Showing papers on "Stochastic process published in 1968"
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01 Jan 1968
TL;DR: This chapter discusses filter theory, applications, and applications of filter theory and modeling techniques for free flight and powered flight navigation, and error analyses and sub-optimal modeling.
Abstract: Part I. Theory: Ordinary differential equations and stability Random processes and stochastic models Observability and controllability Filtering theory Global theory of filtering Stochastic stability Optimal filtering for correlated noise processes Approximate optimal non-linear filtering Optimum filtering for discrete time random processes Stochastic control Open questions and historical comments Part II. Applications: Application to navigation Applications of filter theory and modeling techniques Free flight and powered flight navigation Error analyses and sub-optimal modeling Errors in the filtering process Appendix A. Least squares curve fitting Appendix B. Probability review References Appendix C. The Riccati equation and its bounds Appendix D. Further references Index.
445 citations
TL;DR: In this paper, an ergodic theory for subadditive stochastic processes was developed for the percolation theory of stationary sequences, which is a complete generalization of the classical law of large numbers for stationary sequences.
Abstract: SUMMARY An ergodic theory is developed for the subadditive processes introduced by Hammersley and Welsh (1965) in their study of percolation theory. This is a complete generalization of the classical law of large numbers for stationary sequences. 1. SUBADDITIVE PROCESSES IN an important paper Hammersley and Welsh (1965) introduced the concept of a subadditive stochastic process, and they have shown how such processes arise naturally in various contexts, but particularly in the study of random flows in lattices. They have shown that one may expect these processes to exhibit a certain ergodic behaviour, and have taken the first steps towards the construction of an ergodic theory like the classical one for averages of stationary sequences. If T is any subset of the real line, a subadditive process x on T is a collection of (real) random variables xst(s, t E T, s < t) with the property that
432 citations
251 citations
TL;DR: In this article, an empirical stochastic process for two-sample problems is defined and its weak convergence is studied, based upon an identity which relates the two sample empirical process to the more usual one-sample empirical process.
Abstract: An empirical stochastic process for two-sample problems is defined and its weak convergence studied. The results are based upon an identity which relates the two-sample empirical process to the more usual one-sample empirical process. Based on this identity a relatively simple proof of a Chernoff-Savage theorem is obtained. The $c$-sample analogues of these results are also included.
168 citations
TL;DR: In this paper, a nonstationary Gaussian filtered shot-noise process with a second-order filter is proposed for stochastic simulation of strong-motion earthquakes, and the member functions of the mathematical model and their associated linear response spectra are compared with the corresponding information obtained from the recorded accelerograms.
Abstract: Eight strong-motion accelerograms recorded on firm ground and at moderate epicentral distances are studied for the purpose of developing a reasonable stochastic representation of the time history of ground accelerations during strong earthquakes. The results indicate that over durations which are of interest in structural response calculations, nonstationary random processes are needed for the description of the records. A nonstationary Gaussian filtered shot-noise process is examined for this purpose. The member functions of the mathematical model and their associated linear response spectra are compared with the corresponding information obtained from the recorded accelerograms. On this basis, a nonstationary Gaussian filtered shot-noise process with a second-order filter is proposed for stochastic simulation of strong-motion earthquakes.
135 citations
112 citations
108 citations
TL;DR: Linear nonautonomous random system stability, presenting theorem, two corollaries and second order examples as mentioned in this paper, and showing that linear non-autonomous system stability can be maintained.
Abstract: Linear nonautonomous random system stability, presenting theorem, two corollaries and second order examples
107 citations
TL;DR: In this article, a class of stochastic pursuit-evasion differential games between two linear dynamic systems is studied, where one of the players has imperfect (noisy) knowledge of the states of the two systems.
Abstract: The solution for a class of stochastic pursuit-evasion differential games between two linear dynamic systems is given. This class includes the classical interception game in Euclidean space. The performance index which is optimized is quadratic, and one of the two players has imperfect (noisy) knowledge of the states of the two systems. The "certainty-equivalence principle' or, equivalently, the technique of separating the estimator and the controller which characterizes the standard stochastic control problem is shown to be applicable to this class of differential games.
