WSC 2008

WSC 2008 Final Abstracts

Analysis Methodology Track

Monday 10:30:00 AM 12:00:00 PM
Comparison Using Simulation

Chair: Shane Henderson (Cornell University)

Comparing Two Systems: Beyond Common Random Numbers
Samuel M. T. Ehrlichman and Shane G. Henderson (Cornell University)

Suppose one wishes to compare two closely related systems via stochastic simulation. Common random numbers (CRN) involves using the same streams of uniform random variates as inputs for both systems to sharpen the comparison. One can view CRN as a particular choice of copula that gives the joint distribution of the inputs of both systems. We discuss the possibility of using more general copulae, including simple examples that show how this can outperform CRN.

Run-Length Variability of Two-Stage Multiple Comparisons with the Best for Steady-state Simulations and Its Implications for Choosing First-Stage Run Lengths
Marvin K Nakayama (New Jersey Institute of Technology)

We analyze the asymptotic behavior of two-stage procedures for multiple comparisons with the best (MCB) for comparing the steady-state means of alternative systems using simulation. The two procedures we consider differ in how they estimate the variance parameters of the alternatives in the first stage. One procedure uses a consistent estimator, and the other employs an estimator based on one of Schruben's standardized time series (STS) methods. While both methods lead to mean total run lengths that are of the same asymptotic order of magnitude, the limiting variability of the run lengths is strictly smaller for the method based on a consistent variance estimator. We also provide some analysis showing how to choose the first-stage run length.

Comparison of Bayesian Priors for Highly Reliable Limit Models
Roy R Creasey (Longwood University), Preston White (University of Virginia) and Linda B Wright and Cheryl F Davis (Longwood University)

Limit standards are probability interval requirements for proportions. Simulation literature has focused on finding the confidence interval of the population proportion, which is inappropriate for limit standards. Further, some Frequentist approaches cannot be utilized for highly reliable models, or models which produce no or few non-conforming trials. Bayesian methods provide approaches that can be utilized for all limit standard models. We consider a methodology developed for Bayesian reliability analysis, where historical data is used to define the a priori distribution of proportions p, and the customer desired a posteriori maximum probability is utilized to determine sample size for a replication.

Monday 1:30:00 PM 3:00:00 PM
Efficient Ranking and Selection Procedures I

Chair: John Shortle (George Mason University)

A Preliminary Study of Optimal Splitting for Rare-Event Simulation
John F Shortle and Chun-Hung Chen (George Mason University)

Efficiency is a big concern when using simulation to estimate rare-event probabilities, since a huge number of simulation replications may be needed in order to obtain a reasonable estimate of such a probability. Furthermore, when multiple designs must be compared, and each design requires simulation of a rare event, then the total number of samples across all designs can be prohibitively high. This paper presents a new approach to enhance the efficiency for rare-event simulation. Our approach is developed by integrating the notions of level splitting and optimal computing budget allocation. The goal is to determine the optimal numbers of simulation runs across designs and across a number of splitting levels so that the variance of the rare-event estimator is minimized.

A New Perspective on Feasibility Determination
Roberto Szechtman (Naval Postgraduate School) and Enver Yucesan (INSEAD)

We consider the problem of feasibility determination in a stochastic setting. In particular, we wish to determine whether a system belongs to a given set G based on a performance measure estimated through Monte Carlo simulation. Our contribution is two-fold: (i) we characterize fractional allocations that are asymptotically optimal; and (ii) we provide an easily implementable algorithm, rooted in stochastic approximation theory, that results in sampling allocations that provably achieve in the limit the same performance as the optimal allocations. The finite-time behavior of the algorithm is also illustrated on two small examples.

Restricted Subset Selection
E Jack Chen (BASF Corporation)

This paper develops procedures for electing a set of normal populations with unknown means and unknown variances in order that the final subset of selected populations satisfies the following requirements: with probability at least P*, the selected subset will contain a population or "only and all" of those populations whose mean lies less than the distance d* from the smallest mean. The size of the selected subset is random, however, at most m populations will finally be chosen. A restricted subset attempts to exclude populations that are deviated more than d* from the smallest mean. Here P*, d*, and m are users specified parameters. The procedure can be used when the unknown variances across populations are unequal. An experimental performance evaluation demonstrates the validity and efficiency of these restricted subset selection procedures.

