Antithetic Variates Research Articles

On spherical Monte Carlo simulations for multivariate normal probabilities

The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. In this paper we propose a spherical Monte Carlo method with both theoretical analysis and numerical simulation. We start by writing the multivariate normal probability via an inner radial integral and an outer spherical integral using the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo simulations on some probabilities confirm these claims.

A benchmark study on intelligent sampling techniques in Monte Carlo simulation

In recent years, new, intelligent and efficient sampling techniques for Monte Carlo simulation have been developed. However, when such new techniques are introduced, they are compared to one or two existing techniques, and their performance is evaluated over two or three problems. A literature survey shows that benchmark studies, comparing the performance of several techniques over several problems, are rarely found. This article presents a benchmark study, comparing Simple or Crude Monte Carlo with four modern sampling techniques: Importance Sampling Monte Carlo, Asymptotic Sampling, Enhanced Sampling and Subset Simulation; which are studied over six problems. Moreover, these techniques are combined with three schemes for generating the underlying samples: Simple Sampling, Latin Hypercube Sampling and Antithetic Variates Sampling. Hence, a total of fifteen sampling strategy combinations are explored herein. Due to space constrains, results are presented for only three of the six problems studied; conclusions, however, cover all problems studied. Results show that Importance Sampling using design points is extremely efficient for evaluating small failure probabilities; however, finding the design point can be an issue for some problems. Subset Simulation presented very good performance for all problems studied herein. Although similar, Enhanced Sampling performed better than Asymptotic Sampling for the problems considered: this is explained by the fact that in Enhanced Sampling the same set of samples is used for all support points; hence a larger number of support points can be employed without increasing the computational cost. Finally, the performance of all the above techniques was improved when combined with Latin Hypercube Sampling, in comparison to Simple or Antithetic Variates sampling.

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.

General Theory of Antithetic Time Series

A generalized antithetic time series theory for exponentially derived antithetic random variables is developed. The correlation function between a generalized gamma distributed random variable and its pth exponent is derived. We prove that the correlation approaches minus one as the exponent approaches zero from the left and the shape parameter approaches infinity.

A Fast Monte Carlo Algorithm for Estimating Value at Risk and Expected Shortfall

Risk management today focuses heavily on estimating the location and conditional expectation of the left tail of the probability distribution for returns or portfolio value. The Holy Grail in derivatives pricing is a closed-form valuation equation such as in the Black–Scholes model, which takes a small number of input parameters and produces the exact arbitrage-free properties of the target portfolio, including value-at-risk (VaR) and expected shortfall (ES). But closed-form solutions are rare and largely limited to highly idealized markets. Lattice-based approximation techniques are available for more general settings, but they also have serious constraints. When all else fails, there is Monte Carlo simulation. Simulation always works, in principle, but the amount of calculation required in practice can be tremendous, which provides a strong incentive to find ways to speed up the process. Antithetic variates, control variates, and importance sampling are all helpful. In this article, the authors propose a new technique for estimating VaR and ES that is simple but remarkably powerful. Their first step is to determine which underlying risk factor is the most important. Next, for each simulated value of this primary factor, they simulate values for the remaining factors, requiring that every path generated exceed the VaR threshold. By not computing numerous paths that do not end up in the tail, the procedure can achieve the same accuracy as standard Monte Carlo simulation but several orders of magnitude faster.

Quasi-Monte Carlo Simulation and Variance Reduction Techniques Substantially Reduce Computational Requirements of Patient-Level Simulation Models: an Application to a Discrete Event Simulation Model.

Objectives: Patient-level simulation models provide increased flexibility to overcome the limitations of cohort-based approaches in health-economic analysis. However, computational requirements of reaching convergence is a notorious barrier. The objective was to assess the impact of using quasi-monte carlo simulation (QMCS) and variance reduction techniques (VRTs) on computational requirements. Methods: A recently published discrete event simulation model assessing the cost-effectiveness of an adjunctive antipsychotic treatment for depression was used. The following VRTs were implemented: antithetic variables, common random numbers (CRN) and the combination (Anti-CRN). In addition, QMCS was conducted using the Sobol low discrepancy sequence. The minimal number of patients required to reach equal precision as the reference situation of 1,000,000 simple monte carlo simulations (MCS) was recorded. Precision was defined by the standard error (SE) of the incremental net monetary benefit (INMB) at a willingness to pay of € 20,000 per quality adjusted life year gained. VRT simulations were replicated 100 times. INMB estimates were compared with the reference situation using mean squared error (MSE), mean absolute error (MAE) and percentage of under- and overestimations. Results: Reference INMB (SE) was € 1,413 (76). The average number of patients required to reach reference precision were 929,628, 35,692, 41,683 and 36,803 for antithetic variables, CRN, Anti-CRN and Sobol respectively. This implied a computation time reduction ranging between 7% and 96% compared to simple MCS. MSE was 346,036, 16,314, 155,950 and 7,475 respectively. MAE was 588, 105, 387 and 86 respectively. Antithetic variables and Anti-CRN structurally underestimated INMB (99% and 100%). CRN marginally overestimated INMB in 76 replications. Conclusions: QMCS and VRT reduce computational requirements in terms of simulated patients and computational time up to 96%, enhancing the practical feasibility of patient-level simulation models. This particularly applies to Sobol and CRN. Antithetic variables should be used with caution and its structural bias warrants further research.

