1

Simulation Modeling and Analysis

Input Modeling

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

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

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

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

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).

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.

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.

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]

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.

11

Parameter Estimation

• Once a distribution family has been determined, its parameters must be estimated.

• Sample Mean and Sample Standard Deviation.

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

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)

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.

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

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.

17

Chi-Square Test - contd

• Example: Exponential distribution.

• Example: Weibull distribution.

• Example: Normal distribution.

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.

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!

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).

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)/12

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)

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.

24

AR(1) model

• For a time series model

Xt = + (Xt-1 - ) + t

where

t are normal with mean = 0 and var = 2

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.

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

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.