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Showing papers on "Stochastic process published in 1972"


Journal ArticleDOI
TL;DR: In this article, the authors presented an efficient method for digital simulation of general homogeneous processes as a series of cosine functions with weighted amplitudes, almost evenly spaced frequencies, and random phase angles.

1,460 citations


Book
01 Jan 1972
TL;DR: Stochastic Processes: Description and Definition.
Abstract: Preface. Stochastic Processes: Description and Definition. Markov chains. Irreducible Markov Chains with Ergodic States. Branching Processes and Other Special Topics. Statistical Inference for Markov Chains. Applied Markov Chains. Simple Markov Processes. Statistical Inference for Simple Markov Processes. Applied Markov Processes. Renewal Processes. Stationary Processes and Time Series Analysis. Simulation and Markov Chain Monte Carlo. Answers to Selected Exercises. Appendix. Author Index. Subject Index.

448 citations


Journal ArticleDOI
01 Nov 1972
TL;DR: The dynamic behavior of randomly connected analog neuron-like elements that process pulse-frequency modulated signals is investigated from the macroscopic point of view and it is shown that a stable oscillation exists in such a net?in contrast with the fact that no stable oscillations exist in a net of statistically symmetric structure.
Abstract: The dynamic behavior of randomly connected analog neuron-like elements that process pulse-frequency modulated signals is investigated from the macroscopic point of view. By extracting two statistical parameters, the macroscopic state equations are derived in terms of these parameters under some hypotheses on the stochastics of microscopic states. It is shown that a random net of statistically symmetric structure is monostable or bistable, and the stability criteria are explicitly given. Random nets consisting of many different classes of elements are also analyzed. Special attention is paid to nets of randomly connected excitatory and inhibitory elements. It is shown that a stable oscillation exists in such a net?in contrast with the fact that no stable oscillations exist in a net of statistically symmetric structure even if negative as well as positive synaptic weights are permitted at a time. The results are checked by computer-simulated experiments.

336 citations


Book
01 Jan 1972
TL;DR: This text provides an introduction to elementary probability theory and stochastic processes, and shows how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management science, the physical and social sciences, and operations research.
Abstract: Ross's classic bestseller, Introduction to Probability Models, has been used extensively by professionals and as the primary text for a first undergraduate course in applied probability. It provides an introduction to elementary probability theory and stochastic processes, and shows how probability theory can be applied to the study of phenomena in fields such as engineering, computer science, management science, the physical and social sciences, and operations research. With the addition of several new sections relating to actuaries, this text is highly recommended by the Society of Actuaries. A new section (3.7) on COMPOUND RANDOM VARIABLES, that can be used to establish a recursive formula for computing probability mass functions for a variety of common compounding distributions. A new section (4.11) on HIDDDEN MARKOV CHAINS, including the forward and backward approaches for computing the joint probability mass function of the signals, as well as the Viterbi algorithm for determining the most likely sequence of states. Simplified Approach for Analyzing Nonhomogeneous Poisson processes Additional results on queues relating to the (a) conditional distribution of the number found by an M/M/1 arrival who spends a time t in the system; (b) inspection paradox for M/M/1 queues (c) M/G/1 queue with server breakdown Many new examples and exercises.

326 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that many important reliability measures related to stationary random motion require the knowledge of two spectral parameters which depend on the first few moments of the reduced spectral density function.
Abstract: It is shown that many important reliability measures related to stationary random motion require the knowledge of two spectral parameters which depend on the first few moments of the reduced spectral density function The first is a characteristic frequency, the second a unitless measure of the variability in the frequency content, ie, of the bandwidth or the dispersion of the spectral density about its central frequency For general stationary random processes, the spectral parameters are simply related to the mean square values of the process, its envelope, and their respective time derivatives For Gaussian processes, other statistical properties, eg, average barrier crossing rates, clump sizes, and maximum response characteristics, are importantly related to these parameters A derivation is given for the spectral moments of the stationary response of damped linear multidegree-of-freedom systems for which classical modal analysis is possible