76 citations
TL;DR: Various models of impulse processes are discussed relative to random sampling of random processes and closed form expressions are calculated for the spectral density of s(t) and the sampled process under combinations of the sampling errors.
Abstract: Impulse processes relative to random sampling of random processes, deriving expression for spectral density
69 citations
TL;DR: In this article, a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain is discussed.
Abstract: In this paper we discuss a counter system whose output is a stochastic point process such that the time intervals between pairs of successive events form a first order Markov chain. Such processes may be regarded as next, in order of complexity, in a hierarchy of stochastic point processes, to “renewal” processes, which latter have been studied extensively. The main virtue of the particular system which is studied here is that virtually all its important statistical properties can be obtained in closed form and that it is physically realizable as an electronic device. As such it forms the basis for a laboratory generator whose output may be used for experimental work involving processes of this kind. Such statistical properties as the one and two-dimensional probability densities for the time intervals are considered in both the stationary and nonstationary state and also discussed are corresponding properties of the successive numbers arising in the stores of the counter system. In particular it is shown that the degree of coupling between successive time intervals may be adjusted in practice without altering the one dimensional probability density for the interval lengths. It is pointed out that operation of the counter system may also be regarded as a problem in queueing theory involving one server alternately serving two queues. A generalization of the counter system, whose inputs are normally a pair of statistically independent Poisson processes, to the case where one of the inputs is a renewal process is considered and leads to some interesting functional equations.
TL;DR: The efficiency of learning for an m-state automaton in terms of expediency and convergence, under two distinct types of reinforcement schemes: one based on penalty probabilities and the other on penalty strengths, is discussed.
Abstract: A stochastic automaton responds to the penalties from a random environment through a reinforcement scheme by changing its state probability distribution in such a way as to reduce the average penalty received. In this manner the automaton is said to possess a variable structure and the ability to learn. This paper discusses the efficiency of learning for an m-state automaton in terms of expediency and convergence, under two distinct types of reinforcement schemes: one based on penalty probabilities and the other on penalty strengths. The functional relationship between the successive probabilities in the reinforcement scheme may be either linear or nonlinear. The stability of the asymptotic expected values of the state probability is discussed in detail. The conditions for optimal and expedient behavior of the automaton are derived. Reduction of the probability of suboptimal performance by adopting the Beta model of the mathematical learning theory is discussed. Convergence is discussed in the light of variance analysis. The initial learning rate is used as a measure of the overall convergence rate. Learning curves can be obtained by solving nonlinear difference equations relating the successive expected values. An analytic expression concerning the convergence behavior of the linear case is derived. It is shown that by a suitable choice of the reinforcement scheme it is possible to increase the separation of asymptotic state probabilities.
TL;DR: The coding theorem and weak converse of the coding theorem for averaged semicontinuous stationary channels and for almost periodic discrete channels whose phases are statistically known were proved in this paper.
Abstract: Coding theorem and weak converse of the coding theorem are proved for averaged semicontinuous stationary channels and for almost periodic discrete channels, whose phases are statistically known. Explicit formulas for the capacities are given. The strong converses of the coding theorems do not hold.
TL;DR: The Combinatorial Methods in the Theory of Stochastic Processes (CMLP) as mentioned in this paper is a generalization of the theory of stochastic processes, which is used in this paper.
Abstract: (1968). Combinatorial Methods in the Theory of Stochastic Processes. Technometrics: Vol. 10, No. 3, pp. 630-631.
TL;DR: In this paper, the authors describe statistics of the time-averaged squared sound pressure, as a function of frequency, and apply these statistics to reverberant rooms at high frequencies, such that the modal overlap (ratio of modal bandwidth to the average spacing between modes) is larger than about 3, and regions sufficiently removed from the source that the reverberant field prevails.
