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 open exp_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.dst to 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 p
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
P(A|p)
0.000
0.001
0.004
0.012
0.027
0.039 p
0.8
0.825
0.85
0.875
0.9
0.925
0.95
0.975
P(A|p)
0.027
0.031
0.035
0.038
0.039
0.037
0.032
0.02

10. 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
discrete distribution:
where {pi} is the p.m.f. with parameter 
continuous distribution:
where f(x; ) is the density at x with parameter 

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. Is the distribution to represent the data points appropriate?
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
n sample values x1, ..., xn assumed to be from F
k exhaustive categories for the domain of F
oi = observed frequency of x1, ..., xn in the ith category
ei = expected frequency of x1, ..., xn in the ith category

17. 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 
decision:
if , reject H0; otherwise, accept H0.

18. Goodness-of-Fit Test
Example: The lives of 40 batteries are shown below.
Category i:
Frequency oi 2 1 4 15 10 5 3
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[( )/0.7 < Z < ( )/0.7]
= P(-2.93 < Z <-2.21) =
e1 = (40)  0.5.
Similarly, we can calculate other expected frequencies:
ei:

20. Goodness-of-Fit Test
Similarly, we can calculate other expected frequencies:
ei:
Since some ei’s are smaller than 5, we combine some categories and get the following
Category i:
Frequency oi
Frequency ei 7
8.5 15
10.3 10
10.7 8
10.5

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

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