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

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.

Lehmer Random Number Generators Lehmer’s algorithm: an iterative equation produces a stream of random numbers. Requires 3 inputs: m, a, and x0. m = modulus, a large fixed prime number a = multiplier, a fixed positive integer < m x0 = initial seed, a positive integer < m Produces integers in the range (1, m-1)

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.

Marsaglia’s Discovery He observed a lattice structure when consecutive random numbers were plotted as overlapping ordered pairs. ((x0, x1, x2,…, xn), (x1, x2,…, xn+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.

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. The assumption being made is that generators that yield a distribution of distances similar to the purely random points will be better random number generators.

Overlapping vs. Non-overlapping Pairs Considering distance between consecutive pairs of random numbers, points can be overlapping or non-overlapping. Overlapping: (xi, xi+1), (xi+1, xi+2) Non-overlapping: (xi, xi+1), (xi+2, xi+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.

Non-overlapping Theoretical Distribution If we assume X1, X2, X3, X4 are IID U(0,1) random variables, we can find the distance between (X1, X2) and (X3, X4) by:

Non-overlapping Theoretical Distribution The cumulative distribution, F(x), of D.

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

Hypothesis Testing Kolmogorov-Smirnov (KS) test KS test statistic is the largest distance between F(x) and F(x) for all values of x. A large value indicates a poor fit, thus rejecting H0. Cramer-von Mises (CVM) test Integral of the squared differences of the theoretical and empirical distribution. If this value is larger than the tabulated value, reject H0. Anderson-Darling (AD) test Designed to detect discrepancies in the tails of distributions by giving weights to the differences. Weights are largest for F(x) close to 1 (right tail) and F(x) close to 0 (left tail). ^ KS test - Compares the closeness between F and F-hat. AD test – a drawback of KS test is that the same weight is given to the differences for every value of x ^ ^

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.

Results Interesting cases are when a multiplier is rejected by only 1 or 2 of the 3 tests. See a = 3 in table.

Distances connecting pairs Random number pairs Distances connecting pairs F(x) (solid) vs. F(x) (dotted) ^ Good Suspect Bad

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.

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.