Download Pseudo-Random Number Generation

Simulation Modeling and
Analysis
Pseudo-Random Numbers
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Outline
• Properties of Random Numbers
• Generating Random Numbers
• Testing Random Numbers
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Properties of Random Numbers
• Key Properties
– Uniformity
– Independence
• Density function (continuous!)
f(x) = {1 for 0 < x < 1, 0 otherwise
• Moments
E(R) = 1/2 V(R) = 1/12
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Generating Random Numbers
• Random Numbers vs Pseudo-random
Numbers
• Requirements of a RNG routine
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Speed
Portable
Long Cycle
Replicable RN
Uniform and Independent RN’s
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Random Number Generation
• Linear Congruential Method
X i+1 = (a Xi + c) mod m
Ri = Xi/m
• Note: Only values from the set
I = {0,1/m,2/m,…,(m-1)/m}
are obtained
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Random Number Generation -contd
• Longest Possible Period (P)
– If m = 2b and |c| > 0 , P = m
– If m = 2b and c = 0 , P = m/4
– If m = prime and c = 0 , P = m-1
• Example:
X i+1 = (75 Xi ) mod (231-1)
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Random Number Generation -contd
• Combined Congruential Generators. Two
distinct congruential generators can be
combined to obtain PRN’s with longer
periods.
X i+1 = ((-1)j-1 Xi,j ) mod (m1 - 1)
Ri = Xi/m1, Xi > 0 ; Ri = (m1-1)/m1, Xi > 0
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Testing Random Numbers
• Null Hypotheses
H0 : Ri ~ U[0,1] ; H0 : Ri ~ independent
• Tests
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Frequency test
Runs test
Autocorrelation test
Gap test
Poker test
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Kolmogorov-Smirnov Frequency Test
1.- Arrange data in increasing value
2.- Compute D+, D- and D
3.- Find critical Dc (Handout) for given a
4.- Accept or reject the null hypothesis.
5.- Example: Stat::Fit
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Chi-Square Frequency Test
• The Chi static compares observed
frequencies of occurrence of PRN’s in
selected subdomains against expected
frequencies derived from the U distribution
function. See Stat::Fit
X02 = n (Oi - Ei)2/Ei
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Runs Testing
• Run: sequence of similar events
• Runs up and runs down (independence)
– Maximum number of runs (N numbers) = N-1
– mean = (2N-1)/3; variance = (16N-29)/90
– Test hypothesis against normal distribution.
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Runs Testing -contd
• Runs above and below the mean
– Maximum number of runs (N numbers, n1
above and n2 below the mean) = n1+n2
– mean = 2 n1 n2/N + 1/2
– variance = 2 n1 n2 (2 n1 n2 - N)/N2 (N-1)
– Test hypothesis against normal distribution.
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Runs Testing -contd
• Runs length
– Test hypothesis against Chi square distribution
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Autocorrelation Testing
• Seek the autocorrelation between every m
numbers (I.e. dependence)
• Null Hypothesis H0 : im = 0
• Note: If values are uncorrelated, im has
normal distribution. So, test hypothesis
against normal distribution.
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Gap Testing
• Gap: Interval of recurrence of same digit.
• Monitor Frequency of gaps and test
1.- Specify the cdf F(x) = 1-0.9 x+1
2.- Arrange the observed gaps into S(x)
3.- Find D and Dc
4.- Accept or reject the null hypothesis.
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Poker Test
• Frequency of repetition of certain digits in a
series
• Null hypothesis is tested againts the Chisquare distribution.
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