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 1  Outline  input analysis  input analyzer of ARENA  parameter estimation  maximum likelihood estimator  goodness of fit  randomness  independence.

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Presentation on theme: " 1  Outline  input analysis  input analyzer of ARENA  parameter estimation  maximum likelihood estimator  goodness of fit  randomness  independence."— Presentation transcript:

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

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

3  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  4  Input Analyzer of ARENA  which distribution to use and what parameters for the distribution  Start /Rockwell Software/Arena 7.0/Input AnalyzerRockwell Software/Arena 7.0/Input Analyzer  Choose File/New  Choose File/Data File/Use Existing to open exp_mean_10.txt exp_mean_10.txt  Fit for a particular distribution, or Fit/Fit All

5  5  Criterion for Fitting in Input Analyzer  n: total number of sample points  a i : actual # of sample points in ith interval  e i : expected # of sample points in ith interval  sum of square error to determine the goodness of fit

6  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  7  Generate Random Variates by Input Analyzer  new file in Input Analyzer  Choose File/Data file/Generate New  select the desirable distribution  output expo.dstexpo.dst  changing expo.dst to expo.txt

8  8  Parameter Estimation  two common methods  maximum likelihood estimators  method of moments

9  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 P(A|p)P(A|p) p P(A|p)P(A|p)

10  10  Maximum Likelihood Estimators  let  be the parameter to be estimated from sample values x 1,..., x n  set up the likelihood function in   choose  to maximize the likelihood function ✦ discrete distribution: where {p i } is the p.m.f. with parameter  ✦ continuous distribution: where f(x;  ) is the density at x with parameter 

11  11  Examples of Maximum Likelihood Estimators  Bernoulli Distribution  Exponential Distribution

12  12  Method of Moments  kth moment of X: E(X k )  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  13  Goodness-of-Fit Test Is the distribution to represent the data points appropriate?

14  14  General Idea of Hypothesis Testing  coin tosses  H 0 : P(head) = 1  H 1 : P(head)  1  tossed twice, both being head; accept H 0 ?  tossed 5 times, all being head; accept H 0 ?  tossed 50 times, all being head; accept H 0 ?  to believe (or disbelieve) based on evidence  internal “model” of the statistic properties of the mechanism that generates evidence

15  15  Theory and Main Idea of the  2 Goodness of Fit Test  (X 1, X 2,..., X k ) ~ Multinomial (n; p 1, p 2,..., p k )

16  16  Goodness-of-Fit Test  test the underlying distribution of a population  H 0 : the underlying distribution is F  H 1 : the underlying distribution is not F  Goodness-of-Fit Test  n sample values x 1,..., x n assumed to be from F  k exhaustive categories for the domain of F  o i = observed frequency of x 1,..., x n in the ith category  e i = expected frequency of x 1,..., x n in the ith category

17  17  Goodness-of-Fit Test  “better” to have e i = e j for i not equal to j  for this method to work, e i  5  choose significant level   decision: if, reject H 0 ; otherwise, accept H 0.

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

19  19  Solution: First calculate the expected frequencies under the hypothesis: Goodness-of-Fit Test For category 1: P(1.45 < X < 1.95) = P[( )/0.7 < Z < ( )/0.7] = P(-2.93 < Z <-2.21) = e 1 = (40)  0.5. Similarly, we can calculate other expected frequencies: e i :

20  20  Since some e i ’s are smaller than 5, we combine some categories and get the following Goodness-of-Fit Test Category i:Frequency o i Frequency e i Similarly, we can calculate other expected frequencies: e i :

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

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

23  23  Test for Randomness  graphical techniques  run test (not discussed)  run up and run down test (not discussed)

24  24  Background  random variables X 1, X 2, …. (assumption X i  constant)  if X 1, X 2, … being i.i.d.  j-lag covariance Cov(X i, X i+j )  c j = 0  V(X i )  c 0  j-lag correlation  j  c j /c 0 = 0

25  25  Graphical Techniques  estimate j-lag correlation from sample  check the appearance of the j-lag correlation


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