This document outlines various topics related to pseudo-random number generators (PRNGs). It begins by discussing uniform PRNGs and the goal of approximating independent and uniformly distributed random variables. It then discusses linear congruential generators and multiplicative congruential generators as examples of uniform PRNGs. It notes some weaknesses of these generators, such as short periods and poor distribution in high dimensions. Finally, it briefly discusses statistical tests that can be used to validate PRNGs, such as the gap test and spectral test.
Unit 3 random number generation, random-variate generationraksharao
This document discusses random number generation and random variate generation. It covers:
1) Properties of random numbers such as uniformity, independence, maximum density, and maximum period.
2) Techniques for generating pseudo-random numbers such as the linear congruential method and combined linear congruential generators.
3) Tests for random numbers including Kolmogorov-Smirnov, chi-square, and autocorrelation tests.
4) Random variate generation techniques like the inverse transform method, acceptance-rejection technique, and special properties for distributions like normal, lognormal, and Erlang.
This document discusses random number generation and properties of pseudo-random numbers. It covers techniques for generating pseudo-random numbers like linear congruential methods and combined congruential methods. It also discusses hypothesis tests that can be used to test for uniformity and independence of random numbers, such as the frequency test, Kolmogorov-Smirnov test, chi-square test, runs test, and autocorrelation test.
This presentation discusses methods for generating pseudo-random numbers and testing their randomness. It introduces the midsquare method as the first arithmetic generator but notes its tendency to generate numbers that approach zero. The linear congruential method and combined linear congruential generators are presented as improved approaches. Statistical tests for randomness like the Kolmogorov-Smirnov test and chi-square test are also summarized to evaluate the uniformity and independence of generated random numbers.
This document discusses tests for random number generation, including the autocorrelation test, gap test, and poker test. The autocorrelation test examines dependence between numbers in a sequence. The gap test analyzes the length of gaps between numbers that fall within a given range. The poker test categorizes groups of five consecutive numbers based on arrangements like pairs, three of a kind, etc. and applies a chi-squared test to assess randomness.
Variance reduction techniques (VRTs) can increase the statistical efficiency of simulations by reducing the variances of random variable outputs without changing their expectations, allowing for greater precision with less simulation time. Common VRTs include common random numbers, antithetic variates, control variates, indirect estimation, and conditioning. Common random numbers are used when comparing alternative system configurations, while antithetic variates induce negative correlation between separate simulation runs to offset observations. Control variates take advantage of correlations between random variables, and indirect estimation and conditioning substitute exact analytical solutions for estimates in queueing models.
The document discusses amortized analysis, which averages the time required to perform a sequence of operations over all operations. It describes three methods of amortized analysis: aggregate analysis, accounting analysis, and potential analysis. As an example, it analyzes the amortized cost of operations on a dynamic table using these three methods and shows that the amortized cost of insertion and deletion is O(1), even though some operations may have higher actual costs when triggering expansions or contractions of the table.
Unit 3 random number generation, random-variate generationraksharao
This document discusses random number generation and random variate generation. It covers:
1) Properties of random numbers such as uniformity, independence, maximum density, and maximum period.
2) Techniques for generating pseudo-random numbers such as the linear congruential method and combined linear congruential generators.
3) Tests for random numbers including Kolmogorov-Smirnov, chi-square, and autocorrelation tests.
4) Random variate generation techniques like the inverse transform method, acceptance-rejection technique, and special properties for distributions like normal, lognormal, and Erlang.
This document discusses random number generation and properties of pseudo-random numbers. It covers techniques for generating pseudo-random numbers like linear congruential methods and combined congruential methods. It also discusses hypothesis tests that can be used to test for uniformity and independence of random numbers, such as the frequency test, Kolmogorov-Smirnov test, chi-square test, runs test, and autocorrelation test.
This presentation discusses methods for generating pseudo-random numbers and testing their randomness. It introduces the midsquare method as the first arithmetic generator but notes its tendency to generate numbers that approach zero. The linear congruential method and combined linear congruential generators are presented as improved approaches. Statistical tests for randomness like the Kolmogorov-Smirnov test and chi-square test are also summarized to evaluate the uniformity and independence of generated random numbers.
This document discusses tests for random number generation, including the autocorrelation test, gap test, and poker test. The autocorrelation test examines dependence between numbers in a sequence. The gap test analyzes the length of gaps between numbers that fall within a given range. The poker test categorizes groups of five consecutive numbers based on arrangements like pairs, three of a kind, etc. and applies a chi-squared test to assess randomness.
Variance reduction techniques (VRTs) can increase the statistical efficiency of simulations by reducing the variances of random variable outputs without changing their expectations, allowing for greater precision with less simulation time. Common VRTs include common random numbers, antithetic variates, control variates, indirect estimation, and conditioning. Common random numbers are used when comparing alternative system configurations, while antithetic variates induce negative correlation between separate simulation runs to offset observations. Control variates take advantage of correlations between random variables, and indirect estimation and conditioning substitute exact analytical solutions for estimates in queueing models.