Abstract: Spatial fluctuations of sound in reverberant rooms are examined theoretically for the case of a fixed source and movable receiver. The approach is to describe statistics of the time‐averaged squared sound pressure, as a function of frequency. These statistics apply to reverberant rooms at high frequencies, such that the modal overlap (ratio of modal bandwidth to the average spacing between modes) is larger than about 3, and for regions sufficiently removed from the source that the reverberant field prevails. Within these bounds of frequency and space, squared sound pressure may be profitably viewed as a stochastic process over frequency. At each frequency in this range, squared sound pressure is a random variable obeying an exponential probability law over space. The family of random variables obtained by considering all frequencies in this range defines a stochastic process. The process is employed to derive formulas for the spatial variance of squared sound pressure for the cases of single tone, multitone, warbletone, and narrow‐band noise excitation. Experimental confirmation and suggested applications are given in many instances. In general, the variance is small when the product of bandwidth and reverberation time is large, as long as reverberation time is not so large as to prevent high modal overlap.
TL;DR: In this article, the authors developed a similar model for controlling the output of a dam whose random input depends on a homogeneous Wiener process, and the main object is to determine this function.
Abstract: A previous paper [2] was concerned with the determination of optimal policies for restocking an inventory which is continuous!y depleted by a random process of demands. The purpose of the present paper is to develop a similar model for controlling the output of a dam whose random input depends on a homogeneous Wiener process. This reversal of the roles of input and output does not, by itself, change the character of the problem. But the consideration of set-up costs for ordering replacements, which leads to inventory policies of the (s,S) type, has no counterpart here. It is natural to regard the dam as a device for smoothing out random fluctuations in a flow of water and, under utility assumptions which reflect this attitude, it follows that the optimal output rate is a continuous function of the level of water in the reservoir. Our main object is to determine this function.
TL;DR: In this article, the authors considered two linearly interconnected linear birth and death processes and found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f) g of the random variables involved is not obtainable by standard methods, although one can obtain moments of the variables from these equations.
Abstract: Summary Two cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f) g of the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution for g. This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends to g as n -+ co in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates Xi and pi, i = 1,2, interconnected linearly with transition rates v and 6 (see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) 1 = 6 = 0, with the remaining rates time dependent and (ii) 2 -= 6 = 0, with the remaining rates constant, explicit solutions for g are obtained and studied.
TL;DR: In this paper, the authors deal with the discrete-time linear minimum-variance filtering of nonstationary random processes, where the dynamics of the signal and colored noise processes are represented by a combined random process model.
Abstract: The following deals with the discrete-time linear minimum-variance filtering of nonstationary random processes. The dynamics of the signal and colored noise processes are represented by a combined random process model.[1] Some of the measurement elements contain additional white noise, others do not. Similar to the continuous-time case of Bryson and Johansen,[3] the white-noise-free measurements will be used to reduce the order of the Kalman filter,[1],[2].
TL;DR: A new approach is presented for estimating stochastic signals based on the use of Markov processes and state-variable concepts, and a general model for analog communication via randomly time-varying channels is given and related to the state-vector estimation problem.
Abstract: A new approach is presented for estimating stochastic signals. The approach is based on the use of Markov processes and state-variable concepts. Equations are presented for approximate minimum mean-square error estimates of a Markovian state vector observed in a signal in which it is imbedded nonlinearly. A general model for analog communication via randomly time-varying channels is given and related to the state-vector estimation problem. The model includes as special cases such linear and nonlinear modulation schemes as AM, PM, FM, and PM _{n} /PM; and such continuous channels as Rayleigh and Rician channels, fixed channels with memory, and diversity channels. The state-variable approach leads automatically to physically realizable demodulators whose outputs are approximate MMSE estimates of the stochastic message and, if desired, the channel disturbances. Special consideration is given to angle modulation schemes.
TL;DR: A discrete time, nonlinear system composed of an integrator preceded by a binary quantizer with integrated negative feedback, which can model a tracking loop or a single integrating delta modulation communication system, is discussed with regard to the input-output statistics for two types of input processes: independent inputs and independent increments inputs.
