Showing papers on "Stochastic process published in 1977"

Statistics of random processes

Abstract: 1. Essentials of Probability Theory and Mathematical Statistics.- 2. Martingales and Related Processes: Discrete Time.- 3. Martingales and Related Processes: Continuous Time.- 4. The Wiener Process, the Stochastic Integral over the Wiener Process, and Stochastic Differential Equations.- 5. Square Integrable Martingales and Structure of the Functionals on a Wiener Process.- 6. Nonnegative Supermartingales and Martingales, and the Girsanov Theorem.- 7. Absolute Continuity of Measures corresponding to the Ito Processes and Processes of the Diffusion Type.- 8. General Equations of Optimal Nonlinear Filtering, Interpolation and Extrapolation of Partially Observable Random Processes.- 9. Optimal Filtering, Interpolation and Extrapolation of Markov Processes with a Countable Number of States.- 10. Optimal Linear Nonstationary Filtering.

Statistical-physical models of electromagnetic interference

Abstract: Most man-made and natural electromagnetic interference, or "noise," 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 any adequate spectral management program. Accordingly, this study 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 (narrow band vis-a-vis the receiver), Class B (broad band 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, vis;P1 (? > ?o)A,B, (and the associated probability densities, pdf's w1(?)A,B,[1]) of the envelope are obtained; (the phase is shown to be uniformly distributed in (0, 2?). 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.

Synergetics: An Introduction. Nonequilibrium Phase Transitions and Self- Organization in Physics, Chemistry and Biology

