Showing papers on "Stochastic process published in 2003"

Gaussian processes in machine learning

Abstract: We give a basic introduction to Gaussian Process regression models. We focus on understanding the role of the stochastic process and how it is used to define a distribution over functions. We present the simple equations for incorporating training data and examine how to learn the hyperparameters using the marginal likelihood. We explain the practical advantages of Gaussian Process and end with conclusions and a look at the current trends in GP work.

Reliability and statistics in geotechnical engineering

Abstract: Preface. Part I. 1 Introduction - uncertainty and risk in geotechnical engineering. 1.1 Offshore platforms. 1.2 Pit mine slopes. 1.3 Balancing risk and reliability in a geotechnical design. 1.4 Historical development of reliability methods in civil engineering. 1.5 Some terminological and philosophical issues. 1.6 The organization of this book. 1.7 A comment on notation and nomenclature. 2 Uncertainty. 2.1 Randomness, uncertainty, and the world. 2.2 Modeling uncertainties in risk and reliability analysis. 2.3 Probability. 3 Probability. 3.1 Histograms and frequency diagrams. 3.2 Summary statistics. 3.3 Probability theory. 3.4 Random variables. 3.5 Random process models. 3.6 Fitting mathematical pdf models to data. 3.7 Covariance among variables. 4 Inference. 4.1 Frequentist theory. 4.2 Bayesian theory. 4.3 Prior probabilities. 4.4 Inferences from sampling. 4.5 Regression analysis. 4.6 Hypothesis tests. 4.7 Choice among models. 5 Risk, decisions and judgment. 5.1 Risk. 5.2 Optimizing decisions. 5.3 Non-optimizing decisions. 5.4 Engineering judgment. Part II. 6 Site characterization. 6.1 Developments in site characterization. 6.2 Analytical approaches to site characterization. 6.3 Modeling site characterization activities. 6.4 Some pitfalls of intuitive data evaluation. 6.5 Organization of Part II. 7 Classification and mapping. 7.1 Mapping discrete variables. 7.2 Classification. 7.3 Discriminant analysis. 7.4 Mapping. 7.5 Carrying out a discriminant or logistic analysis. 8 Soil variability. 8.1 Soil properties. 8.2 Index tests and classification of soils. 8.3 Consolidation properties. 8.4 Permeability. 8.5 Strength properties. 8.6 Distributional properties. 8.7 Measurement error. 9 Spatial variability within homogeneous deposits. 9.1 Trends and variations about trends. 9.2 Residual variations. 9.3 Estimating autocorrelation and autocovariance. 9.4 Variograms and geostatistics. Appendix: algorithm for maximizing log-likelihood of autocovariance. 10 Random field theory. 10.1 Stationary processes. 10.2 Mathematical properties of autocovariance functions. 10.3 Multivariate (vector) random fields. 10.4 Gaussian random fields. 10.5 Functions of random fields. 11 Spatial sampling. 11.1 Concepts of sampling. 11.2 Common spatial sampling plans. 11.3 Interpolating random fields. 11.4 Sampling for autocorrelation. 12 Search theory. 12.1 Brief history of search theory. 12.2 Logic of a search process. 12.3 Single stage search. 12.4 Grid search. 12.5 Inferring target characteristics. 12.6 Optimal search. 12.7 Sequential search. Part III. 13 Reliability analysis and error propagation. 13.1 Loads, resistances and reliability. 13.2 Results for different distributions of the performance function. 13.3 Steps and approximations in reliability analysis. 13.4 Error propagation - statistical moments of the performance function. 13.5 Solution techniques for practical cases. 13.6 A simple conceptual model of practical significance. 14 First order second moment (FOSM) methods. 14.1 The James Bay dikes. 14.2 Uncertainty in geotechnical parameters. 14.3 FOSM calculations. 14.4 Extrapolations and consequences. 14.5 Conclusions from the James Bay study. 14.6 Final comments. 15 Point estimate methods. 15.1 Mathematical background. 15.2 Rosenblueth's cases and notation. 15.3 Numerical results for simple cases. 15.4 Relation to orthogonal polynomial quadrature. 15.5 Relation with 'Gauss points' in the finite element method. 15.6 Limitations of orthogonal polynomial quadrature. 15.7 Accuracy, or when to use the point-estimate method. 15.8 The problem of the number of computation points. 15.9 Final comments and conclusions. 16 The Hasofer-Lind approach (FORM). 16.1 Justification for improvement - vertical cut in cohesive soil. 16.2 The Hasofer-Lind formulation. 16.3 Linear or non-linear failure criteria and uncorrelated variables. 16.4 Higher order reliability. 16.5 Correlated variables. 16.6 Non-normal variables. 17 Monte Carlo simulation methods. 17.1 Basic considerations. 17.2 Computer programming considerations. 17.3 Simulation of random processes. 17.4 Variance reduction methods. 17.5 Summary. 18 Load and resistance factor design. 18.1 Limit state design and code development. 18.2 Load and resistance factor design. 18.3 Foundation design based on LRFD. 18.4 Concluding remarks. 19 Stochastic finite elements. 19.1 Elementary finite element issues. 19.2 Correlated properties. 19.3 Explicit formulation. 19.4 Monte Carlo study of differential settlement. 19.5 Summary and conclusions. Part IV. 20 Event tree analysis. 20.1 Systems failure. 20.2 Influence diagrams. 20.3 Constructing event trees. 20.4 Branch probabilities. 20.5 Levee example revisited. 21 Expert opinion. 21.1 Expert opinion in geotechnical practice. 21.2 How do people estimate subjective probabilities? 21.3 How well do people estimate subjective probabilities? 21.4 Can people learn to be well-calibrated? 21.5 Protocol for assessing subjective probabilities. 21.6 Conducting a process to elicit quantified judgment. 21.7 Practical suggestions and techniques. 21.8 Summary. 22 System reliability assessment. 22.1 Concepts of system reliability. 22.2 Dependencies among component failures. 22.3 Event tree representations. 22.4 Fault tree representations. 22.5 Simulation approach to system reliability. 22.6 Combined approaches. 22.7 Summary. Appendix A: A primer on probability theory. A.1 Notation and axioms. A.2 Elementary results. A.3 Total probability and Bayes' theorem. A.4 Discrete distributions. A.5 Continuous distributions. A.6 Multiple variables. A.7 Functions of random variables. References. Index.