296 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of feedback control of a time-invariant uncertain system subject to state constraints over an infinite-time interval and study the behavior of the region of n -step reachability as n tends to infinity.
Abstract: In this paper we consider some aspects of the problem of feedback control of a time-invariant uncertain system subject to state constraints over an infinite-time interval. The central question that we investigate is under what conditions can the state of the uncertain system be forced to stay in a specified region of the state space for all times by using feedback control. At the same time we study the behavior of the region of n -step reachability as n tends to infinity. It is shown that in general this region may exhibit instability as we pass to the limit, and that under a compactness assumption this region converges to a steady state. A special case involving a linear finite-dimensional system is examined in more detail. It is shown that there exist ellipsoidal regions in state space where the state can be confined by making use of a linear time-invariant control law, provided that the system is stabilizable. Such control laws can be calculated efficiently through the solution of a recursive matrix equation of the Riccati type.

289 citations


Book
01 Jan 1972

258 citations


Journal ArticleDOI
TL;DR: In this paper, an efficient and practical method of simulating stationary and non-stationary random envelope processes is presented, in which the stationary envelope processes are simulated by using the fast Fourier transform while the nonstationary envelope process is simulated as the square root of the sum of a series of cosine functions with random phase angles.

195 citations


Journal ArticleDOI
TL;DR: In this article, a computerized Monte Carlo simulation is presented for calculating the statistical properties of the eigenvalues of a spring supported beam-column, where the axial load is equal to 27 % of the fundamental buckling load and the distributions of material and geometric properties are uncorrelated.
Abstract: A computerized Monte Carlo simulation is presented for calculating the statistical properties of the eigenvalues of a spring supported beam-column. The spring supports and axial force are treated as random variables ; the distributions of material and geometric properties are considered to be correlated homogeneous random functions. Each sample distribution is generated using a new method for simulating multivariate homogeneous random processes having a specified cross-spectral density matrix. This method of solution is used to investigate the accuracy of the perturbation method for calculating the variance of the nth vibration and buckling eigenvalues. Numerical results are presented for the case where the axial load is equal to 27 % of the fundamental buckling load and the distributions of material and geometric properties are uncorrelated. The perturbation method is shown to be acceptable for limited ranges of the statistical variations of properties.

191 citations


01 Jan 1972
TL;DR: In this paper, the authors present a set of requirements for a theory of probability measures on a function space, which are based on the notion of limit theorems for stochastic processes.
Abstract: Looking at a stochastic process as a function plucked at random may be interesting in its own right, but there is a statistical reason for doing so: a large class of limit theorems for functions of partial sums of a sequence of random variables or for statistics which are functions of empirical processes may be derived. To achieve such practicality, we are led to ask a number of requirements of a theory of probability measures on a function space. We enumerate a few. (i) To begin with, our function space S must contain the realizations of processes of interest to us and should contain those which are defined in their simplest forms with empirical observations. (ii) The a-field aJ of subsets of S over which our probability measures are defined must contain events whose probability we wish to know. Some of these events may be defined by topological or analytical qualifications; for example, the classes of continuous functions in S or integer valued functions in S. (iii) Processes under investigation must be measurable functions from the underlying probability space over which they are defined; that is, they must induce probability measures on -. Moreover, we should have convenient rules in order to determine whether a process is measurable. (iv) Each real valued and measurable function h on S transforms a stochastic process or random function X into a statistic or random variable h(X). Statistics of interest to us should be so induced by measurable functions of stochastic processes. In addition, we must have criteria, preferably easy to apply, to verify the measurability of real functions h on S and to determine if h(X) is a random variable. (v) For a sequence of stochastic processes {X,,} a mode of convergence should be defined so that we may infer, for a large class of functions {h}, that the statistic h(X5) converges weakly to h(Xo); that is, the distribution function of h(X") converges to that of h(Xo) at each continuity point of the latter. In addition, we need facile standards to determine for a function h when such convergence holds.

