1. Validating a Random Number Generator Based on: A Test of Randomness Based on the Consecutive Distance Between Random Number Pairs By: Matthew J. Duggan, John H. Drew, Lawrence M. Leemis Presented By: Sarah Daugherty MSIM 852 Fall 2007

2. Daugherty MSIM 852 Fall 2007 Introduction  Random numbers are critical to Monte Carlo simulation, discrete event simulation, and bootstrapping  There is a need for RNG with good statistical properties.  One of the most popular methods for generating random numbers in a computer program is a Lehmer RNG.

3. Daugherty MSIM 852 Fall 2007 Lehmer Random Number Generators  Lehmer’s algorithm: an iterative equation produces a stream of random numbers.  Requires 3 inputs: m, a, and x 0. m = modulus, a large fixed prime number a = multiplier, a fixed positive integer < m x 0 = initial seed, a positive integer < m  Produces integers in the range (1, m-1)

4. Daugherty MSIM 852 Fall 2007 Problem  Lehmer RNG are not truly random With carefully chosen m and a, it’s possible to generate output that is “random enough” from a statistical point of view.  However, still considered good generators because their output can be replicated, they’re portable, efficient, and thoroughly documented.  Marsaglia (1968) discovered too much regularity in Lehmer RNG’s.

5. Daugherty MSIM 852 Fall 2007 Marsaglia’s Discovery  He observed a lattice structure when consecutive random numbers were plotted as overlapping ordered pairs. ((x 0, x 1, x 2,…, x n ), (x 1, x 2,…, x n+1 )) Lattice created using m = 401, a = 23. Does not appear to be random at all; BUT a degree of randomness MAY be hidden in it.

6. Daugherty MSIM 852 Fall 2007 Solution  Find the hidden randomness in the order in which the points are generated.  The observed distribution of the distance between consecutive RN’s should be close to the theoretical distance.  Develop a test based on these distances. Hoping to observe that points generally are not generated in order along a plane or in a regular pattern between planes.

7. Daugherty MSIM 852 Fall 2007 Overlapping vs. Non-overlapping Pairs  Considering distance between consecutive pairs of random numbers, points can be overlapping or non-overlapping. Overlapping: (x i, x i+1 ), (x i+1, x i+2 ) Non-overlapping: (x i, x i+1 ), (x i+2, x i+3 )  Both approaches are valid.  The non-overlapping case is mathematically easier in that the 4 numbers represented are independent therefore the 2 points they represent are also independent.

8. Daugherty MSIM 852 Fall 2007 Non-overlapping Theoretical Distribution  If we assume X 1, X 2, X 3, X 4 are IID U(0,1) random variables, we can find the distance between (X 1, X 2 ) and (X 3, X 4 ) by:

9. Daugherty MSIM 852 Fall 2007 Non-overlapping Theoretical Distribution  The cumulative distribution, F(x), of D.

10. Daugherty MSIM 852 Fall 2007 Goodness-of-Fit Test  Now we can compare our theoretical distribution against the Lehmer generator.  Convert the distances between points into an empirical distribution, F(x), which will allow us to perform a hypothesis test. ^ ^ N(x) = # of values that do not exceed x n = # of distances collected ^ ^

11. Daugherty MSIM 852 Fall 2007 Classification of Results  Based on results of 3 hypothesis tests (KS, CVM, and AD tests), each RNG can be classified as: Good – the null hypothesis was not rejected in any test. Suspect – the null hypothesis was rejected in 1 or 2 of the tests. Bad – the null hypothesis was rejected in all 3 tests.

12. Daugherty MSIM 852 Fall 2007 Results  Interesting cases are when a multiplier is rejected by only 1 or 2 of the 3 tests. See a = 3 in table.

13. Daugherty MSIM 852 Fall 2007 Random number pairs Distances connecting pairs F(x) (solid) vs. F(x) (dotted) ^ GoodSuspectBad

14. Daugherty MSIM 852 Fall 2007 Summary  A test of randomness was developed for Lehmer RNG’s based on distance between consecutive pairs of random numbers.  Since some multipliers are rejected by only one or two of the 3 hypothesis tests, the distance between parallel hyperplanes should not be used as the only basis for a test of randomness. The order in which pairs are generated is a second factor to consider.

15. Daugherty MSIM 852 Fall 2007 Critique  Potential – limited. Many other tests exist for validating RNG’s.  Impact – minimal. Frequently used RNG’s use a modulus much larger than the m=401 used here.  Overall – paper is well written; in it’s current state, this test is a justified addition to collection of tests for RNG’s.  Future – use larger modulus; improve theoretical distribution by improving numerical calculations of integral for cdf; test other non-Lehmer generators such as additive linear, composite, or quadratic.