Abstract: A discrete time, nonlinear system composed of an integrator preceded by a binary quantizer with integrated negative feedback, which can model a tracking loop or a single integrating delta modulation communication system, is discussed with regard to the input-output statistics for two types of input processes: independent inputs and independent increments inputs. A recursion on time for the joint distribution of input and output is obtained for the independent inputs process and explicitly solved for the time asymptotic distribution, when it exists. The solution is examined in greater detail for the special case of IID normal inputs. When the system is excited by a process of independent increments, the asymptotic behavior of the input and output (they diverge) is of less interest than that of the difference between input and output, the tracking error. A recursion in time for the characteristic function of the error is developed and the time asymptotic solution found, The tracking error is interpreted by decomposition into static and dynamic parts, and an exponential bound to its distribution is provided. The particular case of normal increments input is discussed in additional detail.
TL;DR: First- and second-order stochastic gradient algorithms are developed for suitably approximating the unknown density and distribution functions of a random vector from a sequence of independent samples.
Abstract: First- and second-order stochastic gradient algorithms are developed for suitably approximating the unknown density and distribution functions of a random vector from a sequence of independent samples. The mean-square-error criterion and the integral-square-error criterion are used in the approximations. The rates of convergence and the approximation error are also evaluated.
TL;DR: In this article, a general theory on the time correlation function and the spectral density for the random motion in molecular system is presented, and the dependence of these functions on the intermolecular potential and on the rate of fluctuation of the potential.
Abstract: A general theory on the time‐correlation function and the spectral density for the random motion in molecular system is presented. We study the dependence of these functions on the intermolecular potential and on the rate of fluctuation of the potential. It is shown that the fluctuating intermolecular interactions or potentials not only introduce the randomness to molecular motion but also characterize important details of the randomness. Here we assume that the fluctuation can be regarded as a Gaussian random process contributed from the perturbers around the molecule. Therefore, the theory may be applied to the molecular motions in the liquid state. As an example of application of the present theory we discuss the collapse of multiplet structure in magnetic resonance spectrum of two‐spin system; Anderson's results for the exchange narrowing of spectrum is derived as a special case of our formulation. The correlation functions and the spectral densities for the random reorientation of diatomic molecules ...
TL;DR: In this article, the authors provided sufficient conditions on the absolute value of the excitation applied to the column problem in order to insure mean-square global stability, which is applicable for any general continuous random process.
Abstract: This paper deals with Lyapunov-type analysis of the dynamic stability of a linear elastic column subjected to an axial stochastic load. Within the past decade, the interest in stability of stochastic systems of differen tial equations has rapidly increased, as indicated by the large number of papers on the subject. The intent of this paper is to provide sufficient conditions on the absolute value of the excitation applied to the column problem in order to insure mean-square global stability. Although this problem has been investigated by several authors by considering the concept of almost sure stability, the bounds provided are related to only the mean value of the absolute value of the excitation, or, at best, the standard deviation of a Gaussian process with zero mean. In the present paper, we are able to relate the required bounds, for mean-square global stability, to the mean and variances of the excitation. The bounds are applicable for any general continuous random process. The sufficient bounds are obtained by using a Lyapunov type of approach, introduced by Bertram and Sarachik and extended here for stability in the mean-square sense.
TL;DR: This paper presents a general approach to the derivation of series expansions of second-order wide-sense stationary mean-square continuous random process valid over an infinite-time interval based on the integral representation of such processes.
Abstract: This paper presents a general approach to the derivation of series expansions of second-order wide-sense stationary mean-square continuous random process valid over an infinite-time interval. The coefficients of the expansion are orthogonal and convergence is in the mean-square sense. The method of derivation is based on the integral representation of such processes. It covers both the periodic and the aperiodic cases. A constructive procedure is presented to obtain an explicit expansion for a given spectral distribution.
TL;DR: An asymptotically valid approximation is obtained for the simple stochastic epidemic in a large population of size N, using the following approach.
