1 Outline input analysis input analyzer of ARENA parameter estimation maximum likelihood...

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OutlineOutline

input analysis input analyzer of ARENA parameter estimation

maximum likelihood estimator

goodness of fit randomness independence of factors homogeneity of data

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Topics in SimulationTopics in Simulation

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

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Input AnalysisInput 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?

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Input Analyzer of ARENAInput 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

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Criterion for Fitting in Input AnalyzerCriterion 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

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2)-(

n

eai

ii

2

i

ii

n

e

n

a

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pp-values in Input Analyzer-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

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Generate Random Variates Generate Random Variates by Input Analyzerby 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

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Parameter EstimationParameter Estimation

two common methods maximum likelihood estimators method of moments

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Idea of Idea of Maximum Likelihood EstimatorsMaximum 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 0 0 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

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Maximum Likelihood EstimatorsMaximum 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

)()...()(1

nxx ppL✦ discrete distribution:

where {pi} is the p.m.f. with parameter

);()...;()( 1 nxfxfL✦ continuous distribution:

where f(x; ) is the density at x with parameter

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Examples of Examples of Maximum Likelihood EstimatorsMaximum Likelihood Estimators

Bernoulli Distribution Exponential Distribution

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Method of Moments 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

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Goodness-of-Fit TestGoodness-of-Fit Test

Is the distribution to represent Is the distribution to represent

the data points appropriate?the data points appropriate?

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General Idea of Hypothesis TestingGeneral 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

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Theory and Main Idea of Theory and Main Idea of the the 22 Goodness of Fit Test Goodness of Fit Test

(X1, X2, ..., Xk) ~ Multinomial (n; p1, p2, ..., pk)

2

1

2

1)(

knk

i i

iik np

npXQ

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Goodness-of-Fit TestGoodness-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 2

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22 ~

)(

k

k

i i

ii

e

eo

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Goodness-of-Fit TestGoodness-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 H0.22

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Example: The lives of 40 batteries are shown below.

Goodness-of-Fit TestGoodness-of-Fit Test

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.

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Solution: First calculate the expected frequencies under the hypothesis:

Goodness-of-Fit TestGoodness-of-Fit Test

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

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Since some ei’s are smaller than 5, we combine some categories and get the following

Goodness-of-Fit TestGoodness-of-Fit Test

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

Similarly, we can calculate other expected frequencies:

ei: 0.5 2.1 5.9 10.3 10.7 7.0 3.5

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calculate statistics: 24

2

1

( )3.5.i i

i i

o e

e

Goodness-of-Fit TestGoodness-of-Fit Test

set the level of significance: = 0.05. degrees of freedom: k-1=3.

20.05 7.815.

accept because 22

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Test for RandomnessTest for Randomness

Do the data points behave like

random variates from i.i.d.

random variables?

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Test for RandomnessTest for Randomness

graphical techniques run test (not discussed)

run up and run down test (not discussed)

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BackgroundBackground

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

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Graphical TechniquesGraphical Techniques

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

jn

xxxxc

jn

ijii

j

1))((

ˆ1

)(ˆ 1

2

2

n

xxs

n

ii

n