Selecting Input Probability Distribution. Introduction need to specify probability distributions of...

Selecting Input Probability Distribution

2

Introduction

• need to specify probability distributions of random inputs

–processing times at a specific machine

– interarrival times of customers/pieces

–demand size• evaluate data sets (if available)

• failure to choose the correct distribution can affect the accuracy of the model’s results!

040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I

Assessing Sample Independencecorrelation plot

scatter diagram

040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I 3

4

Assessing Sample Independence

• important assumption

–observations are supposed to be independent

• graphical techniques for informally assessing whether data are independent

– correlation plot

– scatter diagram


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correlation plot

• graph of sample correlation

– estimate of the true correlation between two observations that are j

observations apart in time

– if observations X1, X2, … , Xn are independent

then ½j = 0 for j = 1, 2, …, n-1

- estimates won’t be exactly zero, even if Xi’s are independent, since its

an observation of a random variable

- if estimates differ from 0 by a significant amount, then its strong

evidence that the Xi’s are not independent040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I

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correlation plot (example)


0 2 4 6 8 10 12 14 16

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

correlation plot

j

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correlation plot (example)


0 2 4 6 8 10 12 14 16 18 20

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

correlation plot

j

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scatter diagram

• plot of pairs (Xi, Xi+1)

– if Xi’s are independent, one would expect the points (Xi, Xi+1) to be

scattered randomly throughout the first quadrant of the plane

–nature of scattering depends on underlying distribution of the Xi’s

– if Xi’s are positively (negatively) correlated, points will tend to lie along a

line with positive (negative) slope


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scatter diagram (example)


0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

scatter plot

Xi

Xi+1

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scatter diagram (example 2)


0 10 20 30 40 50 600

10

20

30

40

50

60

scatter plot

Xi

Xi+1

Specifying Distributionuseful distributions

use values directly

define empirical distribution

fit theoretical distribution


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useful probability distribution

• parameters of continuous distributions

– location parameter °• x-axis location• usually the midpoint (mean for normal distribution) or lower endpoint• also called “shift”-parameter• changes in ° shift the distribution left or right without changing it

otherwise

–scale parameter ¯• determines scale (unit) of measurement• standard deviation ¾ for normal distribution• changes in ¯ compress or expand the associated distribution without

altering its basic form


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useful probability distribution

• parameters of continuous distributions

–shape parameter ®• determines basic form or shape of a distribution within the general

family of distributions of interest• a change in ® generally alters a distribution’s properties (skewness)

more fundamentally than a change in location or scale


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Approaches to specify distribution

• if data collection on an input random variable is possible

–use data values directly in simulation (trace driven)• only reproduces what happened• seldom enough data to make all simulation runs• useful for model validation

–define empirical distribution• at least (for continuous data) any value between min and max• no values outside the range can be generated • may have irregularities

– fit to theoretical distribution• preferred method• easy to change



use values directly




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Uniform U(a,b)

• application

used as a “first” model for a quantity that is felt to be randomly varying

between a and b about which little else is known


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exponential distribution exp(¸)

• application

– interarrival times of entities to a system that occur at a constant rate

– time to failure of a piece of equipment• parameters

– scale parameter ¸ > 0


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gamma(k, µ)

• application

– time to complete some task (customer service, machine repair)• parameters

– shape parameter k > 0

– scale parameter µ > 0


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weibull(k, ¸)

• application

– time to complete some task, time to failure of a piece of equipment

–used as a rough model in absence of data• parameters

– shape parameter k > 0, scale parameter ¸ > 0


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normal N(¹, ¾2)

• application

–errors of various types

–quantities that are the sum of a large number of other quantities• parameters

– location parameter -1 < ¹ < 1 scale parameter ¾ > 0


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triangular (a,b,m)

• application

–used as a rough model in absence of data

–a, b, m are real numbers (a < m < b)• location parameter a• scale parameterb-a• shape parameter m


22

poisson(¸)

• application

–number of events that occur in an interval of time when events are

occurring at a constant rate

–number of items demanded from inventory



use values directly




24

Empirical Distributions

• use observed data themselves to specify distribution directly

–generate random variables from empirical distribution

– (if no theoretical distribution can be fitted)

