Showing papers on "Stochastic process published in 1976"

Method for Random Vibration of Hysteretic Systems

Abstract: Based on a Markov-vector formulation and a Galerkin solution procedure, a new method of modeling and solution of a large class of hysteretic systems (softening or hardening, narrow or wide-band) under random excitation is proposed. The excitation is modeled as a filtered Gaussian shot noise allowing one to take the nonstationarity and spectral content of the excitation into consideration. The solutions include time histories of joint density, moments of all order, and threshold crossing rate; for the stationary case, autocorrelation, spectral density, and first passage time probability are also obtained. Comparison of results of numerical example with Monte-Carlo solutions indicates that the proposed method is a powerful and efficient tool.

Stochastic Realization Algorithms

Abstract: Publisher Summary The progress of mathematical methods for signal processing and the simultaneous progress of digital data processing hardware have generated new interest in Markovian models, which have been known for a long time. Although many papers have appeared on filtering, detection, or control using such models, very little attention has been given to (1) the study of the properties of such models, (2) the set of all models that can represent a given stochastic process, and (3) the design of efficient algorithms to compute such models. This chapter deals with the theoretical problem of studying all Markovian models corresponding to a given stochastic process. The study given in the chapter appears to have interest from a practical point of view by sorting out peculiar Markovian models, such as the statistical filter of the process. This study and the related constructing proofs also lead to the design of efficient algorithms.

Statistical analysis of non-stationary series of events in a data base system

Abstract: Central problems in the performance evaluation of computer systems are the description of the behavior of the system and characterization of the workload. One approach to these problems comprises the interactive combination of data-analytic procedures with probability modeling. This paper describes methods, both old and new, for the statistical analysis of non-stationary univariate stochastic point processes and sequences of positive random variables. Such processes arefr equently encountered in computer systems. As an illustration of the methodology an analysis is given of the stochastic point process of transactions initiated in a running data base system. On theb asis of the statistical analysis, a non-homogeneous Poissonp rocess model for the transaction initiation process is postulated for periods of high system activity and found to be an adequate characterization of the data. For periods of lower system activity, the transaction initiation process has a complex structure, with more clustering evident. Overall models of this type have application to the validation of proposed data base subsystem models.

Concepts and Methods in Stochastic Control

Abstract: Publisher Summary This chapter discusses the concepts and methods in stochastic control. In most control or multistage decision processes arising in engineering, economic, or biological systems, there are inherent uncertainties. These uncertainties prevent the exact determination of the effect of the controller's actions and deterministic control theory cannot be used. In many problems, the uncertainties that could arise in the system, as well as in the observations made on the system could be modeled as stochastic processes. These problems could be treated using stochastic control theory. The chapter discusses the problem of control of dynamic stochastic systems. The state of the type of system that is considered evolves according to the non-linear stochastic difference equations. The performance index is taken as the expected value of a cost function and is to be minimized by a sequence of controls. The choice of these controls, which is a multistage decision process, constitutes the stochastic control problem.

Prediction error identification methods for stationary stochastic processes

Abstract: The strong consistency of a general class of prediction error identification methods for stationary stochastic processes is demonstrated. In particular, the strong consistency of the maximum likelihood method for stationary Gaussian processes [4], [5] and of the quadratic loss prediction error method for stationary stochastic processes [1]-[3] follow as special cases of the general result.

Stochastic theory of minimal realization

Abstract: This paper exploits the concept of a predictor space in the minimal realization problem for systems generating an analytic impulse response matrix. The predictor space constructed, by stochastic input and output processes forms the state space for the stochastic system representation, where a system is represented by the basis of the predictor space and the innovation process of input. The minimal realization problem is then solved for a given analytic impulse response matrix by defining a stochastic system driven by white noise whose input-output covariance equals the given impulse response matrix. It is shown that the coefficient matrices of the stochastic system representation constitute a solution to the minimal realization problem for the deterministic system with given impulse response matrix. The paper provides a unifying overview to many aspects of the realization problem and its algorithms.

Dissipation and fluctuations in nonequilibrium thermodynamics

Abstract: A relationship between dissipation and fluctuations is described which leads to a unified theory of irreversible processes far from equilibrium. The development is based on the principle that dissipation and fluctuations are caused by elementary molecular processes. This permits the formulation of a canonical form for the rate of dissipation of the extensive variables. The canonical form depends on the thermodymanic quantities which are conjugate to the extensive variables, and it is shown that the canonical form leads to the customary transport equations for a variety of linear and nonlinear relaxation processes. Because fluctuations are also caused by molecular events, this formulation of dissipation can be used to examine deviations from the average. The theory associates a nonstationary, Markov stochastic process with fluctuations away from the conditionally averaged extensive variables. This description of nonequilibrium thermodynamics does not require the entropy to be introduced, and for rate proce...

Backwards Markovian models for second-order stochastic processes (Corresp.)

