User's Guide: Testing:Test Results

Test Results

Serial Test

The serial test measures correlation between adjacent elements in a random number sequence. We multiply the double precision random numbers by d and truncate to get an integer in [0,d). We consider n subsequences of 2 random numbers each. There are d² possible subsequences, and each should be equally likely if the distribution were truly uniform. If we interleave two streams with adjacent elements of the sequence coming from the same stream, then again, each subsequence will be equally likely if the individual streams are uniform and the corresponding elements of each stream are uncorrelated. The Chisquare test can then be applied to these sequences to determine the likelihood of the sequence coming from a uniform distribution.

We tested a set of 100 streams individually, without interleaving. We considered 10⁵ pairs of adjacent random integers in [0,100). We then interleaved 100 pairs of these streams and performed this test again. In both the cases, acceptable values of the Chisquare distribution were obtained. We plan to conduct larger tests later.

We illustrate the results of the test qualitatively below. We consider a million random integers in [0,256) from two streams. Then we form pairs from corresponding elements of each stream. There are 256² such possible values for these pairs. We determine the frequencies of each possible pair. We then subtract the expected frequency and scale the result to a value in [0,256). In the plot below, these scaled values are treated as the pixel intensities. The horizontal axis represents the values from the first stream and the vertical axis those from the second stream.

48 bit Linear Congruential Generator with prime addend	Modified Lagged Fibonacci Generator

If any patterns could be observed in the figures above, then this would indicate correlations between corresponding elements of the two streams.

Overview of Empirical Test Results

The numbers in the table are the total number of random numbers tested.

	Sequential					Parallel
Test	48 bit LCG	LFG	MLFG	64 bit LCG	CMRG	PMLCG	48 bit LCG	LFG	MLFG	64 bit LCG	CMRG	PMLCG
Collisions	1012	1012	1012	1012	1012	1012	1012	1012	1012	1012	1012	1012
Coupon	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011
Equidistribution	1011	1011	1011	1011	1011	1011	Not needed
Gap	1011	1013	1011	1011	1011	1011	1012	1012	1011	1011	1011	1011
Maximum of t	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011
Permutations	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011
Poker	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011
Runs up	1011	1011	1011	1011	1011	1011	1012	1012	1011	1011	1011	1011
Serial	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011
Sum of distributions	Not needed						1011	1011	1011	1011	1011	1011
Ising: Metropolis	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011
Ising: Wolff	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011	1011
Random Walk	108	108	108	108	108	108	108	108	108	108	108	108