Showing papers on "Stochastic process published in 1970"

A Stochastic Model for Flood Analysis

Abstract: A stochastic model, based on the recent developments in the theory of extreme values, is presented to describe and analyze excessive streamflows. The model is a particular stochastic process x(t) defined as the maximum term among a random number of random observations in an interval of time [0, t]. Since the number of hydrograph peaks in [0, t] that exceed a certain level x0 and the magnitudes of these peaks are random variables, the foregoing model seems to conform well to the flood phenomenon. The passage time T(x) of the process x(t) relevant to the risk evaluation in the design of hydraulic structures is also considered. The results obtained are applied on the 72-year record of the Susquehanna River at Wilkes-Barre, Pennsylvania. Theoretical and observed results agree reasonably well.

The innovations approach to detection and estimation theory

Abstract: Given a stochastic process, its innovations process will be defined as a white Gaussian noise process obtained from the original process by a causal and causally invertible transformation. The significance of such a representation, when it exists, is that statistical inference problems based on observation of the original process can be replaced by simpler problems based on white noise observations. Seven applications to linear and nonlinear least-squares estimation. Gaussian and non-Gaussian detection problems, solution of Fredholm integral equations, and the calculation of mutual information, will be described. The major new results are summarized in seven theorems. Some powerful mathematical tools will be introduced, but emphasis will be placed on the considerable physical significance of the results.

Extreme value theory for a class of discrete distributions with applications to some stochastic processes

Abstract: Let ξn be the maximum of a set of n independent random variables with common distribution function F whose support consists of all sufficiently large positive integers. Some of the classical asymptotic results of extreme value theory fail to apply to ξn for such F and this paper attempts to find weaker ones which give some description of the behaviour of ξn as n → ∞. These are then applied to the extreme value theory of certain regenerative stochastic processes.

On the entropy of context-free languages

Abstract: The information theoretical concept of the entropy (channel capacity) of context-free languages and its relation to the structure generating function is investigated in the first part of this paper. The achieved results are applied to the family of pseudolinear grammars. In the second part, relations between context-free grammars, infinite labelled digraphs and infinite nonnegative matrices are exhibited. Theorems on the convergence parameter of infinite matrices are proved and applied to the evaluation of the entropy of certain context-free languages. Finally, a stochastic process is associated with any context-free language generated by a deterministic labelled digraph, such that the stochastic process is equivalent to the language in the sense that both have the same entropy.

A martingale analogue of Kolmogorov's law of the iterated logarithm

Abstract: SummaryKolmogorov's law of the iterated logarithm is extended to the martingale case.

Linear and nonlinear filtering

Abstract: This paper provides a review of the theory of filtering for stochastic processes. In particular, the sequential theory of linear filtering is reviewed as well as the theory of nonlinear filtering.

Multireservoir Operation Studies

Abstract: An implicit stochastic process is compared to possible alternative analytical processes. An implicit stochastic process refers to a sequence of steps consisting of streamflow synthesis, deterministic optimization, and multivariate analysis of deterministic optimization results. The comparison is incomplete in that the performance of the multivariate analysis of results is not within the scope of this paper. Some formulations based on the method of Dantzig-Wolfe decomposition are shown to be reasonably accurate representations of a nonlinear multireservoir deterministic optimization problem. One of these methods is shown to consume only moderate amounts of computer time for a problem of distinctly nontrivial size. The research issues yet to be resolved in achieving the overall objective (i.e., a multireservoir operation rule for flow regulation and energy production) are identified.

The vibrating string forced by white noise

Abstract: The equation of the vibrating string forced by white noise is formally solved, using stochastic integrals with respect to a plane Brownian motion, and it is proved that a certain process associated to the energy is a martingale. Then Doob's martingale inequality is used to furnish some probability bounds for the energy.

Random Operator Equations in Mathematical Physics. I

Abstract: For stochastic differential equations arising in physical problems, the objectives, limitations, and restrictive assumptions of the various methods are studied and some promising new methods are derived which eliminate various limitations and allow treatment of a wide class of applications in physics. (Among these, will be an adequate treatment of the propagation of an electromagnetic wave in a random continuum or a random d'Alembertian operator without assumptions of ``small randomness'' and other restrictions.)

Filtering for Linear Distributed Parameter Systems

Abstract: Nonlinear filtering for linear parabolic distributed parameter systems with white noise, considering stochastic boundary value problem

Markov Models for Flow Regulation

Abstract: The problem of defining alternative operating policies for a single multipurpose reservoir was examined through the use of several pairs of discrete stochastic linear- and dynamic-programming models. The net flows into the reservoir were assumed to be serially correlated, their probabilistic sequence defined by first-order Markov chains. Each linear programming model was shown to correspond to a dynamic programming model. The solutions and computational efficiencies of each of the models were compared using a simplified numerical example based on an actual reservoir operating problem. Although the policies obtained from each pair of corresponding models were identical, the time required to solve the dynamic programming models was less than that required for the linear programming models.

