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![(1) Linear Congruential Method
The linear congruential method, initially proposed by
Lehmer [1951], produces a sequence of integers, x1, x2, ...
between zero and (m – l) by following a recursive
relationship:
(X0) is the initial value, it is called the seed
(a) is called the multiplier
(c) is the increment
(m) is the modulus
If c ≠ 0, then the form is called the mixed congruential
method, when c = 0, the form is known as the
multiplicative congruential method.
Xi+1 = (aXi + c) mod m where i = 0, I, 2, ...](https://image.slidesharecdn.com/lec6-250930070406-42521d99/85/Lec6-computer-simulation-Modeling-and-Simulation-pdf-9-320.jpg)
![Characteristics of a Good Generator
Maximum Density
Such that the values assumed by Ri, i = 1,2,…, leave no large
gaps on [0,1]
Problem: Instead of continuous, each Ri is discrete
Solution: a very large integer for modulus m
Approximation appears to be of little consequence
Maximum Period
To achieve maximum density and avoid cycling.
Achieve by: proper choice of a, c, m, and X0.
Most digital computers use a binary representation of
numbers
Speed and efficiency are aided by a modulus, m, to be (or close
to) a power of 2.](https://image.slidesharecdn.com/lec6-250930070406-42521d99/85/Lec6-computer-simulation-Modeling-and-Simulation-pdf-15-320.jpg)
![Example
Use the chi-square test at 0.05 to test for whether the data
shown next are uniformly distributed.
O1 =(0, 0.1] = {0.06, 0.01, 0.05, 0.05, 0.1, 0.02, 0.05, 0.05} = 8
E1 = {0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1} = 10](https://image.slidesharecdn.com/lec6-250930070406-42521d99/85/Lec6-computer-simulation-Modeling-and-Simulation-pdf-20-320.jpg)
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- 1.
- 2. Random Numbers andVariates Introduction to Random Number Generation Important Considerations for Random Number Generation Techniques for Generating Random Numbers Linear Congruential Method Combined Linear Congruential Generators Chi-SquareTest Acceptance-RejectionTechnique
- 3. Random Number Generation Random numbers are a necessary basic ingredient in the simulation of almost all discrete systems. Most computer languages have a subroutine, object, or function that will generate a random number. Similarly, simulation languages generate random numbers that are used to generate event times and other random variables.
- 4. Important Considerations forRandom Number Generation (1) The routine should be fast. Individual computations are inexpensive, but simulation could require many millions of random numbers. The total cost can be managed by selecting a computationally efficient method of random number generation. (2) The routine of generating random numbers should be portable to different computers and programming languages. This is desirable so that the simulation program will produce the same results wherever it is executed.
- 5. Important Considerations (3)The routine of generating random numbers should have a sufficiently long cycle. The cycle length, or period, represents the length of the random number sequence before previous numbers begin to repeat themselves in an earlier order. (4) The random numbers should be replicable. Given the starting point (or conditions) it should be possible to generate the same set of random numbers completely independent of the system that is being simulated. This is helpful for debugging purposes and is a means of facilitating comparisons between systems.
- 6. Important Considerations (5)Most important, the generated random numbers should closely approximate the ideal statistical properties of uniformity and independence. (See Properties of Random Numbers)
- 7. Properties of RandomNumbers
- 8. Techniques for GeneratingRandom Numbers Linear Congruential Method (LCM). Combined Linear Congruential Generators (CLCM).
- 9. (1) Linear CongruentialMethod The linear congruential method, initially proposed by Lehmer [1951], produces a sequence of integers, x1, x2, ... between zero and (m – l) by following a recursive relationship: (X0) is the initial value, it is called the seed (a) is called the multiplier (c) is the increment (m) is the modulus If c ≠ 0, then the form is called the mixed congruential method, when c = 0, the form is known as the multiplicative congruential method. Xi+1 = (aXi + c) mod m where i = 0, I, 2, ...
- 10. (1) Linear CongruentialMethod To use the linear congruential method to generate a sequence of random numbers with x0 , a , c, and m = 100. The integer values X generated between 0 and 99. Random numbers Ri between 0 and 1 can be generated by ,...... 3 , 2 , 1 i m X R i i Xi+1 = (aXi + c) mod 100 where i = 0, I, 2, ...
- 11. Note The seedfor a linear congruential random-number generator: Is the integer value X0 that initializes the random-number sequence. Any value in the sequence can be used to “seed” the generator.
- 12. Example 1 Generaterandom numbers using linear congruential method when X0 = 27, a = 17, c = 43 and m = 100. The Xi and Ri values are X1 = (17*27+43) mod 100 = 502 mod 100 = 2 R1 = 2/100 = 0.02 X2 = (17*2+32) mod 100 = 77 R2 = 77/100 = 0.77 X3 = (17*77+32) mod 100 = 52 R3 = 52/100 = 0.52 …
- 13. Example 2 Letm = 102 = 100, a = 19, c = 0, and X0 = 63, and generate a seqnence of random numbers. The Xi and Ri values are X1 = (19*63) mod 100 = 1197 mod 100 = 97 R1 = 97/100 = 0.97 X2 = (19*97) mod 100 = 1843 mod 100 = 43 R2 = 43/100 = 0.43 X3 = (19*43) mod 100 = 817 mod 100 = 17 R3 = 17/100 = 0.17 …
- 14. Example 3 Usingthe multiplicative congruential method (c = 0), find the period (stop at repreating) of the generator at X0 = 1, 2, 3, and 4 for a = 13, m = 26 = 64 At X0 = 1, P = 16 At X0 = 2, P = 8 At X0 = 3, P = 16 At X0 = 4, P = 4 Show the characteristics For good generators
- 15. Characteristics of aGood Generator Maximum Density Such that the values assumed by Ri, i = 1,2,…, leave no large gaps on [0,1] Problem: Instead of continuous, each Ri is discrete Solution: a very large integer for modulus m Approximation appears to be of little consequence Maximum Period To achieve maximum density and avoid cycling. Achieve by: proper choice of a, c, m, and X0. Most digital computers use a binary representation of numbers Speed and efficiency are aided by a modulus, m, to be (or close to) a power of 2.
- 16. Tests for RandomNumbers Two Categories Testing for uniformity (FrequencyTest) Testing for independence (Out of scope)
- 17. Tests for RandomNumbers When to use these tests: If a well-known simulation languages are used, it is probably unnecessary to test. If the generator is not explicitly known or documented, e.g., spreadsheet programs (EXCEL), symbolic/numerical calculators, tests should be applied to many sample numbers.
- 18. Frequency Tests Testof uniformity Kolmogorov-Smirnov test Chi-square test
- 19. (2) Chi-square Test Chi-square test measures the degree of agreement between the distribution of a sample of generated random numbers and the theoretical uniform distribution. The chi-square is valid only for large samples, say N >= 50 (no. of classes 5-10) Oi is the observed number in the ith class. Ei is the expected number in the ith class n is the number of classes. For the uniform distribution, Ei, the expected number in each class is given by Ei = N/n (Null hypothesis: H0) n i i i i E E O 1 2 2 0
- 20. Example Use thechi-square test at 0.05 to test for whether the data shown next are uniformly distributed. O1 =(0, 0.1] = {0.06, 0.01, 0.05, 0.05, 0.1, 0.02, 0.05, 0.05} = 8 E1 = {0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.1} = 10
- 21. Example The tablecontains the essential computations. The test uses n = 10 intervals of equal length, namely [0, 0.1), [0.1, 0.2), ……, [0.9, l.0). The value of 2 is 3.4 < {2 0.05,9} = 16.919