Uniform Random Number Generator Research Articles

Auto Tuning PID Controller based on Improved Genetic Algorithm for Reverse Osmosis Plant

An optimal control of Reverse Osmosis (RO) plant is studied in this paper utilizing the auto tuning concept in conjunction with PID controller. A control scheme composing an auto tuning stochastic technique based on an improved Genetic Algorithm (GA) is proposed. For better evaluation of the process in GA, objective function defined newly in sense of root mean square error has been used. Also in order to achieve better performance of GA, more pureness and longer period of random number generation in operation are sought. The main improvement is made by replacing the uniform distribution random number generator in conventional GA technique to newly designed hybrid random generator composed of Cauchy distribution and linear congruential generator, which provides independent and different random numbers at each individual steps in Genetic operation. The performance of newly proposed GA tuned controller is compared with those of conventional ones via simulation. Keywords—Genetic Algorithm, Auto tuning, Hybrid random number generator, Reverse Osmosis, PID controller.

Scalable parallel multiple recursive generators of large order

To speed up the process of performing a large statistical simulation study, it is natural and common to divide the large-scale simulation task into several relatively independent sub-tasks in a way that these sub-tasks can be handled by individual processors in parallel. To obtain a good overall simulation result by synthesizing results from these sub-tasks, it is crucial that good parallel random number generators are used. Thus, designing suitable and independent uniform random number generators for the sub-tasks has become a very important issue in large-scale parallel simulations. Two commonly used uniform random number generators, linear congruential generator (LCG) and multiple recursive generator (MRG), have served as backbone generators for some parallel random number generators constructed in the past. We will discuss some general construction methods. A systematic leapfrog method to automatically choose different multipliers for LCGs to have the maximum-period and a method to construct many maximum-period MRGs from a single MRG are available in the literature. In this paper, we propose to combine both approaches to generate different MRGs “randomly”, quickly and automatically, while retaining the maximum-period property.

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.

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.

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.

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.

A Fuzzy Rotor Resistance Updating Scheme for an IFOC Induction Motor Drive

(, )is the (, ) ij element of the inverse of the B-matrix, and Pm net( ) is the net power injection at location m. Note that mi repre- sents all () ms that are connected to i. The power flow, fi j S (, , ) ,i s de- fined as positive to avoid negative wheeling charges. e is a very small positive number used to prevent singularity. This model is the extension of the gaming model the author devel- oped earlier (3). We present a simple case study using the model. The test system is the IEEE 14-bus power flow test system (4). This system has two generators that are located at nodes one and two. Assume that generator one is owned by producer one and generator two by producer two. Linear nodal demand functions are used with ai 0( ) as the intercept of the price axis (see Table 2). Slopes of the linear demand functions are all one. The wheeling cost for each line is derived using the follow- ing formula: wh i j (, ) . =+ 10 uniform (-0.1, +0.1) where uniform() is a uniform random number generator. The solution technique for this gaming problem is similar to the one used in (3). That is, the mixed complementarity programming (MCP) technique is used. Due to the in- troduction of losses, however, the gaming problem becomes highly nonlinear (see (11)) and more difficult to solve. The successive approx- imation method is used to resolve the difficulty. The first approxima- tion starts from the gaming model incorporating the GLP method. In the successive iterations that follow, (11) is approximated using the fol- lowing CLP formula,

Characterizing Cavities in Model Inclusion Fullerenes: A Comparative Study

The fullerene-82 cavity is selected as a model system in order to test several methods for characterizing inclusion molecules. The methods are based on different technical foundations such as a square and triangular tessellation of the molecular surface, spherical tessellation of the molecular surface, numerical integration of the atomic volumes and surfaces, triangular tessellation of the molecular surface, and cubic lattice approach to the molecular volume. Accurate measures of the molecular volume and surface area have been performed with the pseudorandom Monte Carlo (MCVS) and uniform Monte Carlo (UMCVS) methods. These calculations serve as a reference for the rest of the methods. The SURMO2 method does not recognize the cavity and may not be convenient for intercalation compounds. The programs that detect the cavities never exceed 1% deviation relative to the reference value for molecular volume and 5% for surface area. The GEPOL algorithm, alone or combined with TOPO, shows results in good agreement with those of the UMCVS reference. The uniform random number generator provides the fastest convergence for UMCVS and a correct estimate of the standard deviations. The effect of the internal cavity on the solvent-accessible surfaces has been calculated. Fullerene-82 is compared with fullerene-60 and -70.

