Showing papers on "Stochastic process published in 1983"

Random fields, analysis and synthesis

Abstract: Random variation over space and time is one of the few attributes that might safely be predicted as characterizing almost any given complex system. Random fields or "distributed disorder systems" confront astronomers, physicists, geologists, meteorologists, biologists, and other natural scientists. They appear in the artifacts developed by electrical, mechanical, civil, and other engineers. They even underlie the processes of social and economic change. The purpose of this book is to bring together existing and new methodologies of random field theory and indicate how they can be applied to these diverse areas where a "deterministic treatment is inefficient and conventional statistics insufficient." Many new results and methods are included.After outlining the extent and characteristics of the random field approach, the book reviews the classical theory of multidimensional random processes and introduces basic probability concepts and methods in the random field context. It next gives a concise amount of the second-order analysis of homogeneous random fields, in both the space-time domain and the wave number-frequency domain. This is followed by a chapter on spectral moments and related measures of disorder and on level excursions and extremes of Gaussian and related random fields.After developing a new framework of analysis based on local averages of one-, two-, and n-dimensional processes, the book concludes with a chapter discussing ramifications in the important areas of estimation, prediction, and control. The mathematical prerequisite has been held to basic college-level calculus.

Markov Random Field Texture Models

Abstract: We consider a texture to be a stochastic, possibly periodic, two-dimensional image field. A texture model is a mathematical procedure capable of producing and describing a textured image. We explore the use of Markov random fields as texture models. The binomial model, where each point in the texture has a binomial distribution with parameter controlled by its neighbors and ``number of tries'' equal to the number of gray levels, was taken to be the basic model for the analysis. A method of generating samples from the binomial model is given, followed by a theoretical and practical analysis of the method's convergence. Examples show how the parameters of the Markov random field control the strength and direction of the clustering in the image. The power of the binomial model to produce blurry, sharp, line-like, and blob-like textures is demonstrated. Natural texture samples were digitized and their parameters were estimated under the Markov random field model. A hypothesis test was used for an objective assessment of goodness-of-fit under the Markov random field model. Overall, microtextures fit the model well. The estimated parameters of the natural textures were used as input to the generation procedure. The synthetic microtextures closely resembled their real counterparts, while the regular and inhomogeneous textures did not.

Introduction to Random Vibrations

Abstract: This is the first comprehensive text on its subject to appear since the 1960s. It incorporates classical material with the many significant developments in the field and is the only up-to-date introduction currently available."Introduction to Random Vibrations "presents a brief review of probability theory, a concise treatment of random variables and random processes (including normal, Poisson, and Markov processes), and a comprehensive exposition of the theory of random vibrations.It contains a number of noteworthy features. Linear systems theory is introduced with a high degree of generality in order to demonstrate its elegance and range of applicability. The response of discrete and continuous linear systems to random excitations is then developed within this framework. The chapter on the response of nonlinear systems represents a unified view of the topic, incorporating some major recent formulations. The discrete-state approach, which has emerged as a powerful technique, is utilized in the treatment of a number of random process properties, among them level crossings, peaks, envelopes, and first-passage times. The Stieltje integral representation of random processes is introduced in order to simplify the presentation of stationary and nonstationary random processes and response statistics.In addition to the opening review of probability and set theory, appendices review relevant topics in Fourier analysis and ordinary differential equations. Both these reviews and exercises included with the chapters will be useful to students using the book as a course text and to practitioners using it as a reference.This book is third in The MIT Press Series in Structural Mechanics, edited by Max Irvine.

Stochastic finite element analysis of simple beams

Abstract: A method of stochastic finite element analysis is developed for solving a variety of engineering mechanics problems in which physical properties exhibit one-dimensional spatial random variation. The method is illustrated by evaluating the second-order statistics of the deflection of a beam whose rigidity varies randomly along its axis. A key component of the approach is a new treatment of the correlation structure of the random material property in terms of the variance function and its principal parameter, the scale of fluctuation. The methodology permits efficient evaluation of the matrix of covariances between local spatial averages associated with pairs of finite elements. Numerical results are presented for a cantilever beam, with deformation controlled by shear, subjected to a concentrated force at its free end or to a uniformly distributed load.