The document discusses amortized analysis, which averages the time required to perform a sequence of operations over all operations. It describes three methods of amortized analysis: aggregate analysis, accounting analysis, and potential analysis. As an example, it analyzes the amortized cost of operations on a dynamic table using these three methods and shows that the amortized cost of insertion and deletion is O(1), even though some operations may have higher actual costs when triggering expansions or contractions of the table.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
This document discusses using Monte Carlo simulation to price European call options. It begins by noting the uncertainty in underlying asset prices makes option pricing difficult. It then outlines the Monte Carlo simulation procedure of generating random price paths, calculating payoffs, and averaging. Key assumptions are presented for the stock price model and payoff calculation. Simulation results are shown for different numbers of simulations and compared to the Black-Scholes model, with error decreasing as the number of simulations increases.
This document outlines key concepts related to constructing confidence intervals for estimating population means and proportions. It discusses how to calculate confidence intervals when the population standard deviation is known or unknown. Specifically, it provides the formulas and assumptions for constructing confidence intervals for a population mean using the normal and t-distributions. It also outlines how to calculate confidence intervals for a population proportion using the normal approximation. Examples are provided to demonstrate how to construct 95% confidence intervals for a mean and proportion based on sample data.
Operating System - Monitors (Presentation)Experts Desk
This document discusses monitors and their use in interprocess communication and synchronization. It contains the following key points:
1. Monitors provide mutual exclusion and condition variables to avoid race conditions when processes access shared resources. They allow processes to block when they cannot proceed.
2. Semaphores can be used to implement monitors, with a binary semaphore controlling entry to the monitor and additional semaphores for condition variables.
3. Monitors can also implement semaphores and messages, providing a higher-level construct than semaphores for synchronization between processes. Counters and linked lists are used to track semaphore values and message queues.
The k-means clustering algorithm is an unsupervised machine learning algorithm that groups unlabeled data points into k number of clusters. It works by first selecting k random cluster centroids and then assigns each data point to its nearest centroid, forming k clusters. It then recalculates the positions of the centroids and reassigns data points in an iterative process until centroids stabilize. The optimal number of clusters k can be determined using the elbow method by plotting the within-cluster sum of squares against k and selecting the k value at the point of inflection of the curve, resembling an elbow.
This presentation discusses about the following topics:
Truth values and tables,
Fuzzy propositions,
Formation of rules decomposition of rules,
Aggregation of fuzzy rules,
Fuzzy reasoning‐fuzzy inference systems
Overview of fuzzy expert system‐
Fuzzy decision making.
This document discusses randomized algorithms. It begins by listing different categories of algorithms, including randomized algorithms. Randomized algorithms introduce randomness into the algorithm to avoid worst-case behavior and find efficient approximate solutions. Quicksort is presented as an example randomized algorithm, where randomness improves its average runtime from quadratic to linear. The document also discusses the randomized closest pair algorithm and a randomized algorithm for primality testing. Both introduce randomness to improve efficiency compared to deterministic algorithms for the same problems.
Output analysis for simulation models / Elimination of initial BiasTilakpoudel2
This document discusses output analysis for simulation models and eliminating initial bias. It explains that initializing a simulation with an empty or idle state can cause initialization bias, skewing early results. Two common approaches to address this are to start simulations in a more representative state or remove an initial warmup phase of results. The ideal is to start from the true steady state distribution, while typically the warmup phase is eliminated by restarting after a pilot run determines how long the initial bias persists.
Introduction to Automata Languages and ComputationAmey Kerkar
This document provides an introduction to automata theory. It defines automata as self-operating machines that follow predetermined sequences and discusses how automata theory is closely related to formal language theory. The document also outlines some key contributors to automata theory, including Alan Turing who invented the Turing machine, Warren McCulloch and Walter Pitts who developed finite state machine diagrams, and Noam Chomsky who extended automata hierarchy.
1) A Petri net is a graphical modeling tool used to represent systems with concurrent, asynchronous components. It consists of places, transitions, and arcs between them.
2) The document discusses using Petri nets to model resource sharing between two processors to avoid deadlocks. It defines deadlocks and describes the banker's algorithm for deadlock avoidance.
3) The banker's algorithm works by ensuring the system is always in a safe state where all processes can complete even if maximum resources are requested. It maintains data structures to track available, allocated, and needed resources.
The document discusses algorithms for solving the mutual exclusion problem in multithreaded programs. It begins by describing two inadequate algorithms for two threads that fail to guarantee deadlock freedom. It then presents Peterson's algorithm and Kessels' single-writer algorithm, proving they satisfy mutual exclusion, deadlock freedom, and starvation freedom for two threads. The document also discusses using tournament algorithms and the filter algorithm to generalize two-thread solutions to work for multiple threads by having threads progress through levels like a tournament bracket.