Abstract: A state-space model of a second-order random process is a representation as a linear combination of a set of state-variables which obey first-order linear differential equations driven by an input process that is both white and uncorrelated with the initial values of the state-variables. Such a representation is often called a Markovian representation. There are applications in which it is useful to consider time running backwards and to obtain corresponding backwards Markovian representations. Except in one very special circumstance, these backwards representations cannot be obtained simply by just reversing the direction of time in a forwards Markovian representation. We show how this problem can be solved, give some examples, and also illustrate how the backwards model can be used to clarify certain least squares smoothing formulas.

Stochastic processes in estimation theory

Abstract: We describe the role of various stochastic processes, especially martingales and related concepts, in estimation theory. It is shown, in the simplest context, that in nonlinear estimation theory martingales play the same fundamental role as uncorrelation and white noise do in linear estimation.

Introduction to Random Fields

Abstract: One means of generalizing denumerable stochastic processes {x n } with time parameter set ℕ = {0, 1, ... } is to consider random fields {x t }, where t takes on values in an arbitrary countable parameter set T. Roughly, a random field with denumerable state space S is described by a probability measure μ on the space Ω = S T of all configurations of values from S on the generalized time set T. In this chapter we discuss certain extensions of Markov chains, called Markov fields which have been important objects of study in the recent development of probability theory. Only some of the highlights of this rich theory will be covered; we concentrate especially on the case T = ℤ = the integers, where the connections with classical Markov chain theory are deepest.

The statistical analysis of space-time point processes

Abstract: A space-time point process is a stochastic process having as realizations points with random coordinates in both space and time. We define a general class of space-time point processes which we term {\em analytic}. These are point processes that have only finite numbers of points in finite time intervals, absolutely continuous joint-occurrence distributions, and for which points do not occur with certainty in finite time intervals. Analytic point processes possess an intensity determined by the past of the point process. As a class, analytic point processes remain closed under randomization by a parameter. The problem we consider is that of estimating a random parameter of an observed space-time point process. This parameter may be drawn from a function space and can, therefore, model a random variable, random process, or random field that influences the space-time point process. Feedback interactions between the point process and the randomizing parameter are included. The conditional probability measure of the parameter given the observed space-time point process is a sufficient statistic for forming estimates satisfying a wide variety of performance criteria. A general representation for this conditional measure is developed, and applications to filtering, smoothing, prediction, and hypothesis testing are given.

Un théorème de représentation pour les martingales discontinues

Abstract: We consider a fixed integer-valued random measure (also called a random point process). We represent any local martingale as the sum of a stochastic integral with respect to this random measure, and of a local martingale which does not jump on the support of the random measure.

Locally Finite Random Sets: Foundations for Point Process Theory

Abstract: The foundations of point process theory are surveyed. An abstract theory motivated by applications in stochastic geometry is presented. It is shown that it is sufficient to know only which sets are measurable and which are bounded in the basic space, where we use countability hypotheses rather than topological assumptions. (The sole exception is in the construction of probabilities where pseudo-topological hypotheses are needed.) It is shown that there are close connections with the random set theories of Kendall and Matheron.

Linear regulator design for stochastic systems by a multiple time-scales method

Abstract: This short paper develops a hierarchically structured, suboptimal controller for a linear stochastic system composed of fast and slow subsystems. It is proved that the controller is optimal in the limit as the separation of time scales of the subsystems becomes infinite. The methodology is illustrated by design of a controller to suppress the phugoid and short-period modes of the longitudinal dynamics of the F-8 aircraft.

Stochastic space-time and quantum theory

Abstract: Much of quantum mechanics may be derived if one adopts a very strong form of Mach's principle such that in the absence of mass, space-time becomes not flat but stochastic. This is manifested in the metric tensor which is considered to be a collection of stochastic variables. The stochastic-metric assumption is sufficient to generate the spread of the wave packet in empty space. If one further notes that all observations of dynamical variables in the laboratory frame are contravariant components of tensors, and if one assumes that a Lagrangian can be constructed, then one can obtain an explanation of conjugate variables and also a derivation of the uncertainty principle. Finally, the superposition of stochastic metrics and the identification of $\sqrt{\ensuremath{-}g}$ in the four-dimensional invariant volume element $\sqrt{\ensuremath{-}g}\mathrm{dV}$ as the indicator of relative probability yields the phenomenon of interference as will be described for the two-slit experiment.

Stochastic Road Inputs and Vehicle Response

Abstract: SUMMARY During the 1950's increasing awareness of the available theory of stochastic processes, together with the wider availability of digital computers, brought to automobile engineers a new and powerful technique for treating the response of vehicles to the irregular undulations of roads. But while a road profile may reasonably be regarded as a realisation of a stochastic process, the theory of stochastic processes brings practical advantages only where the processes concerned satisfy certain stringent criteria The paper explains the basis of the standard spectral techniques which are available for the description and analysis of stochastic processes, and emphasises the restrictions implied by their acceptance. Progress towards the present state of the art is indicated by reference to work published over the last 25 years; this work has established that a stationary gaussian stochastic process does provide a satisfactory basic model for a road surface, and it has now reached a state of considerable sop...