Abstract: 1. Goal.- 1.1 Order and Disorder: Some Typical Phenomena.- 1.2 Some Typical Problems and Difficulties.- 1.3 How We Shall Proceed.- 2. Probability.- 2.1 Object of Our Investigations: The Sample Space.- 2.2 Random Variables.- 2.3 Probability.- 2.4 Distribution.- 2.5 Random Variables with Densities.- 2.6 Joint Probability.- 2.7 Mathematical Expectation E(X), and Moments.- 2.8 Conditional Probabilities.- 2.9 Independent and Dependent Random Variables.- 2.10 Generating Functions and Characteristic Functions.- 2.11 A Special Probability Distribution: Binomial Distribution.- 2.12 The Poisson Distribution.- 2.13 The Normal Distribution (Gaussian Distribution).- 2.14 Stirling's Formula.- 2.15 Central Limit Theorem.- 3. Information.- 3.1 Some Basic Ideas.- 3.2 Information Gain: An Illustrative Derivation.- 3.3 Information Entropy and Constraints.- 3.4 An Example from Physics: Thermodynamics.- 3.5 An Approach to Irreversible Thermodynamics.- 3.6 Entropy-Curse of Statistical Mechanics?.- 4. Chance.- 4.1 A Model of Brownian Movement.- 4.2 The Random Walk Model and Its Master Equation.- 4.3 Joint Probability and Paths. Markov Processes. The Chapman-Kolmogorov Equation. Path Integrals.- Sections with an asterisk in the heading may be omitted during a first reading..- 4.4 How to Use Joint Probabilities. Moments. Characteristic Function. Gaussian Processes.- 4.5 The Master Equation.- 4.6 Exact Stationary Solution of the Master Equation for Systems in Detailed Balance.- 4.7 The Master Equation with Detailed Balance. Symmetrization, Eigenvalues and Eigenstates.- 4.8 Kirchhoff's Method of Solution of the Master Equation.- 4.9 Theorems about Solutions of the Master Equation.- 4.10 The Meaning of Random Processes, Stationary State, Fluctuations, Recurrence Time.- 4.11 Master Equation and Limitations of Irreversible Thermodynamics.- 5. Necessity.- 5.1 Dynamic Processes.- 5.2 Critical Points and Trajectories in a Phase Plane. Once Again Limit Cycles.- 5.3 Stability.- 5.4 Examples and Exercises on Bifurcation and Stability.- 5.5 Classification of Static Instabilities, or an Elementary Approach to Thorn's Theory of Catastrophes.- 6. Chance and Necessity.- 6.1 Langevin Equations: An Example.- 6.2 Reservoirs and Random Forces.- 6.3 The Fokker-Planck Equation.- 6.4 Some Properties and Stationary Solutions of the Fokker-Planck-Equation.- 6.6 Time-Dependent Solutions of the Fokker-Planck Equation.- 6.6 Solution of the Fokker-Planck Equation by Path Integrals.- 6.7 Phase Transition Analogy.- 6.8 Phase Transition Analogy in Continuous Media: Space-Dependent Order Parameter.- 7. Self-Organization.- 7.1 Organization.- 7.2 Self-Organization.- 7.3 The Role of Fluctuations: Reliability or Adaptibility? Switching.- 7.4 Adiabatic Elimination of Fast Relaxing Variables from the Fokker-Planck Equation.- 7.5 Adiabatic Elimination of Fast Relaxing Variables from the Master Equation.- 7.6 Self-Organization in Continuously Extended Media. An Outline of the Mathematical Approach.- 7.7 Generalized Ginzburg-Landau Equations for Nonequilibrium Phase Transitions.- 7.8 Higher-Order Contributions to Generalized Ginzburg-Landau Equations.- 7.9 Scaling Theory of Continuously Extended Nonequilibrium Systems.- 7.10 Soft-Mode Instability.- 7.11 Hard-Mode Instability.- 8. Physical Systems.- 8.1 Cooperative Effects in the Laser: Self-Organization and Phase Transition.- 8.2 The Laser Equations in the Mode Picture.- 8.3 The Order Parameter Concept.- 8.4 The Single-Mode Laser.- 8.5 The Multimode Laser.- 8.6 Laser with Continuously Many Modes. Analogy with Superconductivity.- 8.7 First-Order Phase Transitions of the Single-Mode Laser.- 8.8 Hierarchy of Laser Instabilities and Ultrashort Laser Pulses.- 8.9 Instabilities in Fluid Dynamics: The Benard and Taylor Problems.- 8.10 The Basic Equations.- 8.11 The Introduction of New Variables.- 8.12 Damped and Neutral Solutions (R ? Rc).- 8.13 Solution Near R = Rc (Nonlinear Domain). Effective Langevin Equations.- 8.14 The Fokker-Planck Equation and Its Stationary Solution.- 8.15 A Model for the Statistical Dynamics of the Gunn Instability Near Threshold.- 8.16 Elastic Stability: Outline of Some Basic Ideas.- 9. Chemical and Biochemical Systems.- 9.1 Chemical and Biochemical Reactions.- 9.2 Deterministic Processes, Without Diffusion, One Variable.- 9.3 Reaction and Diffusion Equations.- 9.4 Reaction-Diffusion Model with Two or Three Variables: The Brusselator and the Oregonator.- 9.5 Stochastic Model for a Chemical Reaction Without Diffusion. Birth and Death Processes. One Variable.- 9.6 Stochastic Model for a Chemical Reaction with Diffusion. One Variable.- 9.7 Stochastic Treatment of the Brusselator Close to Its Soft-Mode Instability.- 9.8 Chemical Networks.- 10. Applications to Biology.- 10.1 Ecology, Population-Dynamics.- 10.2 Stochastic Models for a Predator-Prey System.- 10.3 A Simple Mathematical Model for Evolutionary Processes.- 10.4 A Model for Morphogenesis.- 10.5 Order Parameters and Morphogenesis.- 10.6 Some Comments on Models of Morphogenesis.- 11. Sociology and Economics.- 11.1 A Stochastic Model for the Formation of Public Opinion.- 11.2 Phase Transitions in Economics.- 12. Chaos.- 12.1 What is Chaos?.- 12.2 The Lorenz Model. Motivation and Realization.- 12.3 How Chaos Occurs.- 12.4 Chaos and the Failure of the Slaving Principle.- 12.5 Correlation Function and Frequency Distribution.- 12.6 Discrete Maps, Period Doubling, Chaos, Intermittency.- 13. Some Historical Remarks and Outlook.- References, Further Reading, and Comments.

Statistical-Physical Models of Electromagnetic Interference

Abstract: Most man-made and natural electromagnetic interference, or "noise," 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 any adequate spectral management program. Accordingly, this study 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 (narrow band vis-a-vis the receiver), Class B (broad band 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, vis;P1 (? > ?o)A,B, (and the associated probability densities, pdf's w1(?)A,B,[1]) of the envelope are obtained; (the phase is shown to be uniformly distributed in (0, 2?). 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.

Linear estimation and stochastic control

Abstract: Linear estimation theory. Hilbert space. Stochastic processes. Hilbert space. Othogonal increments processes. Linear stochastic control. Dynamic programming. Stochastic linear regulator. Separation principle.

