Uniform Random Number Research Articles

On the generation of Tikhonov variates

A novel, simple and efficient method for the generation of Tikhonov (a.k.a. von Mises) random variates is proposed. In the proposed method, circular variates of a prescribed Tikhonov distribution pT(x;alpha,xi) are generated via the transformation of variates selected randomly, on a one-for-one basis, from a bank of K distinct Cauchy and Gaussian generators. The mutually exclusive probabilities of sampling from each of the Cauchy or Gaussian generators, as well as the variance and half-width parameters that specify the latter, are derived directly from the Cauchy, Gaussian and Tikhonov characteristic functions, all of which are either known or given in closed form. The proposed random mixture technique is extremely efficient in that a single pair of uniform random numbers is consumed in the generation of each Tikhonov (or von Mises) sample, regardless of the prescribed concentration and centrality parameters (alpha, xi), all requiring neither the rejection of samples, nor the repetitive evaluation of computationally demanding functions. Additional attractive features of the method are as follows. By construction, the first (dominant) N circular moments of Tikhonov variates generated with the proposed random mixture technique are the ones that best approximate their corresponding theoretical values, with errors measured exactly. The exact distribution of generated Tikhonov variates is determined analytically, and its (Kullback-Leibler) divergence to the exact Tikhonov PDF is shown also analytically to be negligible. Finally, the technique establishes a connection between Tikhonov and Gaussian variates which can be exploited, e.g., in the generation of piecewise-continuous pseudo-random functions with Tikhonov-distributed outcomes.

Some Comments on C. S. Wallace's Random Number Generators

We outline some of Chris Wallace's contributions to pseudo-random number generation. In particular, we consider his idea for generating normally distributed variates without relying on a source of uniform random numbers, and compare it with more conventional methods for generating normal random numbers. Implementations of Wallace's idea can be very fast (approximately as fast as good uniform generators). We discuss the statistical quality of the output, and mention how certain pitfalls can be avoided.

Effects of Initialization on Rule Extraction in Structural Learning

This paper studies how our previously proposed initialization effects the rule extraction of neural networks by structural learning with forgetting. The proposed initialization consists of two steps: (1) initializing weights of hidden units so that their separation hyperplanes should pass through the center of an input pattern set and (2) initializing those of output units to zero. From simulation results on Boolean function discovery problems with 5 and 7 inputs, it has been confirmed that the proposed initialization yields a simpler network structure and higher rule extraction ability than the conventional initialization giving uniform random number to all the initial weights of the network.

Resolution-stationary random number generators

Besides speed and period length, the quality of uniform random number generators (RNGs) is usually assessed by measuring the uniformity of their point sets, formed by taking vectors of successive output values over their entire period length. For F 2 -linear generators, the commonly adopted measures of uniformity are based on the equidistribution of the most significant bits of the output. In this paper, we point out weaknesses of these measures and introduce generalizations that also give importance to the low-order (less significant) bits. These measures look at the equidistribution obtained when we permute the bits of each output value in a certain way. In a parameter search for good generators, a quality criterion based on these new measures of equidistribution helps to avoid generators that fail statistical tests targeting their low-order bits. We also introduce the notion of resolution-stationary generators, whose point sets are invariant under a multiplication by certain powers of 2, modulo 1. For such generators, least significant bits have the same equidistribution properties as the most significant ones. Tausworthe generators have this property. We finally show how an arbitrary F 2 -linear generator can be made resolution-stationary by adding an appropriate linear transformation to the output. This provides new efficient ways of implementing high-quality and long-period Tausworthe generators.

Implementing quality control on a random number stream to improve a stochastic weather generator

AbstractFor decades, stochastic modellers have used computerized random number generators to produce random numeric sequences fitting a specified statistical distribution. Unfortunately, none of the random number generators we tested satisfactorily produced the target distribution. The result is generated distributions whose mean even diverges from the mean used to generate them, regardless of the length of run. Non‐uniform distributions from short sequences of random numbers are a major problem in stochastic climate generation, because truly uniform distributions are required to produce the intended climate parameter distributions. In order to ensure generation of a representative climate with the stochastic weather generator CLIGEN within a 30‐year run, we tested the climate output resulting from various random number generators. The resulting distributions of climate parameters showed significant departures from the target distributions in all cases. We traced this failure back to the uniform random number generators themselves. This paper proposes a quality control approach to select only those numbers that conform to the expected distribution being retained for subsequent use. The approach is based on goodness‐of‐fit analysis applied to the random numbers generated. Normally distributed deviates are further tested with confidence interval tests on their means and standard deviations. The positive effect of the new approach on the climate characteristics generated and the subsequent deterministic process‐based hydrology and soil erosion modelling are illustrated for four climatologically diverse sites. Copyright © 2007 John Wiley & Sons, Ltd.

