1. Simulation Modeling and Analysis
Input Modeling 1 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 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 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 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 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 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 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 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 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 10

11. Parameter Estimation
Once a distribution family has been determined, its parameters must be estimated.
Sample Mean and Sample Standard Deviation. 11 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 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 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 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 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 16

17. Chi-Square Test - contd
Example: Exponential distribution.
Example: Weibull distribution.
Example: Normal distribution. 17 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 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 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 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 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 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 23

24. t are normal with mean = 0 and var = 2
AR(1) model
For a time series model
Xt =  +  (Xt-1 - ) + t
where
t are normal with mean = 0 and var = 2
 24 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 25

26. Xt =  Xt-1 + t with prob
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 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 27