Lecture _10 -- Part 1

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Modeling & Simulation

Lecture 10

Part 1

Generating Random

Variates

Instructor:

Eng. Ghada Al-MashaqbehThe Hashemite UniversityComputer Engineering Department

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The Hashemite University 2

Outline

Introduction.

Characteristics of random variatesgeneration algorithms.

General approaches for randomvariates generation. Inverse transform method.

Composition.Convolution.

Acceptance rejection method.

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Generating Random Variates

Say we have fitted an exponentialdistribution to interarrival times ofcustomers

Every time we anticipate a new customer

arrival (place an arrival event on theevents list), we need to generate arealization of the arrival times

We know how to generate unit uniform.

Can we use this to generate exponential?(And other distributions)

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Generating Random VariatesAlgorithms Many algorithms exist in the literature to transform the

generated random number into a variate from a specifieddistribution.

Such algorithm or approach depends on the useddistribution: Whether it is discrete or continuous.

Can meet the algorithm requirements. But if you have more than one algorithm that can

be used which one to use: Accuracy: whether the algorithm can produce exact

variates from the distribution (i.e. with high accuracy) or

it is just an estimation. Efficiency: in terms of needed storage and execution time. Complexity: you must find a trade off between complexity

and performance (i.e. accuracy). Some specific technical issues: some algorithms needs

RNG other than the U(0,1), so pay attention.

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Two Types of Approaches

First we will explore the general approachesfor random variates generation which canbe applied for continuous, discrete, andmixed distributions.

Two types: Direct

Obtain an analytical expression Inverse transform

Requires inverse of the distribution function Composition & Convolution

For special forms of distribution functions

Indirect Acceptance-rejection method

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Inverse-Transform Method

Will be examined for three cases: Continuous distributions. Discrete distributions. And mixed distributions (combination of both

continuous and discrete ones). The main requirement is:

Continuous case: The underlying distributioncdf (F(x)) must have an inverse in either: Closed form. Or can be calculated numerically with a good

accuracy level.

For the discrete case: always apply, norestrictions.

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Inverse Transform Continuous Case I

Conditions: F(x) must be continuous and strictly increasing

(i.e. F(x1) < F(x2) if x1 < x2).

F(x) must have an inverse F-1(x).

Algorithm:

)(Return.2

)1,0(~Generate1.

1UFX

UU

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Inverse Transform Continuous Case II

Proof that the returned X has the desireddistribution F(x)

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Example: Weibull Distribution I

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Example: Weibull Distribution II

-- Density functionof Weibulldistribution.

-- F(x) is shown onthe next slide.

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Example: Weibull Distribution II

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Example: Weibull Distribution III

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Inverse Transform Discrete Case I

Here you have discrete values of x, pmf p(xi),and a cdf F(x).

Algorithm

So, the above algorithm returns xi if and only if

Proof:

Requires search technique to find the closest xvalue, many suggestions can be introduced: Start with the largest p(xi).

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Inverse Transform Discrete Case II

-- Unlike the continuous case, the discrete inversetransform can be applied for any discrete distribution butit could be not the most efficient one.

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Inverse Transform DiscreteCase -- Example

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Inverse Transform: Generalization

Algorithm that can be used for both continuousand discrete distributions is:

The above algorithm can be used with mixed

distributions that have discontinuities in its F(x). Have a look at the example shown in Figure 8.5

in the textbook (pp. 430).

UxFxXUU

)(:minReturn2.

)1,0(~Generate.1

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Inverse Transform More!

Disadvantages: Must evaluate the inverse of the distribution

function May not exist in closed form

Could still use numerical methods

May not be the fastest way

Advantages: Needs only one random number to generate a

random variate value. Ease of generating truncated distributions

(redefine F(X) on a smaller finite period of x).

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Composition

It is based on having a weighted sum of distributions(called convex combination).

That is a distribution that has a from (or can bedecomposed as follows):

Most of the time in such cases it is very difficult to find theinverse of the distribution (i.e. you cannot use the inversetransform method).

Each Fj in the above formulation is a distribution function

found in F(x) where Fjs can be different. You can also have a composed pdf function f(x) similar to

F(x) but remember we are working on F(x). So, the trick is to find Fjs that are easy to use in variates

generation where you can apply geometry to decomposeF(x).

11

1where,)()(j

jj

jj

pxFpxF

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Composition Algorithm The composition method is performed in two steps:

First: select one of the composed cdfs to work with. Use the inverse transform to generate the variate from this

cdf.

The first step is choosing Fjwith probabilitypj. Such selection can be done using the discrete inverse

transform method where P(Fj

)=pj

Step 1 Algorithm1. Generate a positive random integer, such that P(J=j)=pj2. ReturnXwith distribution Fj

Also, to obtain the second step you can use the inversetransform method (or any other method you want).

Step 2 Algorithm: U1 is used to select Fj U2 is used to obtain X from the selected Fj.

As, you see you must generate at least two randomnumbers U1 and U2 to obtain X using the compositionmethod

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How to decompose a complex F(x)?

Using geometry to decompose F(x) include twoapproaches:

Either divide F(x) vertically so you split the interval ofx over which the whole F(x) is defined.

Here you do not alter Fjs at all over each interval.

See example 1 on the next slides. Or divide F(x) horizontally the same x interval for all

Fjs which is the same of the original F(x).

See example 2 on the next slides.

Your next step is to definepjassociated with

each Fj. Hint: always treat pjas the area portion under Fj

from the total area under F(X) but you mustcompensate for it.

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Composition Example 1

We will solve it using two methods:

-- inverse transform.

-- composition.

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Composition Example 1 cont.

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Composition Example 2

Try to solve it using the inverse transform method and seehow it is easier to solve it using composition.

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The Hashemite University25


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Convolution

So, you need one random number U then transform it to each Y.

Do not get confused between convolution and composition: In composition: you express the cdf of X as a weighted sum

of other distribution functions.

In convolution: you express X itself as a sum of otherdistribution functions.

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Convolution -- Example

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Acceptance-Rejection Method I

Specify a function that majorizes the density

Now work with the new density function r(x)

Algorithm

xxfxt ),()(

1.Stepback togoOtherwise

.return),()(If3.

oftindependenGenerate.2

densityithGenerate1.

YXYtYfU

YU

rwY

dxxtcc

xtxr ).(where,

)()(

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Acceptance-Rejection Method II

Note the following: f(x) and r(x) are density functions since they integrate

to 1 over their total interval.

t(x) is not a density function.

The acceptance rejection method is an indirectmethod since it works on f(x) indirectly throughboth r(x) and t(x).

Step 1 in the previous algorithm use one of thedirect methods for random variates generation

that we have learned before.

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Acceptance-Rejection Method III

The probability of acceptance of Y in step 3 =1/c.

So, as c decreases, and so the area under t(x) issmall this will reduce the number of iterations of

this algorithm. Smaller c means that t(x) is very close to f(x)

(has a very similar shape).

So, your task is to find a suitable t(x) (has many

details to resemble f(x)) but at the same timenot too complex to generate Y from it (or fromr(x)).

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Acceptance-Rejection -- Example

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Acceptance-Rejection Example cont.

Exercise: Generate 3 random variates for this example usingany LCG you want.

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Acceptance-Rejection Example Better t(x)

-- See the completesolution of theexample at yourtextbook.

-- Now how togenerate Y from r(x)?

Need compositionthen inversetransform.

-- So, three methodsare involved here.

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Additional Notes

The lecture covers the followingsections from the textbook:

Chapter 8

Sections:

8.1,

8.2 (8.2.1 8.2.4)

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Lecture _10 -- Part 1