Showing papers on "Stochastic process published in 1990"

Stochastic Finite Elements: A Spectral Approach

Abstract: Representation of stochastic processes stochastic finite element method - response representation stochastic finite element method - response statistics numerical examples.

Probability: Theory and Examples

Abstract: This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.

Random vibration and statistical linearization

Abstract: Interest in the study of random vibration problems using the concepts of stochastic process theory has grown rapidly due to the need to design structures and machinery which can operate reliably when subjected to random loads, for example winds and earthquakes. This is the first comprehensive account of statistical linearization - powerful and versatile methods with related techniques allowing the solution of a very wide variety of practical non-linear random vibration problems. The principal value of these methods is that unlike other analytical methods, they are readily generalized to deal with complex mechanical and structural systems and complex types of excitation such as earthquakes.

Multiscaling properties of spatial rainfall and river flow distributions

Abstract: Two common properties of empirical moments shared by spatial rainfall, river flows, and turbulent velocities are identified: namely, the log-log linearity of moments with spatial scale and the concavity of corresponding slopes with respect to the order of the moments. A general class of continuous multiplicative processes is constructed to provide a theoretical framework for these observations. Specifically, the class of log-Levy-stable processes, which includes the lognormal as a special case, is analyzed. This analysis builds on some mathematical results for simple scaling processes. The general class of multiplicative processes is shown to be characterized by an invariance property of their probability distributions with respect to rescaling by a positive random function of the scale parameter. It is referred to as (strict sense) multiscaling. This theory provides a foundation for studying spatial variability in a variety of hydrologic processes across a broad range of scales.

Simulation of random fields via local average subdivision

Abstract: A fast and accurate method of generating realizations of a homogeneous Gaussian scalar random process in one, two, or three dimensions is presented. The resulting discrete process represents local averages of a homogeneous random function defined by its mean and covariance function, the averaging being performed over incremental domains formed by different levels of discretization of the field. The approach is motivated first by the need to represent engineering properties as local averages (since many properties are not well defined at a point and show significant scale effects), and second to be able to condition the realization easily to incorporate known data or change resolution within sub‐regions. The ability to condition the realization or increase the resolution in certain regions is an important contribution to finite element modeling of random phenomena. The Ornstein‐Uhlenbeck and fractional Gaussian noise processes are used as illustrations.

Likelihood ratio gradient estimation for stochastic systems

Abstract: Consider a computer system having a CPU that feeds jobs to two input/output (I/O) devices having different speeds. Let t be the fraction of jobs routed to the first I/O device, so that 1 - t is the fraction routed to the second. Suppose that a = a(t) is the steady-sate amount of time that a job spends in the system. Given that t is a decision variable, a designer might wish to minimize a(t) over t. Since a(·) is typically difficult to evaluate analytically, Monte Carlo optimization is an attractive methodology. By analogy with deterministic mathematical programming, efficient Monte Carlo gradient estimation is an important ingredient of simulation-based optimization algorithms. As a consequence, gradient estimation has recently attracted considerable attention in the simulation community. It is our goal, in this article, to describe one efficient method for estimating gradients in the Monte Carlo setting, namely the likelihood ratio method (also known as the efficient score method). This technique has been previously described (in less general settings than those developed in this article) in [6, 16, 18, 21]. An alternative gradient estimation procedure is infinitesimal perturbation analysis; see [11, 12] for an introduction. While it is typically more difficult to apply to a given application than the likelihood ratio technique of interest here, it often turns out to be statistically more accurate.In this article, we first describe two important problems which motivate our study of efficient gradient estimation algorithms. Next, we will present the likelihood ratio gradient estimator in a general setting in which the essential idea is most transparent. The section that follows then specializes the estimator to discrete-time stochastic processes. We derive likelihood-ratio-gradient estimators for both time-homogeneous and non-time homogeneous discrete-time Markov chains. Later, we discuss likelihood ratio gradient estimation in continuous time. As examples of our analysis, we present the gradient estimators for time-homogeneous continuous-time Markov chains; non-time homogeneous continuous-time Markov chains; semi-Markov processes; and generalized semi-Markov processes. (The analysis throughout these sections assumes the performance measure that defines a(t) corresponds to a terminating simulation.) Finally, we conclude the article with a brief discussion of the basic issues that arise in extending the likelihood ratio gradient estimator to steady-state performance measures.