TestU01

We introduce TestU01 , a software library implemented in the ANSI C language, and offering a collection of utilities for the empirical statistical testing of uniform random number generators (RNGs). It provides general implementations of the classical statistical tests for RNGs, as well as several others tests proposed in the literature, and some original ones. Predefined tests suites for sequences of uniform random numbers over the interval (0, 1) and for bit sequences are available. Tools are also offered to perform systematic studies of the interaction between a specific test and the structure of the point sets produced by a given family of RNGs. That is, for a given kind of test and a given class of RNGs, to determine how large should be the sample size of the test, as a function of the generator's period length, before the generator starts to fail the test systematically. Finally, the library provides various types of generators implemented in generic form, as well as many specific generators proposed in the literature or found in widely used software. The tests can be applied to instances of the generators predefined in the library, or to user-defined generators, or to streams of random numbers produced by any kind of device or stored in files. Besides introducing TestU01 , the article provides a survey and a classification of statistical tests for RNGs. It also applies batteries of tests to a long list of widely used RNGs.

High Quality Uniform Random Number Generation Using LUT Optimised State-transition Matrices

This paper presents a family of uniform random number generators designed for efficient implementation in Lookup table (LUT) based FPGA architectures. A generator with a period of 2 k ???1 can be implemented using k flip-flops and k LUTs, and provides k random output bits each cycle. Each generator is based on a binary linear recurrence, with a state-transition matrix designed to make best use of all available LUT inputs in a given FPGA architecture, and to ensure that the critical path between all registers is a single LUT. This class of generator provides a higher sample rate per area than LFSR and Combined Tausworthe generators, and operates at similar or higher clock-rates. The statistical quality of the generators increases with k, and can be used to pass all common empirical tests such as Diehard, Crush and the NIST cryptographic test suite. Theoretical properties such as global equidistribution can also be calculated, and best and average case statistics shown. Due to the large number of random bits generated per cycle these generators can be used as a basis for generators with even higher statistical quality, and an example involving combination through addition is demonstrated.

A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

We introduce and study a randomized quasi-Monte Carlo method for the simulation of Markov chains up to a random (and possibly unbounded) stopping time. The method simulates n copies of the chain in parallel, using a (d+1)-dimensional, highly uniform point set of cardinality n, randomized independently at each step, where d is the number of uniform random numbers required at each transition of the Markov chain. The general idea is to obtain a better approximation of the state distribution, at each step of the chain, than with standard Monte Carlo. The technique can be used in particular to obtain a low-variance unbiased estimator of the expected total cost when state-dependent costs are paid at each step. It is generally more effective when the state space has a natural order related to the cost function.We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial. The variance can be reduced by factors of several thousands in some cases. We prove bounds on the convergence rate of the worst-case error and of the variance for special situations where the state space of the chain is a subset of the real numbers. In line with what is typically observed in randomized quasi-Monte Carlo contexts, our empirical results indicate much better convergence than what these bounds guarantee.

An efficient stochastic diffusion algorithm for modeling second messengers in dendrites and spines

Intracellular signaling pathways, which encompass both biochemical reactions and second messenger diffusion, interact non-linearly with neuronal membrane properties in their role as essential intermediaries for synaptic plasticity and neuromodulation. Computational modeling is a productive approach for investigating these phenomena; however, most current strategies for modeling neurons exclude signaling pathways. To overcome this deficiency, a new algorithm is presented to simulate stochastic diffusion in a highly efficient manner. The gain in speed is obtained by considering collections of molecules, instead of tracking the movement of individual molecules. The probability of a molecule leaving a spatially discrete compartment is used to create a lookup table that stores the probability of k m molecules leaving the compartment as a function of the total number of molecules in the compartment. During the simulation, the number of molecules leaving the compartment is determined using a uniform random number as an index into the lookup table. Simulations illustrate the accuracy of this algorithm by comparing it with the theoretical solution for deterministic diffusion. Additional simulations show how spines on a dendritic branch compartmentalize diffusible molecules. The efficiency of the algorithm is sufficient to allow simulation of second messenger pathways in a multitude of spines on an entire neuron.