Monday 3:30:00 PM 5:00:00 PM
Efficient Ranking and Selection Procedures II

Chair: Chun-Hung Chen (George Mason University)

An Efficient Ranking and Selection Procedure for a Linear Transient Mean Performance Measure
Douglas J. Morrice (The University of Texas at Austin) and Mark W. Brantley and Chun-Hung Chen (George Mason University)

We develop a Ranking and Selection procedure for selecting the best configuration based on a transient mean performance measure. The procedure extends the OCBA approach to systems whose means are a function of some other variable such as time. In particular, we characterize this as a prediction problem and imbed a regression model in the OCBA procedure. In this paper, we analyze the linear case and discuss a number of extensions. Additionally, we provide some motivating examples for this approach.

Update on Economic Approach to Simulation Selection Problems
Stephen E. Chick (INSEAD) and Noah Gans (Wharton)

This paper summarizes new analytical and empirical results for the economic approach to simulation selection problems that we introduced two years ago. The approach seeks to help managers to maximize the expected net present value (NPV) of system design decisions that are informed by simulation. It considers the time value of money, the cost of simulation sampling, and the time and cost of developing simulation tools. This economic approach to decision making with simulation is therefore an alternative to the statistical guarantees or probabilistic convergence results of other commonly-used approaches to simulation optimization. Empirical results are promising. This paper also retracts a claim that was made regarding the existence of Gittins' indices for these problems - their existence remains an open question.

The Knowledge-Gradient Stopping Rule for Ranking and Selection
Peter Frazier and Warren Buckler Powell (Princeton University)

We consider the ranking and selection of normal means in a fully sequential Bayesian context. By considering the sampling and stopping problems jointly rather than separately, we derive a new composite stopping/sampling rule. The sampling component of the derived composite rule is the same as the previously introduced LL1 sampling rule, but the stopping rule is new. This new stopping rule significantly improves the performance of LL1 as compared to its performance under the best other generally known adaptive stopping rule, EOC Bonf, outperforming it in every case tested.

Tuesday 8:30:00 AM 10:00:00 AM
Efficient Simulation Techniques

Chair: Pirooz Vakili (Boston University)

Monotonicity and Stratification
Gang Zhao and Pirooz Vakili (Boston University)

In utilizing the technique of stratification, the user needs to first partition/stratify the sample space; the next task is to determine how to allocate samples to strata. How to best perform the second task is well understood and analyzed and there are effective and generic recipes for sample allocation. Performing the first task, on the other hand, is generally left to the user who has limited guidelines at her/his disposal. We review explicit and implicit stratification approaches considered in the literature and discuss their relevance to simulation studies. We then discuss the different ways in which monotonicity plays a role in optimal stratification.

Control Variate Technique: A Constructive Approach
Tarik Borogovac and Pirooz Vakili (Boston University)

The technique of control variates requires that the user identify a set of variates that are correlated with the estimation variable and whose means are known to the user. We relax the known mean requirement and instead assume the means are to be estimated. We argue that this strategy can be beneficial in parametric studies, analyze the properties of controlled estimators, and propose a class of generic and effective controls in a parametric estimation setting. We discuss the effectiveness of the estimators via analysis and simulation experiments.

Efficient Simulation for Tail Probabilities of Gaussian Random Field
Robert J. Adler (Technion-Israel Institute of Technology) and Jose H. Blanchet and Jingchen Liu (Columbia University)

We are interested in computing tail probabilities for the maxima of Gaussian random fields. In this paper, we discuss two special cases: random fields defined over a finite number of distinct point and fields with finite Karhunen-Loeve expansions. For the first case we propose an importance sampling estimator which yields asymptotically zero relative error. Moreover, it yields a procedure for sampling the field conditional on it having an excursion above a high level with a complexity that is uniformly bounded as the level increases. In the second case we propose an estimator which is asymptotically optimal. These results serve as a first step analysis of rare-event simulation for Gaussian random fields.