An introduction to stochastic processes with applications to biology

Abstract: Review of Probability Theory and an Introduction to Stochastic Processes Introduction Brief Review of Probability Theory Generating Functions Central Limit Theorem Introduction to Stochastic Processes An Introductory Example: A Simple Birth Process Discrete-Time Markov Chains Introduction Definitions and Notation Classification of States First Passage Time Basic Theorems for Markov Chains Stationary Probability Distribution Finite Markov Chains An Example: Genetics Inbreeding Problem Monte Carlo Simulation Unrestricted Random Walk in Higher Dimensions Biological Applications of Discrete-Time Markov Chains Introduction Proliferating Epithelial Cells Restricted Random Walk Models Random Walk with Absorbing Boundaries Random Walk on a Semi-Infinite Domain General Birth and Death Process Logistic Growth Process Quasistationary Probability Distribution SIS Epidemic Model Chain Binomial Epidemic Models Discrete-Time Branching Processes Introduction Definitions and Notation Probability Generating Function of Xn Probability of Population Extinction Mean and Variance of Xn Environmental Variation Multitype Branching Processes Continuous-Time Markov Chains Introduction Definitions and Notation The Poisson Process Generator Matrix Q Embedded Markov Chain and Classification of States Kolmogorov Differential Equations Stationary Probability Distribution Finite Markov Chains Generating Function Technique Interevent Time and Stochastic Realizations Review of Method of Characteristics Continuous-Time Birth and Death Chains Introduction General Birth and Death Process Stationary Probability Distribution Simple Birth and Death Processes Queueing Process Population Extinction First Passage Time Logistic Growth Process Quasistationary Probability Distribution An Explosive Birth Process Nonhomogeneous Birth and Death Process Biological Applications of Continuous-Time Markov Chains Introduction Continuous-Time Branching Processes SI and SIS Epidemic Processes Multivariate Processes Enzyme Kinetics SIR Epidemic Process Competition Process Predator-Prey Process Diffusion Processes and Stochastic Differential Equations Introduction Definitions and Notation Random Walk and Brownian Motion Diffusion Process Kolmogorov Differential Equations Wiener Process Ito Stochastic Integral Ito Stochastic Differential Equation (SDE) First Passage Time Numerical Methods for SDEs An Example: Drug Kinetics Biological Applications of Stochastic Differential Equations Introduction Multivariate Processes Derivation of Ito SDEs Scalar Ito SDEs for Populations Enzyme Kinetics SIR Epidemic Process Competition Process Predator-Prey Process Population Genetics Process Appendix: Hints and Solutions to Selected Exercises Index Exercises and References appear at the end of each chapter.

