Download Pseudorandom Number Generators Outline Pseudorandom

Outline
Pseudorandom Number
Generators
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M. Sami Fadali
University of Nevada
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Are they random?
How do we generate “random”
numbers for simulation?
How do we generate “random”
numbers for experiments?
MATLAB
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Pseudorandom Numbers
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Example
Appear random.
Deterministic.
Needed for experiments and
simulations.
MATLAB
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Obtain a sample realization of a GaussMarkov process.
Record of duration
.
Play record in a loop (i.e. periodic).
Appears random but clearly
deterministic.
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Spectrum
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Shift Register
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Discrete (line) spectrum for a periodic signal.
Lines of spectrum are
apart.
Envelope of lines.
Y
்
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1.
ଶ
ଶ
ଶ
2.
ଶ
Generate binary sequence using a shift
register with feedback.
Each clock pulse:
Shift right.
Add XOR(aright,amiddle) to left.
+
ଶ
Y
்
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output
ଶ
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11 1 1 1
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Is it random?
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Plots of Binary Waveform
Repeats after 2n1 bits.
Almost half ones and half zeros (e.g n=10,
512 ones and 512 zeros).
Fixed sequence for fixed initial seed and
feedback.
Completely deterministic but appears random.
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Seed = {1, 1, 1, 1, 1}
Sequence={1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, …}
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Mermistor
Uniform Pseudorandom
Mermistor: true random number generator.
 Used for low power logic.
 See IEEE Spectrum July 2012
http://spectrum.ieee.org/semiconductors/me
mory/a-memristor-true-randomnumbergenerator/?utm_source=techalert&utm_mediu
m=email&utm_campaign=071912
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rand: generates arrays of random numbers
uniformly distributed in the interval (0,1)
 Z = rand(m,n) or Z = rand([m n])
returns an m-by-n matrix of random entries
Z = rand % returns a scalar.
Z ~U[0,1], uniform over the interval.
 Y = (max-min)*Z + min % Y ~U[min, max]
 height= 1/(max  min)
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Theorem (Hogg et al., p. 288)
If the random variable
then,
for any continuous distribution function ,
ିଵ
the random variable
.
 Assume that
is monotone increasing
to simplify the proof (one-one).
 Use the theorem to generate many
distributions using the uniform
distribution.
Proof
௎
ିଵ
ி ௫
1
௎
0.8
0.6
଴
0.4
0.2
0
-1
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-0.5
0
0.5
1
1.5
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Example: Exponential Distrib.
Gaussian Pseudorandom
randn: Normally distributed random numbers
and arrays.
 Z = randn(m,n) or Z = randn([m n])
returns an m-by-n matrix of random entries.
 Z = randn
% returns a scalar.
Z ~N (0,1),
Y = sigma*Z + mean, Y ~N(mean, sigma2)
ି௫
X
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Inverse of Exponential Distribution
ିଵ
X
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Statistics Toolbox
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Gaussian
Generates random numbers with other
distributions using rand/randn.
 Commands in the form xxxrnd
 xxx=name of distribution
Examples: chi2rnd (Chi-square), binornd
(binomial), poissrnd (Poisson), etc.
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Z ~N(mean, sigma2)
>>mean=1;sigma=2;m=2;n=1;
>>Z = normrnd(mean, sigma, m,n)
Z=
0.1349
-2.3312
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Random
Uniform
Uses the name of the distribution as a
parameter (Normal, Poisson, Chi2)
>> rn = random('Normal',1,4,2,3)
% mean 1, var =4, 2 by 3 matrix
rn =
0.6174 2.1776 3.8573
-2.3294 -4.3447 7.4942
Z ~U[min, max]
 height= 1/(max  min)
>> min=1.1;max=2;
random('Uniform',min,max,1,3)
% 1 by 3 array with uniform pdf
ans =
1.9551 1.3080 1.6462
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Interesting Web Sites
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Conclusion
http://www.agner.org/random/
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Programs for generating uniform and nonuniform
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distributions.
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J. E. Gentle, Random Number Generation and Monte
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Carlo Methods, Springer-Verlag, NY, 2003.
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Pseudorandom not random.
Monte Carlo simulation.
Application in physics, engineering,
biology, etc.
http://www.mathworks.com/moler/random.pdf
MATLAB random number generator: for all commands
 Controlling the random number generator: rng
>> rng(sd) % sd = nonnegative integer seed
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