Tuesday 10:30:00 AM 12:00:00 PM
Input Modeling

Chair: Jack Chen (BASF Corporation)

Functional Data Analysis for Non Homogeneous Poisson Processes
Fermín Mallor and Martín Gastón (Public University of Navarre) and Teresa León (University of Valencia)

In this paper we intend to illustrate how Functional Data Analysis (FDA) can be very useful for simulation input modelling. In particular, we are interested in the estimation of the cumulative mean function of a non-homogeneous Poisson Process (NHPP). Both parametric and nonparametric methods have been developed to estimate it from observed independent streams of arrival times. As far as we know, these data have not been analyzed as functional data. The basic idea underlying of FDA is treating a functional observation as a single datum rather than as a large set of data on its own. A considerable effort is being made in order to adapt some standard statistical methods for functional data, for instance Principal Components Analysis, ANOVA, classification techniques, boot-strap confidence bands, or outlier detection. We have studied a set of real data making use of these techniques and obtaining very good results.

Reliable Simulation with Input Uncertainties Using an Interval-Based Approach
Ola G. Batarseh and Yan Wang (University of Central Florida)

Uncertainty associated with input parameters and models in simulation has gained attentions in recent years. The sources of uncertainties include lack of data and lack of knowledge about physical systems. In this paper, we present a new reliable simulation mechanism to help improve simulation robustness when significant uncertainties exist. The new mechanism incorporates variabilities and uncertainties based on imprecise probabilities, where the statistical distribution parameters in the simulation are intervals instead of precise real numbers. The mechanism generates random interval variates to model the inputs. Interval arithmetic is applied to simulate a set of scenarios simultaneously in each simulation run. To ensure that the interval results bound those from the traditional real-valued simulation, a generic approach is also proposed to specify the number of replications in order to achieve the desired robustness. This new reliable simulation mechanism can be applied to address input uncertainties to support robust decision making.

Smooth Flexible Models of Nonhomogeneous Poisson Processes Using One or More Process Realizations
Michael E Kuhl and Shalaka C Deo (Rochester Institute of Technology) and James R Wilson (North Carolina State University)

We develop and evaluate a semiparametric method to estimate the mean-value function of a nonhomogeneous Poisson process (NHPP) using one or more process realiza-tions observed over a fixed time interval. To approximate the mean-value function, the method exploits a specially formulated polynomial that is constrained in least-squares estimation to be nondecreasing so the corresponding rate function is nonnegative and smooth (continuously differentiable). An experimental performance evaluation for two typical test problems demonstrates the method’s ability to yield an accurate fit to an NHPP based on a single process realization. A third test problem shows how the method can estimate an NHPP based on multiple realizations of the process.

Tuesday 1:30:00 PM 3:00:00 PM

Chair: Russell Cheng (University of Southampton)

Stochastic Kriging for Simulation Metamodeling
Barry L Nelson, Jeremy Staum, and Bruce Ankenman (Northwestern University)

We extend the basic theory of kriging, as applied to the design and analysis of deterministic computer experiments, to the stochastic simulation setting. Our goal is to provide flexible, interpolation-based metamodels of simulation output performance measures as functions of the controllable design or decision variables. To accomplish this we characterize both the intrinsic uncertainty inherent in a stochastic simulation and the extrinsic uncertainty about the unknown response surface. We use tractable examples to demonstrate why it is critical to characterize both types of uncertainty, derive general results for experiment design and analysis, and present a numerical example that illustrates the stochastic kriging method.

Selecting the Best Linear Simulation Metamodel
Russell Cheng (University of Southampton)

We consider the output of a simulation model of a system about which little is initially known. This output is often dependent on a large number of factors. It is helpful, in examining the behaviour of the system, to find a statistical metamodel containing only those factors most important in influencing this output. The problem is therefore one of selecting a parsimonious metamodel that includes only a subset of the factors, but which nevertheless adequately describes the behaviour of the output. The total number of possible submodels from which we are choosing grows exponentially with the number of factors, so a full examination of all possible submodels rapidly becomes intractable. We show how resampling can provide a simple solution to the problem, by allowing potentially good submodels to be rapidly identified. This resampling approach also allows a systematic statistical comparison of good submodels to be made.