170 citations


Journal ArticleDOI
TL;DR: The queues formed by aircraft in stacks awaiting landing clearance have usually been treated either by machine simulation, or analytically as stochastic processes with time-independent transition probabilities, but the present paper regards the queue-developing process as strongly time-dependent, often with a diurnal (24-hour) periodicity.
Abstract: The queues formed by aircraft in stacks awaiting landing clearance have usually been treated either by machine simulation, or analytically as stochastic processes with time-independent transition probabilities (possessing stationary solutions). In contrast to such methods, the present paper regards the queue-developing process in question as strongly time-dependent, often with a diurnal (24-hour) periodicity. The formulation and treatment are entirely analytic and make use of machines only to solve the equations for the probabilities, by economical deterministic steps, using the coefficients as given in tabular form. Time-varying Poisson arrivals are assumed, and also an upper limit to queue length. Two laws of servicing are used: Poisson and fixed service time; these extremes are found to lead to numerically close results in the realistic case. This situation contrasts with the much cruder approximation of deterministic flow models. The stochastic equations belong to well studied types of differential or...


Journal ArticleDOI
01 Mar 1972
TL;DR: In this paper, the authors studied the distribution of the duration of strikes in the United Kingdom and found that the fit of this distribution to the observed observations is very close, with few exceptions, very close.
Abstract: SUMMARY This paper is a study of the nature of the frequency distribution of the duration of strikes in the United Kingdom as recorded by the Department of Employment. The settlement of a strike is regarded as a probabilistic process and the duration of a strike is treated as an observation on a random variable. A model for this random duration is created by supposing that at each point of time after the commencement of a strike there exists an index of the difference between the parties to the dispute. This index is itself regarded as a one-dimensional stochastic process in continuous time and space and determines the duration of the strike by the first point of time at which the difference index passes through an absorbing barrier representing "agreement". The duration of a strike then becomes the First Passage time of a stochastic process to a single absorbing barrier. As a first approximation the index process is assumed to be simple Brownian motion with drift and in consequence the duration of a strike has the Inverse Gaussian for its probability distribution. The fit of this distribution to the observations is shown to be, with few exceptions, very close.

Proceedings ArticleDOI
01 May 1972
TL;DR: A variant of the Kolmogorov concept of complexity which yields a common theory of finite and infinite random sequences and some concepts of effective tests which are proved to be equivalent are established.
Abstract: We propose a variant of the Kolmogorov concept of complexity which yields a common theory of finite and infinite random sequences. The process complexity does not oscillate. We establish some concepts of effective tests which are proved to be equivalent.

Journal ArticleDOI
TL;DR: In this article, a non-linear and non-quadratic estimator of the realizations of a homogeneous random field ζ(t,x1,x2) is proposed.
Abstract: Summary The estimation of spectra of random stationary processes is an important part of the statistics of random processes. There are several books on spectral analysis, e.g. Blackman & Tukey, Hannan, and Jenkins & Watts. As a rule, spectral estimators are quadratic functions of the realizations. Recently Capon suggested a new method for estimation of spectra of random fields, in which a non-quadratic function of the realization is used: he considered a homogeneous random field ζ(t,x1,x2), i.e. one which is stationary in time and space and a random function of the time and space co-ordinates t, x1, x2. For the sake of expository convenience we shall consider ordinary stationary processes of time only, ζ(t); the generalization of our results to the case of random fields is easy. Comparison of the conventional spectral estimator and the ‘high-resolution’ estimator for an artificial example showed that the latter has less smoothing effect on the true spectrum (Capon). This was later confirmed by examples using real data (Capon). However, it was not clear whether for a finite realization the high-resolution estimator distorted the true spectrum, i.e. whether it behaved for example like a conventional estimator raised to some power. In the present paper we introduce and study a new class of spectral estimators which are generally non-linear and non-quadratic functionals of the realizations. These estimators include the conventional and high-resolution ones, for which we shall give the approximate distributions. We derive under rather general conditions the limiting distribution of the new class of estimators, and illustrate them with several examples. As a matter of fact, these new estimators are weighted means of the eigenvalues of the covariance matrix, e.g. the arithmetic mean, geometric mean, and so on.

Journal ArticleDOI
TL;DR: In this paper, a two-state Markov process for barrier passage statistics is proposed, where first passage probability depends on the first three area moments of the process power spectral density, and the concept of a time-dependent power spectrum conveniently describes the frequency decomposition of the response of an oscillator suddenly exposed to broadband stationary excitation.
Abstract: A two-state Markov process for barrier passage statistics provides a more realistic model than the traditional Poisson process, especially for the response of a lightly-damped oscillator to broad-bank excitation. For high barrier levels the two give similar results. With the Markov model, first passage probability depends on the first three area moments of the process power spectral density. The concept of a time-dependent power spectrum conveniently describes the frequency decomposition of the response of an oscillator suddenly exposed to broad-band stationary excitation. Analytical expressions for time-dependent spectral moments lead to an evolutionary power spectral density shape parameter and improved first-passage results, especially for lightly-damped oscillators and short durations.