Polynomial Chaos in Stochastic Finite Elements

Abstract: A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of a convergent orthogonal expansion of the process. The solution process is viewed as an element in the Hilbert space of random functions, in which a sequence of projection operators is identified as the polynomial chaos of consecutive orders. Thus, the solution process is represented by its projections onto the spaces spanned by these polynomials

Probability Concepts in Electric Power Systems

Abstract: Role of probability models in power system engineering concepts and theorems of probability random variables functions of random variables stochastic processes decision analysis reliability probabilistic structural design and analysis of transmission systems preventive maintenance, inspection and replacement probabilistic load flow probabilistic short circuit analysis probabilistic power system stability Monte Carlo simulation elements of acceptance sampling.

Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems

Abstract: 1 Weak Convergence- 0 Outline of the Chapter- 1 Basic Properties and Definitions- 2 Examples- 3 The Skorohod Representation- 4 The Function Space Ck [0, T]- 5 The Function Space Dk [0, T]- 6 Measure Valued Random Variables and Processes- 2 Stochastic Processes: Background- 0 Outline of the Chapter- 1 Martingales- 2 Stochastic Integrals and Ito's Lemma- 3 Stochastic Differential Equations: Bounds- 4 Controlled Stochastic Differential Equations: Existence of Solutions- 5 Representing a Martingale as a Stochastic Integral- 6 The Martingale Problem- 7 Jump-Diffusion Processes- 8 Jump-Diffusion Processes: The Martingale Problem Formulation- 3 Controlled Stochastic Differential Equations- 0 Outline of the Chapter- 1 Controlled SDE's: Introduction- 2 Relaxed Controls: Deterministic Case- 3 Stochastic Relaxed Controls- 4 The Martingale Problem Revisited- 5 Approximations, Weak Convergence and Optimality- 4 Controlled Singularly Perturbed Systems- 0 Outline of the Chapter- 1 Problem Formulation: Finite Time Interval- 2 Approximation of the Optimal Controls and Value Functions- 3 Discounted Cost and Optimal Stopping Problems- 4 Average Cost Per Unit Time- 5 Jump-Diffusion Processes- 6 Other Approaches- 5 Functional Occupation Measures and Average Cost Per Unit Time Problems- 0 Outline of the Chapter- 1 Measure Valued Random Variables- 2 Limits of Functional Occupation Measures for Diffusions- 3 The Control Problem- 4 Singularly Perturbed Control Problems- 5 Control of the Fast System- 6 Reflected Diffusions- 7 Discounted Cost Problem- 6 The Nonlinear Filtering Problem- 0 Outline of the Chapter- 1 A Representation of the Nonlinear Filter- 2 The Filtering Problem for the Singularly Perturbed System- 3 The Almost Optimality of the Averaged Filter- 4 A Counterexample to the Averaged Filter- 5 The Near Optimality of the Averaged Filter- 6 A Repair and Maintainance Example- 7 Robustness of the Averaged Filters- 8 A Robust Computational Approximation to the Averaged Filter- 9 The Averaged Filter on the Infinite Time Interval- 7 Weak Convergence: The Perturbed Test Function Method- 0 Outline of the Chapter- 1 An Example- 2 The Perturbed Test Function Method: Introduction- 3 The Perturbed Test Function Method: Tightness and Weak Convergence- 4 Characterization of the Limits- 8 Singularly Perturbed Wide-Band Noise Driven Systems- 0 Outline of the Chapter- 1 The System and Noise Model- 2 Weak Convergence of the Fast System- 3 Convergence to the Averaged System- 4 The Optimality Theorem- 5 The Average Cost Per Unit Time Problem- 9 Stability Theory- 0 Outline of the Chapter- 1 Stability Theory for Jump-Diffusion Processes of Ito Type- 2 Singularly Perturbed Deterministic Systems: Bounds on Paths- 3 Singularly Perturbed Ito Processes: Tightness- 4 The Linear Case- 5 Wide Bandwidth Noise- 6 Singularly Perturbed Wide Bandwidth Noise Driven Systems- 10 Parametric Singularities- 0 Outline of the Chapter- 1 Singularly Perturbed Ito Processes: Weak Convergence- 2 Stability- References- List of Symbols

