Showing papers on "Stochastic simulation published in 2009"

The pseudo-marginal approach for efficient Monte Carlo computations

Abstract: We introduce a powerful and flexible MCMC algorithm for stochastic simulation. The method builds on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139--1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method. Theoretical results are given describing the convergence properties of the proposed method, and simple numerical examples are given to illustrate the promising empirical characteristics of the technique. Interesting comparisons with a more obvious, but inexact, Monte Carlo approximation to the marginal algorithm, are also given.

The pseudo-marginal approach for efficient Monte Carlo computations

Abstract: We introduce a powerful and flexible MCMC algorithm for stochastic simulation. The method builds on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139-1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method. Theoretical results are given describing the convergence properties of the proposed method, and simple numerical examples are given to illustrate the promising empirical characteristics of the technique. Interesting comparisons with a more obvious, but inexact, Monte Carlo approximation to the marginal algorithm, are also given.

Hybrid stochastic simplifications for multiscale gene networks

Abstract: Background Stochastic simulation of gene networks by Markov processes has important applications in molecular biology. The complexity of exact simulation algorithms scales with the number of discrete jumps to be performed. Approximate schemes reduce the computational time by reducing the number of simulated discrete events. Also, answering important questions about the relation between network topology and intrinsic noise generation and propagation should be based on general mathematical results. These general results are difficult to obtain for exact models.

Approximations of Stochastic Hybrid Systems

Abstract: This paper develops a notion of approximation for a class of stochastic hybrid systems that includes, as special cases, both jump linear stochastic systems and linear stochastic hybrid automata. Our approximation framework is based on the recently developed notion of the so-called stochastic simulation functions. These Lyapunov-like functions can be used to rigorously quantify the distance or error between a system and its approximate abstraction. For the class of jump linear stochastic systems and linear stochastic hybrid automata, we show that the computation of stochastic simulation functions can be cast as a tractable linear matrix inequality problem. This enables us to compute the modeling error incurred by abstracting some of the continuous dynamics, or by neglecting the influence of stochastic noise, or even the influence of stochastic discrete jumps.

Biochemical simulations: stochastic, approximate stochastic and hybrid approaches

Abstract: Computer simulations have become an invaluable tool to study the sometimes counterintuitive temporal dynamics of (bio-)chemical systems In particular, stochastic simulation methods have attracted increasing interest recently In contrast to the well-known deterministic approach based on ordinary differential equations, they can capture effects that occur due to the underlying discreteness of the systems and random fluctuations in molecular numbers Numerous stochastic, approximate stochastic and hybrid simulation methods have been proposed in the literature In this article, they are systematically reviewed in order to guide the researcher and help her find the appropriate method for a specific problem

Economic Analysis of Simulation Selection Problems

Abstract: Ranking and selection procedures are standard methods for selecting the best of a finite number of simulated design alternatives based on a desired level of statistical evidence for correct selection. But the link between statistical significance and financial significance is indirect, and there has been little or no research into it. This paper presents a new approach to the simulation selection problem, one that maximizes the expected net present value of decisions made when using stochastic simulation. We provide a framework for answering these managerial questions: When does a proposed system design, whose performance is unknown, merit the time and money needed to develop a simulation to infer its performance? For how long should the simulation analysis continue before a design is approved or rejected? We frame the simulation selection problem as a “stoppable” version of a Bayesian bandit problem that treats the ability to simulate as a real option prior to project implementation. For a single proposed system, we solve a free boundary problem for a heat equation that approximates the solution to a dynamic program that finds optimal simulation project stopping times and that answers the managerial questions. For multiple proposed systems, we extend previous Bayesian selection procedures to account for discounting and simulation-tool development costs.

