Stochastic simulation

About: Stochastic simulation is a research topic. Over the lifetime, 4431 publications have been published within this topic receiving 120543 citations.

Exact Stochastic Simulation of Coupled Chemical Reactions

Abstract: There are two formalisms for mathematically describing the time behavior of a spatially homogeneous chemical system: The deterministic approach regards the time evolution as a continuous, wholly predictable process which is governed by a set of coupled, ordinary differential equations (the “reaction-rate equations”); the stochastic approach regards the time evolution as a kind of random-walk process which is governed by a single differential-difference equation (the “master equation”). Fairly simple kinetic theory arguments show that the stochastic formulation of chemical kinetics has a firmer physical basis than the deterministic formulation, but unfortunately the stochastic master equation is often mathematically intractable. There is, however, a way to make exact numerical calculations within the framework of the stochastic formulation without having to deal with the master equation directly. It is a relatively simple digital computer algorithm which uses a rigorously derived Monte Carlo procedure to numerically simulate the time evolution of the given chemical system. Like the master equation, this “stochastic simulation algorithm” correctly accounts for the inherent fluctuations and correlations that are necessarily ignored in the deterministic formulation. In addition, unlike most procedures for numerically solving the deterministic reaction-rate equations, this algorithm never approximates infinitesimal time increments df by finite time steps At. The feasibility and utility of the simulation algorithm are demonstrated by applying it to several well-known model chemical systems, including the Lotka model, the Brusselator, and the Oregonator.

Random variables and stochastic processes

Abstract: An electromagnetic pulse counter having successively operable, contact-operating armatures. The armatures are movable to a rest position, an intermediate position and an active position between the main pole and the secondary pole of a magnetic circuit.

Non-uniform random variate generation

Abstract: This is a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods. Authors’ address: School of Computer Science, McGill University, 3480 University Street, Montreal, Canada H3A 2K6. The authors’ research was sponsored by NSERC Grant A3456 and FCAR Grant 90-ER-0291. 1. The main paradigms The purpose of this chapter is to review the main methods for generating random variables, vectors and processes. Classical workhorses such as the inversion method, the rejection method and table methods are reviewed in section 1. In section 2, we discuss the expected time complexity of various algorithms, and give a few examples of the design of generators that are uniformly fast over entire families of distributions. In section 3, we develop a few universal generators, such as generators for all log concave distributions on the real line. Section 4 deals with random variate generation when distributions are indirectly specified, e.g, via Fourier coefficients, characteristic functions, the moments, the moment generating function, distributional identities, infinite series or Kolmogorov measures. Random processes are briefly touched upon in section 5. Finally, the latest developments in Markov chain methods are discussed in section 6. Some of this work grew from Devroye (1986a), and we are carefully documenting work that was done since 1986. More recent references can be found in the book by Hörmann, Leydold and Derflinger (2004). Non-uniform random variate generation is concerned with the generation of random variables with certain distributions. Such random variables are often discrete, taking values in a countable set, or absolutely continuous, and thus described by a density. The methods used for generating them depend upon the computational model one is working with, and upon the demands on the part of the output. For example, in a ram (random access memory) model, one accepts that real numbers can be stored and operated upon (compared, added, multiplied, and so forth) in one time unit. Furthermore, this model assumes that a source capable of producing an i.i.d. (independent identically distributed) sequence of uniform [0, 1] random variables is available. This model is of course unrealistic, but designing random variate generators based on it has several advantages: first of all, it allows one to disconnect the theory of non-uniform random variate generation from that of uniform random variate generation, and secondly, it permits one to plan for the future, as more powerful computers will be developed that permit ever better approximations of the model. Algorithms designed under finite approximation limitations will have to be redesigned when the next generation of computers arrives. For the generation of discrete or integer-valued random variables, which includes the vast area of the generation of random combinatorial structures, one can adhere to a clean model, the pure bit model, in which each bit operation takes one time unit, and storage can be reported in terms of bits. Typically, one now assumes that an i.i.d. sequence of independent perfect bits is available. In this model, an elegant information-theoretic theory can be derived. For example, Knuth and Yao (1976) showed that to generate a random integer X described by the probability distribution {X = n} = pn, n ≥ 1, any method must use an expected number of bits greater than the binary entropy of the distribution, ∑

COPASI---a COmplex PAthway SImulator

Abstract: Motivation: Simulation and modeling is becoming a standard approach to understand complex biochemical processes. Therefore, there is a big need for software tools that allow access to diverse simulation and modeling methods as well as support for the usage of these methods. Results: Here, we present COPASI, a platform-independent and user-friendly biochemical simulator that offers several unique features. We discuss numerical issues with these features; in particular, the criteria to switch between stochastic and deterministic simulation methods, hybrid deterministic--stochastic methods, and the importance of random number generator numerical resolution in stochastic simulation. Availability: The complete software is available in binary (executable) for MS Windows, OS X, Linux (Intel) and Sun Solaris (SPARC), as well as the full source code under an open source license from http://www.copasi.org. Contact: mendes@vbi.vt.edu

Stochastic Simulation

Abstract: One fifth (4 of 20) of the research articles published in the Journal of Educational Statistics in 1988 include simulation studies that justify or illustrate the authors' conclusions. A similar fraction (6 of 33) of the articles in the 1988 volume of Psychometrika include simulations; comparable proportions could be expected in other journals at the boundary of theoretical statistics and social/psychological applications. Due in part to the complexity of the problems tackled today and in part to the availability of cheap, powerful computing—by no means independent influences—simulation and Monte Carlo methods have become both necessary and practical tools for statisticians and applied workers in quantitative areas of education and psychology. Simulation has become popular—not only in the quantitative social sciences, but in all of the mathematical sciences from physics to operations research to number theory—because it is almost always easy to do. This ease of use makes the simulation experimenter vulnerable to two common pitfalls. Selection of the basic source of "random numbers" is often passive: Whatever is available in the computer's standard subroutine library is used. However, the fact that a pseudo-random number generator appears in a popular software package or operating system is hardly reason to trust it, as is shown by the infamous RANDU generator, once popular on IBM mainframes and PDP mini-computers, and by the generators burned into RAM on today's PCs. Simulation design and reporting also deserve special care. Some attempt must be made to assess the accuracy of the simulation estimates: One should accurately estimate and report SE (6) as well as 6. In addition, enough detail should be reported that the interested reader can replicate the study and check the results, just as with other experiments. Yet these considerations are also easy to overlook. Brian D. Ripley's Stochastic Simulation is a short, yet ambitious, survey of modern simulation techniques. Three themes run throughout the book. First, one shoud not take basic simulation subroutines for granted, especially on minior microcomputers where they tend to be poor implementations, implementations of poor algorithms, or both. Second, design of experiments, or variance reduction as it is known in this field, deserves greater consideration. Third, modern methods make it possible to simulate and analyze processes that are dependent over time, and using such processes opens the door to new simulation techniques, such as simulated annealing in optimization. Ripley intends this book to be a "comprehensive guide," and it is indeed most accurately described as a researcher's handbook with examples and