Testing Random Number Generators


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1 Testing Random Number Generators Raj Jain Washington University Saint Louis, MO Audio/Video recordings of this lecture are available at:
2 Overview 1. Chisquare test 2. KolmogorovSmirnov Test 3. Serialcorrelation Test 4. Twolevel tests 5. Kdimensional uniformity or kdistributivity 6. Serial Test 7. Spectral Test 272
3 Testing RandomNumber Generators Goal: To ensure that the random number generator produces a random stream.! Plot histograms! Plot quantilequantile plot! Use other tests! Passing a test is necessary but not sufficient! Pass Good Fail Bad! New tests Old generators fail the test! Tests can be adapted for other distributions 273
4 ChiSquare 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 o i = Observed frequency for ith cell e i = Expected frequency! D=0 Exact fit! D has a chisquare distribution with k1 degrees of freedom. Compare D with χ 2 [1α; k1] Pass with confidence α if D is less 274
5 ! 1000 random numbers with x 0 = 1! Example 27.1! Observed difference = ! Observed is Less Accept IID U(0, 1) 275
6 ChiSquare for Other Distributions! Errors in cells with a small e i affect the chisquare statistic more! Best when e i 's are equal. Use an equiprobable 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 kr1 (in place of k1), 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 276
7 KolmogorovSmirnov Test! Developed by A. N. Kolmogorov and N. V. Smirnov! Designed for continuous distributions! Difference between the observed CDF (cumulative distribution function) F o (x) and the expected cdf F e (x) should be small. 277
8 KolmogorovSmirnov Test! K + = maximum observed deviation below the expected cdf! K  = minimum observed deviation below the expected cdf! K + < K [1α;n] and K  < K [1α;n] Pass at α level of significance.! Don't use max/min of Fe(x i )F o (x i )! Use F e (x i+1 )F o (x i ) for K ! For U(0, 1): F e (x)=x! F o (x) = j/n, where x > x 1, x 2,..., x j
9 Example Random numbers using a seed of x 0 =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,
10 Example 27.2 (Cont) The normalized numbers obtained by dividing the sequence by 31 are: , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,
11 Example 27.2 (Cont)! K [0.9;n] value for n = 30 and a = 0.1 is ! Observed<Table Pass 2711
12 Chisquare vs. KS K S Test 2712
13 SerialCorrelation Test! Nonzero covariance Dependence. The inverse is not true! R k = Autocovariance at lag k = Cov[x n, x n+k ]! For large n, R k is normally distributed with a mean of zero and a variance of 1/[144(nk)]! 100(1α)% confidence interval for the autocovariance is: For k 1 Check if CI includes zero! For k = 0, R 0 = variance of the sequence Expected to be 1/12 for IID U(0,1) 2713
14 Example 27.3: Serial Correlation Test 10,000 random numbers with x 0 =1: 2714
15 Example 27.3 (Cont)! All confidence intervals include zero All covariances are statistically insignificant at 90% confidence
16 TwoLevel 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 twolevel tests Use Chisquare test on n samples of size k each and then use a Chisquare test on the set of n Chisquare statistics so obtained Chisquare on Chisquare test.! Similarly, KS on KS! Can also use this to find a ``nonrandom'' segment of an otherwise random sequence
17 ! kdimensional Uniformity kdistributivity! Chisquare uniformity in one dimension Given two real numbers a 1 and b 1 between 0 and 1 such that b 1 > a 1! This is known as 1distributivity property of u n.! The 2distributivity is a generalization of this property in two dimensions: For all choices of a 1, b 1, a 2, b 2 in [0, 1], b 1 >a 1 and b 2 >a
18 ! kdistributed if: kdistributivity (Cont)! For all choices of a i, b i in [0, 1], with b i >a i, i=1, 2,..., k.! kdistributed sequence is always (k1)distributed. The inverse is not true.! Two tests: 1. Serial test 2. Spectral test 3. Visual test for 2dimensions: Plot successive overlapping pairs of numbers 2718
19 ! Tausworthe sequence generated by: Example 27.4! The sequence is k distributed for k up to d /l e, that is, k=1.! In two dimensions: Successive overlapping pairs (x n, x n+1 ) 2719
20 Example 27.5! Consider the polynomial:! Better 2distributivity than Example