Engineering Genetic Circuits

Abstract: An Engineer's Guide to Genetic Circuits Chemical Reactions Macromolecules Genomes Cells and Their Structure Genetic Circuits Viruses Phage lambda: A Simple Genetic Circuit Learning Models Experimental Methods Experimental Data Cluster Analysis Learning Bayesian Networks Learning Causal Networks Experimental Design Differential Equation Analysis A Classical Chemical Kinetic Model Differential Equation Simulation Qualitative ODE Analysis Spatial Methods Stochastic Analysis A Stochastic Chemical Kinetic Model The Chemical Master Equation Gillespie's Stochastic Simulation Algorithm Gibson/Bruck's Next Reaction Method Tau-Leaping Relationship to Reaction Rate Equations Stochastic Petri-Nets Phage lambda Decision Circuit Example Spatial Gillespie Reaction-Based Abstraction Irrelevant Node Elimination Enzymatic Approximations Operator Site Reduction Statistical Thermodynamical Model Dimerization Reduction Phage lambda Decision Circuit Example Stoichiometry Amplification Logical Abstraction Logical Encoding Piecewise Models Stochastic Finite-State Machines Markov Chain Analysis Qualitative Logical Models Genetic Circuit Design Assembly of Genetic Circuits Combinational Logic Gates PoPS Gates Sequential Logic Circuits Future Challenges Solutions to Selected Problems References Glossary Index Sources and Problems appear at the end of each chapter.

Kinetic Modeling of Biological Systems

Abstract: The dynamics of how the constituent components of a natural system interact defines the spatio-temporal response of the system to stimuli. Modeling the kinetics of the processes that represent a biophysical system has long been pursued with the aim of improving our understanding of the studied system. Due to the unique properties of biological systems, in addition to the usual difficulties faced in modeling the dynamics of physical or chemical systems, biological simulations encounter difficulties that result from intrinsic multi-scale and stochastic nature of the biological processes.This chapter discusses the implications for simulation of models involving interacting species with very low copy numbers, which often occur in biological systems and give rise to significant relative fluctuations. The conditions necessitating the use of stochastic kinetic simulation methods and the mathematical foundations of the stochastic simulation algorithms are presented. How the well-organized structural hierarchies often seen in biological systems can lead to multi-scale problems and the possible ways to address the encountered computational difficulties are discussed. We present the details of the existing kinetic simulation methods and discuss their strengths and shortcomings. A list of the publicly available kinetic simulation tools and our reflections for future prospects are also provided.

Better simulation metamodeling: the why, what, and how of stochastic kriging

Abstract: Stochastic kriging is a methodology recently developed for metamodeling stochastic simulation. Stochastic kriging can partake of the behavior of kriging and of generalized least squares regression. This advanced tutorial explains regression, kriging, and stochastic kriging as metamodeling methodologies, emphasizing the consequences of misspecified models for global metamodeling. It provides an exposition of how to choose parameters in stochastic kriging and how to build a metamodel with it given simulation output, and discusses future research directions to enhance stochastic kriging.

The multinomial simulation algorithm for discrete stochastic simulation of reaction-diffusion systems.

Abstract: The Inhomogeneous Stochastic Simulation Algorithm (ISSA) is a variant of the stochastic simulation algorithm in which the spatially inhomogeneous volume of the system is divided into homogeneous subvolumes, and the chemical reactions in those subvolumes are augmented by diffusive transfers of molecules between adjacent subvolumes. The ISSA can be prohibitively slow when the system is such that diffusive transfers occur much more frequently than chemical reactions. In this paper we present the Multinomial Simulation Algorithm (MSA), which is designed to, on the one hand, outperform the ISSA when diffusive transfer events outnumber reaction events, and on the other, to handle small reactant populations with greater accuracy than deterministic-stochastic hybrid algorithms. The MSA treats reactions in the usual ISSA fashion, but uses appropriately conditioned binomial random variables for representing the net numbers of molecules diffusing from any given subvolume to a neighbor within a prescribed distance. Simulation results illustrate the benefits of the algorithm.