The information in this slide is very useful for me to do the assignment regarding the simulation in which we have to report together with the presentation...
This document provides an introduction to NP-completeness, including: definitions of key concepts like decision problems, classes P and NP, and polynomial time reductions; examples of NP-complete problems like satisfiability and the traveling salesman problem; and approaches to dealing with NP-complete problems like heuristic algorithms, approximation algorithms, and potential help from quantum computing in the future. The document establishes NP-completeness as a central concept in computational complexity theory.
This document provides an overview of input modeling for simulation. It discusses the four main steps: 1) collecting real system data, 2) identifying the probability distribution, 3) estimating distribution parameters, and 4) evaluating goodness of fit. Common distributions are identified like Poisson, normal, exponential. Methods for identifying the distribution include histograms and Q-Q plots. Goodness of fit can be tested using chi-square and Kolmogorov-Smirnov tests. The document also discusses modeling non-stationary processes, selecting distributions without data, and multivariate/time-series input models.
John likes all foods, apples and chicken are foods, anything that does not kill someone who eats it is a food, Bill eats peanuts and is still alive so peanuts are food, and Sue eats everything that Bill eats. This document translates statements about people and foods into logical forms using predicates and quantifiers, and then expresses them in conjunctive normal form.
L03 ai - knowledge representation using logicManjula V
The document discusses knowledge representation using predicate logic. It begins by reviewing propositional logic and its semantics using truth tables. It then introduces predicate logic, which can represent properties and relations using predicates with arguments. It discusses representing knowledge in predicate logic using quantifiers, predicates, and variables. It also covers inferencing in predicate logic using techniques like forward chaining, backward chaining, and resolution. An example problem is presented to illustrate representing a problem and solving it using resolution refutation in predicate logic.
A pseudo random number generator (PRNG) is a mechanism for generating random numbers that appear random but are determined by an algorithm. PRNGs are important in cryptography as they are used to generate keys, initialization vectors, and other random values needed for encryption. Good PRNGs should produce numbers that are evenly distributed, unpredictable, and have a long repeating cycle. The RSA algorithm is one example of a PRNG that uses exponentiation modulo a large prime number to generate a stream of pseudorandom bits.
This presentation on Pseudo Random Number Generator enlists the different generators, their mechanisms and the various applications of random numbers and pseudo random numbers in different arenas.
FellowBuddy.com is an innovative platform that brings students together to share notes, exam papers, study guides, project reports and presentation for upcoming exams.
We connect Students who have an understanding of course material with Students who need help.
Benefits:-
# Students can catch up on notes they missed because of an absence.
# Underachievers can find peer developed notes that break down lecture and study material in a way that they can understand
# Students can earn better grades, save time and study effectively
Our Vision & Mission – Simplifying Students Life
Our Belief – “The great breakthrough in your life comes when you realize it, that you can learn anything you need to learn; to accomplish any goal that you have set for yourself. This means there are no limits on what you can be, have or do.”
Like Us - https://www.facebook.com/FellowBuddycom
This document discusses using Monte Carlo simulation to price European call options. It begins by noting the uncertainty in underlying asset prices makes option pricing difficult. It then outlines the Monte Carlo simulation procedure of generating random price paths, calculating payoffs, and averaging. Key assumptions are presented for the stock price model and payoff calculation. Simulation results are shown for different numbers of simulations and compared to the Black-Scholes model, with error decreasing as the number of simulations increases.
This document outlines key concepts related to constructing confidence intervals for estimating population means and proportions. It discusses how to calculate confidence intervals when the population standard deviation is known or unknown. Specifically, it provides the formulas and assumptions for constructing confidence intervals for a population mean using the normal and t-distributions. It also outlines how to calculate confidence intervals for a population proportion using the normal approximation. Examples are provided to demonstrate how to construct 95% confidence intervals for a mean and proportion based on sample data.
Operating System - Monitors (Presentation)Experts Desk
This document discusses monitors and their use in interprocess communication and synchronization. It contains the following key points:
1. Monitors provide mutual exclusion and condition variables to avoid race conditions when processes access shared resources. They allow processes to block when they cannot proceed.
2. Semaphores can be used to implement monitors, with a binary semaphore controlling entry to the monitor and additional semaphores for condition variables.
3. Monitors can also implement semaphores and messages, providing a higher-level construct than semaphores for synchronization between processes. Counters and linked lists are used to track semaphore values and message queues.
The k-means clustering algorithm is an unsupervised machine learning algorithm that groups unlabeled data points into k number of clusters. It works by first selecting k random cluster centroids and then assigns each data point to its nearest centroid, forming k clusters. It then recalculates the positions of the centroids and reassigns data points in an iterative process until centroids stabilize. The optimal number of clusters k can be determined using the elbow method by plotting the within-cluster sum of squares against k and selecting the k value at the point of inflection of the curve, resembling an elbow.
