陳 逸群 D937817

1. Batch Size EffectsandMethod of Minimal-MSE Linear Combinationin the Analysis of Simulation Output 陳逸群 D937817

2. Reference Papers • Bruce Schmeiser. 1981. Batch Size Effects in the Analysis of Simulation Output. Operations Research • Bruce Schmeiser.1988. Minimal-MSE Linear Combination of Variance Estimators of the Sample Mean. 1988 Winter Simulation Conference • Bruce Schmeiser. 2004. Simulation Output Analysis: A Tutorial Based on One Research Thread. 2004 Winter Simulation Conference

3. Source of randomness Random number Output variates Point estimators Random variates U Y G X The value of the performance measure Background • All simulation experiments that we consider can be view as • In general, μ is used to measure the performance and is used to be the point estimator for μ • But, how to judge quality of the point estimator?

4. Performance measure Quality measure Correlated data iid data Regerative ARMA STS.A. OBM NBM Batch Size Effect m* Minimal-MSE Linear Combination Background

5. X1 X2 Xn-1 Xn …….… X1 X(k-1)m+1 Xm+1 Xm+2 X2 X(k-1)m+2 X2m Xkm Xm … … …… … … Introduction: Batching • Batching: Transforming correlated observations into fewer uncorrelated and normally distributed batch means (Conway, 1963) when k ≦ k* batch size: m ; number of batches: k Non-overlapping Batching

6. Introduction : Batching • For n observations, point estimator of μ is • For each batch means, the sample variance is • is an estimator of So, a (1-α)100% C.I. is ,where is the half width confidence interval

7. Introduction : Batch Size Effects • The primary question is how to select the appropriate batch size m under constant n • Tradeoff issue • Less batches are used, half width is much longer, more information lost • More batches are used (k>k*), independency and nonnormality will not be hold • The most used criterion for evaluating m • Half width • Expected value of half width • Standard deviation of half width • Coefficient of variation of half width • Probability of coverage

8. Properties of the Half Width • When k ≦ k*, the assumption of negligible dependence of nonnormality is satisfied, then • For sample variance Sk2, then

9. Properties of the Half Width • Define a bias ratio , then we have • For the expected h.w., • For the variance of the h.w., • For the C.V. of the h.w.,

10. Result: Half Width • When n remains constant, changing k does affect the properties of

11. Result: Half Width (cont’) • For , ↘ as k↗ for all α. The rate of decrease is much larger for small values of k than for large values. is robust for k≧10. • For , ↘ as k↗ for all α. The rate of decrease is much larger for small values of k than for large values. • For C.V., k<10 is to be avoided because of high variability and for k>30 is worthwhile.

12. Result: Half Width (cont’) • Less than 10 batches yield wide unstable interval; additional batches beyond 30 yield little additional improvement

13. Properties of Coverage • Coverage Function ( smaller values, better performance ) which is more comprehensive than half width because considers the performance of and , and • Use δ to demonstrate the difference between μand μ1

14. Result: Coverage • When α=0.01, 0.05, 0.1, comparison of β by number of batches k

15. Result: Coverage (cont’) • The decrease in due to incrementing k by one. • Decreases as k increases • Decreases as α increases(additional batches have more value when α is small) • Is small when • With any α, then decrease is very small when k>30

16. Conclusion • Using a small number of batches will allow the simulation run to be terminated earlier, and the research result can be used to determine the penalty for using a specific number of batches. • 10≦k≦30 is reasonable for most simulation situations due to • k＜10, 1. more loss in information; 2. high variability • k＞30, 1. worthwhile; 2. independency and nonnormality might be invalid • Both limits are subjective but will be useful for practitioners when selecting the number of batches

17. Introduction: Minimal-MSE Linear Combination

18. Minimal-MSE Problems • Two cases of linear combination of estimators are studied • Constrained: The sum of weights is required to be 1 • Unconstrained: The sum of weights is not required to be 1

19. Minimal-MSE Problems (Constrained Weights)

20. Minimal-MSE Problems (Constrained Weights)

21. Minimal-MSE Problems (Constrained Weights) • Proposition 1

22. Minimal-MSE Problems (Constrained Weights)

23. Minimal-MSE Problems (Constrained Weights)

24. Minimal-MSE Problems (Constrained Weights)

25. Minimal-MSE Problems (Unconstrained Weights)

26. Minimal-MSE Problems (Unconstrained Weights)

27. Minimal-MSE Problems (Unconstrained Weights)

28. Numerical Study • The numerical study is to see how much the use of linear combination can improve • The component estimators are NBM, OBM, STS.A, NBM+ STS.A optimal batch size and their properties of individual estimators are presented first as follows

29. Numerical Study • The individual component estimators that comprise the optimal linear combinations

30. Numerical Study • Linear combinations composed of two different types estimators with constrained and unconstrained weights

31. Conclusion • The formula for mse-optimal linear combinations are the functions of biases, variances, and covariances of the component estimators • The choice of component estimator type is relatively unimportant