21 Serial Test! Goal: To test for uniformity in two dimensions or higher.! In two dimensions, divide the space between 0 and 1 into K 2 cells of equal area x n +1 x n 2721
22 Serial Test (Cont)! Given {x 1, x 2,, x n }, use n/2 nonoverlapping pairs (x 1, x 2 ), (x 3, x 4 ), and count the points in each of the K 2 cells.! Expected= n/(2k 2 ) points in each cell.! Use chisquare test to find the deviation of the actual counts from the expected counts.! The degrees of freedom in this case are K 21.! For kdimensions: use ktuples of nonoverlapping values.! ktuples must be nonoverlapping.! Overlapping number of points in the cells are not independent chisquare test cannot be used! In visual check one can use overlapping or nonoverlapping.! In the spectral test overlapping tuples are used.! Given n numbers, there are n1 overlapping pairs, n/2 nonoverlapping pairs
23 Spectral Test! Goal: To determine how densely the ktuples {x 1, x 2,, x k }can fill up the kdimensional hyperspace.! The ktuples from an LCG fall on a finite number of parallel hyperplanes.! Successive pairs would lie on a finite number of lines! In three dimensions, successive triplets lie on a finite number of planes
24 Example 27.6: Spectral Test Plot of overlapping pairs! All points lie on three straight lines.! Or: 2724
25 Example 27.6 (Cont)! In three dimensions, the points (x n, x n1, x n2 ) for the above generator would lie on five planes given by: Obtained by adding the following to equation Note that k+k 1 will be an integer between 0 and
26 Spectral Test (More)! Marsaglia (1968): Successive ktuples obtained from an LCG fall on, at most, (k!m) 1/k parallel hyperplanes, where m is the modulus used in the LCG.! Example: m = 2 32, fewer than 2,953 hyperplanes will contain all 3tuples, fewer than 566 hyperplanes will contain all 4 tuples, and fewer than 41 hyperplanes will contain all 10 tuples. Thus, this is a weakness of LCGs.! Spectral Test: Determine the max distance between adjacent hyperplanes.! Larger distance worse generator! In some cases, it can be done by complete enumeration 2726
27 Example 27.7! Compare the following two generators:! Using a seed of x 0 =15, first generator:! Using the same seed in the second generator: 2727
28 Example 27.7 (Cont)! Every number between 1 and 30 occurs once and only once Both sequences will pass the chisquare test for uniformity 2728
29 ! First Generator: Example 27.7 (Cont) 2729
30 Example 27.7 (Cont)! 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+c 1 and y=ax+c 2 is given by! The distance between the above lines is or
31 ! Second Generator: Example 27.7 (Cont) 2731
32 Example 27.7 (Cont)! 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 2distributivity.! The set with a larger distance may not always be the set with fewer lines
33 Example 27.7 (Cont)! Either overlapping or nonoverlapping ktuples can be used. " With overlapping ktuples, we have k times as many points, which makes the graph visually more complete.the number of hyperplanes and the distance between them are the same with either choice.! With serial test, only nonoverlapping ktuples should be used.! For generators with a large m and for higher dimensions, finding the maximum distance becomes quite complex. See Knuth (1981) 2733
34 Summary 1. Chisquare test is a onedimensional test Designed for discrete distributions and large sample sizes 2. KS test is designed for continuous variables 3. Serial correlation test for independence 4. Two level tests find local nonuniformity 5. kdimensional uniformity = kdistributivity tested by spectral test or serial test 2734
35 Homework 27! Submit detailed answer to Exercise Print 10,000 th number also
36 Exercise 27.1 Generate 10,000 numbers using a seed of x 0 =1 in the following generator: Classify the numbers into ten equal size cells and test for uniformity using the chisquare test at 90% confidence
37 Exercise 27.2 Generate 15 numbers using a seed of x 0 =1 in the following generator: Perform a KS test and check whether the sequence passes the test at a 95% confidence level
38 Exercise 27.3 Generate 10,000 numbers using a seed of x 0 =1 in the following LCG: Perform the serial correlation test of randomness at 90% confidence and report the result
39 Exercise 27.4 Using the spectral test, compare the following two generators Which generator has a better 2distributivity? 2739
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