A new class of highly efficient exact stochastic simulation algorithms for chemical reaction networks

Abstract: We introduce an alternative formulation of the exact stochastic simulation algorithm (SSA) for sampling trajectories of the chemical master equation for a well-stirred system of coupled chemical reactions. Our formulation is based on factored-out, partial reaction propensities. This novel exact SSA, called the partial-propensity direct method (PDM), is highly efficient and has a computational cost that scales at most linearly with the number of chemical species, irrespective of the degree of coupling of the reaction network. In addition, we propose a sorting variant, SPDM, which is especially efficient for multiscale reaction networks.

Application of Multiple Point Geostatistics to Non-stationary Images

Abstract: Simulation of flow and solute transport through aquifers or oil reservoirs requires a precise representation of subsurface heterogeneity that can be achieved by stochastic simulation approaches. Traditional geostatistical methods based on variograms, such as truncated Gaussian simulation or sequential indicator simulation, may fail to generate the complex, curvilinear, continuous and interconnected facies distributions that are often encountered in real geological media, due to their reliance on two-point statistics. Multiple Point Geostatistics (MPG) overcomes this constraint by using more complex point configurations whose statistics are retrieved from training images. Obtaining representative statistics requires stationary training images, but geological understanding often suggests a priori facies variability patterns. This research aims at extending MPG to non-stationary facies distributions. The proposed method subdivides the training images into different areas. The statistics for each area are stored in separate frequency search trees. Several training images are used to ensure that the obtained statistics are representative. The facies probability distribution for each cell during simulation is calculated by weighting the probabilities from the frequency trees. The method is tested on two different object-based training image sets. Results show that non-stationary training images can be used to generate suitable non-stationary facies distributions.

Two ways to handle dependent uncertainties in multi-criteria decision problems

Abstract: We consider multi-criteria group decision-making problems, where the decision makers (DMs) want to identify their most preferred alternative(s) based on uncertain or inaccurate criteria measurements. In many real-life problems the uncertainties may be dependent. In this paper, we focus on multicriteria decision-making (MCDM) problems where the criteria and their uncertainties are computed using a stochastic simulation model. The model is based on decision variables and stochastic parameters with given distributions. The simulation model determines for the criteria a joint probability distribution, which quantifies the uncertainties and their dependencies. We present and compare two methods for treating the uncertainty and dependency information within the SMAA-2 multi-criteria decision aid method. The first method applies directly the discrete sample generated by the simulation model. The second method is based on using a multivariate Gaussian distribution. We demonstrate the methods using a decision support model for a retailer operating in the deregulated European electricity market.

Identification of random shapes from images through polynomial chaos expansion of random level set functions

Abstract: In this paper, an efficient method is proposed for the identification of random shapes in a form suitable for numerical simulation within the extended stochastic finite element method (X-SFEM). The method starts from a collection of images representing different outcomes of the random shape to identify. The key point of the method is to represent the random geometry in an implicit manner using the level set technique. In this context, the problem of random geometry identification is equivalent to the identification of a random level set function, which is a random field. This random field is represented on a polynomial chaos (PC) basis and various efficient numerical strategies are proposed in order to identify the coefficients of its PC decomposition. The performance of these strategies is evaluated through some ‘manufactured’ problems and useful conclusions are provided. The propagation of geometrical uncertainties in structural analysis using the X-SFEM is finally examined. Copyright © 2009 John Wiley & Sons, Ltd.

