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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



  • 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-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


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.


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)

  • 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
  • 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








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
  • 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
  • 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:
  • 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
  • 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
  • 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.




Use inversion



Is CDF a sum

of other CDFs?

Use composition

Is pdf a sum

of other pdfs?


Use Composition

summary cont d
Summary (cont’d)

Is the

variate a sum of other



Use convolution

Is the

variate related to other



Use characterization


a majorizing function



Use rejection


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