Large deviations and rare events in the study of stochastic algorithms

Abstract: New asymptotics formulas for the mean exit time from an almost stable domain of a discrete-time Markov process are obtained. An original fast simulation method is also proposed. The mathematical background involves the large deviation theorems and approximations by a diffusion process. We are chiefly concerned with the classical Robbins-Monroe algorithm. The validity of the results are tested on examples from the ALOHA system (a satellite type communication algorithm).

Optimization over Time. Dynamic Programming and Stochastic Control. Volume 1

Abstract: of the models analyzed include the spectral representation of solution vectors, limiting states, the covariance matrix of the elements of the state composition vector, the means and variances of sojourn times, and expanding and contracting systems, rather than methods of statistical estimation. In the discussion of discrete time models, the illustrations are drawn primarily from the study of social and occupational mobility, and for continuous time models, they are drawn from the field of educational and manpower planning. The remaining five chapters treat control theory for Markov models, models for duration and size, models for social systems with fixed class sizes, and simple and general epidemic models for the diffusion of news, rumors, and ideas. Simple epidemic models are birth process models that assume that infection is an irreversible state, so given either a constant individual rate of transmission or a single constant source of transmission, the entire population is eventually infected. General epidemic models allow for the duration of infection to be a random variable. The book ends with a full, up-to-date bibliography, an author index, and a subject index. In summary, Bartholomew gives an excellent introduction to many types of stochastic processes and a broad range of applications for modeling and planning social systems. The applications of stochastic processes to studying social mobility and flows of personnel within organizations receive much more extended treatment here than in other introductory treatments of applied stochastic processes. The "Complements" section at the end of each chapter is a useful overview of recent research investigations in many other areas of application that apply or extend the models presented in the chapter. Since no problems for solution are contained and the exposition is often informal, some teachers may wish to supplement the book with more traditional stochastic process textbooks, one good choice being Karlin and Taylor (1975). There seem to be few typographical errors.

Plasma transport in stochastic magnetic fields. Part 3. Kinetics of test particle diffusion

Abstract: A discussion is given of test particle transport in the presence of specified stochastic magnetic fields, with particular emphasis on the collisional limit. Certain paradoxes and inconsistencies in the literature regarding the form of the scaling laws are resolved by carefully distinguishing a number of physically distinct correlation lengths, and thus identifying several collisional subregimes. The common procedure of averaging the conventional fluid equations over the statistics of a random field is shown to fail in some important cases because of breakdown of the Chapman-Enskog ordering in the presence of a stochastic field component with short autocorrelation length. A modified perturbation theory is introduced which leads to a Kubo-like formula valid in all collisional regimes. The direct-interaction approximation is shown to fail in the interesting limit in which the orbit exponentiation length LK appears explicitly. A higher-order renormalized kinetic theory in which LK appears naturally is discussed and used to rederive more systematically the results of the heuristic scaling arguments.

A stochastic theory of fatigue crack propagation

Abstract: A new mathematical theory is proposed to analyze the propagation of fatigue crack based on the concepts of fracture mechanics and random processes. The time-dependent crack size is approximated by a Markov process. Analytical expressions are obtained for the probability distribution of crack size at any given time and the probability distribution of the random time at which a given crack size is reached, conditional on the knowledge of the initial crack size. Examples are given to illustrate the application of the theory, and the results are compared with available experimental data.

Second-order convergence analysis of stochastic adaptive linear filtering

Abstract: The convergence of an adaptive filtering vector is studied, when it is governed by the mean-square-error gradient algorithm with constant step size. We consider the mean-square deviation between the optimal filter and the actual one during the steady state. This quantity is known to be essentially proportional to the step size of the algorithm. However, previous analyses were either heuristic, or based upon the assumption that successive observations were independent, which is far from being realistic. Actually, in most applications, two successive observation vectors share a large number of components and thus they are strongly correlated. In this work, we deal with the case of correlated observations and prove that the mean-square deviation is actually of the same order (or less) than the step size of the algorithm. This result is proved without any boundedness or barrier assumption for the algorithm, as it has been done previously in the literature to ensure the nondivergence. Our assumptions are reduced to the finite strong-memory assumption and the finite-moments assumption for the observation. They are satisfied in a very wide class of practical applications.

A Course in the Theory of Stochastic Processes

Abstract: (1983). A Course in the Theory of Stochastic Processes. Technometrics: Vol. 25, No. 1, pp. 116-116.