Journal ArticleDOI
01 Jan 1972
TL;DR: In this article, a Monte Carlo approach is proposed to solve structural problems of this kind and deals, as an example, with the stress-wave propagation through a random structure under impact loading.
Abstract: In recent years, a considerable amount of effort has been made on the part of structural engineers to model the excitation to structures as a random process and to analyze the structural response so that Structural behaviors under nondeterministic disturbances can be predicted with certain probability statements. However, much less work has been accomplished dealing with structures with a random property. This appears to be due to the fact that the governing equations of motion for structures with spatial statistical variation of material property involve stochastic parameters and are usually extremely difficult to solve. This paper introduces a Monte Carlo approach to solve structural problems of this kind and deals, as an example, with the stress-wave propagation through a random structure under impact loading. Through this is example, total compatibility of the proposed approach with the infinite element analysis, which is capable of taking into consideration the effects of irregular boundarie...

Journal Article
TL;DR: In this paper, the enhancement of images that are characterized only by statistical data where the picture contains additive noise is considered, and the results are extended to obtain the smoothing of the data.
Abstract: The enhancement of images that are characterized only by statistical data where the picture contains additive noise is considered. The random process representing the output of the scanner is characterized by the output of a dynamic system with white noise input. The dynamic system describes a first-order vector-Markov process. The procedure of Kalman filtering is then utilized to recursively determine the minimum mean-square error estimate of the image. The result is then extended to obtain the smoothing of the data.

Journal ArticleDOI
TL;DR: In this article, an algorithm is given to estimate the noise covariance matrices for a linear, discrete, time-varying stochastic system, and it is found that the lagged products of the innovations sequence is also linear in these parameters.

Journal ArticleDOI
TL;DR: In this paper, Green's functions are derived for Laplace transformed master equations (here described in the language of hopping excitons) on finite chains with either periodic or free-end boundary conditions, and with either a disruptive (substitutional impurity) or a nondisruptive quencher.
Abstract: Four separate but related contributions to the theory of quenched stochastic processes in one dimension are presented. First, Green's functions are derived for Laplace transformed master equations (here described in the language of, but not restricted to, hopping excitons) on finite chains with either periodic or free‐end boundary conditions, and with either a disruptive (substitutional impurity) or a nondisruptive quencher. We solve these problems in spectral form for short‐range quenching with arbitrary quencher location and quenching rate parameter Qo. Second, the analogous random walk situations are treated. The solution of the generating function (finite‐difference analog of the Laplace transform) equation is identical to that of the Laplace‐transformed master equation with a disruptive quencher, but not with a nondisruptive quencher. Unlike the master equation case, slowly damped oscillations of the random walk chain excitation function can exist. Other differences also exist and are discussed; thes...

Journal ArticleDOI
TL;DR: The theory of multiplicative stochastic processes is contrasted with the theory of additive processes as discussed by the authors, and it is suggested that multiplicative processes lead to a conceptual foundation for none-quilibrium thermodynamics and nonequilibrium statistical mechanics of marked generality.
Abstract: The theory of multiplicative stochastic processes is contrasted with the theory of additive stochastic processes. The case of multiplicative factors which are purely random, Gaussian, stochastic processes is treated in detail. In a spirit originally introduced by theoretical work in nuclear magnetic resonance and greatly extended by Kubo, dissipative behavior is demonstrated, on the average, for dynamical equations which do not show dissipative behavior without averaging. It is suggested that multiplicative stochastic processes lead to a conceptual foundation for nonequilibrium thermodynamics and nonequilibrium statistical mechanics, of marked generality.

