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

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

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?

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

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

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

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

Parameter Estimation two common methods maximum likelihood estimators method of moments

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

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 

Examples of Maximum Likelihood Estimators Bernoulli Distribution Exponential Distribution

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

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

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

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

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

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.

Goodness-of-Fit Test Example: The lives of 40 batteries are shown below. Category i: Frequency oi 1.45-1.95 2 1.95-2.45 1 2.45-2.95 4 2.95-3.45 15 3.45-3.95 10 3.95-4.45 5 4.45-4.95 3 Test the hypothesis that the battery lives are approximately normally distributed with μ = 3.5 and σ = 0.7.

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

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 some ei’s are smaller than 5, we combine some categories and get the following Category i: Frequency oi Frequency ei 1.45-2.95 7 8.5 2.95-3.45 15 10.3 3.45-3.95 10 10.7 3.95-4.95 8 10.5

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

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

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

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

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