Random Number Generation for the New Century

Use of empirical studies based on computer-generated random numbers has become a common practice in the development of statistical methods, particularly when the analytical study of a statistical procedure becomes intractable. The quality of any simulation study depends heavily on the quality of the random number generators. Classical uniform random number generators have some major defects—such as the (relatively) short period length and the lack of higher-dimension uniformity. Two recent uniform pseudo-random number generators (MRG and MCG) are reviewed. They are compared with the classical generator LCG. It is shown that MRG/MCG are much better random number generators than the popular LCG. Special forms of MRG/MCG are introduced and recommended as the random number generators for the new century. A step-by-step procedure for constructing such random number generators is also provided.

Daily Streamflow Simulation Using Recession Characteristics

Time series of daily streamflows are studied with particular emphasis on the behavior of recessions. Long-term and periodic characteristics are modeled through spectral analysis. Gamma autoregressive models are used to simulate time intervals D between consecutive low flows and the conditional duration d of recessions. A similar model is also used for generating the magnitudes of low flows, conditional on D. A nonnormal ARIMA process is adopted for simulating a recession backward in time over a time interval d commencing with the associated low flow. Transformation to stationarity of the recession process is by means of a difference-ratio, which is the difference between the current and antecedent streamflows divided by their average. The rising limb of the hydrograph is simulated using a ranked uniform random number generator. Application is made to four river basins ranging in size from the Po at Pontelagoscuro (70,090 km2) in northern Italy to the Arzino (123 km2) in northeast Italy. Hydrograph characteristics and the distributions of average, low, and peak flows are maintained.

Miscellanea. A note on generating correlated binary variables

SUMMARY It is important to be able to generate correlated binary data in an efficient, easily programmed manner for, among other things, the generation of large bootstrap samples. In this note nothing more than the basic ingredient of the uniform random number generator is required for simulating binary data from the most commonly occurring correlation structures.

Characterizing cavities in model inclusion molecules: a comparative study

We have selected fullerene-60 and -70 cavities as model systems in order to test several methods for characterizing inclusion molecules. The methods are based on different technical foundations such as a square and triangular tessellation of the molecule taken as a unitary sphere, spherical tessellation of the molecular surface, numerical integration of the atomic volumes and surfaces, triangular tessellation of the molecular surface, and a cubic lattice approach to a molecular space. Accurate measures of the molecular volume and surface area have been performed with the pseudo-random Monte Carlo (MCVS) and uniform Monte Carlo (UMCVS) methods. These calculations serve as a reference for the rest of the methods. The SURMO2 and MS methods have not recognized the cavities and may not be convenient for intercalation compounds. The programs that have detected the cavities never exceed 5% deviation relative to the reference values for molecular volume and surface area. The GEPOL algorithm, alone or combined with TOPO, shows results in good agreement with those of the UMCVS reference. The uniform random number generator provides the fastest convergence for UMCVS and a correct estimate of the standard deviations. The effect of the internal cavity on the accessible surfaces has been calculated.

Guest editors' introduction: special issue on uniform random number generation

Guest editors' introduction: special issue on uniform random number generation

Designing a Uniform Random Number Generator with Approximate Asymptotic Randomness

Designing a Uniform Random Number Generator with Approximate Asymptotic Randomness

The weighted spectral test

In this article, we present a new approach to assessing uniform random number generators, the weighted spectral test , or diaphony . In contrast to the usual spectral test the weighted spectral test is not limited to random number generators with a lattice structure. Its computational complexity is O (s·N 2 ) for any point set of cardinality N in the s-dimensional unit cube. As the main results of this article, we prove an analog of the classical inequality of Erdös-Turán-Koksma, present the necessary tools to transcribe known discrepancy bounds into bounds for diaphony, and provide bounds for the diaphony of multiplicative congruetial pseudorandom numbers. The last section contains numerical results.

Tests based on sum-functions of spacings for uniform random numbers

We examine the idea of testing uniform random number generators via two goodness-of-fit statistics: the sum of the logarithms and the sum of squares of overlapping m-spacings. The first statistic is related to an estimator of the entropy of a density and is good to detect clustering, whereas the second one, known as Greenwood's statistic for m=1,is optimal in terms of Pitman efficiency,in certain setups,among sum-functions of m-spacings. These statistics are asymptotically normally distributed.We evaluate the distance between the standard normal distribution and that of the standardized statistics,as a function of m and of the sample size,when standardization is done using either the asymptotic or the exact (for finite sample size) mean and variance. We then report on experiments with these statistics to detect defects in some popular random number generators.