Journal ArticleDOI
01 Jan 1972
TL;DR: In this paper, the authors extended the stationary random envelope definition to the envelope of nonstationary random processes possessing evolutionary power spectral densities and derived the density, the joint density function, the moment function, and the crossing rate of a level of the non-stationary envelope process.
Abstract: The definition of stationary random envelope proposed by Cramer and Leadbetter, is extended to the envelope of nonstationary random process possessing evolutionary power spectral densities. The density function, the joint density function, the moment function, and the crossing rate of a level of the nonstationary envelope process are derived. Based on the envelope statistics, approximate solutions to the first excursion probability of nonstationary random processes are obtained. In particular, applications of the first excursion probability to the earthquake engineering problems are demonstrated in detail.

Journal ArticleDOI
TL;DR: In this paper, a stochastic method of analysis of offshore towers subjected separately to random sea waves and to strong motion earthquakes is presented. Butts et al. used the Pierson-Moskowitz wave height spectrum along with linear wave theory to define a stationary random sea state as caused by wind generated surface waves.


Journal ArticleDOI
TL;DR: In this paper, a notion of linear dependence of one stochastic process upon another is introduced, and studied in the case of symmetric stable processes, and the problem of imbedding one Lq space into another is shown to be related to this study.

Journal ArticleDOI
TL;DR: In this article, a stochastic model for the exciton motion which comprises both the coherent and the incoherent motion is presented. But the model is restricted to the case of two molecules and does not take into account the influence of the exchange interaction integral.
Abstract: We use a stochastic model for the exciton motion which comprises both the coherent and the incoherent motion. The incoherent part is taken care of by a stochastic process which allows the local excitation energy and the transition matrix element to fluctuate by means of a Markovian process. The interaction between the spins and their surroundings is described by the usual spin-Hamiltonian which is, however, simplified to a spin 1/2 particle (instead of the triplet state). In the present paper we solve exactly the two limiting cases of completely coherent and incoherent motion (for two molecules). In the incoherent case the influence of the exchange interaction integral is taken into account by perturbation theory. We find expressions which are immediately comparable with ESR-experimental data. This comparison and additional information derived from optical absorption measurements allow us to determine all free parameters of our model uniquely. In particular, the fluctuations of the exchange interaction integral (with strength γ1) play an important role. From these parameters we may furthermore calculate the correlation time of the proton spin resonance in agreement with experimental data. The results show clearly that at room temperature in anthracene crystals the exciton undergoes a hopping process.

Journal ArticleDOI
TL;DR: The time of occurrence T(t) of the largest momentary flood exceedance in some time interval (0, t] is a stochastic process as discussed by the authors, where the counting process for exceedances is a nonhomogeneous Poissonian process.
Abstract: The time of occurrence T(t) of the largest momentary flood exceedance in some time interval (0, t] is a stochastic process. The one-dimensional distribution function and the mathematical expectation of T(t) are determined for the case in which individual flood exceedances are independent, identically distributed random variables and the counting process for exceedances is a nonhomogeneous Poissonian process. Rather good agreement between observed and theoretical distributions for two rivers indicates that these assumptions are not extremely restrictive.

01 Jul 1972
TL;DR: In this article, a class of bilinear estimation problems involving single-degree-of-freedom rotation is formulated and resolved, and both continuous and discrete time estimation problems are considered.
Abstract: : A class of bilinear estimation problems involving single-degree-of- freedom rotation is formulated and resolved. Both continuous and discrete time estimation problems are considered. Error criteria, probability distributions, and optimal estimates on the circle are studies. An effective synthesis procedure for continuous time estimation is provided, and a generalization to estimation on arbitrary abelian Lie groups is included. An intrinsic difference between the discrete and continuous problems is discussed, and the complexity of the equations in the discrete time case is analyzed in this setting. Applications of these results to a number of practical problems including FM demodulation and frequency stability are examined.

Journal ArticleDOI
TL;DR: In this paper, a method is developed by which the input leading to the highest possible response in an interval of time can be determined for a class of non-linear systems, where the input, if deterministic, is constrained to have a known finite energy (or norm) in the interval under consideration.

Journal ArticleDOI
Yu-Chi Ho1
TL;DR: In this article, the explicit feedback control law for the singular linear quadratic-gaussian stochastic control problem is derived and the interesting implication of the control law in terms of information pattern is discussed.
Abstract: The explicit feedback control law for the singular linear-quadratic-gaussian stochastic control problem is derived. The interesting implication of the control law in terms of information pattern is discussed.