A cautionary note on the use of simulation procedures for analyzing contingency tables containing small expected cell frequencies

A comparison of a simple simulation procedure and exact tests for tables used in psychiatric genetic studies is performed, with focus on tables with small expected cell counts. The study shows that naive simulation procedures using uniform random numbers, could lead to conservative results, as compared with Fisher exact test for contingency tables, thus discarding as non-significant tables that are significant according to the exact test. Exact tests are recommended as an alternative to naive simulation for evaluating the statistical significance of contingency tables with small expected cell counts.

Monte Carlo Simulation of Stress Corrosion Cracking in Structural Metal Materials

According to laboratory accelerated test data, stress corrosion cracking (SCC) in structural metal materials occurs by initiation and coalescence of micro cracks, subcritical crack growth, multiple large crack formation and final failure under the combination of materials, stress and corrosive environment. In this paper, a computer simulation model for the process of SCC has been proposed. The procedure is as follows: The possible number of crack initiations is set for a given space and the initiation times for all cracks are assigned by random numbers based on exponential distribution. The sites and length of the cracks are assigned by uniform random numbers and normal random numbers, respectively. The coalescence of cracks and the subcritical crack growth are determined based on the fracture mechanics. The simulation is terminated when the maximum crack length reaches a critical value or all of the possible number of cracks is initiated. The results obtained in this paper indicate the applicability of the present model to predict the SCC behavior in real structures based on the laboratory accelerated test data.

Improved long-period generators based on linear recurrences modulo 2

Fast uniform random number generators with extremely long periods have been defined and implemented based on linear recurrences modulo 2. The twisted GFSR and the Mersenne twister are famous recent examples. Besides the period length, the statistical quality of these generators is usually assessed via their equidistribution properties. The huge-period generators proposed so far are not quite optimal in this respect. In this article, we propose new generators of that form with better equidistribution and “bit-mixing” properties for equivalent period length and speed. The state of our new generators evolves in a more chaotic way than for the Mersenne twister. We illustrate how this can reduce the impact of persistent dependencies among successive output values, which can be observed in certain parts of the period of gigantic generators such as the Mersenne twister.

Generation of physical random number using frequency-modulated LC oscillation circuit with shot noise

The authors previously published a physical method for the generation of random numbers by a variable capacitance parametron, which can generate uniform random numbers without periodicity. It is considered possible to generate random numbers with an ordinary oscillator if the oscillation frequency is made unstable. As a method of introducing instability, frequency modulation of an LC oscillator with shot noise is considered. The present paper describes experiments with an LC oscillator to which shot noise is applied. An evaluation of statistical randomness (FIPS140-2) is carried out for the generated number sequence as cryptographic random numbers. It is shown that generation of random numbers is possible. © 2005 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 88(5): 12–19, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20149

A Comment on the Implementation of the Ziggurat Method

We show that the short period of the uniform random number generator in the published implementation of Marsaglia and Tsang's Ziggurat method for generating random deviates can lead to poor distributions. Changing the uniform random number generator used in its implementation fixes this issue.

Phase-ordering simulation of one-dimensional conserved kink system

The kink-antikink kinetics of one-dimensional phase ordering under conserved order parameter dynamics is studied numerically. The average domain size is found to grow logarithmically, and the distribution of domain size and order parameter correlation function are shown to satisfy a scaling relation. The two-time autocorrelation function follows a power law of A (t(0))(t) approximately t(-lambda) , where lambda depends on the start time of the calculation t(0) . If t(0) is in the scaling regime, lambda takes a constant value of 3.0. Thus the scaling functions are sensitive to the initial configuration of domains. When the initial kink positions are given by uniform random numbers, the scaling functions agree with those obtained by cell dynamical system simulation.