A Blocking Markov Chain Monte Carlo Method for Inverse Stochastic Hydrogeological Modeling

Abstract: An adequate representation of the detailed spatial variation of subsurface parameters for underground flow and mass transport simulation entails heterogeneous models. Uncertainty characterization generally calls for a Monte Carlo analysis of many equally likely realizations that honor both direct information (e.g., conductivity data) and information about the state of the system (e.g., piezometric head or concentration data). Thus, the problems faced is how to generate multiple realizations conditioned to parameter data, and inverse-conditioned to dependent state data. We propose using Markov chain Monte Carlo approach (MCMC) with block updating and combined with upscaling to achieve this purpose. Our proposal presents an alternative block updating scheme that permits the application of MCMC to inverse stochastic simulation of heterogeneous fields and incorporates upscaling in a multi-grid approach to speed up the generation of the realizations. The main advantage of MCMC, compared to other methods capable of generating inverse-conditioned realizations (such as the self-calibrating or the pilot point methods), is that it does not require the solution of a complex optimization inverse problem, although it requires the solution of the direct problem many times.

Upper semicontinuity of random attractors for non-compact random dynamical systems.

Abstract: The upper semicontinuity of random attractors for non-compact random dynamical systems is proved when the union of all perturbed random attractors is precompact with probability one. This result is applied to the stochastic Reaction-Diusion with white noise defined on the entire space R n .

Refining the weighted stochastic simulation algorithm

Abstract: The weighted stochastic simulation algorithm (wSSA) recently introduced by Kuwahara and Mura [J Chem Phys 129, 165101 (2008)] is an innovative variation on the stochastic simulation algorithm (SSA) It enables one to estimate, with much less computational effort than was previously thought possible using a Monte Carlo simulation procedure, the probability that a specified event will occur in a chemically reacting system within a specified time when that probability is very small This paper presents some procedural extensions to the wSSA that enhance its effectiveness in practical applications The paper also attempts to clarify some theoretical issues connected with the wSSA, including its connection to first passage time theory and its relation to the SSA

Extended Stochastic Petri Nets for Model-Based Design of Wetlab Experiments

Abstract: This paper introduces extended stochastic Petri nets to model wetlab experiments. The extentions include read and inhibitor arcs, stochastic transitions with freestyle rate functions as well as several deterministically timed transition types: immediate firing, deterministic firing delay, and scheduled firing. The extensions result into non-Markovian behaviour, which precludes analytical analysis approaches. But there are adapted stochastic simulation analysis (SSA) methods, ready to deal with the extended behaviour. Having the simulation traces, we apply simulative model checking of PLTL, a linear-time temporal logic (LTL) in a probabilistic setting. We present some typical model components, demonstrating the suitability of the introduced Petri net class for the envisaged application scenario. We conclude by looking briefly at a classical example of prokaryotic gene regulation, the lac operon case.

Copula-Based Multivariate Input Models for Stochastic Simulation

Abstract: As large-scale discrete-event stochastic simulation becomes a tool that is used routinely for the design and analysis of stochastic systems, the need for input-modeling support with the ability to represent complex interactions and interdependencies among the components of multivariate time-series input processes is more critical than ever. Motivated by the failure of independent and identically distributed random variables to represent such input processes, a comprehensive framework called Vector-Autoregressive-To-Anything (VARTA) has been introduced for multivariate time-series input modeling. Despite its flexibility in capturing a wide variety of distributional shapes, we show that VARTA falls short in representing dependence structures that arise in situations where extreme component realizations occur together. We demonstrate that it is possible to extend VARTA to work for such dependence structures via the use of the copula theory, which has been used primarily for random vectors in the simulation input-modeling literature, for multivariate time-series input modeling. We show that our copula-based multivariate time-series input model, which includes VARTA as a special case, allows the development of statistically valid fitting and fast sampling algorithms well suited for driving large-scale stochastic simulations.

Spatial stochastic frontier models: accounting for unobserved local determinants of inefficiency

Abstract: This paper analyzes the productivity of farms across 370 municipalities in the Center-West region of Brazil. A stochastic frontier model with a latent spatial structure is proposed to account for possible unknown geographical variation of the outputs. The paper compares versions of the model that include the latent spatial effect in the mean of output or as a variable that conditions the distribution of inefficiency, include or not observed municipal variables, and specify independent normal or conditional autoregressive priors for the spatial effects. The Bayesian paradigm is used to estimate the proposed models. As the resultant posterior distributions do not have a closed form, stochastic simulation techniques are used to obtain samples from them. Two model comparison criteria provide support for including the latent spatial effects, even after considering covariates at the municipal level. Models that ignore the latent spatial effects produce significantly different rankings of inefficiencies across agents.