This presentation discusses about the following topics:
Truth values and tables,
Fuzzy propositions,
Formation of rules decomposition of rules,
Aggregation of fuzzy rules,
Fuzzy reasoning‐fuzzy inference systems
Overview of fuzzy expert system‐
Fuzzy decision making.
This document discusses randomized algorithms. It begins by listing different categories of algorithms, including randomized algorithms. Randomized algorithms introduce randomness into the algorithm to avoid worst-case behavior and find efficient approximate solutions. Quicksort is presented as an example randomized algorithm, where randomness improves its average runtime from quadratic to linear. The document also discusses the randomized closest pair algorithm and a randomized algorithm for primality testing. Both introduce randomness to improve efficiency compared to deterministic algorithms for the same problems.
Output analysis for simulation models / Elimination of initial BiasTilakpoudel2
This document discusses output analysis for simulation models and eliminating initial bias. It explains that initializing a simulation with an empty or idle state can cause initialization bias, skewing early results. Two common approaches to address this are to start simulations in a more representative state or remove an initial warmup phase of results. The ideal is to start from the true steady state distribution, while typically the warmup phase is eliminated by restarting after a pilot run determines how long the initial bias persists.
Introduction to Automata Languages and ComputationAmey Kerkar
This document provides an introduction to automata theory. It defines automata as self-operating machines that follow predetermined sequences and discusses how automata theory is closely related to formal language theory. The document also outlines some key contributors to automata theory, including Alan Turing who invented the Turing machine, Warren McCulloch and Walter Pitts who developed finite state machine diagrams, and Noam Chomsky who extended automata hierarchy.
1) A Petri net is a graphical modeling tool used to represent systems with concurrent, asynchronous components. It consists of places, transitions, and arcs between them.
2) The document discusses using Petri nets to model resource sharing between two processors to avoid deadlocks. It defines deadlocks and describes the banker's algorithm for deadlock avoidance.
3) The banker's algorithm works by ensuring the system is always in a safe state where all processes can complete even if maximum resources are requested. It maintains data structures to track available, allocated, and needed resources.
The document discusses algorithms for solving the mutual exclusion problem in multithreaded programs. It begins by describing two inadequate algorithms for two threads that fail to guarantee deadlock freedom. It then presents Peterson's algorithm and Kessels' single-writer algorithm, proving they satisfy mutual exclusion, deadlock freedom, and starvation freedom for two threads. The document also discusses using tournament algorithms and the filter algorithm to generalize two-thread solutions to work for multiple threads by having threads progress through levels like a tournament bracket.
The information in this slide is very useful for me to do the assignment regarding the simulation in which we have to report together with the presentation...
This document provides an introduction to NP-completeness, including: definitions of key concepts like decision problems, classes P and NP, and polynomial time reductions; examples of NP-complete problems like satisfiability and the traveling salesman problem; and approaches to dealing with NP-complete problems like heuristic algorithms, approximation algorithms, and potential help from quantum computing in the future. The document establishes NP-completeness as a central concept in computational complexity theory.
This document provides an overview of input modeling for simulation. It discusses the four main steps: 1) collecting real system data, 2) identifying the probability distribution, 3) estimating distribution parameters, and 4) evaluating goodness of fit. Common distributions are identified like Poisson, normal, exponential. Methods for identifying the distribution include histograms and Q-Q plots. Goodness of fit can be tested using chi-square and Kolmogorov-Smirnov tests. The document also discusses modeling non-stationary processes, selecting distributions without data, and multivariate/time-series input models.
John likes all foods, apples and chicken are foods, anything that does not kill someone who eats it is a food, Bill eats peanuts and is still alive so peanuts are food, and Sue eats everything that Bill eats. This document translates statements about people and foods into logical forms using predicates and quantifiers, and then expresses them in conjunctive normal form.
L03 ai - knowledge representation using logicManjula V
The document discusses knowledge representation using predicate logic. It begins by reviewing propositional logic and its semantics using truth tables. It then introduces predicate logic, which can represent properties and relations using predicates with arguments. It discusses representing knowledge in predicate logic using quantifiers, predicates, and variables. It also covers inferencing in predicate logic using techniques like forward chaining, backward chaining, and resolution. An example problem is presented to illustrate representing a problem and solving it using resolution refutation in predicate logic.
A pseudo random number generator (PRNG) is a mechanism for generating random numbers that appear random but are determined by an algorithm. PRNGs are important in cryptography as they are used to generate keys, initialization vectors, and other random values needed for encryption. Good PRNGs should produce numbers that are evenly distributed, unpredictable, and have a long repeating cycle. The RSA algorithm is one example of a PRNG that uses exponentiation modulo a large prime number to generate a stream of pseudorandom bits.