Extreme values of independent stochastic processes

Abstract: The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

A dynamic model for social networks

Abstract: A continuous‐time binary‐matrix‐valued Markov chain is used to model the process by which social structure effects individual behavior. The model is developed in the context of sociometric networks of interpersonal affect. By viewing the network as a time‐dependent stochastic process it is possible to construct transition intensity equations for the probability that choices between group members will change. These equations can contain parameters for structural effects. Empirical estimates of the parameters can be interpreted as measures of structural tendencies. Some elementary processes are described and the application of the model to cross‐sectional data is explained in terms of the steady state solution to the process.

The Exit Problem for Randomly Perturbed Dynamical Systems

Abstract: The cumulative effect on dynamical systems, of even very small random perturbations, may be considerable after sufficiently long times. For example, even if the corresponding deterministic system h...

Stochastic process for extrapolating concrete creep

Abstract: Creep of concrete is modeled as a process with independent increments of locally gamma distribution. The process is transformed to a stationary gamma process. The mean prediction agrees with the deterministic double power law established previously. Infinite divisibility of the increment distribution is assumed. This is justified by additivity of deformations and of stresses, and also by considerations of the microscopic mechanism of creep, assuming creep to be due to migrations of widely spaced solid particles along micropore passages whose length is statistically distributed. The treatment of creep as a stochastic process allows extracting considerable information from measurements even on one specimen, although a greater number of specimens is preferable. The main use of the model is in extrapolation of short time creep data into long times, and calculation of confidence limits. Methods of determining process parameters from creep test data are given. Monte Carlo simulations demonstrate reasonable agreement with test data.

Error Probability of Asynchronous Spread Spectrum Multiple Access Communication Systems

Abstract: Several approaches for the evaluation of upper and lower bounds on error probability of asynchronous spread spectrum multiple access communication systems are presented. These bounds are obtained by utilizing an isomorphism theorem in the theory of moment spaces. From this theorem, we generate closed, compact, and convex bodies, where one of the coordinates represents error probability, while the other coordinate represents a generalized moment of the multiple access interference random variable. Derivations for the second moment, fourth moment, single exponential moment, and multiple exponential moment are given in terms of the partial cross correlations of the codes used in the system. Error bounds based on the use of these moments are obtained. By using a sufficient number of terms in the multiple exponential moment, upper and lower error bounds can be made arbitrarily tight. In that case, the error probability equals the multiple exponential moment of the multiple access interference random variable. An example using partial cross correlations based on codes generated from Gold's method is presented.

A Bayesian comparison of different classes of dynamic models using empirical data

Abstract: This paper deals with the Bayesian methods of comparing different types of dynamical structures for representing the given set of observations. Specifically, given that a given process y(\cdot) obeys one of r distinct stochastic or deterministic difference equations each involving a vector of unknown parameters, we compute the posterior probability that a set of observations {y(1),...,y(N)} obeys the i th equation, after making suitable assumptions about the prior probability distribution of the parameters in each equation. The difference equations can be nonlinear in the variable y but should be linear in the parameter vector in it. Once the posterior probability is known, we can find a decision rule to choose between the various structures so as to minimize the average value of a loss function. The optimum decision rule is asymptotically consistent and gives a quantitative explanation for the "principle of parsimony" often used in the construction of models from empirical data. The decision rule answers a wide variety of questions such as the advisability of a nonlinear transformation of data, the limitations of a model which yields a perfect fit to the data (i.e., zero residual variance), etc. The method can be used not only to compare different types of structures but also to determine a reliable estimate of spectral density of process. We compare the method in detail with the hypothesis testing method, and other methods and give a number of illustrative examples.

Stochastic Models of Intermediate State Interaction in Second Order Optical Processes –Stationary Response. II–

Abstract: A general theory of transient light scattering is developed on the basis of our stochastic theory of intermediate state interaction. The transient behaviors of various components of the second orde...

Statistical-physical models of electromagnetic interference

Abstract: Most man-made and natural electromagnetic interference, or "noise," 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 any adequate spectral management program. Accordingly, this study 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 (narrow band vis-a-vis the receiver), Class B (broad band 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, vis;P1 (? > ?o)A,B, (and the associated probability densities, pdf's w1(?)A,B,[1]) of the envelope are obtained; (the phase is shown to be uniformly distributed in (0, 2?). 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.