Learning Applied to Successive Approximation Algorithms

Abstract: A linear reinforcement learning technique is proposed to provide a memory and thus accelerate the convergence of successive approximation algorithms. The learning scheme is used to update weighting coefficients applied to the components of the correction terms of the algorithm. A direction of the search approaching the direction of a "ridge" will result in a gradient peak-seeking method which accelerates considerably the convergence to a neighborhood of the extremum. In a stochastic approximation algorithm the learning scheme provides the required memory to establish a consistent direction or search insensitive to perturbations introduced by the random variables involved. The accelerated algorithms and the respective proofs of convergence are presented. Illustrative examples demonstrate the validity of the proposed algorithms.

Solutions for some diffusion processes with two barriers

Abstract: Use is often made of the Wiener and Ornstein-Uhlenbeck (O.U.) processes in various applications of stochastic processes to problems of engineering interest. These applications frequently involve the presence of barriers. Although mathematical methods for solving Kolmogorov's forward equation for the above processes have previously been discussed ([1], [2]), many solutions for problems with two barriers do not seem to be available in the literature. Instead, one finds solutions for unrestricted processes or simulation used in place of analytical solutions in various applications ([3], [4], [5]). In this paper, solutions of Kolmogorov's forward equations in the presence of constant absorbing and/or reflecting barriers are obtained by means of separation of variables. This enables one to obtain expressions for the probability density functions for first passage times when absorbing barriers are present. The solution for the O.U. process is used to obtain a result of Breiman's [6] concerning first passage times.

Analysis of a decision-directed receiver with unknown priors

Abstract: A decision-directed receiver (DDR) uses previous outputs (decisions) to estimate unknown parameters and, on the basis of these estimates, modifies the detector structure for subsequent decisions Although the DDR is less complex to instrument than other adaptive schemes, inherent in the decision-directed approach is the possibility of a runaway This occurs when the detector commits a sequence of decision errors resulting in a degradation of parameter estimates, which, in turn, results in a further deterioration of detector performance Because of the dependencies introduced by the learning process, runaway is difficult to analyze In this paper, a DDR with unknown a priori probabilities is considered The priors are estimated by the relative frequency of decisions of that event For binary detection, it is shown that there is a positive probability of a runaway (the estimates converge to 1 or 0), which equals 1 if the signal-to-noise ratio is below a critical value A tight bound on the probability of a runaway is obtained by approximating the learning process by a random walk with independent increments The analysis demonstrates that a runaway is quite improbable even for moderate signal-to-noise ratios The analysis is extended to multiple signals and to the situation where the estimates of the priors are updated continuously through exponential weighting rather than allowed to converge

Random Processes with Evolutionary Power

Abstract: In a recent paper, Hammond demonstrated a use of the evolutionary power spectral density, which was originally introduced by Priestley, for slowly varying random processes appearing in engineering applications. The purpose herein is to show that the form of input-output relationships derived by Hammond under certain assumed asymptotic forms of the responses is also valid without such restrictions, and that it can be derived in a much more straightforward fashion.

Simple stochastic networks: Some problems and procedures

Abstract: : The paper provides an analysis of 'stochastic networks,' that is networks whose underlying link times are expressed as random variables. Analytic and Monte Carlo methods for finding the distribution function (or parameters thereof) of the maximal time through such a network are discussed. The first section, which considers analytical procedures, emphasizes networks whose link times are given by probability functions from an exponential family. It is shown that the Markov property of the exponential distribution may be utilized to simplify analytic computations. In the second section Monte Carlo techniques are described which provide useful information on project completion time distributions. It is argued that these easily applied techniques are more economical (fewer computations are required) than naive simulation. Throughout the paper, we suggest a blending of the analytical and simulation procedures into an efficient overall method for studying stochastic networks. (Author)