This presentation on Pseudo Random Number Generator enlists the different generators, their mechanisms and the various applications of random numbers and pseudo random numbers in different arenas.
Pseudorandom number generators powerpointDavid Roodman
This document summarizes and tests four different pseudorandom number generators. Generator 1, which uses modular exponentiation, produces well-distributed numbers and differences but is too computationally intensive. Generator 2, which uses a sine wave function, generates numbers that are clustered on the sides and has differences centered around 0. Generator 3, which uses exponential functions, has near-perfect number distribution but a limited difference range of two values. Generator 4, based on logarithms, fails to produce a consistent distribution and has differences that converge to a single value. In conclusion, creating effective pseudorandom number generators is very challenging.
This document describes the design and implementation of a PRBS (pseudorandom bit sequence) generator module using a linear feedback shift register (LFSR). It includes the theoretical background of LFSRs, a 4-bit example, hardware implementation on a breadboard and printed circuit board, and results showing the output sequences for different feedback configurations. The generator can be extended to output any desired 8-bit sequence using a parallel-to-serial converter. Maximum randomness was achieved with feedback from the 1st and 2nd shift registers.
Phylogenetic models and MCMC methods for the reconstruction of language historyRobin Ryder
The document summarizes a phylogenetic model and Markov chain Monte Carlo (MCMC) methods for reconstructing language history from linguistic data. The model treats languages as diverging over time like species in a phylogenetic tree. MCMC is used to infer rates of language change and divergence times. Analysis of Indo-European language data strongly supported an Anatolian root dating to around 8000 years ago, rather than the alternative Kurgan hypothesis. The methods were shown to be robust even with simulated borrowing between languages.
This document discusses pseudo-noise (PN) sequences, which are random-looking bit sequences that repeat periodically and have useful properties for applications like code division multiple access (CDMA) networks. It outlines a 15-stage PN generator using a shift register, describes the properties of equal probability of 1s and 0s and high auto-correlation. It also discusses how PN sequences are used for data detection through correlation and includes a MATLAB code example to generate a PN sequence.
This document discusses methods for generating and testing random numbers. There are two main types of random number generators discussed: combined generators and inversive generators. Combined generators work by combining the outputs of two or more simpler random number generators. They are useful for simulating highly reliable systems or complex networks. The document also discusses how to test random numbers using the Kolmogorov-Smirnov test and runs tests. The Kolmogorov-Smirnov test compares the cumulative distribution function of observed values to expected values, while runs tests examine the arrangements of values in a sequence. Both can be used to determine if a random number generator is producing independent and identically distributed values.
This document summarizes an algorithm for the fast inversion of the normal cumulative distribution function (CDF) based on Marsaglia's method. It represents numbers in a way that allows sample points of the inverse CDF to be extracted via bit manipulation for very fast evaluation. The algorithm tabulates sample values and uses quadratic interpolation between them, indexing the values in a way that can be computed with one bit operation. This allows for inverse CDF computation in less than 5 CPU instructions on most machines.
This document provides an overview of Markov chain Monte Carlo (MCMC) methods. It begins with motivations for using MCMC, such as dealing with latent variable models where the likelihood function is intractable. It then covers random variable generation techniques before introducing the key MCMC algorithms: the Metropolis-Hastings algorithm and the Gibbs sampler. The document outlines the remaining topics to be covered, which include Monte Carlo integration, notions of Markov chains, and further advanced topics.
Omiros' talk on the Bernoulli factory problemBigMC
This document summarizes previous work on simulating events of unknown probability using reverse time martingales. It discusses von Neumann's solution to the Bernoulli factory problem where f(p)=1/2. It also summarizes the Keane-O'Brien existence result, the Nacu-Peres Bernstein polynomial approach, and issues with implementing the Nacu-Peres algorithm at large n due to the large number of strings involved. It proposes developing a reverse time martingale approach to address these issues.
This document discusses various methods for estimating normalizing constants that arise when evaluating integrals numerically. It begins by noting there are many computational methods for approximating normalizing constants across different communities. It then lists the topics that will be covered in the upcoming workshop, including discussions on estimating constants using Monte Carlo methods and Bayesian versus frequentist approaches. The document provides examples of estimating normalizing constants using Monte Carlo integration, reverse logistic regression, and Xiao-Li Meng's maximum likelihood estimation approach. It concludes by discussing some of the challenges in bringing a statistical framework to constant estimation problems.
Statistics (1): estimation, Chapter 2: Empirical distribution and bootstrapChristian Robert
The document discusses the bootstrap method and its applications in statistical inference. It introduces the bootstrap as a technique for estimating properties of estimators like variance and distribution when the true sampling distribution is unknown. This is done by treating the observed sample as if it were the population and resampling with replacement to create new simulated samples. The bootstrap then approximates characteristics of the sampling distribution, allowing inferences like confidence intervals to be constructed.
