Hurst's Rescaled Range Statistical Analysis for Pseudorandom Number Generators used in Physical Simulations

The rescaled range statistical analysis (R/S) is proposed as a new method to detect correlations in pseudorandom number generators used in Monte Carlo simulations. In an extensive test it is demonstrated that the RS analysis provides a very sensitive method to reveal hidden long run and short run correlations. Several widely used and also some recently proposed pseudorandom number generators are subjected to this test. In many generators correlations are detected and quantified.Comment: 12 pages, 12 figures, 6 tables. Replaces previous version to correct citation [19

A Statistical Evaluation of Algorithms for Independently Seeding Pseudo-Random Number Generators of Type Multiplicative Congruential (Lehmer-Class).

To be effective, a linear congruential random number generator (LCG) should produce values that are (a) uniformly distributed on the unit interval (0,1) excluding endpoints and (b) substantially free of serial correlation. It has been found that many statistical methods produce inflated Type I error rates for correlated observations. Theoretically, independently seeding an LCG under the following conditions attenuates serial correlation: (a) simple random sampling of seeds, (b) non-replicate streams, (c) non-overlapping streams, and (d) non-adjoining streams. Accordingly, 4 algorithms (each satisfying at least 1 condition) were developed: (a) zero-leap, (b) fixed-leap, (c) scaled random-leap, and (d) unscaled random-leap. Note that the latter satisfied all 4 independent seeding conditions. To assess serial correlation, univariate and multivariate simulations were conducted at 3 equally spaced intervals for each algorithm (N=24) and measured using 3 randomness tests: (a) the serial correlation test, (b) the runs up test, and (c) the white noise test. A one-way balanced multivariate analysis of variance (MANOVA) was used to test 4 hypotheses: (a) omnibus, (b) contrast of unscaled vs. others, (c) contrast of scaled vs. others, and (d) contrast of fixed vs. others. The MANOVA assumptions of independence, normality, and homogeneity were satisfied. In sum, the seeding algorithms did not differ significantly from each other (omnibus hypothesis). For the contrast hypotheses, only the fixed-leap algorithm differed significantly from all other algorithms. Surprisingly, the scaled random-leap offered the least difference among the algorithms (theoretically this algorithm should have produced the second largest difference). Although not fully supported by the research design used in this study, it is thought that the unscaled random-leap algorithm is the best choice for independently seeding the multiplicative congruential random number generator. Accordingly, suggestions for further research are proposed

A Search for Good Pseudo-random Number Generators : Survey and Empirical Studies

In today's world, several applications demand numbers which appear random but are generated by a background algorithm; that is, pseudo-random numbers. Since late $19^{th}$ century, researchers have been working on pseudo-random number generators (PRNGs). Several PRNGs continue to develop, each one demanding to be better than the previous ones. In this scenario, this paper targets to verify the claim of so-called good generators and rank the existing generators based on strong empirical tests in same platforms. To do this, the genre of PRNGs developed so far has been explored and classified into three groups -- linear congruential generator based, linear feedback shift register based and cellular automata based. From each group, well-known generators have been chosen for empirical testing. Two types of empirical testing has been done on each PRNG -- blind statistical tests with Diehard battery of tests, TestU01 library and NIST statistical test-suite and graphical tests (lattice test and space-time diagram test). Finally, the selected $29$ PRNGs are divided into $24$ groups and are ranked according to their overall performance in all empirical tests

A Comparative Study of Some Pseudorandom Number Generators

We present results of an extensive test program of a group of pseudorandom number generators which are commonly used in the applications of physics, in particular in Monte Carlo simulations. The generators include public domain programs, manufacturer installed routines and a random number sequence produced from physical noise. We start by traditional statistical tests, followed by detailed bit level and visual tests. The computational speed of various algorithms is also scrutinized. Our results allow direct comparisons between the properties of different generators, as well as an assessment of the efficiency of the various test methods. This information provides the best available criterion to choose the best possible generator for a given problem. However, in light of recent problems reported with some of these generators, we also discuss the importance of developing more refined physical tests to find possible correlations not revealed by the present test methods.Comment: University of Helsinki preprint HU-TFT-93-22 (minor changes in Tables 2 and 7, and in the text, correspondingly

Analysis of Random Number Generators Using Monte Carlo Simulation

Revisions are almost entirely in the introduction and conclusion. Results are unchanged, however the comments and recommendations on different generators were changed, and more references were added.Comment: Email: [email protected] 16 pages, Latex with 1 postscript figure. NPAC technical report SCCS-52

Generation of pseudo-random numbers

Practical methods for generating acceptable random numbers from a variety of probability distributions which are frequently encountered in engineering applications are described. The speed, accuracy, and guarantee of statistical randomness of the various methods are discussed

Hardware Accelerated Scalable Parallel Random Number Generation

The Scalable Parallel Random Number Generators library (SPRNG) is widely used due to its speed, quality, and scalability. Monte Carlo (MC) simulations often employ SPRNG to generate large quantities of random numbers. Thanks to fast Field-Programmable Gate Array (FPGA) technology development, this thesis presents Hardware Accelerated SPRNG (HASPRNG) for the Virtex-II Pro XC2VP30 FPGAs. HASPRNG includes the full set of SPRNG generators and provides programming interfaces which hide detailed internal behavior from users. HASPRNG produces identical results with SPRNG, and it is verified with over 1 million consecutive random numbers for each type of generator. The programming interface allows a developer to use HASPRNG the same way as SPRNG. HASPRNG introduces 4-70 times faster execution than the original SPRNG. This thesis describes the implementation of HASPRNG, the verification platform, the programming interface, and its performance

Pseudo-random number generators for Monte Carlo simulations on Graphics Processing Units

Basic uniform pseudo-random number generators are implemented on ATI Graphics Processing Units (GPU). The performance results of the realized generators (multiplicative linear congruential (GGL), XOR-shift (XOR128), RANECU, RANMAR, RANLUX and Mersenne Twister (MT19937)) on CPU and GPU are discussed. The obtained speed-up factor is hundreds of times in comparison with CPU. RANLUX generator is found to be the most appropriate for using on GPU in Monte Carlo simulations. The brief review of the pseudo-random number generators used in modern software packages for Monte Carlo simulations in high-energy physics is present.Comment: 31 pages, 9 figures, 3 table

Fundamental Change in Random Number Generation�

The current state of the art in computer random number generation uses a method developed over thirty-five years ago. Although much has been done to improve the sequences generated by these methods, they still have serious problems. As the results become more "random", the methodology becomes more complex. Perhaps this explains the number of poor generators in use today. The following study is an attempt to develop a fundamental change in the methodology of random number generation in an effort to both simplify and improve current methods.Industrial Engineering and Managemen