Stochastic transport in disordered systems

Abstract: We develop a theory of stochastic transport in disordered media, starting from a linear master equation with random transition rates. A Green function formalism is employed to reduce the basic equation to a form suitable for the construction of a class of effective medium approximations (EMAs). The lowest order EMA, developed in detail here, corresponds to recent approximations proposed by Odagaki and Lax [Phys. Rev. B 24, 5284 (1981], Summerfield [Solid State Commun. 39, 401 (1981)], and Webman [Phys. Rev. Lett. 47, 1496 (1981)]. It yields an effective transition rate Wm which can be identified as the memory kernel of a generalized master equation, and used to define an associated continuous‐time random walk on a uniform lattice. The long‐time behavior of the mean‐squared displacement arising from an initially localized state can be found from Wm, as can diffusion constants in any case where the long‐time behavior of the system is diffusive. Detailed calculations are presented for seven lattice systems i...

Stochastic Network Equilibrium with Multiple Vehicle Types and Asymmetric, Indefinite Link Cost Jacobians

Abstract: This paper discusses a family of general link cost functions that can be used to model multimodal transportation network equilibrium problems. The family includes as a special case the currently favored family of monotonically increasing functions but does not necessarily have to have a symmetric or semi-definite Jacobian. In this way multimodal networks can be modeled somewhat more realistically. The paper also allows stochastic link costs for some or all the links and modes. It is shown that under mild conditions the equilibrium exists and is unique, but more importantly, that there is a simple, single-equilibrium algorithm that is proven to converge for stochastic networks with the link cost family discussed in the paper. Finally, it is pointed out that many variable demand problems also exhibit unique equilibria, but that there are some which do not. An example which illustrates this fact is given.

A Probabilistic Approach to Power System Stability Analysis

Abstract: Power system stability analysis is usually performed in a deterministic framework in which the time domain response of the power system is studied for certain specific disturbances to determine the adequacy of the system. However, the occurrence of disturbances and their attendant protective switching sequences are random processes and it would be more meaningful to determine the probability of stability for a power system. An approach for such a determination is presented in this paper. The probability of steady state stability is relatively easier to determine because of the linearization of the system equations. The probability of transient stability, on the other hand, is much more difficult to obtain analytically because of the nonlinear transformations required. However, a Monte Carlo simulation method to determine transient stability probability is shown to be feasible.

On the extension of the Kramers theory of chemical relaxation to the case of nonwhite noise

Abstract: The first step of our approach consists of relating the generalized Brownian motion in a double‐well potential to a suitable time‐independent Fokker–Planck operator implying that an arbitrary large number of ‘‘virtual’’ variables be used Then, to simplify the solution of this multidimensional Fokker–Planck equation, we develop a procedure of adiabatic elimination of the fastly relaxing variables As a significant feature of this reduction scheme, we point out that no limitation on the number of the virtual variables is implied The explicit form of the first correction term to the Smoluchowski equation is also shown to depend on whether or not the stochastic force is white Via a comparison with the analytical results of Grote and Hynes’ theory [J Chem Phys 73, 2715 (1980)] it is argued that the ‘‘exact’’ approach and the ‘‘reduction’’ procedure can be regarded as being complementary to one another

Statistics of continuous trajectories in quantum mechanics: Operation-valued stochastic processes

Abstract: A formalism developed in previous papers for the description of continual observations of some quantities in the framework of quantum mechanics is reobtained and generalized, starting from a more axiomatic point of view. The statistics of the observations of continuous state trajectories is treated from the beginning as a generalized stochastic process in the sense of Gel'fand. An effect-valued measure and an operation-valued measure on the σ-algebra generated by the cylinder sets in the space of trajectories are introduced. The properties of the characteristic functional for the “operation-valued stochastic process” are discussed and, through a suitableansatz, a significant class of such processes is explicitly constructed, which contains the examples of the preceding papers as particular cases.

A stochastic-dynamic model for the spatial structure of forecast error statistics

Abstract: The present investigation is concerned with the presentation of a simplified model of the spatial structure of forecast error statistics, a comparison of the model with actual numerical weather prediction results, and the extent to which simplifying assumptions made in the model are justified. A stochastic-dynamic model is derived for the spatial structure of the global atmospheric mass-field forecast error. The model states that the relative potential vorticity of the forecast error is random. The covariance function of the model's solutions is found to be governed by a simple deterministic equation. The agreement between the stochastic model and actual mass-field forecast errors fields for 12-36 h periods validates the assumptions on which the model is derived. Within this period, the difference between the potential voriticity fields of the atmosphere and of the numerical forecasts used in the comparison is well represented by white noise.

