Uniform Random Number Research Articles

Average Case Subquadratic Exact and Heuristic Procedures for the Traveling Salesman 2-OPT Neighborhood

We describe an exact algorithm for finding the best 2-OPT move that, experimentally, was observed to be much faster than the standard quadratic approach for a large part of a best-improvement local search convergence starting at a random tour. To analyze its average-case complexity, we introduce a family of heuristic procedures and discuss their complexity when applied to a random tour in graphs whose edge costs are either uniform random numbers in [0, 1] or Euclidean distances between random points in the plane. We prove that, for any probability p, there is a heuristic in the family that can find the best 2-OPT move with probability at least p in average-time [Formula: see text]) for uniform instances and [Formula: see text] for Euclidean instances. The exact algorithm is then proved to be even faster in the sense that in those instances in which a heuristic finds the best move, the exact algorithm finds it in a smaller time. We give empirical evidence that a slight variant of our algorithm finds the best move in [Formula: see text] time on both types of instances, achieving the best possible performance for this particular problem. Computational experiments are reported to show the effectiveness of our algorithms, both in best-improvement and in first-improvement 2-OPT local search. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms: Discrete. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0169 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0169 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Generalized likelihood ratio method for stochastic models with uniform random numbers as inputs

We propose a new unbiased stochastic gradient estimator for a family of stochastic models driven by uniform random numbers as inputs. Dropping the requirement that the tails of the density of the input random variables decay smoothly, the estimator extends the applicability of the generalized likelihood ratio (GLR) method. We demonstrate the new estimator for several general classes of input random variates, including independent inverse transform random variates and dependent input random variables governed by an Archimedean copula. We show how the new estimator works in settings such as density estimation, and we illustrate applications to credit risk derivatives. Numerical experiments substantiate broad applicability and flexibility in dealing with discontinuities in the sample performance.

Depth-based vessel position fixing by means of a neural network

A depth-based vessel position fixing method on the basis of a neural network is proposed. The network takes as an input a sequence of depth values measured by an echo-sounder and predicts vessel latitude and longitude for the moment of the latest depth measurement. The neural network has a fully-connected feedforward architecture with several layers which satisfies conditions of the universal approximation in compliance with the Stone-Weierstrass theorem. The Adamax algorithm for the neural network training with controlling a maximum value of position error at each epoch is implemented. Modeling is conducted with the Python programming language and the Tensorflow library. The model surface of seabed is performed as a second-order polynomial. Training samples on the basis of virtual soundings at the coordinate net knots with the space resolution not worse than one cable are obtained. After samples obtaining the training of the neural network is conducted. A validation set is not used. Several neural networks are trained. They have different number of hidden layers and different number of neurons per each hidden layer. After training the test procedure is performed. Test samples are generated in the assumption that a vessel is moving along meridians which are not used at the stage of the preliminary soundings survey. The cases of mean and random test meridians are considered. The random meridians are obtained with a uniform random number generator. As the result, all the tested neural networks have shown approximately identical navigational accuracy which is close to the accuracy for the training set.

On random irregular subgraphs

AbstractLet be a ‐regular graph on vertices. Frieze, Gould, Karoński, and Pfender began the study of the following random spanning subgraph model . Assign independently to each vertex of a uniform random number , and an edge of is an edge of if and only if . Addressing a problem of Alon and Wei, we prove that if , then with high probability, for each nonnegative integer , there are vertices of degree in .

Variable-Length Resolvability for General Sources and Channels.

We introduce the problem of variable-length (VL) source resolvability, in which a given target probability distribution is approximated by encoding a VL uniform random number, and the asymptotically minimum average length rate of the uniform random number, called the VL resolvability, is investigated. We first analyze the VL resolvability with the variational distance as an approximation measure. Next, we investigate the case under the divergence as an approximation measure. When the asymptotically exact approximation is required, it is shown that the resolvability under two kinds of approximation measures coincides. We then extend the analysis to the case of channel resolvability, where the target distribution is the output distribution via a general channel due to a fixed general source as an input. The obtained characterization of channel resolvability is fully general in the sense that, when the channel is just an identity mapping, it reduces to general formulas for source resolvability. We also analyze the second-order VL resolvability.

