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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 619 testing random number generators

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
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
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
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
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
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
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 test1
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
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
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 d1
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
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
Example 27.3: Serial Correlation Test

10,000 random numbers with x0=1:


Example 27 3 cont d
Example 27.3 (cont’d)

  • All confidence intervals include zero  All covariances are statistically insignificant at 90% confidence.


Two level tests
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-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-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
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
Example 27.5

  • Consider the polynomial:

  • Better 2-distributivity than Example 27.4


Serial test

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
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
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


Example 27 6 spectral test

Plot of overlapping pairs

All points lie on three straight lines.

Or:

Example 27.6: Spectral Test


Example 27 6 cont d
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
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
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
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 d1
Example 27.7 (cont’d)

  • First Generator:


Example 27 7 cont d2
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 d3
Example 27.7 (cont’d)

  • Second Generator:


Example 27 7 cont d4
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 d5
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
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



Overview1
Overview

  • Inverse transformation

  • Rejection

  • Composition

  • Convolution

  • Characterization


Random variate generation1
Random-Variate Generation

  • General Techniques

  • Only a few techniques may apply to a particular distribution

  • Look up the distribution in Chapter 29


Inverse transformation

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
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
Example 28.2

  • The packet sizes (trimodal) probabilities:

  • The CDF for this distribution is:


Example 28 2 cont d
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
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 21
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
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:


Cpe 619 testing random number generators

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
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
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
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
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.


Summary1

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
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
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