Domain-wall dynamics and Barkhausen effect in metallic ferromagnetic materials. I, Theory

Abstract: The Barkhausen effect (BE) in metallic ferromagnetic systems is theoretically investigated by a Langevin description of the stochastic motion of a domain wall in a randomly perturbed medium. BE statistical properties are calculated from approximate analytical solutions of the Fokker–Planck equation associated with the Langevin model, and from computer simulations of domain‐wall motion. It is predicted that the amplitude probability distribution P0(Φ) of the B flux rate Φ should obey the equation P0(Φ)∝Φc−1 exp(−cΦ/〈Φ〉), with c>0. This result implies scaling properties in the intermittent behavior of BE at low magnetization rates, which are described in terms of a fractal structure of fractal dimension D<1. Analytical expressions for the B power spectrum are also derived. Finally, the extension of the theory to the case where many domain walls participate in the magnetization process is discussed.

A Monte-Carlo algorithm for path planning with many degrees of freedom

Abstract: A stochastic technique is described for planning collision-free paths of robots with many degrees of freedom (DOFs). The algorithm incrementally builds a graph connecting the local minima of a potential function defined in the robot's configuration space and concurrently searches the graph until a goal configuration is attained. A local minimum is connected to another one by executing a random motion that escapes the well of the first minimum, succeeded by a gradient motion that follows the negated gradient of the potential function. All the motions are executed in a grid shown through the robot's configuration space. The random motions are implemented as random walks which are known to converge toward Brownian motions when the steps of the walks tend toward zero. The local minima graph is searched using a depth-first strategy with random backtracking. In the technique, the planner does not explicitly represent the local-minima graph. The path-planning algorithm has been fully implemented and has run successfully on a variety of problems involving robots with many degrees of freedom. >

A stochastic model for the motion of particle pairs in isotropic high-reynolds-number turbulence, and its application to the problem of concentration variance

Abstract: A new stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence is proposed. The model is three-dimensional and its formulation takes account of recent improvements in the understanding of one-particle models. In particular the model is designed so that if the particle pairs are initially well mixed in the fluid, they will remain so. In contrast to previous models, the new model leads to a prediction for the particle separation probability density function which is in qualitative agreement with inertial subrange theory. The values of concentration variance from the model show encouraging agreement with experimental data. The model results suggest that, at large times, the intensity of concentration fluctuations (i.e. standard deviation of concentration divided by mean concentration) tends to zero in stationary conditions and to a constant in decaying turbulence.

Stochastic backscatter in a subgrid-scale model: Plane shear mixing layer

Abstract: The traditional Smagorinsky subgrid‐scale viscosity (CSλ)2S has been supplemented by the addition of stochastic backscatter. The random acceleration is derived from a vector potential Cb‖S δt‖3/2(λ/δt)2g. Here S is the local strain rate, λ is the grid resolution length scale, δt is the time step, and g is a unit random Gaussian. It is found that values CS=0.2 for the Smagorinsky constant and Cb=0.4 for the backscatter constant give a robust calculation of the two‐dimensional shear mixing layer with the observed growth rate and with realistic emergence of random coherent eddy structures.