WALLAX: A memristor-based Gaussian random number generator

Generating Gaussian random numbers is essential in many applications such as cryptography, games, and computer simulations. Although software Gaussian Random Number Generators (GRNG) are widely used, hardware designs have been explored for their faster speed and lower computational cost. However, hardware GRNGs usually occupy large silicon areas when implemented in Complementary Metal Oxide Semiconductor (CMOS) technology, especially for their essential Uniform Random Number Generator (URNG) part. Here, we present a memristor-based GRNG, named WALLAX, conceived from the Wallace method to generate random numbers iteratively. This GRNG circuit benefits not only from the fully parallel analog-based Vector Matrix Multiplication (VMM) feature of memristive crossbars but also harness the intrinsic stochastic switching behaviour of the memristive devices to efficiently produce truly random numbers. The vector-matrix multiplication of WALLAX is implemented on the memristive crossbar, while its random fetching step is realized by a URNG based on the stochastic switching nature of memristors. WALLAX successfully passes all the tests in the NIST 800-22 randomness test suite with 105 numbers generated and five goodness-of-fit tests with various pool sizes and effectively reduces the power and area consumption by 68.78% and 70.0% compared to digital implementations of the same GRNG method. The impact brought by memristor non-idealities is investigated by simulating the proposed structure with 1000 pools under various scenarios. Wire resistance and the stuck of state, each result in a 2.2% and 12.3% reduction in test pass rate within the tested range, respectively.

Stochastic modelling of wind speed over northern states in Nigeria

This paper presents a stochastic modeling of wind speed over sixteen (16) Northern states in Nigeria using thirty-seven years (1984-2020) wind speed real time data. A Markov Chain Model was developed for the monthly wind speed state across study locations. In order to obtain the Markov chain transitional probabilities, the wind speed data was categories into various states using the Beaufort wind scale. It was observed that only the first four description of wind speed state A, B, C and D exist in the study locations. Uniform random states were also formed by generating uniform random number. The comparison of monthly simulated and actual wind speed state clearly shows that the model simulated over six months correctly across study locations except Niger. Given a current wind speed state conditions, the stochastic models available in this paper can be adapted to generate future wind speed state condition. The understanding of wind speed state helps in wind turbine design and selection of wind farm sites for wind energy generation.

Monte Carlo and Quasi–Monte Carlo Density Estimation via Conditioning

Estimating the unknown density from which a given independent sample originates is more difficult than estimating the mean in the sense that, for the best popular nonparametric density estimators, the mean integrated square error converges more slowly than at the canonical rate of [Formula: see text]. When the sample is generated from a simulation model and we have control over how this is done, we can do better. We examine an approach in which conditional Monte Carlo yields, under certain conditions, a random conditional density that is an unbiased estimator of the true density at any point. By averaging independent replications, we obtain a density estimator that converges at a faster rate than the usual ones. Moreover, combining this new type of estimator with randomized quasi–Monte Carlo to generate the samples typically brings a larger improvement on the error and convergence rate than for the usual estimators because the new estimator is smoother as a function of the underlying uniform random numbers. Summary of Contribution: Stochastic simulation is commonly used to estimate the mathematical expectation of some output random variable X together with a confidence interval for this expectation. But the simulations usually provide information to do much more, such as estimating the entire distribution (or density) of X. Histograms are routinely provided by standard simulation software, but they are very primitive density estimators. Kernel density estimators perform better, but they are trickier to use, have bias, and their mean square error converges more slowly than the canonical rate of O(1/n) with n independent samples. In this paper, we explain how to construct unbiased density estimators that converge at the canonical rate and even much faster when combined with randomized quasi–Monte Carlo. The key idea is to use conditional Monte Carlo to hide appropriate information and obtain a computable (random) conditional density, which acts (under certain conditions) as an unbiased density estimator. Moreover, this sample density is typically smoother than the classic density estimators as a function of the underlying uniform random numbers, so it can get along much better with randomized quasi–Monte Carlo methods. This offers an opportunity to further improve the O(1/n) rate. We observe rates near O(1/n2) on some examples, and we give conditions under which this type of rate provably holds. The proposed approach is simple, easy to implement, and extremely effective, so it provides a significant addition to the stochastic simulation toolbox.

Molecular dynamics studies of entropic elasticity of condensed lattice networks connected with uniform functionality f = 4.