The Godunov-inverse iteration: A fast and accurate solution to the symmetric tridiagonal eigenvalue problem

We present a new algorithm for computing eigenvectors of real symmetric tridiagonal matrices based on Godunov's two-sided Sturm sequence method and inverse iteration, which we call the Godunov-inverse iteration. We use eigenvector approximations computed recursively from two-sided Sturm sequences as starting vectors in inverse iteration, replacing any nonnumeric elements of these approximate eigenvectors with uniform random numbers. We use the left-hand bounds of the smallest machine presentable eigenvalue intervals found by the bisection method as inverse iteration shifts, while staying within guaranteed error bounds. In most test cases convergence is reached after only one or two iterations, producing accurate residuals.

First and second order Markov chain models for synthetic generation of wind speed time series

Hourly wind speed time series data of two meteorological stations in Malaysia have been used for stochastic generation of wind speed data using the transition matrix approach of the Markov chain process. The transition probability matrices have been formed using two different approaches: the first approach involves the use of the first order transition probability matrix of a Markov chain, and the second involves the use of a second order transition probability matrix that uses the current and preceding values to describe the next wind speed value. The algorithm to generate the wind speed time series from the transition probability matrices is described. Uniform random number generators have been used for transition between successive time states and within state wind speed values. The ability of each approach to retain the statistical properties of the generated speed is compared with the observed ones. The main statistical properties used for this purpose are mean, standard deviation, median, percentiles, Weibull distribution parameters, autocorrelations and spectral density of wind speed values. The comparison of the observed wind speed and the synthetically generated ones shows that the statistical characteristics are satisfactorily preserved.

A system of high-dimensional, efficient, long-cycle and portable uniform random number generators

We propose a system of multiple recursive generators of moduluspand orderkwhere all nonzero coefficients of the recurrence are equal. The advantage of this property is that a single multiplication is needed to compute the recurrence, so the generator would run faster than the general case. Forp= 231− 1, the most popular modulus used, we provide tables of specific parameter values yielding maximum period for recurrence of orderk= 102 and 120. Forp= 231− 55719 andk= 1511, we have found generators with a period length approximately 1014100.5.

Continuous random variate generation by fast numerical inversion

The inversion method for generating nonuniform random variates has some advantages compared to other generation methods, since it monotonically transforms uniform random numbers into non-uniform random variates. Hence, it is the method of choice in the simulation literature. However, except for some simple cases where the inverse of the cumulative distribution function is a simple function we need numerical methods. Often inversion by "brute force" is used, applying either very slow iterative methods or linear interpolation of the CDF and huge tables. But then the user has to accept unnecessarily large errors or excessive memory requirements, that slow down the algorithm. In this article, we demonstrate that with Hermite interpolation of the inverse CDF we can obtain very small error bounds close to machine precision. Using our adaptive interval splitting method, this accuracy is reached with moderately sized tables that allow for a fast and simple generation procedure.

GENERATING REPRESENTATIVE SEQUENCES OF DAILY PRECIPITATION FOR AGRICULTURAL SIMULATIONS

Generated weather that represents alternative realizations of a particular historical record is often needed forcomputer simulation of agricultural or agronomic impacts. The stochastic component of generated weather is based onrandom numbers. This study shows that relatively short sequences of uniform random numbers, as used to generate dailyprecipitation, may not display sufficient uniformity to reproduce the distribution of the target historical precipitation recordor of a seasonal precipitation forecast. The magnitude of the discrepancy becomes an issue in applications that requireaccurate replication of the historical record or forecast. A procedure is proposed to test sequences of uniform random numbersbefore their use and retain only those sequences that display a high degree of uniformity as required by the weather generationmodel. Generated precipitation with and without testing for uniformity of random numbers shows that the discrepancybetween the two can be important particularly for simulation duration less than 50 years, as are often involved in practicalwater resources and agricultural applications. The use of tested random numbers leads to transition probabilities and aprecipitation distribution that are generally closer in agreement with those of the historical record. The effectiveness of themethod is illustrated by comparing generated daily precipitation based on tested and nontested random number forthe Kingfisher, Oklahoma, historical precipitation record. The increased representativeness of the generated precipitationusing tested random numbers allows for shorter simulation duration of agricultural models and greater capability tosimulate subtle changes in precipitation such as those associated with seasonal precipitation forecasts.