CPE 619 Testing Random-Number Generators

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# CPE 619 Testing Random-Number Generators - PowerPoint PPT Presentation

CPE 619 Testing Random-Number Generators. Aleksandar Milenković The LaCASA Laboratory Electrical and Computer Engineering Department The University of Alabama in Huntsville http://www.ece.uah.edu/~milenka http://www.ece.uah.edu/~lacasa. Overview. Chi-square test Kolmogorov-Smirnov Test

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### CPE 619Testing Random-Number Generators

Aleksandar Milenković

The LaCASA Laboratory

Electrical and Computer Engineering Department

The University of Alabama in Huntsville

http://www.ece.uah.edu/~milenka

http://www.ece.uah.edu/~lacasa

Overview
• Chi-square test
• Kolmogorov-Smirnov Test
• Serial-correlation Test
• Two-level tests
• K-dimensional uniformity or k-distributivity
• Serial Test
• Spectral Test
Testing Random-Number Generators

Goal: To ensure that the random number generator produces a

random stream

• Plot histograms
• Plot quantile-quantile plot
• Use other tests
• Passing a test is necessary but not sufficient
• Pass ¹ GoodFail  Bad
• New tests  Old generators fail the test
• Tests can be adapted for other distributions
Chi-Square Test
• Most commonly used test
• Can be used for any distribution
• Prepare a histogram of the observed data
• Compare observed frequencies with theoretical
• k = Number of cells
• oi = Observed frequency for ith cell
• ei = Expected frequency
• D=0 Þ Exact fit
• D has a chi-square distribution with k-1 degrees of freedom.

Þ Compare D with c2[1-a; k-1] Pass with confidence a if D is less

Example 27.1
• 1000 random numbers with x0 = 1
• Observed difference = 10.380
• Observed is Less  Accept IID U(0, 1)
Chi-Square for Other Distributions
• Errors in cells with a small ei affect the chi-square statistic more
• Best when ei\'s are equal
• Þ Use an equi-probable histogram with variable cell sizes
• Combine adjoining cells so that the new cell probabilities are approximately equal
• The number of degrees of freedom should be reduced to k-r-1 (in place of k-1), where r is the number of parameters estimated from the sample
• Designed for discrete distributions and for large sample sizes only  Lower significance for finite sample sizes and continuous distributions
• If less than 5 observations, combine neighboring cells
Kolmogorov-Smirnov Test
• Developed by A. N. Kolmogorov and N. V. Smirnov
• Designed for continuous distributions
• Difference between the observed CDF (cumulative distribution function) Fo(x) and the expected cdf Fe(x) should be small
Kolmogorov-Smirnov Test
• K+ = maximum observed deviation below the expected cdf
• K- = minimum observed deviation below the expected cdf
• K+ < K[1-a;n] and K- < K[1-a;n]Þ Pass at a level of significance
• Don\'t use max/min of Fe(xi)-Fo(xi)
• Use Fe(xi+1)-Fo(xi) for K-
• For U(0, 1): Fe(x)=x
• Fo(x) = j/n, where x > x1, x2, ..., xj-1
Example 27.2
• 30 Random numbers using a seed of x0=15:
• The numbers are:14, 11, 2, 6, 18, 23, 7, 21, 1, 3, 9, 27, 19, 26, 16, 17, 20, 29, 25, 13, 8, 24, 10, 30, 28, 22, 4, 12, 5, 15.
Example 27.2 (cont’d)
• The normalized numbers obtained by dividing the sequence by 31 are:0.45161, 0.35484, 0.06452, 0.19355, 0.58065, 0.74194, 0.22581, 0.67742, 0.03226, 0.09677, 0.29032, 0.87097, 0.61290, 0.83871, 0.51613, 0.54839, 0.64516, 0.93548, 0.80645, 0.41935, 0.25806, 0.77419, 0.32258, 0.96774, 0.90323, 0.70968, 0.12903, 0.38710, 0.16129, 0.48387.
Example 27.2 (cont’d)
• K[0.9;n] value for n = 30 and a = 0.1 is 1.0424
• Observed<Table Pass
Serial-Correlation Test
• Nonzero covariance Þ Dependence. The inverse is not true
• Rk = Autocovariance at lag k = Cov[xn, xn+k]
• For large n, Rk is normally distributed with a mean of zero and a variance of 1/[144(n-k)]
• 100(1-a)% confidence interval for the autocovariance is:

For k1 Check if CI includes zero

• For k = 0, R0= variance of the sequence Expected to be 1/12 for IID U(0,1)
Example 27.3: Serial Correlation Test

10,000 random numbers with x0=1:

Example 27.3 (cont’d)
• All confidence intervals include zero  All covariances are statistically insignificant at 90% confidence.
Two-Level Tests
• If the sample size is too small, the test results may apply locally, but not globally to the complete cycle.
• Similarly, global test may not apply locally
• Use two-level tests

Þ Use Chi-square test on n samples of size k each and then use a Chi-square test on the set of n Chi-square statistics so obtained

Þ Chi-square on Chi-square test.