Non‐Markovian theory of activated rate processes. I. Formalism

Abstract: The escape of a particle from a potential well is treated using a generalized Langevin equation (GLE) in the low friction limit. The friction is represented by a memory kernel and the random noise is characterized by a finite correlation time. This non‐Markovian stochastic equation is reduced to a Smoluchowski diffusion equation for the action variable of the particle and explicit expressions are obtained for the drift and diffusion terms in this equation in terms of the Fourier coefficients of the deterministic trajectory (associated with the motion without coupling to the heat bath) and of the Fourier transform of the friction kernel. The latter (frequency dependent friction) determines the rate of energy exchange with the heat bath. The resulting energy (or action) diffusion equation is used to determine the rate of achieving the critical (escape) energy. Explicit expressions are obtained for a Morse potential. These results for the escape rate agree with those from stochastic trajectories based on the...

Stochastic analysis of one‐dimensional steady state unsaturated flow: A Comparison of Monte Carlo and Perturbation Methods

Abstract: The stochastic nature of moisture content in a soil profile under steady state, unsaturated infiltration is examined, where the saturated hydraulic conductivity is taken as a stationary stochastic process. Two different techniques are employed in determining the stochastic output. These are Monte Carlo simulations and an analytic derivation from a first-order perturbation solution. In comparing the two techniques, the liability of the perturbation expressions is investigated if the Monte Carlo results are assumed to exactly represent the stochastic nature of the moisture content. In four test examples a relatively good comparison is obtained between the results of the two techniques. The analytic nature of the perturbation solution then readily provides information concerning the general stochastic properties of the moisture content, and in this way, additional Monte Carlo runs are avoided. From this analysis it is shown that the moisture content is a stationary stochastic process at distances far from boundary regions. The distance from the boundary to regions of stationarity is dependent upon soil properties and the boundary conditions imposed.

Generalized Langevin dynamics simulations with arbitrary time‐dependent memory kernels

Abstract: An algorithm is described that allows dynamical simulations to be performed based on generalized Langevin equations with arbitrary, time‐dependent memory kernels. Test simulations show that good results are obtained for kernels with distinctly different forms (e.g., exponential and Gaussian).

Bifurcation and optimal stochastic control

Abstract: : Two questions concerning bifuraction theory and optimal stochastic control are considered. First, in a few examples, we give the interpretation of a bifurcation in terms of optimal stochastic control. Next, we introduce the analogue of the lowest eigenvalue for the nonlinear operator associated with the Hamilton-Jacobi-Bellman equations of Optimal Stochastic Control. (Author)

The Thermodynamic Limit for Long-Range Random Systems

Abstract: Long-range spin systems with random interactions are considered. A simple argument is presented showing that the thermodynamic limit of the free energy exists and depends neither on the specific random configuration nor on the sample shape, provided there is no external field. The argument is valid for both classical and quantum spin systems, and can be applied to (a) spins randomly distributed on a lattice and interacting via dipolar interactions; and (b) spin systems with potentials of the formJ(x 1,x 2)/|x 1 -x 2| αd , where theJ(x 1,x 2) are independent random variables with mean zero,d is the dimension, and α > 1/2. The key to the proof is a (multidimensional) subadditive ergodic theorem. As a corollary we show that, for random ferromagnets, the correlation length is a nonrandom quantity.

Cycloergodic properties of discrete- parameter nonstationary stochastic processes

Abstract: It is shown that a large class of nonstationary discrete parameter stochastic processes possess novel ergodic properties, which are referred to as {\em cycloergodic} properties. Specifically, it is shown that periodic components of time-varying probabilistic parameters can be consistently estimated from time averages on one sample path. The cycloergodic theory developed herein extends and generalizes existing ergodic theory for asymptotically mean stationary and N -stationary (cyclostationary) processes, and is presented in both wide-sense and strict-sense contexts.

Stochastic theory of adiabatic explosion

Abstract: A stochastic description of an exothermic reaction leading to adiabatic explosion is set up. The numerical solution of the master equation reveals the appearance of a long tail and of multiple humps of the probability distribution, which subsist for a certain period of time. During this interval the system displays a markedly chaotic behavior, reflecting the random character of the ignition process. An analytical description of this transient evolution is developed, using a piecewise linear approximation of the transition rates. A comparison with other transient phenomena observed in stochastic theory is carried out.