Abstract: SUMMARY An asymptotically valid approximation is obtained for the simple stochastic epidemic in a large population of size N, using the following approach. For suitably defined variables, the stochastic process tends to a deterministic limit as N ->oo. The moment-generating function of the deterministic process satisfies an easily soluble partial differential equation and can be represented by an eigenfunction expansion. The solution of the corresponding stochastic equation can accordingly be obtained by standard perturbation theory, and the paper presents various results for the first-order perturbations of order N-1.
TL;DR: In this paper, a stochastic model of human learning behavior in a manual control task is described, where a two-position relay controller and a visual display are used to drive the process from an initial state to the null state.
Abstract: A stochastic model of human learning behavior in a manual control task is described. Regulation of the state of a double integral plant to minimize the integrated absolute error is the operator's task. Subjects given this task were instructed to drive the process from an initial state to the null state using a two-position relay controller and a visual display. A subject is conceptualized in the model as a sequential data-processing system. A sensor, a decision maker, and an effector are the three serially connected components making up the system. Each element requires a finite time to either process or transmit information, and thus a delay is incurred between the reception of the visual stimulus and the execution of a motor response. Response decisions are based on the a priori estimate of the probability that the control polarity should be switched, given the current state of the plant. Patterns in the resultant phase trajectory are used as evidence by the decision maker to revise the prior estimate with an algorithm according to Bayes' theorem. Behavior of this model is compared with subject behavior in the motor skill experiment, and the model's characterization of the time-varying random nature of human learning is brought out by this comparison. Also discussed are the applications of the concept of this model to other manual control tasks.
01 Jan 1968
TL;DR: In this paper, a series of isolated stochastic models, partly motivated by apphcations to astronomy, biology, engineering, and physics, are treated. But the main focus of this paper is on the analysis of problems involving combinations of stochastastic processes.
Abstract: In this chapter we treat a series of isolated stochastic models, partly motivated by apphcations to astronomy, biology, engineering, and physics. These processes are formed by compounding various classical processes, including Poisson, branching, and growth processes of the diffusion type. In each case, a secondary process feeds into a primary process whose state variable is the object of study. In Section 1 we will characterize multidimensional Poisson processes and in the following section give an apphcation to astronomy. The concept of multidimensional Poisson processes will play an important part in the formulation of cascade or compound stochastic processes. Some of these will be studied in the later sections of this chapter (e.g., see Section 2).
In Section 3, we examine a stochastic model involving growth and immigration. In Section 4 we formulate a stochastic process of growth involving two types, a normal type and a mutant type. The population of wild (i.e., normal) types grows deterministically while the mutant type grows in accordance with the laws of a Markov branching process. Moreover, each normal type at its death (hfetime is exponentially distributed) changes into a mutant type and then reproduces like any other mutant individual.
Sections 5 and 6 treat stochastic models of growth which take account of the factor of geographical distribution and spread of the population as well as their natural growth behavior.
The Stochastic processes investigated are typical of a large class of general cascade processes. The purpose of this chapter is to introduce the student to the richness of apphcations and subtleties of analysis of problems involving combinations of stochastic processes.
Section 7 is devoted to a review of some deterministic models of population growth, taking account of the age structure of the population.
TL;DR: In this paper, a Robbins-Monro stochastic approximation procedure for identifying a finite memory time-discrete time-stationary linear system from noisy input-output measurements is developed.
Abstract: A Robbins-Monro [1] stochastic approximation procedure for identifying a finite memory time-discrete time-stationary linear system from noisy input-output measurements is developed. Two algorithms are presented to sequentially identify the linear system. The first one is derived, based on the minimization of the mean-square error between the unknown system and a model, and is shown to develop a bias which depends only on the variance of the input signal measurement error. Under the assumption that this variance is known a priori, a second algorithm is developed which, in the limit, identifies the unknown system exactly. The case when the covariance matrix of the random input sequence is not positive definite is also considered.