Adaptive Spectral Methods for Simulation Output Analysis

Abstract

This paper addresses two central problems in simulation methodology: the generation of confidence intervals for the steady state means of the output sequences and the sequential use of these confidence intervals to control the run length. The variance of the sample mean of a covariance stationary process is given approximately by p(0)IN, where pif) is the spectral density atfrequencyfand N is the sample size. In an earlier paper we developed a method of confidence interval generation based on the estimation of p(0) through the least squares fit of a quadratic to the logarithm of the periodogram. This method was applied in a run length Control procedure to a sequence of hatched means. As the run length increased the batch means were rehatched into larger batch sizes so as to limit storage requirements. In this rebatching the shape of the spectral density changes, gradually becoming flat as N increases. Quadratics were chosen as a compromise between small sample bias and large sample stability.In this paper we consider smoothing techniques which adapt to the changing .spectral shape in an attempt to improve both the small and large sample behavior of the method. The techniques considered are polynomial smoothing with the degree selected sequentially using standard regression statistics, polynomial smoothing with the degree selected by cross validation, and smoothing splines with the amount of smoothing determined by cross validation. These techniques were empirically evaluated both for fixed sample sizes and when incorporated into the sequential run length control procedure. For fixed sample sizes they did not improve the small sample behavior and only marginally improved the large sample behavior when compared with the quadratic method. Their performance in the sequential procedure was unsatisfactory. Hence, the straightforward quadratic technique recommended in the earlier paper is still recommended as an effective, practical technique for simulation confidence interval generation and run length control.

Performance evaluation of recent procedures for steady-state simulation analysis

“…Heidelberger and Welch (1981a, 1981b, 1983) develop a spectral method in which they use standard regression Downloaded by [The UC Irvine Libraries] at 16:23 02 November 2014 techniques to estimate the power spectrum given by Equation (6) of the given output process at zero frequency. They estimate γ X by fitting a quadratic polynomial to the logarithm of a smoothed version of the periodogram in Equation (7) for the output process over the frequency range between 0 and 1/2 cycles per time unit (excluding the endpoints), where the smoothing operation consists of averaging nonoverlapping pairs of periodogram values.…”

On the Distribution of Throughput of Transfer Lines

“…Moreover, there is no definitive procedure for choosing the value of the weighting function. Detailed discussions of spectral methods can be found in Heidelberger and Welch [99,97,98]. The recent studies using the spectral method are accomplished by Raatikainen [189] in which a procedure is proposed for estimating quantiles based on the P^ algorithm of Jain and Chlamtac [108], which does not require storing and sorting the observation.…”

“…The recent studies using the spectral method are accomplished by Raatikainen [189] in which a procedure is proposed for estimating quantiles based on the P^ algorithm of Jain and Chlamtac [108], which does not require storing and sorting the observation. In another study, Raatikainen [190] combines the spectral method introduced by Heidelberger and Welch [97] and the method of independent replications for run length control in parallel simulation. The objective is to determine whether a fixed number of independent replications executed in parallel and the spectral method can provide estimates that are accurate enough.…”

Two-Stage Stopping Procedures Based on Standardized Time Series

“…Others have also proposed and empirically studied sequential procedures for use in simulations. In particular, Fishman (1977), Adam (1983), Law and Carson (1979), and Law and Kelton (1982) all consider sequential schemes using batch means; Lavenberg and Sauer (1977) investigate sequential procedures in regenerative simulations; Heidelberger and Welch (1981a, 1981b) use a spectral approach; and Iglehart (1977), in a di erent kind of application, presents selection procedures based on sequential methods. For an overview of many of these methods, see pp.…”