Simulation Modeling and Analysis
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Simulation Modeling and Analysis
Input Modeling
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Outline
• Introduction
• Data Collection
• Matching Distributions with Data
• Parameter Estimation
• Goodness of Fit Testing
• Input Models without Data
• Multivariate and Time Series Input Models
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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
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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
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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
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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).
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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.
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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.
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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]
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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.
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Parameter Estimation
• Once a distribution family has been determined, its parameters must be estimated.
• Sample Mean and Sample Standard Deviation.
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Parameter Estimation -contd
• Suggested Estimators– Poisson: ~ mean– Exponential: ~ 1/mean– Uniform (on [0,b]): b ~ (n+1) max(X)/n– Normal: ~ mean; 2 ~ S2
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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)
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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.
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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
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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.
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Chi-Square Test - contd
• Example: Exponential distribution.
• Example: Weibull distribution.
• Example: Normal distribution.
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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.
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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!
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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).
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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)/12
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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)
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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.
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AR(1) model
• For a time series model
Xt = + (Xt-1 - ) + t
where
t are normal with mean = 0 and var = 2
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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.
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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
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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.