A statistical approach to learning and generalization in layered neural networks

Abstract: A general statistical description of the problem of learning from examples is presented. Learning in layered networks is posed as a search in the network parameter space for a network that minimizes an additive error function of a statistically independent examples. By imposing the equivalence of the minimum error and the maximum likelihood criteria for training the network, the Gibbs distribution on the ensemble of networks with a fixed architecture is derived. The probability of correct prediction of a novel example can be expressed using the ensemble, serving as a measure to the network's generalization ability. The entropy of the prediction distribution is shown to be a consistent measure of the network's performance. The proposed formalism is applied to the problems of selecting an optimal architecture and the prediction of learning curves. >

Maximum-likelihood narrow-band direction finding and the EM algorithm

Abstract: Generalized EM (expectation-maximization) algorithms have been derived for the maximum-likelihood estimation of the direction-of-arrival of multiple narrowband signals in noise. Both deterministic and stochastic signal models are considered. The algorithm for the deterministic model yields estimates of the signal amplitudes, while that for the stochastic model yields estimates of the powers of the signal. Both algorithms have the properties that their limit points are stable and satisfy the necessary maximizer conditions for maximum-likelihood estimators. It is shown via simulation that the maximum-likelihood method allows for the resolution of the directions-of-arrival of signals at angular separation and noise levels for which other high-resolution methods will not work. Algorithm convergence does depend on initial conditions; however, convergence to a global maximum has been observed in simulation when the initial estimates are within a significant fraction if one beamwidth (componentwise) of this maximum. Simulations also show that the deterministic model has a significant impact on the angle estimator performance. >

Second order discretization schemes of stochastic differential systems for the computation of the invariant law

Abstract: We Discretize in Time With Step-Size h a Stochastic Differential Equation Whose Solution has a Unique Invariant Probability Measure is the Solution of the Discretized System, we Give an Estimate of in Terms of h for Several Discretization Methods. In Particular, Methods Which are of Second Order for the Approximation of in Finite Time are Shown to be Generically of Second Order for the Ergodic Criterion(1).

The velocity‐dissipation probability density function model for turbulent flows

Abstract: In probability density function (pdf) methods, statistics of inhomogeneous turbulent flow fields are calculated by solving a modeled transport equation for a one‐point joint probability density function. The method based on the joint pdf of velocity and fluid compositions is particularly successful since the most important processes—convection and reaction—do not have to be modeled. However, this joint pdf contains no length‐scale or time‐scale information that can be used in the modeling of other processes. This deficiency can be remedied by considering the joint pdf of velocity, dissipation, and composition. In this paper, by reference to the known properties of homogeneous turbulence, a modeled equation for the joint pdf of velocity and dissipation is developed. This is achieved by constructing stochastic models for the velocity and dissipation following a fluid particle.

Probabilistic load flow by a multilinear simulation algorithm

Abstract: Load flow analysis is undoubtedly the most useful method of designing and operating power systems. The input data necessary for these studies are best described by random variables, considering the probabilistic nature of loads, generation and networks. The effects of uncertainties on the steady-state behaviour of power systems can be evaluated by a stochastic or probabilistic load flow (PLF) analysis. The authors present a new method for obtaining the PLF solution, by combining Monte Carlo simulation techniques and linearised power flow equations for different system load levels. The performance of the proposed algorithm is illustrated through the IEEE 14-busbar test system, and also through its application in part of the Brazilian network. >

Stochastic discrimination

Abstract: A general method is introduced for separating points in multidimensional spaces through the use of stochastic processes. This technique is called stochastic discrimination.

Stationary Stochastic Models

Abstract: Recursive stochastic equations - weak and strong solutions - ergodic weak solutions - construction of stationary weak and strong solutions - Wald's identity for dependent random variables - model continuity in the presence of renewing epochs - method of metric modification - continuity of weak solutions the case of stationary Markov chains stationary sequences and stationary marked point processes - ergodic marked point processes continuous time models - stochastic processes with an embedded point process - semi-Markov and semi-regenerative processes - continuous time state processes - recursive stochastic equations in continuous time arrival-stationary queuing processes: existence and uniqueness - notations - the system - single server queue - the system with FCFS queueing discipline - the system with cyclic queueing discipline - the single server queue with warming-up - the many server loss system with repeated call attempts - open networks of loss systems relationships between arrival, time and departure stationary queueing processes - Poisson arrivals see time averages (PASTA) - the number of customers - the busy cycle - Takaes' and Pollaczek-Khinchin formulae batch-arrival stationary queuing processes - the system with geometrically distributed batch size - constant batch size - single server queue with batch arrivals - feedback queues - comparing batch delays and customer delays continuity of queueing models further models - the stochastic equation with stationary coefficients - stability of robust filter cleaners - non-contractivity of the filters - a further method for constructing solutions of recursive stochastic equations - the filter with fixed scale - variable scale.