To study the linear region of entropic elasticity, we considered the simplest physical model possible and extracted the linear entropic regime by using the least squares fit and the minimum of the mean absolute error. With regard to the effect of the fluctuation of the strand length Ns, the strand length with fluctuation was set to a form proportional to (1.0 + C (R - 0.5)), where R is a uniform random number between 0 and 1 and C is the amplitude of fluctuation. This form enabled us to analytically calculate the fluctuation dependence of the elastic modulus G. To reveal the linear regions of entropic elasticity as a function of the strand length between neighboring nodes in lattices, molecular dynamics (MD) simulations of condensed lattice networks with harmonic bonds without the excluded volume interactions were performed. Stress-strain curves were estimated by performing uniaxial stretching MD simulations under periodic boundary conditions with a bead number density of 0.85. First, we used a diamond lattice with functionality f = 4. The linear region of the entropic elasticity was found to become larger with the increasing number of beads in a strand Ns. For Ns = 100, the linear region had a strain of up to 8 for a regular diamond lattice. We investigated the effect of strand length fluctuation on the diamond lattice, and we confirmed that the equilibrium shear modulus G increases as the obtained analytical prediction and the linear entropic region in the stress-strain curves becomes narrower with increasing fluctuation of Ns. To investigate the difference in network topology with the same functionality f and uniform strand length Ns, we performed MD simulations on regular networks of the BC-8 structure with f = 4 prepared from the ab initio DFT calculations of carbon at high pressure. We found that the elastic behavior depends on the network connectivity (i.e., topology). This indicates that the network topology plays an important role in the emergence of nonlinearity owing to the crossover from entropic to energetic elasticity.

Design of Gaussian modulator for continuous-variable quantum key distribution

The continuous-variable quantum key distribution experimental system requires the randomness and speed of a Gaussian modulator. We present the hardware design for a Gaussian random number (GRN) generator based on the Box–Muller method, which can be implemented on a field-programmable gate array (FPGA). An external high-speed true random number generator application-specific integrated circuit (ASIC) is used to this end. The ASIC operates faster and more independently compared with other Gaussian algorithms based on a linear feedback shift register. The trigonometric and logarithmic functions are calculated using the nonuniform piecewise linear approximation. The calculation of the Gaussian random modulus and phase value is realized using this hardware. Compared with an arbitrary waveform generator, the GRN can be uploaded to a computer to estimate the security key rate. Furthermore, the GRN generator accurately provides a Gaussian probability density function that passes the Jarque–Bera inspection and Lilliefors tests. The proposed FPGA-based system is demonstrated to convert eight-channel 320-Mbps uniform random numbers to 40-MHz 16-bit GRN in real time. Finally, the control of the optical module and Gaussian modulation is realized via the FPGA-based outputs of four dual-channel, 16-bit, 1-giga samples per second digital-to-analog converters.

Studying the Impact of Initialization for Population-Based Algorithms with Low-Discrepancy Sequences

To solve different kinds of optimization challenges, meta-heuristic algorithms have been extensively used. Population initialization plays a prominent role in meta-heuristic algorithms for the problem of optimization. These algorithms can affect convergence to identify a robust optimum solution. To investigate the effectiveness of diversity, many scholars have a focus on the reliability and quality of meta-heuristic algorithms for enhancement. To initialize the population in the search space, this dissertation proposes three new low discrepancy sequences for population initialization instead of uniform distribution called the WELL sequence, Knuth sequence, and Torus sequence. This paper also introduces a detailed survey of the different initialization methods of PSO and DE based on quasi-random sequence families such as the Sobol sequence, Halton sequence, and uniform random distribution. For well-known benchmark test problems and learning of artificial neural network, the proposed methods for PSO (TO-PSO, KN-PSO, and WE-PSO), BA (BA-TO, BA-WE, and BA-KN), and DE (DE-TO, DE-WE, and DE-KN) have been evaluated. The synthesis of our strategies demonstrates promising success over uniform random numbers using low discrepancy sequences. The experimental findings indicate that the initialization based on low discrepancy sequences is exceptionally stronger than the uniform random number. Furthermore, our work outlines the profound effects on convergence and heterogeneity of the proposed methodology. It is expected that a comparative simulation survey of the low discrepancy sequence would be beneficial for the investigator to analyze the meta-heuristic algorithms in detail.