On up- and downcrossings

Abstract: For the sample functions of the stationary virtual waiting-time process v, of the GI/G/1 queueing system some properties of the number of up- and downcrossings of level v by the v,-process during a busy cycle are investigated. It turns out that the simple fact that this number of upcrossings is equal to that of downcrossings leads in a rather easy way to basic relations for the waiting-time distributions. This approach to the study of the v,-process of the GI G /1 system seems to be applicable to many other types of stochastic processes. As another example of this approach the infinite dam with non-constant release rate is briefly discussed.

The generalized Langevin equation with Gaussian fluctuations

Abstract: It is shown that all statistical properties of the generalized Langevin equation with Gaussian fluctuations are determined by a single, two‐point correlation function. The resulting description corresponds with a stationary, Gaussian, non‐Markovian process. Fokker–Planck‐like equations are discussed, and it is explained how they can lead one to the erroneous conclusion that the process is nonstationary, Gaussian, and Markovian.

Stochastic analysis of a nonequilibrium phase transition: Some exact results

Abstract: A system involving all-or-none transitions away from equilibrium is considered. Under the assumption of spatially homogeneous fluctuations an integral representation of the solution of the master equation is derived, which permits an exact evaluation of the variance in the thermodynamic limit. A systematic perturbative solution of the master equation is also developed. Both approaches yield “classical” exponents describing the divergence of the second-order variance as the instability point is approached on either side. Finally, at the instability point the second-order variance is shown to diverge as the 32 power of the volume.

The effect of a memoryless nonlinearity on the spectrum of a random process

Abstract: It is shown that, for a Gaussian process and for some non-Gaussian processes, any memoryless nonlinearity has a whitening effect in the sense that the output spectrum is smoother and occupies a greater bandwidth than the input spectrum. The ratio between input and output bandwidths is investigated by using several measures of bandwidth. Also, it is shown that classes of nonlinearities exist that are equivalent in the sense of producing the same spectral transformations.

Concerning the validity of the stochastic approach to chemical kinetics

Abstract: A simple argument advanced recently in support of the legitimacy of the stochastic formulation of chemical kinetics has been criticized because it seems to require the imminent collision of widely separated molecules. It is argued here that this criticism is unwarranted because it is based on an incorrect use of probabilities. To illustrate the various probabilistic considerations involved, a detailed analysis is presented of a closely related but mathematically simpler problem: the calculation of the collision probability per unit time for a thermally equilibrized one-dimensional gas of point particles.

Photoelectron Statistics: With Applications to Spectroscopy and Optical Communication

Abstract: 1. Introduction.- I. Tools From Mathematical Statistics.- 2. Statistical Description of Random Variables and Stochastic Processes.- 3. Point Processes.- II. Theory.- 4. The Optical Field: A Stochastic Vector Field or, Classical Theory of Optical Coherence.- 5. Photoelectron Events: A Doubly Stochastic Poisson Process or Theory of Photoelectron Statistics.- III. Applications.- 6. Applications to Optical Communication.- 7. Applications to Spectroscopy.- References.

An investigation of temporal moments of stochastic waves

Abstract: It is proposed to describe the temporal characteristics of a wave propagating in a random medium in terms of its temporal moments. The first two moments are related to the mean arrival time and the mean pulse width. It is shown that the one-position two-frequency mutual coherence function enters in the formulation naturally. The form of the expression suggests expanding the mutual coherence function in a narrow-band expansion whose coefficients can be solved exactly from the parabolic equation that takes into account all multiple scattering effects except the backscattering. A brief survey of the literature shows that the irregularity spectrum, under various conditions, has a power-law dependence. In order to conform to this observation a Bessel function spectrum proposed by Shkarofsky is found convenient to use since it not only reduces to the desired power-law form in the proper range of wavenumber space, but also has all the finite moments. Exact expressions for the mean arrival time and mean square pulse width are obtained; some numerical examples are given. Finally, the effect of noise on these moments is discussed.

Stochastic Models and Fluctuations in Reversal Time of Ambiguous Figures

Abstract: Five probability distributions for the description of temporal fluctuations in the perception of ambiguous figures were fitted to previously obtained experimental results and classified according to their efficiency in describing the data. The gamma, Wiener, and Capocelli-Riciardi distributions showed the highest efficiency, while the chi2 and Taylor-Aldridge distributions showed a very low effiency. Therefore the underlying process may be described either by a simple Poisson model or by a random-walk model. For the gamma distribution there was a strong correlation between the parameters, while for the Wiener distribution this correlation was lower.