This document discusses Bayesian hypothesis testing and some of the challenges associated with it. It makes three key points:
1) There is tension between using posterior probabilities from a loss function approach versus Bayes factors, which eliminate prior dependence but have no direct connection to the posterior.
2) Bayesian hypothesis testing relies on choosing prior probabilities for hypotheses and prior distributions for parameters, which can strongly impact results and are often arbitrary.
3) Common Bayesian testing procedures like using Bayes factors can produce paradoxical results in some cases, like Lindley's paradox where the Bayes factor favors the null hypothesis as sample size increases despite evidence against it.
This document discusses properties of pseudo-random numbers and methods for generating random numbers computationally. It covers:
- Properties of pseudo-random numbers including being continuous between 0 and 1 and uniformly distributed.
- Common methods for generating pseudo-random numbers including table lookup, linear congruential generators (LCG), and feedback shift registers.
- Desirable properties for random number generators including being fast, requiring little memory, having a long cycle or period, and producing numbers that are close to uniform and independent.
Frequency-hopping spread spectrum (FHSS) is a communication scheme where the transmitter and receiver switch between different frequencies according to a known standard. FHSS provides robust communication that is resistant to noise, interference, and multipath effects. It allows multiple networks to operate simultaneously without interfering with each other. FHSS also provides security benefits without additional cost due to the unpredictable hopping between frequencies. Common applications of FHSS include military communications systems and small business radios.
The document describes the design of a 32-bit pseudo random number generator using a linear feedback shift register (LFSR) in a 90nm CMOS technology. The objectives are to design the circuit using MAGIC layout tools, simulate it using HSpice, and verify the results in MATLAB and Java. A 32-bit LFSR generates random numbers by shifting the values through 32 D-flipflops and tapping the outputs. For increased randomness, a "leap forward" technique shifts all values simultaneously instead of one bit at a time.
This document discusses random number generators and reviews Intel's random number generator. It begins with an introduction to random number generation and common pseudorandom number generators like linear congruential generators. It then describes Intel's true random number generator which uses thermal noise from resistors to modulate the frequency of an oscillator. The random bits generated from the clock drift are then processed digitally before being made available through Intel's software library. Empirical and theoretical tests for evaluating random number generators are also summarized.
This document describes a MATLAB code project on an FHSS (Frequency Hopped Spread Spectrum) system. It includes the theory of FHSS, block diagrams of the transmitter and receiver, and an explanation of the improved security method used. The project generates bit sequences, modulates the signal, creates an improved PN sequence, and performs frequency hopping to generate the spread signal. Output plots generated in MATLAB are included and analyzed. The results match the theoretical background. In conclusion, the document demonstrates implementing and analyzing an FHSS system in MATLAB to improve security.
Este algoritmo genera números pseudoaleatorios mediante la multiplicación de dos semillas con un número determinado de dígitos. A partir del producto de las semillas se toman los dígitos centrales para formar el primer número pseudoaleatorio, y luego se multiplican los números generados de forma secuencial para continuar la secuencia, eliminando la semilla más antigua en cada paso.
Monte Carlo methods rely on repeated random sampling to compute results. They generate random samples from a population according to a probability distribution and use them to obtain numerical results. The founders of the Monte Carlo method were J. von Neumann and S. Ulam during the Manhattan Project in the 1940s. Monte Carlo methods can be used to solve multidimensional integrals and have better convergence than classical numerical integration methods for dimensions greater than 4. The variance of Monte Carlo estimates decreases as 1/N, where N is the number of samples, resulting in slow convergence. Variance reduction techniques can improve the convergence rate.
The aim of this talk is to introduce two pieces of software, ``Lattice Builder'' and ``Stochastic Simulation in Java'' (SSJ). They allow to easily conduct experiments that rely on Monte Carlo (MC), quasi-Monte Carlo (QMC), or randomized QMC (RQMC). Lattice Builder is a C++ library designed to efficiently search, produce, and examine rank-1-lattices as well as polynomial lattices. It allows the user to choose from a broad palette of search criteria, types of weights, and construction methods, which can be accessed through a graphical interface as well as through a command line tool. Its structure also facilitates the implementation of own extensions and encourages to combine Lattice Builder with other programming languages.
SSJ is a Java library covering an extensive set of tools for stochastic simulations. It is particularly useful for experiments relying on MC and (R)QMC, ranging from integration problems over RQMC for Markov Chains (Array-RQMC) to density estimation.
This talk gives an introductory tour through the interfaces of Lattice Builder, shows how to integrate it into SSJ, and provides first steps in SSJ based on an example for mean estimation with RQMC.
Higher-order factorization machines (HOFMs) provide a framework for modeling feature interactions of arbitrary order in recommendation systems and link prediction tasks. The key ideas are:
(1) HOFMs express the prediction function as a weighted sum of ANOVA kernels of varying orders, capturing interactions between features.