Stiffness in stochastic chemically reacting systems: The implicit tau-leaping method

Abstract: We show how stiffness manifests itself in the simulation of chemical reactions at both the continuous-deterministic level and the discrete-stochastic level. Existing discrete stochastic simulation methods, such as the stochastic simulation algorithm and the (explicit) tau-leaping method, are both exceedingly slow for such systems. We propose an implicit tau-leaping method that can take much larger time steps for many of these problems.

Fast Evaluation of Fluctuations in Biochemical Networks With the Linear Noise Approximation

Abstract: Biochemical networks in single cells can display large fluctuations in molecule numbers, making mesoscopic approaches necessary for correct system descriptions. We present a general method that allows rapid characterization of the stochastic properties of intracellular networks. The starting point is a macroscopic description that identifies the system's elementary reactions in terms of rate laws and stoichiometries. From this formulation follows directly the stationary solution of the linear noise approximation (LNA) of the Master equation for all the components in the network. The method complements bifurcation studies of the system's parameter dependence by providing estimates of sizes, correlations, and time scales of stochastic fluctuations. We describe how the LNA can give precise system descriptions also near macroscopic instabilities by suitable variable changes and elimination of fast variables.

Improved leap-size selection for accelerated stochastic simulation

Abstract: In numerically simulating the time evolution of a well-stirred chemically reacting system, the recently introduced “tau-leaping” procedure attempts to accelerate the exact stochastic simulation algorithm by using a special Poisson approximation to leap over sequences of noncritical reaction events. Presented here is an improved procedure for determining the maximum leap size for a specified degree of accuracy.

Stochastic Networks and Queues

Abstract: 1 Point Processes- 2 GI/GI/1 FIFO Queues and Random Walks- 3 Limit Theorems for GI/GI/1 Queues- 4 Stochastic Networks and Reversibility- 5 The M/M/1 Queue- 6 The M/M/? Queue- 7 Queues with Poisson Arrivals- 8 Recurrence and Transience of Markov Chains- 9 Resealed Markov Processes and Fluid Limits- 10 Ergodic Theory: Basic Results- 11 Stationary Point Processes- 12 The G/G/1 FIFO Queue- A Martingales- A1 Discrete Time Parameter Martingales- A2 Continuous Time Martingales- A3 The Stochastic Integral for a Poisson Process- A4 Stochastic Differential Equations with Jumps- B Markovian Jump Processes- B2 Global Balance Equations- B3 The Associated Martingales- C Convergence in Distribution- C1 Total Variation Norm on Probability Distributions- C2 Convergence of Stochastic Processes- D An Introduction to Skorohod Problems- D1 Dimension 1- D2 Multi-Dimensional Skorohod Problems- References- Research Papers

Deterministic Approximation of Stochastic Evolution in Games

Abstract: This paper provides deterministic approximation results for stochastic processes that arise when finite populations recurrently play finite games. The processes are Markov chains, and the approximation is defined in continuous time as a system of ordinary differential equations of the type studied in evolutionary game theory. We establish precise connections between the long-run behavior of the discrete stochastic process, for large populations, and its deterministic flow approximation. In particular, we provide probabilistic bounds on exit times from and visitation rates to neighborhoods of attractors to the deterministic flow. We sharpen these results in the special case of ergodic processes.

Stochastic Vehicle Routing with Random Travel Times

Abstract: We consider stochastic vehicle routing problems on a network with random travel and service times. A fleet of one or more vehicles is available to be routed through the network to service each node. Two versions of the model are developed based on alternative objective functions. We provide bounds on optimal objective function values and conditions under which reductions to simpler models can be made. Our solution method embeds a branch-and-cut scheme within a Monte Carlo sampling-based procedure.

Universal discrete denoising: known channel

Abstract: A discrete denoising algorithm estimates the input sequence to a discrete memoryless channel (DMC) based on the observation of the entire output sequence. For the case in which the DMC is known and the quality of the reconstruction is evaluated with a given single-letter fidelity criterion, we propose a discrete denoising algorithm that does not assume knowledge of statistical properties of the input sequence. Yet, the algorithm is universal in the sense of asymptotically performing as well as the optimum denoiser that knows the input sequence distribution, which is only assumed to be stationary. Moreover, the algorithm is universal also in a semi-stochastic setting, in which the input is an individual sequence, and the randomness is due solely to the channel noise. The proposed denoising algorithm is practical, requiring a linear number of register-level operations and sublinear working storage size relative to the input data length.

Fully nonparametric estimation of scalar diffusion models

Abstract: We propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process. We show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter. The asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens. We prove almost sure consistency and weak convergence to mixtures of normal laws, where the mixing variates depend on the chronological local time of the underlying diffusion process, that is the random time spent by the process in the vicinity of a generic spatial point. The estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes.