• Similarly, K-S on K-S
• Can also use this to find a ``nonrandom\'\' segment of an otherwise random sequence.
k-Distributivity
• k-Dimensional Uniformity
• Chi-square Þ uniformity in one dimensionÞ Given two real numbers a1 and b1 between 0 and 1 such that b1> a1
• This is known as 1-distributivity property of un.
• The 2-distributivity is a generalization of this property in two dimensions:

For all choices of a1, b1, a2, b2 in [0, 1], b1>a1 and b2>a2

k-Distributivity (cont’d)
• k-distributed if:
• For all choices of ai, bi in [0, 1], with bi>ai, i=1, 2, ..., k.
• k-distributed sequence is always (k-1)-distributed. The inverse is not true.
• Two tests:
• Serial test
• Spectral test
• Visual test for 2-dimensions: Plot successive overlapping pairs of numbers
Example 27.4
• Tausworthe sequence generated by:
• The sequence is k-distributed for k up to d /le, that is, k=1.
• In two dimensions: Successive overlapping pairs (xn, xn+1)
Example 27.5
• Consider the polynomial:
• Better 2-distributivity than Example 27.4

xn +1

xn

Serial Test
• Goal: To test for uniformity in two dimensions or higher.
• In two dimensions, divide the space between 0 and 1 into K2 cells of equal area
Serial Test (cont’d)
• Given {x1, x2,…, xn}, use n/2 non-overlapping pairs (x1, x2), (x3, x4), … and count the points in each of the K2 cells
• Expected= n/(2K2) points in each cell
• Use chi-square test to find the deviation of the actual counts from the expected counts
• The degrees of freedom in this case are K2-1
• For k-dimensions: use k-tuples of non-overlapping values
• k-tuples must be non-overlapping
• Overlapping  number of points in the cells are not independent chi-square test cannot be used
• In visual check one can use overlapping or non-overlapping
• In the spectral test overlapping tuples are used
• Given n numbers, there are n-1 overlapping pairs, n/2 non-overlapping pairs
Spectral Test
• Goal: To determine how densely the k-tuples {x1, x2, …, xk} can fill up the k-dimensional hyperspace
• The k-tuples from an LCG fall on a finite number of parallel hyper-planes
• Successive pairs would lie on a finite number of lines
• In three dimensions, successive triplets lie on a finite number of planes
Plot of overlapping pairs

All points lie on three straight lines.

Or:

Example 27.6: Spectral Test
Example 27.6 (cont’d)
• In three dimensions, the points (xn, xn-1, xn-2) for the above generator would lie on five planes given by:

Obtained by adding the following to equation

Note that k+k1 will be an integer between 0 and 4.

Spectral Test (More)
• Marsaglia (1968): Successive k-tuples obtained from an LCG fall on, at most, (k!m)1/k parallel hyper-planes, where m is the modulus used in the LCG.
• Example: m = 232, fewer than 2,953 hyper-planes will contain all 3-tuples, fewer than 566 hyper-planes will contain all 4-tuples, and fewer than 41 hyper-planes will contain all 10-tuples. Thus, this is a weakness of LCGs.
• Spectral Test: Determine the max distance between adjacent hyper-planes.
• Larger distanceÞ worse generator
• In some cases, it can be done by complete enumeration
Example 27.7
• Compare the following two generators:
• Using a seed of x0=15, first generator:
• Using the same seed in the second generator:
Example 27.7 (cont’d)
• Every number between 1 and 30 occurs once and only once

Þ Both sequences will pass the chi-square test for uniformity

Example 27.7 (cont’d)
• First Generator:
Example 27.7 (cont’d)
• Three straight lines of positive slope or ten lines of negative slope
• Since the distance between the lines of positive slope is more, consider only the lines with positive slope
• Distance between two parallel lines y=ax+c1 and y=ax+c2 is given by
• The distance between the above lines is or 9.80
Example 27.7 (cont’d)
• Second Generator:
Example 27.7 (cont’d)
• All points fall on seven straight lines of positive slope or six straight lines of negative slope.
• Considering lines with negative slopes:
• The distance between lines is: or 5.76.
• The second generator has a smaller maximum distance and, hence, the second generator has a better 2-distributivity
• The set with a larger distance may not always be the set with fewer lines
Example 27.7 (cont’d)
• Either overlapping or non-overlapping k-tuples can be used
• With overlapping k-tuples, we have k times as many points, which makes the graph visually more complete.The number of hyper-planes and the distance between them are the same with either choice.
• With serial test, only non-overlapping k-tuples should be used.
• For generators with a large m and for higher dimensions, finding the maximum distance becomes quite complex.