• define a continuous piecewise-linear distribution function

– sort Xj’s into increasing order

–X(i) denotes the ith smallest value of all Xj’s


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Empirical Distribution (example)

observation: X1 = 3, X2 = 8, X3 = 18, X4 = 10, X5 = 13, X6 = 6

sorted observation: X(1) = 3, X(2) = 6, X(3) = 8, X(4) = 10, X(5) = 13, X(6) = 18

distribution

F(X(i))

F(X(i)) = (i-1)/(n-1)

F(X(1)) = F(3) = 0/5 = 0

F(X(2)) = F(6) = 1/5

F(X(3)) = F(8) = 2/5

etc…


F(X) if X(i) · X · X(i+1)

F(X) = (i-1)/(n-1) + (X –X(i))/((n-1)*(X(i+1)-X(i))

F(12) = ??interval: X(4) · 12 < X(5)

(n = 6, i = 4)F(12) = 3/5 + 2/(5*3) = 0.68

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Empirical Distribution (example)


2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

x

F(x)


use values directly




28

Necessary Steps for fitting a theoretical distribution

• hypothesize family

– summary statistics

–histogram

–quantile summary & box plots

• estimate parameters

• how representative is fitted distribution?

–Chi-Square Goodness of fit test

–Kolmogorov-Smirnoff Test


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Hypothesizing families of distributions

• first step in selecting a particular input distribution:

–decide upon general family appears to be appropriate

• prior knowledge might be helpful

– service times should never be generated from a normal distribution

WHY????

• approaches


–histograms

–quantile summaries and box plots


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Summary Statistics

some distributions are characterized at least partially by functions of

their true paramters

sample estimate

–estimate for range• minimum X(1)

• maxiumum X(n)

–measure of tendency• mean ¹• median x0.5


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Summary Statistics (cont.)

sample estimate

–measure of variability• variance ¾2

• coefficient of variation cv

–measure of symmetry• skewness n


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Histograms

• graphic estimate of the plot of the density function corresponding to the distribution of data

–density functions tend to have recognizable shapes in many cases

–graphical estimate of a density should provide a good clue to the

distribution that might be tried as a model for the data


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Histograms

• how to

–break up range of values into k disjoint adjacent intervals (same width)

[b0, b1), [b1, b2), …, [bk-1, bk) ¢b = bj – bj-1

– you might want to throw out a few extremely large or small Xi’s to avoid

getting an unwidely-looking histogram plot

– let hj be the proportion of Xi’s that are in the jth interval [bj-1, bj)

–hint: try several values of ¢b and choose the smallest one that gives a

“smooth” histogram040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I

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Histogram (example)

• create 1000 random variables ~N(0,1)

– create histogram


-3-2

.5 -2-1

.5 -1-0

.5 00.

5 11.

5 22.

5 30

20

40

60

80

100

120

140

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

Histogramm

Häufigkeit

Kumuliert %

Klasse

Häu

fig

keit

35

Quantile Summaries

• useful for determining whether the underlying probability density function is skewed to the right or left

– if F(x) is the distribution function for a continuous random variable

–q-quantile of F(x) is that number xq such that F(xq) = q

• median x0.5

• lower/upper quartiles x0.25 / x0.75

• lower/upper octiles x0.125 / x0.875


36

Quantile Summaries

Quantile Depth Sample Values Midpoint

Median i = (n+1)/2 X(i) X(i)

Quartiles j = (floor(i)+1)/2 X(j) X(n-j+1) [X(j) + X[n-j+1)]/2

Octiles k = (floor(j)+1)/2X(k) X(n-k+1) [X(k) + X[n-k+1)]/2

Extremes 1 X(1) X(n) [(X(1) + X(n)]/2

– if the underlying distribution of the Xi’s is symmetric, then the midpoints

should be approximately equal

– if the underlying distribution is skewed to the right (left), then the

midpoints should be increasing (decreasing)040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I

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Box Plots (example)

• graphical representation of quantile summary

– fifty percent of observations fall within the horizontal boundaries of the

box [x0.25, x0.75]


1

0 0.5 1 1.5 2 2.5 3 3.5 4

box plot

38




–histogram







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Estimation of Parameters

• After one ore more candidate families of distributions have been hypothesized we most somehow specify the values of their parameters in order to have a completely specified distributions for possible use in simulation