Use of an Antithetic Variate for Better Location of Upper-Stem Height Measurements with Critical Height and Importance Sampling in Horizontal Line Sampling

Use of an Antithetic Variate for Better Location of Upper-Stem Height Measurements with Critical Height and Importance Sampling in Horizontal Line Sampling

Bias Reduction in Estimating Quantile Sensitivities

Abstract In this paper, we introduce an Infinitesimal Perturbation Analysis (IPA) estimator with jackknifing to estimate quantile sensitivities, and theoretically prove the two-fold jackknife method reduces bias by eliminating the order 1/ n bias in the original IPA estimator. Numerical examples in finance on portfolio return and options pricing are presented to illustrate the superiority of the new estimator over the original IPA estimator, especially for high and low quantile levels. Antithetic variates are used to further reduce the variance.

An antithetic variate to facilitate upper-stem height measurements for critical height sampling with importance sampling

Critical height sampling (CHS) estimates cubic volume per unit area by multiplying the sum of critical heights measured on trees tallied in a horizontal point sample (HPS) by the HPS basal area factor. One of the barriers to practical application of CHS is the fact that trees near the field location of the point-sampling sample point have critical heights that occur quite high on the stem, making them difficult to view from the sample point. To surmount this difficulty, use of the “antithetic variate” associated with the critical height together with importance sampling from the cylindrical shells integral is proposed. This antithetic variate will be u = (1 − b/B), where b is the cross-sectional area at “borderline” condition and B is the tree’s basal area. The cross-sectional area at borderline condition b can be determined with knowledge of the HPS gauge angle by measuring the distance to the sample tree. When the antithetic variate u is used in importance sampling, the upper-stem measurement will be low on tree stems close to the sample point and high on tree stems distant from the sample point, enhancing visibility and ease of measurement from the sample point. Computer simulations compared HPS, CHS, CHS with importance sampling (ICHS), ICHS and an antithetic variate (AICHS), and CHS with paired antithetic varariates (PAICHS) and found that HPS, ICHS, AICHS, and PAICHS were very nearly equally precise and were more precise than CHS. These results are favorable to AICHS, since it should require less time than either PAICHS or ICHS and is not subject to individual-tree volume equation bias.

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.

A new hybrid Monte Carlo simulation for Asian options pricing

The aim of this paper is to present a new hybrid algorithm for pricing financial derivatives in the arithmetic Asian options. In this paper, two variance reduction techniques are combined, the multiple control variates (MCV) and the antithetic variates (AV). We propose an efficient algorithm for pricing arithmetic Asian options based on the AV and the MCV procedures. A detailed numerical study illustrates the efficiency of the proposed algorithm.

Composite Power System Vulnerability Evaluation to Cascading Failures Using Importance Sampling and Antithetic Variates

Large-scale blackouts typically result from cascading failure in power systems operation. Their mitigation in power system planning calls for the development of methods and algorithms that assess the risk of cascading failure due to relay overtripping, short-circuits induced by overgrown vegetation, voltage sags, line and transformer overloading, transient instabilities, voltage collapse, to cite a few. This paper describes such a method based on composite power system reliability evaluation via sequential Monte Carlo simulation. One of the impediments of the study of these phenomena is the prohibitively large computational burden involved by the simulations. To overcome this difficulty, importance sampling technique utilizing the Weibull distribution is applied to power generator outages. Another method combing importance sampling and antithetic variates together is implemented as well. It is shown that both methods noticeably reduce the number of samples that need to be investigated while maintaining the accuracy at a given level. It is found that the combined method outperforms importance sampling to certain extent. To illustrate the developed techniques, two case studies are conducted and analyzed on the IEEE one-area and three-area reliability test system.

Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem

We consider a nonlinear convex stochastic homogenization problem, in astationary setting. Inpractice, the deterministic homogenized energy density isapproximated by a random apparent energy density, obtained by solving thecorrector problem on a truncated domain.&nbsp; &nbspWe show that the technique of antithetic variables can be used to reducethe variance of the computed quantities, and thereby decrease thecomputational cost at equal accuracy. This leads to an efficientapproach for approximating expectations of the apparent homogenizedenergy density and of related quantities.&nbsp; &nbspThe efficiency of the approach is numerically illustrated on severaltest cases. Some elements of analysis are also provided.

Improving Precision and Reducing Runtime of Microscopic Traffic Simulators through Stratified Sampling

This paper examines the application of Latin Hypercube Sampling (LHS) and Antithetic Variables (AVs) to reduce the variance of estimated performance measures from microscopic traffic simulators. LHS and AV allow for a more representative coverage of input probability distributions through stratification, reducing the standard error of simulation outputs. Two methods of implementation are examined, one where stratification is applied to headways and routing decisions of individual vehicles and another where vehicle counts and entry times are more evenly sampled. The proposed methods have wider applicability in general queuing systems. LHS is found to outperform AV, and reductions of up to 71% in the standard error of estimates of traffic network performance relative to independent sampling are obtained. LHS allows for a reduction in the execution time of computationally expensive microscopic traffic simulators as fewer simulations are required to achieve a fixed level of precision with reductions of up to 84% in computing time noted on the test cases considered. The benefits of LHS are amplified for more congested networks and as the required level of precision increases.

European basket option pricing by maximizing over a subset of lower bounds

In this paper we derive a lower bound for the price of a European basket option under the Black and Scholes framework. The bound is based on the conditioning method of [L.C.G. Rogers and Z. Shi, The value of an Asian option, J. of Appl. Prob. 32 (1995), 1077–1088] and [M. Curran, Valuing Asian and portfolio options by conditioning on the geometric mean, Management Science 40(12) (1994), 1705–1711]. We adapt the method of [K.W. Chen and Y.D. Lyuu, Accurate pricing formulas for Asian options, Applied Mathematics and Computation 188 (2007), 1711–1724] applied for Asian options to the case of the European basket options. Using an improved Monte Carlo method based on the antithetic variates, we benchmark our results. We also calculate the lower bound for various parameters involved, compare these results with those obtained by the improved Monte Carlo simulations and conclude that the proposed bound is indeed optimal.

Efficiency improvements for the numerical computation of NLO corrections

In this paper we discuss techniques, which lead to a significant improvement of the efficiency of the Monte Carlo integration, when one-loop QCD amplitudes are calculated numerically with the help of the subtraction method and contour deformation. The techniques discussed are: holomorphic and non-holomorphic division into sub-channels, optimisation of the integration contour, improvement of the ultraviolet subtraction terms, importance sampling and antithetic variates in loop momentum space, recurrence relations.

Screening and selection procedures with control variates and correlation induction techniques

AbstractWe consider the problem of identifying the simulated system with the best expected performance measure when the number of alternatives is finite and small (often &lt; 500). Recently, more research efforts in the simulation community have been directed to develop ranking and selection (R&amp;S) procedures capable of exploiting variance reduction techniques (especially the control variates). In this article, we propose new R&amp;S procedures that can jointly use control variates and correlation induction techniques (including antithetic variates and Latin hypercube sampling). Empirical results and a realistic illustration show that the proposed procedures outperform the conventional procedures using sample means or control variates alone. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012

Reducing the number of MC runs with antithetic and common random fields

In this paper we adapt two variance reduction techniques, namely antithetic variates and common random numbers, to a sequential simulation scheme which uses copulas as spatial dependence functions to simulate Gaussian and non-Gaussian random fields. The resulting antithetic random fields (ARF) are highly negatively correlated, while common random fields (CRF) exhibit strong positive correlation. We further extend the method in such a way that ARF can be constructed not only as pairs of fields, but also as antithetic triplets, quadruples and any n-tuple of higher dimension. If such ARF or CRF are used as input in Monte Carlo frameworks, this negative or positive correlation of the input random fields is propagated through the physical model to a negative or positive correlation of the output. Ultimately, this enables a significant reduction of simulation runs required for convergence of an estimator. The performances of the proposed methods are examined with two typical applications of stochastic hydrogeology.