State Variables and Communication Theory

Abstract: Although state variable concepts are a part of modern control theory, they have not been extensively applied in communication theory. The purpose of this book is to demonstrate how the concepts and methods of state variables can be used advantageously in analyzing a variety of communication theory problems. In contrast to the impulse response and covariance function description of systems and random processes commonly used in the analysis of communication problems, Professor Baggeroer points out that a state variable approach describes these systems and processes in terms of differential equations and their excitation, which is usually a white-noise process. Theoretically, such a description provides a very general characterization on which a large class of systems, possibly time varying and nonlinear, can be modeled. Practically, the state variable approach often provides a more representative physical description of the actual dynamics of the systems involved and, most importantly, can lead to solution techniques that are readily implemented on a computer and that yield specific numerical results.The work focuses on how state variables can be used to solve several of the integral equations that recur in communication theory including, for example, the Kahunen-Loeve theorem, the detection of a known signal in the presence of a colored noise, and the Wiener-Hopf equation. The book is divided into two parts. The first part deals with the development from first principles of the state variable solution techniques for homogeneous and inhomogeneous Fredholm integral solutions. The second part considers two specific applications of the author's integral equation theory: to optimal signal design for colored noise channels, and to linear estimation theory.The main thrust of the material presented in this book is toward finding effective numerical procedures for analyzing complex problems. Professor Baggeroer has combined several different mathematical tools not commonly used together to attack the detection and signal design problems. Numerous examples are presented throughout the book to emphasize the numerical aspects of the author's methods. If the reader is familiar with detection and estimation theory and with deterministic state variable concepts, the ideas, techniques, and results contained in this work will prove highly relevant, if not directly applicable, to a large number of communication theory problems."MIT Research Monograph No. 61"

A stochastic model of longitudinal diffusion in porous media

Abstract: A set of tagged particles carried along by a fluid flow through a porous media is subject to random dispersion. The total distance from the origin, in the direction of the main flow traveled by a tagged particle, is a stochastic process. To apply Lagrangian description of the dispersion the distribution function of the traveled distance for every time t is determined. This distribution is not Gaussian and depends on two parameters which in turn depend on hydraulic conditions. To apply Eulerian description, the distribution function of time a tagged particle to travel a given distance x is derived. This distribution is not Gaussian either, and depends on the same two parameters. The sum of these two distributions is always one. This identity represents the relationship between Lagrangian and Eulerian descriptions. According to experimental results, the theoretical distributions agree reasonably well with observed distributions.

Instantaneous and time-varying spectraߞan introduction

Abstract: Several definitions of 'time-varying', 'short-time' and 'instantaneous' spectra exist. The paper relates these to the time-frequency energy distribution of a signal and to the time-varying power spectrum of a non-stationary random process. The treatment emphasizes physical interpretation, rather than mathematical rigour.

On the distribution of the supremum for stochastic processes

Abstract: © Gauthier-Villars, 1970, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

The capacity of linear channels with additive Gaussian noise

Abstract: The standard method of computing the mutual information between two stochastic processes with finite energy replaces the processes with their Fourier coefficients. This procedure is mathematically justified here for random signals w,(ω) square-integrable in the product space t × ω where t ∊ [O, T] and w is an element of a probability space. A natural notion of the sigma field generated by w, (ω) is presented and it is shown to coincide with the sigma field generated by the random Fourier coefficients of w,(ω) in any complete orthonormal system in L 2 [O, T]. This justifies the use of Fourier coefficients in mutual information computations. Capacity is calculated for finite and infinite-dimensional channels, where the output signal consists of a filter (general Hilbert-Schmidt operator) operating on the input signal with additive Gaussian noise. The finite-dimensional optimal signal is obtained. In the infinite-dimensional case capacity can be approached arbitrarily closely with finite-dimensional inputs. The question of the existence of an infinite-dimensional signal which achieves capacity is considered. There are channels for which no signal achieves capacity. Some results are obtained when the noise coordinates are independent in the eigensystem of the filter.

The Optimal Spacing of Repeated Measurements

Abstract: The treatments in a repeated measurements design can frequently be ordered along some natural continuum (e.g. time), and as a consequence the expected values and covariance structure of the responses can be represented as functions of their positions on the scale. We shall consider rules for spacing the treatments so as to maximize the power of the T2 test for equal treatment effects, as well as general two-sample T2 tests for the equality of mean vectors. These results will be obtained under the assumptions of covariance structures of the kinds associated with Wiener and Markov stochastic processes, and alternative hypothesis mean vectors whose elements are linear or quadratic functions of the scale variable.

Synthesis of stochastic representations of ground motions

Abstract: In this paper, we study a number of stochastic models including stationary, nonstationary, and linear processes for the purpose of simulating earthquake- or explosion-induced ground motions. The important statistical characteristics of each model and their effects on structural systems are investigated in some detail. We obtain expressions for the mean-square response of simple linear mass-spring oscillators to each model. We discuss numerical procedures for time series simulations of these models. The objectives of this paper are (i) to examine and compare the statistical properties and effects on structures of all possible stochastic processes applicable to model ground motions and (ii) to offer engineers a basis for forming their own judgments.

Subordination of infinite-dimensional stationary stochastic processes

Abstract: © Gauthier-Villars, 1970, tous droits réservés. L’accès aux archives de la revue « Annales de l’I. H. P., section B » (http://www.elsevier.com/locate/anihpb) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.