Reliability-Based Design With the Mixture of Random and Interval Variables

Abstract: In Reliability-Based Design (RBD), uncertainties usually imply for randomness. Nondeterministic variables are assumed to follow certain probability distributions. However, in real engineering applications, some of distributions may not be precisely known or uncertainties associated with some uncertain variables are not from randomness. These nondeterministic variables are only known within intervals. In this paper, a method of RBD with the mixture of random variables with distributions and uncertain variables with intervals is proposed. The reliability is considered under the condition of the worst combination of interval variables. In comparison with traditional RBD, the computational demand of RBD with the mixture of random and interval variables increases dramatically. To alleviate the computational burden, a sequential single-loop procedure is developed to replace the computationally expensive double-loop procedure when the worst case scenario is applied directly. With the proposed method, the RBD is conducted within a series of cycles of deterministic optimization and reliability analysis. The optimization model in each cycle is built based on the Most Probable Point (MPP) and the worst case combination obtained in the reliability analysis in previous cycle. Since the optimization is decoupled from the reliability analysis, the computational amount for MPP search is decreased to the minimum extent. The proposed method is demonstrated with a structural design example.Copyright © 2003 by ASME

Stochastic evolution equations with fractional Brownian motion

Abstract: In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed; separate proofs are required for the cases of Hurst parameter above and below 1/2. The particular case of the Laplacian on the circle is discussed in detail.

Insensitive Bandwidth Sharing in Data Networks

Abstract: We represent a data network as a set of links shared by a dynamic number of competing flows. These flows are generated within sessions and correspond to the transfer of a random volume of data on a pre-defined network route. The evolution of the stochastic process describing the number of flows on all routes, which determines the performance of the data transfers, depends on how link capacity is allocated between competing flows. We use some key properties of Whittle queueing networks to characterize the class of allocations which are insensitive in the sense that the stationary distribution of this stochastic process does not depend on any traffic characteristics (session structure, data volume distribution) except the traffic intensity on each route. We show in particular that this insensitivity property does not hold in general for well-known allocations such as max-min fairness or proportional fairness. These results are ilustrated by several examples on a number of network topologies.

Quantitative model of price diffusion and market friction based on trading as a mechanistic random process.

Abstract: We model trading and price formation in a market under the assumption that order arrival and cancellations are Poisson random processes. This model makes testable predictions for the most basic properties of markets, such as the diffusion rate of prices (which is the standard measure of financial risk) and the spread and price impact functions (which are the main determinants of transaction cost). Guided by dimensional analysis, simulation, and mean-field theory, we find scaling relations in terms of order flow rates. We show that even under completely random order flow the need to store supply and demand to facilitate trading induces anomalous diffusion and temporal structure in prices.

The potential role of spectral properties in detecting thresholds in the Earth system: application to the thermohaline circulation

Abstract: A simple two-box model of the hemispheric thermohaline circulation (THC) is considered. The model parameterizes fluctuations in the freshwater forcing by a stochastic process. The dependence of the power spectral density and the lifetime of quasistationary states of the THC on the distance to the bifurcation point, where the THC collapses, is calculated analytically. It is shown that power spectral properties change as the system is moved closer to the bifurcation point. These changes allow an estimate of the distance to the bifurcation point.

A Stochastic Collocation Algorithm for Uncertainty Analysis

Abstract: This report describes a stochastic collocation method to adequately handle a physically intrinsic uncertainty in the variables of a numerical simulation. For instance, while the standard Galerkin approach to Polynomial Chaos requires multi-dimensional summations over the stochastic basis functions, the stochastic collocation method enables to collapse those summations to a one-dimensional summation only. This report furnishes the essential algorithmic details of the new stochastic collocation method and provides as a numerical example the solution of the Riemann problem with the stochastic collocation method used for the discretization of the stochastic parameters.

Capacity of wireless optical communications

Abstract: We consider the ergodic capacity and capacity-versus-outage probability of direct-detection optical communication through a turbulent atmosphere using multiple transmit and receive apertures. We assume shot-noise-limited operation in which detector outputs are doubly stochastic Poisson processes whose rates are proportional to the sum of the transmitted powers, scaled by lognormal random fades, plus a background noise. In the high and low signal-to-background ratio regimes, we show that the ergodic capacity of this fading channel equals or exceeds that for a channel with deterministic path gains. Furthermore, knowledge of these path gains is not necessary to achieve capacity when the signal-to-background ratio is high. In the low signal-to-background ratio regime, path-gain knowledge provides minimal capacity improvement when using a moderate number of transmit apertures. We also develop expressions for the capacity-versus-outage probability in the high and low signal-to-background ratio regimes, by means of a moment-matching approximation to the distribution for the sum of lognormal random variables. Monte Carlo simulations show that these capacity-versus-outage approximations are quite accurate for moderate numbers of apertures.