Transition to chaos for random dynamical systems

Abstract: We study the transition to chaos for random dynamical systems. Near the transition, on the chaotic side, the long-time particle distribution (which is fractal) that evolves from an initial smooth distribution exhibits an extreme form of temporally intermittent bursting whose scaling we investigate. As a physical example, the problem of the distribution of particles floating on the surface of a fluid whose flow velocity has a complicated time dependence is considered.

A Stochastic Approach to the Problem of Upscaling of Conductivity in Disordered Media: Theory and Unconditional Numerical Simulations

Abstract: A general problem in groundwater modeling is how to assign conductivity values to blocks of a numerical model. In this study, a method is proposed to upscale the conductivity measurements observed at a given scale to block conductivity values for arbitrary block size. Assuming a multilognormal distribution for the point conductivities, the block conductivity appears as a random function whose distribution, conditional to point measurements, can be derived. This conditional distribution allows filling the flow simulator blocks with equiprobable realizations of block conductivities locally conditioned to actual data. These realizations could also be conditioned to conductivity values measured at any scale if the size of the measurement support is known. The analytical results are compared to results obtained through a Monte Carlo analysis, and a good agreement was found even for variances of ln (T) larger than 1.

On Arrivals That See Time Averages

Abstract: We investigate when Arrivals See Time Averages ASTA in a stochastic model; i.e., when the stationary distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points arrivals of an associated point process, coincides with the stationary distribution of the observed process. We also characterize the relation between the two distributions when ASTA does not hold. We introduce a Lack of Bias Assumption LBA which stipulates that, at any time, the conditional intensity of the point process, given the present state of the observed process, be independent of the state of the observed process. We show that LBA, without the Poisson assumption, is necessary and sufficient for ASTA in a stationary process framework. Consequently, LBA covers known examples of non-Poisson ASTA, such as certain flows in open Jackson queueing networks, as well as the familiar Poisson case PASTA. We also establish results to cover the case in which the process is observed just after the points, e.g., when departures see time averages. Finally, we obtain a new proof of the Arrival Theorem for product-form queueing networks.

Multifractal properties of snapshot attractors of random maps.

Abstract: We consider qualitative and quantitative properties of ``snapshot attractors'' of random maps. By a random map we mean that the parameters that occur in the map vary randomly from iteration to iteration according to some probability distribution. By a ``snapshot attractor'' we mean the measure resulting from many iterations of a cloud of initial conditions viewed at a single instant (i.e., iteration). In this paper we investigate the multifractal properties of these snapshot attractors. In particular, we use the Lyapunov number partition function method to calculate the spectra of generalized dimensions and of scaling indices for these attractors; special attention is devoted to the numerical implementation of the method and the evaluation of statistical errors due to the finite number of sample orbits. This work was motivated by problems in the convection of particles by chaotic fluid flows.

Stochastic analysis of a three-phase fluidized bed; Fractal approach

Abstract: This paper reports on three-phase fluidized beds that have played important roles in various areas of chemical and biochemical processing. The characteristics of such beds are highly stochastic due to the influence of a variety of phenomena, including the jetting and bubbling of the fluidizing medium and the motion of the fluidized particles. A novel approach based on the concept of fractals, has been adopted to analyze these complicated and stochastic characteristics. Specifically, pressure fluctuations in a gas-liquid-solid fluidized bed under different batch operating conditions have been analyzed in terms of Hurst's rescaled range (R/S) analysis, thus yielding the estimates for the so-called Hurst exponent, H. The time series of the pressure fluctuations has a local fractal dimension of d{sub FL} = 2 {minus} H. An H value of 1/2 signifies that the time series follows Brownian motion; otherwise, it follows fractional Brownian motion (FBM), which has been found to be the case for the three-phase fluidized bed investigated.