(2) Computing the ANOVA kernel and its gradient can be done in linear time using dynamic programming, enabling efficient learning and prediction.
(3) Experiments on link prediction tasks show HOFMs can effectively model higher-order interactions to improve predictions compared to lower-order models like FM.
The document summarizes key concepts from Chapter 8 of the textbook "Fundamentals of Multimedia" on lossy compression algorithms. It introduces lossy compression and discusses distortion measures, rate-distortion theory, quantization techniques including uniform, non-uniform, and vector quantization. It also covers transform coding techniques such as the discrete cosine transform and its use in image compression standards to remove spatial redundancies by transforming pixel values into frequency coefficients.
This document provides an overview of reduced-order models and emulators. It discusses two main approaches: polynomial chaos expansions (PCE) and Gaussian process (GP) emulators. PCE approximates quantities of interest using orthogonal polynomials and provides error bounds, while GP emulators approximate computer model outputs as realizations of random processes. Both rely on designs of experiments, with Latin hypercube designs commonly used. The document compares the approaches and discusses pros and cons, noting they are complementary rather than competing methods. It concludes by emphasizing the importance of accounting for model discrepancy in surrogate modeling.
The document provides an overview of Monte Carlo simulation techniques. It discusses random number generation, methods for computing integrals using Monte Carlo integration, and techniques for reducing variance in Monte Carlo estimates. The lecture covers generating uniform and non-uniform random variables, numerical integration methods, the curse of dimensionality, and compares quasi-Monte Carlo and standard Monte Carlo integration.
Stochastic reaction networks (SRNs) are a particular class of continuous-time Markov chains used to model a wide range of phenomena, including biological/chemical reactions, epidemics, risk theory, queuing, and supply chain/social/multi-agents networks. In this context, we explore the efficient estimation of statistical quantities, particularly rare event probabilities, and propose two alternative importance sampling (IS) approaches [1,2] to improve the Monte Carlo (MC) estimator efficiency. The key challenge in the IS framework is to choose an appropriate change of probability measure to achieve substantial variance reduction, which often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection between finding optimal IS parameters and solving a variance minimization problem via a stochastic optimal control formulation. We pursue two alternative approaches to mitigate the curse of dimensionality when solving the resulting dynamic programming problem. In the first approach [1], we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. As an alternative, we present in [2] a dimension reduction method, based on mapping the problem to a significantly lower dimensional space via the Markovian projection (MP) idea. The output of this model reduction technique is a low dimensional SRN (potentially one dimension) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained via a discrete $L^2$ regression. By solving a resulting projected Hamilton-Jacobi-Bellman (HJB) equation for the reduced-dimensional SRN, we get projected IS parameters, which are then mapped back to the original full-dimensional SRN system, and result in an efficient IS-MC estimator of the full-dimensional SRN. Our analysis and numerical experiments verify that both proposed IS (learning based and MP-HJB-IS) approaches substantially reduce the MC estimator’s variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators. [1] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. Learning-based importance sampling via stochastic optimal control for stochastic reaction net-works. Statistics and Computing 33, no. 3 (2023): 58. [2] Ben Hammouda, C., Ben Rached, N., and Tempone, R., and Wiechert, S. (2023). Automated Importance Sampling via Optimal Control for Stochastic Reaction Networks: A Markovian Projection-based Approach. To appear soon.
Big Data and Small Devices by Katharina MorikBigMine
How can we learn from the data of small ubiquitous systems? Do we need to send the data to a server or cloud and do all learning there? Or can we learn on some small devices directly? Are smartphones small? Are navigation systems small? How complex is learning allowed to be in times of big data? What about graphical models? Can they be applied on small devices or even learned on restricted processors?
Big data are produced by various sources. Most often, they are distributedly stored at computing farms or clouds. Analytics on the Hadoop Distributed File System (HDFS) then follows the MapReduce programming model. According to the Lambda architecture of Nathan Marz and James Warren, this is the batch layer. It is complemented by the speed layer, which aggregates and integrates incoming data streams in real time. When considering big data and small devices, obviously, we imagine the small devices being hosts of the speed layer, only. Analytics on the small devices is restricted by memory and computation resources.
The interplay of streaming and batch analytics offers a multitude of configurations. In this talk, we discuss opportunities for using sophisticated models for learning spatio-temporal models. In particular, we investigate graphical models, which generate the probabilities for connected (sensor) nodes. First, we present spatio-temporal random fields that take as input data from small devices, are computed at a server, and send results to -possibly different — small devices. Second, we go even further: the Integer Markov Random Field approximates the likelihood estimates such that it can be computed on small devices. We illustrate our learning models by applications from traffic management.
