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CPE 619 Testing Random-Number Generators

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

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  1. 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

  2. Overview • Chi-square test • Kolmogorov-Smirnov Test • Serial-correlation Test • Two-level tests • K-dimensional uniformity or k-distributivity • Serial Test • Spectral Test

  3. 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

  4. 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

  5. Example 27.1 • 1000 random numbers with x0 = 1 • Observed difference = 10.380 • Observed is Less  Accept IID U(0, 1)

  6. 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

  7. 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

  8. 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

  9. 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.

  10. 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.

  11. Example 27.2 (cont’d) • K[0.9;n] value for n = 30 and a = 0.1 is 1.0424 • Observed<Table Pass

  12. Chi-square vs. K-S Test

  13. 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)

  14. Example 27.3: Serial Correlation Test 10,000 random numbers with x0=1:

  15. Example 27.3 (cont’d) • All confidence intervals include zero  All covariances are statistically insignificant at 90% confidence.

  16. 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.

  17. 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

  18. 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

  19. 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)

  20. Example 27.5 • Consider the polynomial: • Better 2-distributivity than Example 27.4

  21. 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

  22. 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

  23. 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

  24. Plot of overlapping pairs All points lie on three straight lines. Or: Example 27.6: Spectral Test

  25. 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.

  26. 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

  27. Example 27.7 • Compare the following two generators: • Using a seed of x0=15, first generator: • Using the same seed in the second generator:

  28. 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

  29. Example 27.7 (cont’d) • First Generator:

  30. 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

  31. Example 27.7 (cont’d) • Second Generator:

  32. 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

  33. 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)

  34. 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

  35. Random Variate Generation

  36. Overview • Inverse transformation • Rejection • Composition • Convolution • Characterization

  37. Random-Variate Generation • General Techniques • Only a few techniques may apply to a particular distribution • Look up the distribution in Chapter 29

  38. 1.0 u CDF F(x) 0.5 0 x Inverse Transformation • Used when F-1 can be determined either analytically or empirically

  39. Proof

  40. 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:

  41. Example 28.2 • The packet sizes (trimodal) probabilities: • The CDF for this distribution is:

  42. 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

  43. Applications of the Inverse-Transformation Technique

  44. 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

  45. 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

  46. 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:

  47. 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.

  48. 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

  49. 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

  50. 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

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