Re-exploring the Penney-Ante Game

We propose a single loop diagram and use it to devise a single loop matrix method to computationally solve the Penney-Ante game. This method avoids the nuances of repeated use of conditional probability and Markov chain representations. We remove the limitations of Conway’s trick as applied to a fair coin and generalize the method where the coin is allowed to be biased. A uniform random number generator is used to simulate the game and formulate implicit mathematical relations to explore nontransitivity.

Comparative Analysis of Low Discrepancy Sequence-Based Initialization Approaches Using Population-Based Algorithms for Solving the Global Optimization Problems

Metaheuristic algorithms have been widely used to solve diverse kinds of optimization problems. For an optimization problem, population initialization plays a significant role in metaheuristic algorithms. These algorithms can influence the convergence to find an efficient optimal solution. Mainly, for recognizing the importance of diversity, several researchers have worked on the performance for the improvement of metaheuristic algorithms. Population initialization is a vital factor in metaheuristic algorithms such as PSO and DE. Instead of applying the random distribution for the initialization of the population, quasirandom sequences are more useful for the improvement the diversity and convergence factors. This study presents three new low-discrepancy sequences named WELL sequence, Knuth sequence, and Torus sequence to initialize the population in the search space. This paper also gives a comprehensive survey of the various PSO and DE initialization approaches based on the family of quasirandom sequences such as Sobol sequence, Halton sequence, and uniform random distribution. The proposed methods for PSO (TO-PSO, KN-PSO, and WE-PSO) and DE (DE-TO, DE-WE, and DE-KN) have been examined for well-known benchmark test problems and training of the artificial neural network. The finding of our techniques shows promising performance using the family of low-discrepancy sequences over uniform random numbers. For a fair comparison, the approaches using low-discrepancy sequences for PSO and DE are compared with the other family of low-discrepancy sequences and uniform random number and depict the superior results. The experimental results show that the low-discrepancy sequences-based initialization performed exceptionally better than a uniform random number. Moreover, the outcome of our work presents a foresight on how the proposed technique profoundly impacts convergence and diversity. It is anticipated that this low-discrepancy sequence comparative simulation survey would be helpful for studying the metaheuristic algorithm in detail for the researcher.

Deterministic sampling from uniform distributions with Sierpiński space-filling curves

In this paper the problem of sampling from uniform probability distributions is approached by means of space-filling curves, a topological concept that has found a number of important applications in recent years. Departing from the theoretical fact that they are surjective but not necessarily injective, the investigation focused upon the structure of the distributions obtained when their domains are swept in a uniform and discrete manner, and the corresponding values used to build histograms, that are approximations of their true PDFs. This work concentrates on the real interval [0,1] and the Sierpiński space-filling curve was chosen because of its favorable computational properties. In order to validate the results, the Kullback–Leibler and other divergence measures are used when comparing the obtained distributions in several levels of granularity with other already established sampling methods. In truth, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. In this fashion, sequences resulting from this sort of process are not truly random. Despite this, and to be coherent with the literature, the expression “random number” will be used along the text to mean “pseudo-random number”.

Microtubule Disassembling Through Stathmin Bending Bow: A Molecular Dynamic Study

By this work, molecular modeling has been used to interpret the dynamic instabilities of these macromolecules in their structures. By this investigation, multi-dimension structures of microtubules are fixed in both length and width. Via Monte Carlo simulation, the tubulins have been added from the first side of the tubule towards the opposite side by gradually growing a random position. This method is theoretically accomplished via generating a uniform random number between (0, 1) based on the Monte Carlo approach. Our calculations have been done by proper dimension around 5×10-6 meters of length that consists of 2000 tubulin dimers. The structure growth rates are based on soluble tubulin dimer concentration. Hereby all results were run between 6-12 times in our modeling of any conditions. There have been recorded value numbers, average length, free tubulin concentration, and the important data of thermodynamic parameters for each simulation.