Automated Trace Analysis of Discrete-Event System Models

Abstract: In this paper, we describe a novel technique that helps a modeler gain insight into the dynamic behavior of a complex stochastic discrete event simulation model based on trace analysis. We propose algorithms to distinguish progressive from repetitive behavior in a trace and to extract a minimal progressive fragment of a trace. The implied combinatorial optimization problem for trace reduction is solved in linear time with dynamic programming. We present and compare several approximate and one exact solution method. Information on the reduction operation as well as the reduced trace itself helps a modeler to recognize the presence of certain errors and to identify their cause. We track down a subtle modeling error in a dependability model of a multi-class server system to illustrate the effectiveness of our approach in revealing the cause of an observed effect. The proposed technique has been implemented and integrated in Traviando, a trace analyzer to debug stochastic simulation models.

Numerically Stable Stochastic Simulation Approaches for Solving Dynamic Economic Models

Abstract: We develop numerically stable stochastic simulation approaches for solving dynamic economic models. We rely on standard simulation procedures to simultaneously compute an ergodic distribution of state variables, its support and the associated decision rules. We differ from existing methods, however, in how we use simulation data to approximate decision rules. Instead of the usual least-squares approximation methods, we examine a variety of alternatives, including the least-squares method using SVD, Tikhonov regularization, least-absolute deviation methods, principal components regression method, all of which are numerically stable and can handle ill-conditioned problems. These new methods enable us to compute high-order polynomial approximations without encountering numerical problems. Our approaches are especially well suitable for high-dimensional applications in which other methods are infeasible.

Galerkin spectral method applied to the chemical master equation

Abstract: This study is concerned with the numerical solution of certain stochastic models of chemical reactions Such descriptions have been shown to be useful tools when studying biochemical processes inside living cells where classical deterministic rate equations fail to reproduce actual behavior The main contribution of this thesis lies in its theoretical and practical investigation of different methods for obtaining numerical solutions to such descriptions In a preliminary study, a simple but often quite effective approach to the moment closure problem is examined A more advanced program is then developed for obtaining a consistent representation of the high dimensional probability density of the solution The proposed method gains efficiency by utilizing a rapidly converging representation of certain functions defined over the semi-infinite integer lattice Another contribution of this study, where the focus instead is on the spatially distributed case, is a suggestion for how to obtain a consistent stochastic reaction-diffusion model over an unstructured grid Here it is also shown how to efficiently collect samples from the resulting model by making use of a hybrid method In a final study, a time-parallel stochastic simulation algorithm is suggested and analyzed Efficiency is here achieved by moving parts of the solution phase into the deterministic regime given that a parallel architecture is available Necessary background material is developed in three chapters in this summary An introductory chapter on an accessible level motivates the purpose of considering stochastic models in applied physics In a second chapter the actual stochastic models considered are developed in a multi-faceted way Finally, the current state-of-the-art in numerical solution methods is summarized and commented upon

On the domination of a random walk on a discrete cylinder by random interlacements

Abstract: We consider simple random walk on a discrete cylinder with base a large $d$-dimensional torus of side-length $N$, when $d$ is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length almost of order $N$, at certain random times comparable to the square of the number of sites in the base. We show a domination control in terms of the trace left in similar boxes by random interlacements in the infinite $(d+1)$-dimensional cubic lattice at a suitably adjusted level. As an application we derive a lower bound on the disconnection time of the discrete cylinder, which as a by-product shows the tightness of the laws of the ratio of the square of the number of sites in the base to the disconnection time. This fact had previously only been established when $d$ is at least 17.