Self-sampling Strategies for Multimemetic Algorithms in Unstable Computationa...Rafael Nogueras
This document discusses self-sampling strategies for multimemetic algorithms (MMAs) in unstable computational environments subject to churn. It proposes using probabilistic models to sample new individuals when populations need to be enlarged due to node failures. Experimental results show the bivariate model is superior for high churn, maintaining diversity and convergence better than random strategies. Future work aims to extend these self-sampling strategies to dynamic network topologies and more complex probabilistic models.
- The document discusses representation of stochastic processes in real and spectral domains and Monte Carlo sampling.
- Stochastic processes can be represented in the real (time or space) domain using autocorrelation and variogram functions, and in the spectral domain using power spectral density functions.
- Monte Carlo sampling uses techniques to generate random numbers from a probability density function for random sampling.
This document summarizes research on computing stochastic partial differential equations (SPDEs) using an adaptive multi-element polynomial chaos method (MEPCM) with discrete measures. Key points include:
1) MEPCM uses polynomial chaos expansions and numerical integration to compute SPDEs with parametric uncertainty.
2) Orthogonal polynomials are generated for discrete measures using various methods like Vandermonde, Stieltjes, and Lanczos.
3) Numerical integration is tested on discrete measures using Genz functions in 1D and sparse grids in higher dimensions.
4) The method is demonstrated on the KdV equation with random initial conditions. Future work includes applying these techniques to SPDEs driven
Computational Intelligence for Time Series PredictionGianluca Bontempi
This document provides an overview of computational intelligence methods for time series prediction. It begins with introductions to time series analysis and machine learning approaches for prediction. Specific models discussed include autoregressive (AR), moving average (MA), and autoregressive moving average (ARMA) processes. Parameter estimation techniques for AR models are also covered. The document outlines applications in areas like forecasting, wireless sensors, and biomedicine and concludes with perspectives on future directions.
The document provides guidelines for a presentation by two professors. It includes their names, the presenter's name, date and location of the presentation. It then discusses topics related to algorithms and computation such as deterministic and nondeterministic computation modes, complexity classes like TIME and SPACE, and the time complexity hierarchy theorem. It also mentions random number generators, pseudo-random number generators, and Monte Carlo methods for testing random numbers.
Opening of our Deep Learning Lunch & Learn series. First session: introduction to Neural Networks, Gradient descent and backpropagation, by Pablo J. Villacorta, with a prologue by Fernando Velasco
This document discusses using fractional calculus techniques for wind and solar power generation forecasting. It introduces a simple grid network model with three generation nodes and three load dispatch points. Forecasting the generation at each node separately is important for stability, rather than just aggregating the total generation. Statistical forecasting is formulated as an error minimization problem that can be solved using fractional derivatives. This allows incorporating the non-local nature of renewable generation. The document advocates using techniques like fractional PDEs and scenario analysis for computational forecasting methods. Accuracy data for wind and solar plant-level forecasts is also presented.
2017-03, ICASSP, Projection-based Dual Averaging for Stochastic Sparse Optimi...asahiushio1
We present a variant of the regularized dual averaging (RDA) algorithm for stochastic sparse optimization. Our approach differs from the previous studies of RDA in two respects. First, a sparsity-promoting metric is employed, originated from the proportionate-type adaptive filtering algorithms. Second, the squared-distance function to a closed convex set is employed as a part of the objective functions. In the particular application of online regression, the squared-distance function is reduced to a normalized version of the typical squared-error (least square) function. The two differences yield a better sparsity-seeking capability, leading to improved convergence properties. Numerical examples show the advantages of the proposed algorithm over the existing methods including ADAGRAD and adaptive proximal forward-backward splitting (APFBS).
High-Dimensional Network Estimation using ECLHPCC Systems
Kshitij Khare & Syed Rahman, University of Florida, present at the 2015 HPCC Systems Engineering Summit Community Day. In this presentation, we will discuss the motivation/theory behind CONCORD and its advantages over previous methods. In particular, we will discuss how the CONCORD estimate is superior to the empirical covariance matrix. We will end with an example detailing the implementation and use of the CONCORD algorithm in ECL. An exposure to multivariate statistics is helpful, but not necessary. Attendees should expect to come out with an understanding of sparse covariance estimation, its applications and how to use the CONCORD algorithm in ECL.
IRJET- An Efficient Reverse Converter for the Three Non-Coprime Moduli Set {4...IRJET Journal
This paper proposes a new and efficient reverse converter for converting residue numbers to decimal numbers for the three moduli set {6, 10, 15} which shares the common factor of 5. The proposed converter replaces larger multipliers used in previous converters with smaller multipliers and adders, reducing the hardware requirements. The hardware implementation of the proposed converter is presented and compared to other state-of-the-art converters, showing it performs better with fewer adders and multipliers. The proposed converter efficiently implements reverse conversion for the non-coprime three moduli set while requiring less hardware than previous approaches.
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