Kalman filtering for adaptive nonuniformity correction in infrared focal-plane arrays.

Abstract: A novel statistical approach is undertaken for the adaptive estimation of the gain and bias nonuniformity in infrared focal-plane array sensors from scene data. The gain and the bias of each detector are regarded as random state variables modeled by a discrete-time Gauss–Markov process. The proposed Gauss–Markov framework provides a mechanism for capturing the slow and random drift in the fixed-pattern noise as the operational conditions of the sensor vary in time. With a temporal stochastic model for each detector’s gain and bias at hand, a Kalman filter is derived that uses scene data, comprising the detector’s readout values sampled over a short period of time, to optimally update the detector’s gain and bias estimates as these parameters drift. The proposed technique relies on a certain spatiotemporal diversity condition in the data, which is satisfied when all detectors see approximately the same range of temperatures within the periods between successive estimation epochs. The performance of the proposed technique is thoroughly studied, and its utility in mitigating fixed-pattern noise is demonstrated with both real infrared and simulated imagery.

Multidimensional Residual Analysis of Point Process Models for Earthquake Occurrences

Abstract: Residual analysis methods for examining the fit of multidimensional point process models are applied to point process models for the space–time–magnitude distribution of earthquake occurrences, using, in particular, the multidimensional version of Ogata's epidemic-type aftershock sequence (ETAS) model and a 30-year catalog of 580 earthquakes occurring in Bear Valley, California. One method involves rescaled residuals, obtained by transforming points along one coordinate to form a homogeneous Poisson process inside a random, irregular boundary. Another method involves thinning the point process according to the conditional intensity to form a homogeneous Poisson process on the original, untransformed space. The thinned residuals suggest that the fit of the model may be significantly improved by using an anisotropic spatial distance function in the estimation of the spatially varying background rate. Using rescaled residuals, it is shown that the temporal–magnitude distribution of aftershock activity is not...

Continuous-time random-walk model for financial distributions.

Abstract: We apply the formalism of the continuous-time random walk to the study of financial data. The entire distribution of prices can be obtained once two auxiliary densities are known. These are the probability densities for the pausing time between successive jumps and the corresponding probability density for the magnitude of a jump. We have applied the formalism to data on the U.S. dollar-deutsche mark future exchange, finding good agreement between theory and the observed data.

Fractional reaction-diffusion equation

Abstract: A fractional reaction-diffusion equation is derived from a continuous time random walk model when the transport is dispersive. The exit from the encounter distance, which is described by the algebraic waiting time distribution of jump motion, interferes with the reaction at the encounter distance. Therefore, the reaction term has a memory effect. The derived equation is applied to the geminate recombination problem. The recombination is shown to depend on the intrinsic reaction rate, in contrast with the results of Sung et al. [J. Chem. Phys. 116, 2338 (2002)], which were obtained from the fractional reaction-diffusion equation where the diffusion term has a memory effect but the reaction term does not. The reactivity dependence of the recombination probability is confirmed by numerical simulations.

Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term

Abstract: We study existence and uniqueness of a mild solution in the space of continuous functions and existence of an invariant measure for a class of reaction-diffusion systems on bounded domains of ℝd, perturbed by a multiplicative noise. The reaction term is assumed to have polynomial growth and to be locally Lipschitz-continuous and monotone. The noise is white in space and time if d=1 and coloured in space if d>1; in any case the covariance operator is never assumed to be Hilbert-Schmidt. The multiplication term in front of the noise is assumed to be Lipschitz-continuous and no restrictions are given either on its linear growth or on its degenaracy. Our results apply, in particular, to systems of stochastic Ginzburg-Landau equations with multiplicative noise.

Self-similar anomalous diffusion and Levy-stable laws

Abstract: Stochastic principles for constructing the process of anomalous diffusion are considered, and corresponding models of random processes are reviewed. The self-similarity and the independent-increments principles are used to extend the notion of diffusion process to the class of Levy-stable processes. Replacing the independent-increments principle with the renewal principle allows us to take the next step in generalizing the notion of diffusion, which results in fractional-order partial space–time differential equations of diffusion. Fundamental solutions to these equations are represented in terms of stable laws, and their relationship to the fractality and memory of the medium is discussed. A new class of distributions, called fractional stable distributions, is introduced.