Statistical-Physical Models of Man-made and Natural Radio Noise Part II: First Order Probability Models of the Envelope and Phase

Abstract: Most man-made and natural electromagnetic interferences are highly non-gaussian random processes, whose degrading effects on system performance can be severe, particularly on most conventional systems, which are designed for optimal or near optimal performance against normal noise. In addition, the nature, origins, measurement and prediction of the general EM interference environment are a major concern of an adequate spectral management program. Accordingly, this second study in a continuing series [cf. Middleton, 1974] is devoted to the development of analytically tractable, experimentally verifiable, statistical-physical models of such electromagnetic interference Here, classification into three major types of noise is made: Class A (narrowband vis-a-vis the receiver), Class B (broadband vis-a-vis the receiver), and Class C (=Class A+Class B). First-order statistical models are constructed for the Class A and Class B cases. In particular, the APD (a posteriori probability distribution) or exceedance probability, PD, viz. P1 (e>eo)A,B, and the associated probability densities, pdf’s, w1(e)A,B, of the envelope are obtained; [the phase is shown to be uniformly distributed in (0,2p)]. These results are canonical, i.e., their analytic forms are invariant of the particular noise source and its quantifying parameter values, levels, etc. Class A interference is described by a 3-parameter model, Class B noise by a 6-parameter model. All parameters are deducible from measurement, and like the APD’s and pdf’s, are also canonical in form: their structure is based on the general physics underlying the propagation and reception processes involved, and they, too, are invariant with respect to form and occurrence of particular interference sources. Excellent agreement between theory and experiment is demonstrated, for many types of EM noise, man-made and natural, as shown by a broad spectrum of examples. Results for the moments of these distributions are included, and more precise analytical conditions for distinguishing between Class A, B, and C interference are also given. Methods for estimating the canonical model parameters from experimental data (essentially embodied in the APD) are outlined in some detail, and a program of possible next steps in developing the theory of these highly nongaussian random processes for application to general problems of spectrum management is presented.

Countable-State, Continuous-Time Dynamic Programming with Structure

Abstract: We consider the problem P of maximizing the expected discounted reward earned in a continuous-time Markov decision process with countable state and finite action space. (The reward rate is merely bounded by a polynomial.) By examining a sequence 〈pN〉 of approximating problems, each of which is a semi-Markov decision process with exponential transition rate ΛN, ΛN ↗ ∞, we are able to show that P is obtained as the limit of the PN. The value in representing P as the limit of PN is that structural properties present in each PN persist, in both the finite and the infinite horizon problem. Three queuing optimization models illustrating the method are given.

One‐dimensional random walks of linear clusters

Abstract: A stochastic formalism is developed for the one‐dimensional surface diffusion of atom clusters, with component atoms located in adjacent channels, by representing the diffusion as a random walk of the center of mass (c.m.). Relations between the mean square displacement of the center of mass and the rate constants characterizing c.m. motion are derived for dimers and trimers, starting from the Kolmogorov equation. For dimers in the limit of long diffusion intervals, c.m. rate constants and individual atomic jump rates can be deduced knowing the mean square displacement and the frequency of occurrence of different dimer configurations. This analysis is feasible for trimers only under special conditions; even then, separation into the individual atomic rate processes is not, in general, possible.

Estimation of Optimum Structures and Parameters for Linear Systems

Abstract: The problem of determining a linear state space model for a stochastic process from measured output data contains, as an important part, the problem of choosing of a suitable structure for the model. A criterion is studied which measures the fit between the model and the data as well as the validity of a structure assumption. Hence the parameter and structure estimates can be obtained by minimization of a single criterion. The minimizing model is shown to converge to one with a correct structure and parameter values in this structure.

A Commutation Relation for Wide Sense Stationary Processes

Abstract: For a purely nondeterministic, wide sense stationary random process, two self-adjoint operators T and H are connected with the time domain and spectral representation of the process respectively. It is shown that these two operators satisfy the commutation relation $TH - HT = iI$. Applications of this result are given to the multiplicity and representation theory of the process. A new and more general definition of the concept of rank of a wide sense stationary process is also formulated.

A Statistical Analysis of the Influence of Cyclic Variation on the Formation of Nitric Oxide in Spark Ignition Engines

Abstract: A statistical method is presented to study the influence of cyclic variation of the combustion process on the formation of nitric oxide in spark ignition engines. In the analysis, each engine cycle was treated as a realization of a random experiment. Specifically, the cylinder pressure as a function of crank angle was considered a stochastic process, and the nitric oxide concentration frozen in the expansion stroke was considered a random variable associated with the random experiment. A set of consecutive cylinder pressure-crank angle data furnished the statistical properties of the cylinder pressure of a spark ignition engine at a normal running condition. The stochastic process of cylinder pressure was first approximated by one degree of randomness expression with peak combustion pressure as the characterizing random variable. Secondly, a combustion and nitric oxide kinetics model was used to establish the functional relationship of frozen nitric oxide concentration to cylinder pressure, particularly to the peak combustion pressure. Finally, the probability density function of the nitric oxide concentration was calculated by using the fundamental theorem of functions of random variables.