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Simulation Modeling and Analysis

Simulation Modeling and Analysis. Input Modeling. 1. Outline. Introduction Data Collection Matching Distributions with Data Parameter Estimation Goodness of Fit Testing Input Models without Data Multivariate and Time Series Input Models. 2. Introduction.

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Simulation Modeling and Analysis

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  1. Simulation Modeling and Analysis Input Modeling 1

  2. Outline • Introduction • Data Collection • Matching Distributions with Data • Parameter Estimation • Goodness of Fit Testing • Input Models without Data • Multivariate and Time Series Input Models 2

  3. Introduction • Steps in Developing Input Data Model • Data collection from the real system • Identification of a probability distribution representing the data • Select distribution parameters • Goodness of fit testing 3

  4. Data Collection • Useful Suggestions • Plan, practice, preobserve • Analyze data as it is collected • Combine homogeneous data sets • Watch out for censoring • Build scatter diagrams • Check for autocorrelation 4

  5. Identifying the Distribution • Construction of Histograms • Divide range of data into equal subintervals • Label horizontal and vertical axes appropriately • Determine frequency occurrences within each subinterval • Plot frequencies 5

  6. Physical Basis of Common Distributions • Binomial: Number of successes in n independent trials each of probability p . • Negative Binomial (Geometric): Number of trials required to achieve k successes. • Poisson: Number of independent events occurring in a fixed amount of time and space (Time between events is Exponential). 6

  7. Physical Basis of Common Distributions - contd • Normal: Processes which are the sum of component processes. • Lognormal: Processes which are the product of component processes. • Exponential: Times between independent events (Number of events is Poisson). • Gamma: Many applications. Non-negative random variables only. 7

  8. Physical Basis of Common Distributions - contd • Beta: Many applications. Bounded random variables only. • Erlang: Processes which are the sum of several exponential component processes. • Weibull: Time to failure. • Uniform: Complete uncertainty. • Triangular: When only minimum, most likely and maximum values are known. 8

  9. Quantile-Quantile Plots • If X is a RV with cdf F, the q-quantile of X is the value  such that F() = P(X < ) = q • Raw data {xi} • Data rearranged by magnitude {yj} • Then: yj is an estimate of the (j-1/2)/n quantile of X, i.e. yj ~ F-1[(j-1/2)/n] 9

  10. Quantile-Quantile Plots -contd • If F is a member of an appropriate family then a plot of yj vs. F-1[(j-1/2)/n] is a straight line • If F also has the appropriate parameter values the line has a slope = 1. 10

  11. Parameter Estimation • Once a distribution family has been determined, its parameters must be estimated. • Sample Mean and Sample Standard Deviation. 11

  12. Parameter Estimation -contd • Suggested Estimators • Poisson:  ~ mean • Exponential:  ~ 1/mean • Uniform (on [0,b]): b ~ (n+1) max(X)/n • Normal:  ~ mean; 2 ~ S2 12

  13. Goodness of Fit Tests • Test the hypothesis that a random sample of size n of the random variable X follows a specific distribution. • Chi-Square Test (large n; continuous and discrete distributions) • Kolmogorov-Smirnov Test (small n; continuous distributions only) 13

  14. Chi-Square Test • Statistic 20 = k (Oi - Ei)2/Ei • Follows the chi-square distribution with k-s-1 degrees of freedom (s = d.o.f. of given distribution) • Here Ei = n pi is the expected frequency while Oi is the observed frequency. 14

  15. Chi-Square Test -contd • Steps • Arrange the n observations into k cells • Compute the statistic 20 = k (Oi - Ei)2/Ei • Find the critical value of 2 (Handout) • Accept or reject the null hypothesis based on the comparison • Example: Stat::Fit 15

  16. Chi-Square Test - contd • If the test involves a discrete distribution each value of the RV must be in a class interval unless combined intervals are required. • If the test involves a continuous distribution class intervals must be selected which are equal in probability rather than width. 16

  17. Chi-Square Test - contd • Example: Exponential distribution. • Example: Weibull distribution. • Example: Normal distribution. 17

  18. Kolmogorov-Smirnov Test • Identify the maximum absolute difference D between the values of of the cdf of a random sample and a specified theoretical distribution. • Compare against the critical value of D (Handout). • Accept or reject H0 accordingly • Example. 18

  19. Input Models without Data • When hard data are not available, use: • Engineering data (specs) • Expert opinion • Physical and/or conventional limitations • Information on the nature of the process • Uniform, triangular or beta distributions • Check sensitivity! 19

  20. Multivariate and Time-Series Input Models • If input variables are not independent their relationship must be taken into consideration (multivariable input model). • If input variables constitute a sequence (in time) of related random variables, their relationship must be taken into account (time-series input model). 20

  21. Covariance and Correlation • Measure the linear dependence between two random variables X1 (mean 1, std dev 1) and X2 (mean 2, std dev 2) X1 - 1 = (X2 - 2) +  • Covariance: cov(X1,X2) = E(X1 X2) - 1 2 • Correlation:  = cov(X1,X2)/12 21

  22. Multivariate Input Models • If X1 and X2 are normally distributed and interrelated, they can be modeled by a bivariate normal distribution • Steps • Generate Z1 and Z2 indepedendent standard RV’s • Set X1 = 1 + 1 Z1 • Set X2 = 2 + 2(Z1 + (1-2)1/2 Z2) 22

  23. Time-Series Input Models • Let X1,X2,X3,… be a sequence of identically distributed and covariance-stationary RV’s. The lag-h correlation is h = corr(Xt,Xt+h) = h • If all Xt are normal: AR(1) model. • If all Xt are exponential: EAR(1) model. 23

  24. AR(1) model • For a time series model Xt =  +  (Xt-1 - ) + t where t are normal with mean = 0 and var = 2  24

  25. AR(1) model -contd 1.- Generate X1 from a normal with mean  and variance 2 /(1 - 2). Set t = 2. 2.- Generate t from a normal with mean = 0 and variance 2 . 3.- Set Xt =  +  (Xt-1 - ) + t 4.- Set t = t+1 and go to 2. 25

  26. EAR(1) model • For a time series model Xt =  Xt-1 with prob Xt =  Xt-1 + t with prob where t are exponential with mean = 1/ and  26

  27. EAR(1) model - contd 1.- Generate X1 from an exponential with mean  . Set t = 2. 2.- Generate U from a uniform on [0,1]. If U <  set Xt =  Xt-1 . Otherwise generate from an exponential with mean 1/ and set Xt =  Xt-1 + t 4.- Set t = t+1 and go to 2. 27

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