• maximum –likelihood estimators (MLEs)

–estimator = numerical function of the data

–unknown parameter µ

–hypothesized density function fµ(x)

– likelihood function L(µ)

–estimator is value µ that maximizes Lµ over all permissible values of µ


40

Estimation for Parameters (example)

• exponential distribution with unknown parameter ¯ (µ = ¯)

– f¯(x) = (1/¯) e-x/¯ for x ¸ 0

– likelihood function L(¯)

–we seek value of ¯ that maximizes L(¯) over all ¯ > 0

–easier to work with its logarithm

(maximize l(¯) instead of L(¯))

–maximize: set derivative equal to zero and solve for ¯


41




–histogram







42

Goodness-of-Fit Tests

• Statistical hypothesis tests• used to assess formally whether the observations X1, X2, … Xn are

independent samples form a particular distribution with distribution function

• H0 the Xi’s are IID random variables with distribution function

• be careful: failure to reject H0 should not be interpreted as “accepting H0 as being true”.

• we’ll concentrate on two different ones

– chi-square test

–Kolmogorov-Smirnoff tests040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I

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Chi-Square Goodness-of-Fit Test

• more formal comparison of a histogram with the fitted density or mass function

• how to

–divide range into k adjacent intervals [a0, a1), [a1, a2), …, [ ak-1, ak)

• how to choose number and size of intervals? ! equiprobable

–determine Nj (number of Xi’s in the jth interval [aj-1, aj)

– compute pj (expected proportion of the Xi’s that would fall in the jth

interval if we were sampling from the fitted distribution

–determine test statistic χ² and reject H0 if its too large


44

Chi-Square Goodness-of-Fit Test (cont.)

• case 1: all parameters of the fitted distribution are known

– if H0 is true, Â2 converges in distribution (as n → 1) to a chi-square

distribution with k-1 degrees of freedom

– for large n, a test with approximate level ® is obtained by rejecting H0 if

– upper 1 - ® critical point for a chi-square distribution with k-1 dfs


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Chi-Square Goodness-of-Fit Test (cont.)

•case 2: m parameters had to be estimated to specify fitted distribution

– if H0 is true, then as n ! 1 the distribution function of 2 converges to a

distribution function that lies between the distribution function with k-

1 and k-m-1 degrees of freedom

– the upper 1 - ® critical point of the asymptotic distribution of 2

(in general not known)

– reject H0 if

–do not reject H0 if

–ambiguous situation if• recommendation reject H0 if (conservative)040669 || WS 2008 || Dr. Verena Schmid || PR KFK PM/SCM/TL Praktikum Simulation I

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Kolmogorov-Smirnov Goodness-of-Fit Test

• compares an empirical distribution function with the distribution function of the hypothesized distribution

–not necessary to group data

– valid for any sample size n

– tend to be more powerful than chi-squared tests

–but: only valid if all parameters of the hypothesized distribution are

known and the distribution is continuous


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Kolmogorov-Smirnov Goodness-of-Fit Test (cont.)

• compute tests statistics

–define empirical distribution function

– test statistic Dn corresponds to largest (vertical) distance between

Fn(x) and hypothesized distribution function of


48


• case 1: all parameters of estimated distribution function are known

–distribution of Dn does not depend on (if is continuous)

– reject H0 if

–c1-® (does not depend on n) given in the following table

1 - ® 0.85 0.9 0.95 0.975 0.99

c1-® 1.138 1.224 1.358 1.48 1.628


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• case 2:

–hypothesized distribution is N(¹, ¾2) with both ¹ and ¾2 unknown

(estimated) , estimated distribution function

–Dn is calculated the same way as in case 1 - different critical points

– reject H0 if

–c’1-® (does not depend on n) given in the following table

1 - ® 0.85 0.9 0.95 0.975 0.99

c’1-® 0.775 0.819 0.895 0.955 1.035


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• case 3

–hypothesized distribution is exponentially distributed (exp(¸))

–with ¸ unknown (estimated using )

–estimated distribution function

– reject H0 if

–c’’1-® (does not depend on n) given in the following table

1 - ® 0.85 0.9 0.95 0.975 0.99

c’1-® 0.926 0.990 1.094 1.19 1.308


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