Outline

1. Outline • input analysis • input analyzer of ARENA • parameter estimation • maximum likelihood estimator • goodness of fit • randomness • independence of factors • homogeneity of data

2. Topics in Simulation • knowledge in distributions and statistics • random variate generation • input analysis • output analysis • verification and validation • optimization • variance reduction

3. Input Analysis • statistical tests to analyze data collected and to build model • standard distributions and statistical tests • estimation of parameters • enough data collected? • independent random variables? • any pattern of data? • distribution of random variables? • factors of an entity being independent from each other? • data from sources of the same statistical property?

4. Input Analyzer of ARENA • which distribution to use and what parameters for the distribution • Start /Rockwell Software/Arena 7.0/Input Analyzer • Choose File/New • Choose File/Data File/Use Existing to openexp_mean_10.txt • Fit for a particular distribution, or Fit/Fit All

5. Criterion for Fitting in Input Analyzer • n: total number of sample points • ai: actual # of sample points in ith interval • ei: expected # of sample points in ith interval • sum of square error to determine the goodness of fit

6. p-values in Input Analyzer • Chi Square Test and the Kolmogorov-Smirnov Test in fitting • p-value: • a measure of the probability of getting such a set of sample values from the chosen distribution • the larger the p-value, the better

7. Generate Random Variates by Input Analyzer • new file in Input Analyzer • Choose File/Data file/Generate New • select the desirable distribution • output expo.dst • changing expo.dstto expo.txt

8. Parameter Estimation • two common methods • maximum likelihood estimators • method of moments

9. Idea of Maximum Likelihood Estimators • a coin flipped 10 times, giving 9 heads & then 1 tail • best estimate of p = P(head)? • let A be the event of 9 heads followed by 1 tail

10. discrete distribution: where {pi}is the p.m.f. with parameter  • continuous distribution: where f(x;)is the density at x with parameter  Maximum Likelihood Estimators • let  be the parameter to be estimated from sample values x1, ..., xn • set up the likelihood function in  • choose  to maximize the likelihood function

11. Examples of Maximum Likelihood Estimators • Bernoulli Distribution • Exponential Distribution

12. Method of Moments • kth moment of X: E(Xk) • two ways to express moments • from empirical values • in terms of parameters • estimates of parameters by equating the two ways • Examples: Bernoulli Distribution, Exponential Distribution

13. Goodness-of-Fit Test Is the distribution to represent the data points appropriate?

14. General Idea of Hypothesis Testing • coin tosses • H0:P(head) = 1 • H1:P(head)  1 • tossed twice, both being head; accept H0? • tossed 5 times, all being head; accept H0? • tossed 50 times, all being head; accept H0? • to believe (or disbelieve) based on evidence • internal “model” of the statistic properties of the mechanism that generates evidence

15. Theory and Main Idea of the 2 Goodness of Fit Test • (X1, X2, ..., Xk) ~ Multinomial (n; p1, p2, ..., pk)

16. Goodness-of-Fit Test • test the underlying distribution of a population • H0: the underlying distribution is F • H1: the underlying distribution is not F • Goodness-of-Fit Test • nsample values x1, ..., xn assumed to be from F • k exhaustive categories for the domain of F • oi = observedfrequency of x1, ..., xn in the ith category • ei= expectedfrequency of x1, ..., xn in the ith category

17. decision: if , reject H0; otherwise, accept H0. Goodness-of-Fit Test • “better” to have ei = ej for i not equal to j • for this method to work, ei 5 • choose significant level 

18. Goodness-of-Fit Test Example: The lives of 40 batteries are shown below. Test the hypothesis that the battery lives are approximately normally distributed with μ= 3.5 and σ= 0.7.

19. Goodness-of-Fit Test Solution:First calculate the expected frequencies under the hypothesis: For category 1: P(1.45 < X < 1.95) = P[(1.45-3.5)/0.7 < Z < (1.95-3.5)/0.7] = P(-2.93 < Z <-2.21) = 0.0119. e1= 0.0119(40)  0.5. Similarly, we can calculate other expected frequencies: ei: 0.5 2.1 5.9 10.3 10.7 7.0 3.5

20. Goodness-of-Fit Test Similarly, we can calculate other expected frequencies: ei: 0.5 2.1 5.9 10.3 10.7 7.0 3.5 Since someei’s are smaller than 5, we combine some categories and get the following

21. calculate statistics: • set the level of significance:= 0.05. • degrees of freedom:k-1=3. • accept because Goodness-of-Fit Test

22. Test for Randomness Do the data points behave like random variates from i.i.d. random variables?

23. Test for Randomness • graphical techniques • run test (not discussed) • run up and run down test (not discussed)

24. Background • random variables X1, X2, …. (assumption Xi constant) • if X1, X2, … being i.i.d. • j-lag covariance Cov(Xi, Xi+j) cj = 0 • V(Xi) c0 • j-lag correlation j  cj/c0 = 0

25. Graphical Techniques • estimate j-lag correlation from sample • check the appearance of the j-lag correlation