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

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

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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?

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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

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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

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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

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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

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

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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)

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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

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11 Examples of Maximum Likelihood Estimators Bernoulli Distribution Exponential Distribution

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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

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

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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

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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 )

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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

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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.

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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.

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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 :

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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 :

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

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

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

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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

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

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