High-Speed Interval Type-2 Fuzzy Systems for Dynamic Parameter Adaptation in Harmony Search for Optimal Design of Fuzzy Controllers

Fuzzy systems have become a good solution to the problem of fixed parameters in metaheuristic algorithms, proving their efficiency when performing dynamic parameter adaptations using type-1 and type-2 fuzzy logic. However, the computational cost of type-2 fuzzy systems when using the continuous enhanced Karnik–Mendel (CKM) algorithm for type-reduction, when applied to control and optimization, is too high. Therefore, it is proposed to use an approximation to the CKM algorithm in the type-2 fuzzy system for adjusting the pitch adjustment rate (PArate) parameter in the original harmony search algorithm (HS). The main contribution of this article is to verify that the implementation of the proposed methodology achieves results that are equivalent to the interval type-2 fuzzy system with the CKM algorithm, but in less computing time and also allowing an efficient dynamic parameter adaptation. It is noteworthy that this method is relatively new in the area of metaheuristics algorithms so there is a current interest to work with this methodology. The proposed method was used in optimizing the antecedents and consequents for an interval type-2 fuzzy controller of direct current motor. Experimental results without noise and then with uniform random noise numbers (Gaussian noise) in the controller were obtained to verify that the implementation is efficient when compared to conventional and other existing methods.

An efficient weak Euler–Maruyama type approximation scheme of very high dimensional SDEs by orthogonal random variables

We will introduce Euler–Maruyama approximations given by an orthogonal system in L2[0,1] for high dimensional SDEs, which could be finite dimensional approximations of SPDEs. In general, the higher the dimension is, the more one needs to generate uniform random numbers at every time step in numerical simulation. The schemes proposed in this paper, in contrast, can deal with this problem by generating very few uniform random numbers at every time step. The schemes save time in the simulation of very high dimensional SDEs. In particular, we conclude that an Euler–Maruyama approximation based on the Walsh system is efficient in high dimensions.

Fuzzy Random Variable Generation Using $\alpha$-Cuts

A method for fuzzy random variable generation based on $\boldsymbol {\alpha }$ -cuts and uniform random numbers is proposed in this article. A form of $\boldsymbol {\alpha }$ -invariance is proven for a class of fuzzy numbers (FNs) and exact equations for random variable generators of some well-known singleton-core FNs are provided. Applications to univariate/bivariate FNs, two optimization inventory models, and a discussion of the obtained results are provided.

Difusi Bebas 1D dan 2D dengan Monte Carlo: Perbandingan Distribusi Bilangan Random Normal dan Seragam dengan Box-Müller

Many biological processes in the human body are based on the diffusion system. Diffusion is defined as a process of random movement of the particle whose the direction is from high concentrations to low concentrations. Many of various study of diffusion have been done both experimentally and computationally. Because the particle interaction is stochastic, the Monte Carlo (MC) method is used in performing particle simulations. The main of MC method is the use of random numbers. Many software have provided uniform random number generators. But based on the analytic results, the solution is normal distribution. Therefore, Box-Müller can be used as a transformation of particle distribution. The software used, MATLAB, has a normal random generator. Therefore, the aims of this study is comparing particle distribution of these two different random number generator with MATLAB and showing the impact of timestep parameter to these random number generator. This result can be used as based for the modelling of more complex biological systems.

Model diagnostics for censored regression via randomized survival probabilities

Residuals in normal regression are used to assess a model's goodness-of-fit (GOF) and discover directions for improving the model. However, there is a lack of residuals with a characterized reference distribution for censored regression. In this article, we propose to diagnose censored regression with normalized randomized survival probabilities (RSP). The key idea of RSP is to replace the survival probability (SP) of a censored failure time with a uniform random number between 0 and the SP of the censored time. We prove that RSPs always have the uniform distribution on (0, 1) under the true model with the true generating parameters. Therefore, we can transform RSPs into normally distributed residuals with the normal quantile function. We call such residuals by normalized RSP (NRSP residuals). We conduct simulation studies to investigate the sizes and powers of statistical tests based on NRSP residuals in detecting the incorrect choice of distribution family and nonlinear effect in covariates. Our simulation studies show that, although the GOF tests with NRSP residuals are not as powerful as a traditional GOF test method, a nonlinear test based on NRSP residuals has significantly higher power in detecting nonlinearity. We also compared these model diagnostics methods with a breast-cancer recurrent-free time dataset. The results show that the NRSP residual diagnostics successfully captures a subtle nonlinear relationship in the dataset, which is not detected by the graphical diagnostics with CS residuals and existing GOF tests.