation outputs and with respect to a variety of optimally criteria, the combined procedure is shown to be superior to each of the following conventional correlation-based variance reduction techniques: independent random number streams, common random number streams, contro...

The Art of Simulation

SUMMARY This paper surveys some general issues related to simulation. Random number generation and the simulation of specific probability distributions are discussed. It is shown how statistical methods, especially graphical ones, can be used to analyze data from simulation experiments and to plan such experiments. Finally, the paper discusses how antithetic variables can be used in variance reduction. Throughout the paper the use of computers is considered.

Combined correlation methods for metamodel estimation in multipopulation simulation experiments

In this paper we examine two variance-reduction strategies for estimating the parameters of a simulation metamodel that are based on the reduction of the variances of the mean of the responses across the design points that comprise the simulation experiment through the effective combination of well known variance-reduction techniques. In the first strategy, Combined Method I, we extend a combined approach of antithetic variates and control variates that we have developed in an earlier paper for a single design point simulation experiment. In Combined Method I, we perform h independent paris of simulation runs at each design point; and on the second run of each such pair, we use random number streams to drive the non-control variate components that are antithetic to the corresponding streams of the first run of the pair. The random number streams that drive the control variate components are randomly selected across all simulation runs. In Combined Method II, we extend Combined Method I to incorporate the ...

Discussion on the Meeting on the Gibbs Sampler and Other Markov Chain Monte Carlo Methods

Discussion on the Meeting on the Gibbs Sampler and Other Markov Chain Monte Carlo Methods

Monte Carlo Summation Applied to Product-Form Loss Networks

Loss networks with direct routing have a product-form solution for their equilibrium probabilities. The product-form solution typically involves a normalization constant calling for a multidimensional summation over an astronomical number of states. We propose the application of Monte Carlo summation in order to determine the normalization constant, the blocking probabilities, and the revenue sensitivities. We show that if the proper sampling technique is employed, then the computational effort of Monte Carlo summation is independent of link capacities. We also discuss the application of importance sampling, antithetic variates, and indirect estimation via Little's formula. The method is illustrated with a four-leaf star network supporting multirate traffic.

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.

A note on antithetic variates from discrete distributions

This note outlines a method which produces antithetic variates from discrete distributions, in particular from Poisson distributions. The method provides maximal negative correlation, hence it is highly suitable as a variance reduction technique. The technique is easily extended to produce the full range of possible correlations between pairs of Poisson random variables.

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.

An empirical evaluation of antithetic variates and quasirandom points for simulating stochastic networks

This paper is concerned with simulation procedures for estimating the distribution function of the time to complete stochastic activity networks. A conditional Monte Carlo method using uniformly directed cutsets is considered and combined with antithetic variate sampling and quasirandom points. The efficiency of the two variance reduction techniques is compared using CPU times and mean- square errors. By using three examples the study illustrates the decreasing benefits of using quasirandom points over antithetic variate technique as the network size increases.

Stochastic Algorithms and Bayesian Inference

Complex stochastic systems, large collections of random variables with non-trivial dependence structure, arise in probability modelling in many contexts. Examples include statistical mechanics, geographical epidemiology, pedigrees in genetics, statistical image analysis, and general multi-parameter Bayesian inference. In these contexts, we usually require the marginal distribution of some of the variables, or perhaps its expectation. However, in practice, such computations may not be amenable to either exact numerical calculation or direct simulation, because of the large numbers of variables that may be involved. An appealing alternative approach, which is currently receiving much attention, is the use of dynamic Monte Carlo methods such as the Gibbs sampler (Geman & Geman, 1984). This is an iterative simulation technique, in which each variable is considered in turn: its current value is discarded and replaced by a random number drawn from the conditional distribution of that variable given the current values of all the others. This process is repeated indefinitely: it is easy to see that it defines a Markov chain that converges to the required distribution. We show that this is just one of a family of Markov chains with the same properties, known as Metropolis methods and considered in general terms by Hastings (1970). Different members of this family perform very differently, in terms both of the speed of convergence and of the precision of the resulting estimators of the quantities of interest. The latter aspect of performance can be much improved by a generalization of the idea of antithetic variables. As well as being much cheaper to compute (because they avoid the need to sample from complicated conditional distributions), some Metropolis/Hastings samplers perform much better than the Gibbs sampler in the same situation. Green and Han (1991) discuss these ideas in greater detail, give examples, and also report on some extensive experiments with these methods as applied to an example from Bayesian image analysis.

An improved monte carlo technique for radiation shield design

Reduction of statistical error in Monte Carlo shielding calculations is achieved by using antithetic variates with geometrical surface splitting while solving the radiation transport problems. Numerical results are presented for sample calculations for slab shields of different dimensions. The present method shows great improvement in efficiency for thick slabs and therefore is useful for practical shielding calculations.