Data Enhancement, Smoothing, Reconstruction and Optimization by Kriging Interpolation
Hasan Gunes and Hakki Ergun Cekli (Istanbul Technical University) and Ulrich Rist (Universitaet Stuttgart)

The performance of Kriging for enhancement, smoothing, reconstruction and optimization of a test data set is investigated. Specifically, the ordinary two-dimensional Kriging and 2D line-Kriging interpolation are investigated and compared with the well-known digital filters for data smoothing. We used an analytical 2D synthetic test data with several minima and maxima. Thus, we could perform detailed analyses in a well-controlled manner in order to assess the effectiveness of each procedure. We have demonstrated that Kriging method can be used effectively to enhance and smooth a noisy data set and re-construct large missing regions (black zones) in lost data. It has also been shown that, with the appropriate selection of the correlation function (variogram model) and its correlation parameter, one can control the ‘degree’ of smoothness in a robust way. Finally, we illustrate that Kriging can be a viable ingredient in constructing effective global optimization algorithms in conjunction with simulated annealing.

Tuesday 3:30:00 PM 5:00:00 PM
Output Analysis

Chair: Wheyming Song (National Tsing Hua University)

Skart: A Skewness- and Autoregression-Adjusted Batch-Means Procedure for Simulation Analysis
Ali Tafazzoli (NC State University), James R. Wilson (North Carolina State University), Emily K. Lada (SAS Institute Inc) and Natalie M. Steiger (Maine Business School)

We discuss Skart, an automated batch-means procedure for constructing a skewness- and autoregression-adjusted confidence interval for the steady-state mean of a simulation output process. Skart is a sequential procedure designed to deliver a confidence interval that satisfies user-specified requirements concerning not only coverage probability but also the absolute or relative precision provided by the half-length. Skart exploits separate adjustments to the half-length of the classical batch-means confidence interval so as to account for the effects on the distribution of the underlying Student's t-statistic that arise from nonnormality and autocorrelation of the batch means. Skart also delivers a point estimator for the steady-state mean that is approximately free of initialization bias. In an experimental performance evaluation involving a wide range of test processes, Skart compared favorably with other simulation analysis methods - namely, its predecessors ASAP3, WASSP, and SBatch as well as ABATCH, LBATCH, the Heidelberger-Welch procedure, and the Law-Carson procedure.

A Large Deviations View of Asymptotic Efficiency for Simulation Estimators
Sandeep Juneja (Tata Institute of Fundamental Research) and Peter Glynn (Stanford University)

Consider a simulation estimator alpha(c) based on expending c units of computer time, to estimate a quantity alpha. One measure of efficiency is to attempt to minimize P(|alpha(c) - alpha| > epsilon) for large c. This helps identify estimators with less likelihood of witnessing large deviations. In this article we establish an exact asymptotic for this probability when the underlying samples are independent and a weaker large deviations result under more general dependencies amongst the underlying samples.

Displaying Statistical Point Estimators: The Leading-Digit Procedure
Wheyming T. Song (National Tsing Hua University) and Bruce Schmeiser (Purdue University)

We propose a procedure for reporting a statistical point estimator and its precision for statistical experiments such as simulation experiments. Based on three criteria - loss of statistical information, number of characters required, and likelihood of user misinterpretation - we advocate our procedure for use when reporting many point estimators in tabular form. The procedure discards meaningless digits of the point estimator, and all but the left-most non-zero digit of the standard error. These two resulting values are separated by the ``;'' sign.

Wednesday 8:30:00 AM 10:00:00 AM
Output Analysis and SPC

Chair: Seong-Hee Kim (Georgia Institute of Technology)

The More Plot: Displaying Measures of Risk & Error From Simulation Output
Barry L Nelson (Northwestern University)

The focus on mean (long-run average) performance as the primary output measure produced by simulation experiments diminishes the usefulness of simulation for characterizing risk. Confidence intervals on means are often misinterpreted as measures of future risk, when in fact they are measures of error. We introduce the Measure of Risk & Error (MORE) plot as a way to display and make intuitive the concepts of risk and error and thus support sound experiment design and correct decision making.

