Showing papers on "Stochastic simulation published in 1986"

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, ∑

Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and application

Abstract: We construct asymptotic expansions for the exponential growth rate (Lyapunov exponent) and rotation number of the random oscillator when the noise is large, small, rapidly varying or slowly varying. We then apply our results to problems in the stability of the random oscillator, the spectrum of the one-dimensional random Schr6dinger operator and wave propagation in a one-dimensional random medium.

Non‐Markovian activated rate processes: Comparison of current theories with numerical simulation data

Abstract: We calculate the barrier crossing rate constants for a Brownian particle in a double well potential experiencing a non‐Markovian friction kernel using a full stochastic simulation. We compare the simulation results with recently proposed interpolation formulas which are based on the Grote–Hynes theory and the energy diffusion mechanism. We find that such formulas can fail by orders of magnitude in a physically interesting regime. Slow activation in an effective dynamic double well potential is probably responsible for the deviations observed.

Stochastic models of multi-species kinetics in radiation-induced spurs

Abstract: Diffusion-limited kinetics in multi-species radical clusters, typical of those resulting from the passage of high-energy radiation through a liquid, have been analysed using several theories. A Monte Carlo (MC) model, in which the trajectories of the diffusing reactants are modelled by time-discretised random flights and in which reaction occurs when particles undergo pairwise encounters, is proposed as a measure of reality. The random reaction times (RRT) model, an efficient stochastic simulation technique based on the approximation that interparticle distances evolve independently, is generalised to multi-species spurs and validated by comparison with the full Monte Carlo model. Further comparisons with other analytical models are less favourable and demonstrate the importance of a stochastic rate equation in predicting product ratios. The RRT model is also extended to include reactive products in three different ways, each of which provides an acceptable approximation to the observed (MC) kinetics in all systems tested so far.

Problems in Bayesian analysis of stochastic simulation

Abstract: It is argued that Bayesian methodology is an appropriate tool in certain simulation contexts. Computational problems, specific to simulation applications, are then described in some detail; possible remedies are also outlined.

Diffusion in random structures with a topological bias.

Abstract: We study biased diffusion on a topological random comb with an exponential distribution of dangling ends which is relevant to the essential physics of biased diffusion in random structures such as percolation systems above criticality. By mapping the problem onto a linear chain with a power-law distribution of transition rates we find that above a bias threshold, diffusion is anomalous in two respects: ${d}_{w}$ (the fractal dimension of a random walk) is above 1, and depends continuously upon the magnitude of the bias. Our analytic results are confirmed by extensive computer simulations.

Regression methodology for estimating model prediction error

Abstract: The process of validating a stochastic simulation model involves the comparison of data generated by the model with corresponding data from the real system. Instead of applying statistical tests to...

Examining short-rotation hybrid poplar investments by using stochastic simulation

Abstract: We examined and compared short-rotation hybrid poplar investments using standard discounted cash flow and stochastic simulation. With stochastic simulation, triangular probability density functions...

The importance of non-linearities in large forecasting models with stochastic error processes

Abstract: This paper considers the consequences of the stochastic error process in large non-linear forecasting models. As such models are non-linear, the deterministic forecast is neither the mean nor the mode of the density function of the endogenous variables. Under a specific assumption as to the class of the non-linearity it is shown that the deterministic forecast is actually the vector of marginal medians of the density function. Stochastic simulation techniques are then used to test whether one large forecasting model actually lies within this class.

Errors in Simulation of Random Processes

Abstract: Simulation of random processes is frequently based on discrete approximations of the power spectral density of these processes. This study evaluates effects of these approximations on simulation-based estimates of the mean rate at which a Gaussian process crosses a level. The mean upcrossing rate is a useful probabilistic descriptor for the analysis of fatigue and first-excursion failures.

Generation of continuous random deviates using empirical event count data in digital simulation

Abstract: Simulation of continuous-time systems may be difficult due to lack of data characterizing the continuous random variable involved. This problem may be resolved by using empirical event counts or event per unit time data. When inter-event times may be assumed to be independent, a renewal theory-based algorithm allows the inter-event time distribution to be approximated using event per unit time data. The algorithm is also useful in more general cases. INTRODUCTION In the study of a stochastic system it is important to be able to identify the behavior of those processes which define the behavior of the system itself. In theoretical studies, these systems may have characteristics ana processes which are well defined and mathematically tractable. In practical systems analysis, however, the problem may be much more difficult. In practice, the problem is not one of process selection, but identification. It is identification of the inter-event time process that is the topic of this research. Many processes consist of events that take place at random points in time, where the time between adjacent events is a random variable. Analysis of these processes may require knowledge of the inter-event time distribution. In practice, however, information characterizing this distribution may be very difficult to obtain. In many cases the only information available consists of events per unit time data. Even where inter-event times are observable, data collection may be very expensive; events per unit time data, on the other hand, can usually be collected economically. The resolution of this problem has taken many forms. Since Poissondistributed events per unit time imply negative exponentially distributed times between events, the investigator might attempt to find the Poisson distribution Annual Simulation Symposium

The asymptotic p-stability of composite stochastic systems

Abstract: The sufficient conditions of asymptotic p-stability of non-linear composite stochastic systems are established with the objective of analysing composite systems in their lower order subsystems in t...

Simulation of a random differential equation

Abstract: The problem considered is that of a second-order differential equation with stochastic parameters. Time-dependent random coefficients may assume both positive and negative values when modeling parametric excitations. As there is no exact solution in the case of colored noise random coefficients, one must use approximate techniques. We presently investigate an iterative procedure. The quality of this approximation to the exact solution is verified with a Monte Carlo simulation. The particular example considered is that of a harmonic oscillator with a time-dependent random stiffness and excited by an external random forcing function. After a brief review of the iterative method, and an outline of the design of the Monte Carlo simulation, an extensive parametric study is presented to establish ranges of parameter values for which the approximation is valid. This comparison study leads to a design criterion for the mathematical modeling of structures with parametric uncertainties. We are interested in using information from studies such as this to understand the behavior of large-scale structures.