An Insight into the Data Structure of the Dynamic Batch Means Algorithm with Binary Tree Code

Mingchang Chih

doi:10.3390/math7090791

Abstract

Batching is a well-known method used to estimate the variance of the sample mean in steady-state simulation. Dynamic batching is a novel technique employed to implement traditional batch means estimators without the knowledge of the simulation run length a priori. In this study, we reinvestigated the dynamic batch means (DBM) algorithm with binary tree hierarchy and further proposed a binary coding idea to construct the corresponding data structure. We also present a closed-form expression for the DBM estimator with binary tree coding idea. This closed-form expression implies a mathematical expression that clearly defines itself in an algebraic binary relation. Given that the sample size and storage space are known in advance, we can show that the computation complexity in the closed-form expression for obtaining the indexes c j ( k ) , i.e., the batch mean shifts s , is less than the effort in recursive expression.

Highlights

Consider a sequence {Y1, Y2, · · ·, Yn } representing the n random observations of simulation output from a covariance stationary stochastic process with an unknown mean μ = E(Yi ) and an unknown variance σ2 = var(Yi )
Estimating the variance of the sample mean is a fundamental problem in simulation output analysis
The (100 f )% overlapping batch means estimator (OBM), 0 < f < 1, another form of partial-overlapping batch means (PBM), indicates that (100 f )% overlap exists among all data between two adjacent batches, where f = 1 − (s/m) and 1 < s < m (i.e., 0 ≤ f < 1)

Summary

Introduction

Consider a sequence {Y1 , Y2 , · · · , Yn } representing the n random observations of simulation output from a covariance stationary stochastic process with an unknown mean μ = E(Yi ) and an unknown variance σ2 = var(Yi ). Estimating the variance of the sample mean is a fundamental problem in simulation output analysis. The NBM estimator with batch size m is the special case obtained when s = m. The OBM estimator with batch size m is the special case obtained when s = 1. The PBM estimator with batch size m is the special case obtained when 1 < s < m. The SBM estimator with batch size m is the special case obtained when s > m

Methods

Results

Conclusion