Confidence intervals for quantiles when applying variance-reduction techniques

Quantiles, which are also known as values-at-risk in finance, frequently arise in practice as measures of risk. This article develops asymptotically valid confidence intervals for quantiles estimated via simulation using variance-reduction techniques (VRTs). We establish our results within a general framework for VRTs, which we show includes importance sampling, stratified sampling, antithetic variates, and control variates. Our method for verifying asymptotic validity is to first demonstrate that a quantile estimator obtained via a VRT within our framework satisfies a Bahadur-Ghosh representation. We then exploit this to show that the quantile estimator obeys a central limit theorem (CLT) and to develop a consistent estimator for the variance constant appearing in the CLT, which enables us to construct a confidence interval. We provide explicit formulae for the estimators for each of the VRTs considered.

Wiley encyclopedia of operations research and management science

Use of Lagrange Interpolating Polynomials in RLT. Systems in Series. Parallel Configurations. Spanning trees. Recycling. Split Cuts. Retrial Queues. Horse Racing. MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique). Dual simplex. History of CP. Procurement Contracts. MRP. Just in Time. Push and Pull Production. MULTIVARIATE INPUT MODELING. Operational Research Society. The M/G/1 Queue. Drama Theory. Hyper-heuristics. Antithetic Variates. Music. Edgeworth market games. Operational Risk. Queueing Notation. Environmental impact. Why variance is not risk. Conjugate gradient methods. Birth-and-Death Processes. Surgery Planning and Scheduling. Large-scale linear models. Standby Systems. Billiards/Pool. AXIOMATIC MEASURES OF RISK AND RISK-VALUE MODELS. Age Replacement Policies. PASTA and Related Results. Collaborative Procurement. LP projection algorithms. R&D risk management. Bioterrorism. Vendor-Managed Inventory. Gomory Cuts. Travel demand modeling. System Availability. Infinite horizon problems. Trust. Bayesian Network Classifiers. Integer Programming Duality. Queueing Disciplines. Stakeholder participation. Models and Basic Properties. Parametric LP analysis. Operations Research and Golfing. Ice Hockey. Clustering. Finite-Population Models. Batch Arrivals and Service. Optimal Monitoring Strategies. Basic Polyhedral Theory. Operations Research and Art. Managing portfolios of risks. Measures of risk equity. Branch and Bound Algorithms. Analytics in Retail. Variational Inequalities. Cross-Entropy Method. TSP heuristics. Capacity allocation. Control Variates. Cover Inequalities. Direct Search Methods. NLP Software. Bilinear optimization. Discrete-Time Martingales. Branch and Price. Column generation. IP Preprocessing. Optimal Reliability Allocation. Simplex-based LP solvers. Instance Formats. Clique generalizations. Availability in Stochastic Models. Evolutionary Algorithms. Basic CP theory: Search. Multimethodology. Branch and Cut. Memetic Algorithms. Urban mass transit. Coherent Systems. Subgradient optimization. Klimov's Model. Credit risk assessment. Branch-width and Tangles. Stochastic Hazard Process. Deterministic Global Optimization. Multistage (stochastic) games. Basis Reduction Methods. ACCIDENT PRECURSORS AND WARNING SYSTEMS MANAGEMENT: A BAYESIAN APPROACH TO MATHEMATICAL MODELS. Genetic Algorithms. Newsvendor models. Parallel Systems. Opportunities in Tennis. Job-shop scheduling. Simplex method and complexity. Geometric programming. Evacuation Planning. The Performance? Allocation Games. Scenario Generation. Robustness Analysis. Dynamic auctions. Neuroeconomics and game theory. Common Random Numbers. Selective Support Vector Machines. Semi-Markov Processes. Risk Averse Models. Sampling Methods. Domination problems. Simulation of Rare Events. Introduction to Tabu Search. Basic Interdiction Models. Introductory Concepts. Remanufacturing. PAUSE procedure. Wardrop equilibria. Mass customization. Ellipsoidal algorithms. Benders decomposition. Dynamic vehicle routing. Warranty Modeling. Design for Network Resiliency. Passenger rail transportation. Vehicle Routing Problem. Supply Chain Outsourcing. Lovasz-Schriver Reformulation. Hazard Rate Function. Lot-sizing. Differential games. Regenerative Processes. Guided Local Search. Preferences for repeated gambles. Percolation Theory. Simulated Annealing. Scoring Rules. Mixing Sets. Cricket. Prospect Theory. Combining Forecasts. Graph search techniques. Quasi-Newton methods. Random search algorithms. Supply chain coordination. Service Outsourcing. Markov Renewal Processes. K-out-of-n Systems.