A Distribution-Free Tabular Cusum Chart for Correlated Data with Automated Variance Estimation
Joongsup Jay Lee, Christos Alexopoulos, David Goldsman, Seong-Hee Kim, and Kwok-Leung Tsui (Georgia Institute of Technology) and James R. Wilson (North Carolina State University)

We formulate and evaluate distribution-free statistical process control (SPC) charts for monitoring an autocorrelated process when a training data set is used to estimate the marginal mean and variance of the process as well as its variance parameter (i.e., the sum of covariances at all lags). We adapt variance-estimation techniques from the simulation literature for automated use in DFTC-VE, a distribution-free tabular CUSUM chart for rapidly detecting shifts in the mean of an autocorrelated process. Extensive experimentation shows that our variance-estimation techniques do not seriously degrade the performance of DFTC-VE compared with its performance using exact knowledge of the variance parameter; moreover, the performance of DFTC-VE compares favorably with that of other competing distribution-free SPC charts.

Implementable MSE-Optimal Dynamic Partial-Overlapping Batch Means Estimators for Steady-State Simulations
Wheyming Tina Song and Mingchang Chih (National Tsing Hua University)

Estimating the variance of the sample mean from a stochastic process is essential in assessing the quality of using the sample mean to estimate the population mean which is the fundamental question in simulation experiments. Most existing studies for estimating the variance of the sample mean from simulation output assume simulation run length is known in advance. This paper proposes an implementable batch-size selection procedure for estimating the variance of the sample mean without requiring that the sample size or simulation run length a priori.

Wednesday 10:30:00 AM 12:00:00 PM
QMC Methods in Finance

Chair: Pierre L'Ecuyer (DIRO, Université de Montréal)

Simulation of a Lévy Process by PCA Sampling to Reduce the Effective Dimension
Pierre L'Ecuyer (DIRO, Université de Montréal) and Jean-Sébastien Parent-Chartier and Maxime Dion (Université de Montréal)

For a Lévy process monitored at s observation times, we want to estimate the expected value of some function of the observations by RQMC. For the case of a Brownian motion, PCA sampling has been proposed to reduce the effective dimension of the problem by using an eigen-decomposition of the covariance matrix of the vector of observations. We show how this method applies to other Lévy processes, and we examine its effectiveness in improving RQMC efficiency empirically. The idea is to simulate a Brownian motion at s observation points using PCA, transform its increments into independent uniforms over (0,1), then transform these uniforms again by applying the inverse distribution function of the increments of the Lévy process.

Fast Simulation of Equity-Linked Life Insurance Contracts with a Surrender Option
Carole Bernard and Christiane Lemieux (University of Waterloo)

In this paper, we consider equity-linked life insurance contracts that give their holder the possibility to surrender their policy before maturity. Such contracts can be valued using simulation methods proposed for the pricing of American options, but the mortality risk must also be taken into account when pricing such contracts. Here, we use the least-squares Monte Carlo approach of Longstaff and Schwartz coupled with quasi-Monte Carlo sampling and a control variate in order to construct efficient estimators for the value of such contracts. We also show how to incorporate the mortality risk into these pricing algorithms without explicitly simulating it.

On the Approximation Error in High Dimensional Model Representation
Xiaoqun Wang (Department of Mathematical Sciences, Tsinghua University)

Mathematical models are often described by multivariate functions, which are usually approximated by a sum of lower dimensional functions. A major problem is the approximation error introduced and the factors that affect it. This paper investigates the error of approximating a multivariate function by a sum of lower dimensional functions in high dimensional model representations. Two kinds of approximations are studied, namely, the approximation based on the ANOVA (analysis of variance) decomposition and the approximation based on the anchored decomposition. We prove new theorems for the expected error of approximation based on anchored decomposition when the anchor is chosen randomly and establish the relationship of the expected errors with the global sensitivity indices of Sobol'. The expected error gives indications on how good or how bad could be the approximation based on anchored decomposition and when the approximation is good or bad. Methods for choosing good anchors are presented.