See Knuth (1981)

Summary
• Chi-square test is a one-dimensional testDesigned for discrete distributions and large sample sizes
• K-S test is designed for continuous variables
• Serial correlation test for independence
• Two level tests find local non-uniformity
• k-dimensional uniformity = k-distributivity tested by spectral test or serial test

### Random Variate Generation

Overview
• Inverse transformation
• Rejection
• Composition
• Convolution
• Characterization
Random-Variate Generation
• General Techniques
• Only a few techniques may apply to a particular distribution
• Look up the distribution in Chapter 29

1.0

u

CDF

F(x)

0.5

0

x

Inverse Transformation
• Used when F-1 can be determined either analytically or empirically
Example 28.1
• For exponential variates:
• If u is U(0,1), 1-u is also U(0,1)
• Thus, exponential variables can be generated by:
Example 28.2
• The packet sizes (trimodal) probabilities:
• The CDF for this distribution is:
Example 28.2 (cont’d)
• The inverse function is:
• Note: CDF is continuous from the right the value on the right of the discontinuity is used The inverse function is continuous from the left u=0.7  x=64
Rejection
• Can be used if a pdf g(x) exists such that c g(x)majorizes the pdf f(x) c g(x)>f(x)8x
• Steps:
• 1. Generate x with pdf g(x)
• 2. Generate y uniform on [0, cg(x)]
• 3. If y<f(x), then output x and returnOtherwise, repeat from step 1 Continue rejecting the random variates x and y until y>f(x)
• Efficiency = how closely c g(x) envelopes f(x)Large area between c g(x) and f(x)  Large percentage of (x, y) generated in steps 1 and 2 are rejected
• If generation of g(x) is complex, this method may not be efficient
Example 28.2
• Beta(2,4) density function:
• Bounded inside a rectangle of height 2.11 Steps:
• Generate x uniform on [0, 1]
• Generate y uniform on [0, 2.11]
• If y< 20 x(1-x)3, then output x and returnOtherwise repeat from step 1
Composition
• Can be used if CDF F(x) = Weighted sum of n other CDFs.
• Here, , and Fi\'s are distribution functions.
• n CDFs are composed together to form the desired CDFHence, the name of the technique.
• The desired CDF is decomposed into several other CDFs Also called decomposition
• Can also be used if the pdf f(x) is a weighted sum of n other pdfs:
Steps:
• Generate a random integer I such that:
• This can easily be done using the inverse-transformation method.
• Generate x with the ith pdf fi(x) and return.
Example 28.4
• pdf:
• Composition of two exponential pdf\'s
• Generate
• If u1<0.5, return; otherwise return x=a ln u2.
• Inverse transformation better for Laplace
Convolution
• Sum of n variables:
• Generate n random variate yi\'s and sum
• For sums of two variables, pdf of x = convolution of pdfs of y1 and y2. Hence the name
• Although no convolution in generation
• If pdf or CDF = Sum  Composition
• Variable x = Sum  Convolution
Convolution: Examples
• Erlang-k = åi=1k Exponentiali
• Binomial(n, p) = åi=1n Bernoulli(p) Generated n U(0,1), return the number of RNs less than p
• c2(n) = åi=1n N(0,1)2
• G(a, b1)+G(a,b2)=G(a,b1+b2) Non-integer value of b = integer + fraction
• åi=1n Any = Normal å U(0,1) = Normal
• åi=1m Geometric = Pascal
• åi=12 Uniform = Triangular
Characterization
• Use special characteristics of distributions characterization
• Exponential inter-arrival times  Poisson number of arrivals Continuously generate exponential variates until their sum exceeds T and return the number of variates generated as the Poisson variate.
• The ath smallest number in a sequence of a+b+1 U(0,1) uniform variates has a b(a, b) distribution.
• The ratio of two unit normal variates is a Cauchy(0, 1) variate.
• A chi-square variate with even degrees of freedom c2(n) is the same as a gamma variate g(2,n/2).
• If x1 and x2 are two gamma variates g(a,b) and g(a,c), respectively, the ratio x1/(x1+x2) is a beta variate b(b,c).
• If x is a unit normal variate, em+s x is a lognormal(m, s) variate.

Is CDF

invertible?

Yes

Use inversion

Summary

Yes

Is CDF a sum

of other CDFs?

Use composition

Is pdf a sum

of other pdfs?

Yes

Use Composition

Summary (cont’d)

Is the

variate a sum of other

variates

Yes

Use convolution

Is the

variate related to other

variates?

Yes

Use characterization

Does

a majorizing function

exist?

Yes

Use rejection

No

Use empirical inversion

Homework #6
• Submit answers to exercise 27.1
• Submit answers to exercise 27.4
• Due: Monday, April 7, 2008, 12:45 PM
• Submit a hard copy to instructor