Showing papers on "Stochastic process published in 1974"

The theory of stochastic processes

Abstract: Note: Trad. de : Teoriya sluchainyh protsessov. Bibliogr. a la fin de chaque volume. Index Reference Record created on 2004-09-07, modified on 2016-08-08

Markovian Representation of Stochastic Processes and Its Application to the Analysis of Autoregressive Moving Average Processes

Abstract: The problem of identifiability of a multivariate autoregressive moving average process is considered and a complete solution is obtained by using the Markovian representation of the process. The maximum likelihood procedure for the fitting of the Markovian representation is discussed. A practical procedure for finding an initial guess of the representation is introduced and its feasibility is demonstrated with numerical examples.

Stochastic theory of minimal realization

Abstract: In this paper it is shown that a natural representation of a state space is given by the predictor space, the linear space spanned by the predictors when the system is driven by a Gaussian white noise input with unit covariance matrix. A minimal realization corresponds to a selection of a basis of this predictor space. Based on this interpretation, a unifying view of hitherto proposed algorithmically defined minimal realizations is developed. A natural minimal partial realization is also obtained with the aid of this interpretation.

Complete integrability and stochastization of discrete dynamical systems

Abstract: We use the inverse scattering method to study a system of particles with exponential interaction (the Toda chain) and a set of equations describing induced scattering of plasma oscillations by ions. We show that a Toda chain with an arbitrary number of particles is completely integrable. We develop a scheme to integrate these systems and study the interaction between solitons. We indicate a class of completely integrable discrete systems, that is, systems which can not be stochastized.

Modeling of Continuous Stochastic Processes from Discrete Observations with Application to Sunspots Data

Abstract: Discretization of a continuous autoregressive moving average process at an equispaced sampling interval results in a discrete autoregressive moving average process The relationship between the continuous and the discrete parameters yields a simple method of maximum likelihood estimation of the continuous parameters from a discretely sampled data A technique is described for modeling of continuous processes from discrete observations and is illustrated with analysis of the yearly Wolfer's sunspot numbers data

Nucleation in systems with multiple stationary states

Abstract: We consider a reaction diffusion system, far from equilibrium, which has multiple stationary states (phases) for given ranges of external constraints. If two stable phases are put in contact, then in general one phase annihilates the other and in that process there occurs a single front propagation (soliton). We investigate the macroscopic dynamics of the front structure and velocity for two model systems analytically and numerically, and for general reaction-diffusion systems by a suitable perturbation method. The vanishing of the soliton velocity establishes the analogue of the Maxwell construction used in equilibrium thermodynamics. The problem of nucleation of one phase imbedded in another is studied by a stochastic theory. We show that if the reaction dynamics is derived from a generalized potential function then the macroscopic steady states are extrema of the probability distribution. We use this result to obtain an expression for the critical radius of a nucleating phase and confirm the prediction of the stochastic theory by numerical solution of the deterministic macroscopic kinetics for a model system.

The ergodic decomposition of stationary discrete random processes

Abstract: The ergodic decomposition is discussed, and a version focusing on the structure of individual sample functions of stationary processes is proved for the special case of discrete-time random processes with discrete alphabets. The result is stronger in this case than the usual theorem, and the proof is both intuitive and simple. Estimation-theoretic and information-theoretic interpretations are developed and applied to prove existence theorems for universal source codes, both noiseless and with a fidelity criterion.

An implicit sampling theorem for bounded bandlimited functions

Abstract: A rigorous proof of the 'strong bias tone' scheme is embodied in the implicit sampling theorem. The representation of signals that are sample functions of possible nonstationary random processes being of principal interest, the proof could not directly invoke results from classical analysis, which depend on the existence of the Fourier transform of the function under consideration; rather, it is based on Zakai's (1965) theorem on the series expansion of functions, band-limited under a suitably extended definition. A practical circuit that restores an approximate version of the signal from its sine-wave-crossings is presented and possible improvements to it are discussed.

Wave propagation in a random medium: a complete set of the moment equations with different wavenumbers

Abstract: Propagation of waves in a random medium is studied under the "quasioptics" and the "Markov random process" approximations. Under these assumptions, a Fokker‐Planck equation satisfied by the characteristic functional of the random wave field is derived. A complete set of the moment equations with different transverse coordinates and different wavenumbers is then obtained from the Fokker‐Planck equation of the characteristic functional. The applications of our results to the pulse smearing of the pulsar signal and the frequency correlation function of the wave intensity in interstellar scintillation are briefly discussed.

Propagation of pulse trains through a random medium

Abstract: Applying the parabolic approximation, the equations for two-frequency symmetric and antisymmetric mutual intensity functions for waves propagating through a random medium are derived, including the multiple scattering effects. These functions are applied to derive the general formulas for the covariance functions of narrowband pulses. They are used to compute the signal intensities for pulse trains passing through an ionospheric irregularity slab.

Optimal Filtering of Continuous-Time Stationary Processes by Means of the Backward Innovation Process

Abstract: A new approach to linear least squares estimation of continuous-time (wide sense) stationary stochastic processes is presented. The basic idea is that the relevant estimates can be expressed not only in terms of the usual (forward) innovation process but also in terms of a backward innovation process. The functions determining the optimal filter as well as the error covariance functions are seen to satisfy some differential equations. As an important example the Kalman-Busy filter is considered. It is demonstrated that the optimal gain matrix can be determined from 2mn equations (where n is the dimension of the system and m of the output) rather than $\frac{1}{2}n(n + 1)$ as in the conventional theory. This is an advantage when, as is usually the case, $m \ll n$. These equations were first derived by Kailath, who used a different method. Also they are the continuous-time versions of some equations previously obtained (independently of Kailath) by the author.

Weak Convergence of Multidimensional Empirical Processes for Stationary $\phi$-Mixing Processes

Abstract: For a stationary $\phi$-mixing sequence of stochastic $p(\geqq 1)$-vectors, weak convergence of the empirical process (in the $J_1$-topology on $D^p\lbrack 0, 1 \rbrack)$ to an appropriate Gaussian process is established under a simple condition on the mixing constants $\{\phi_n\}$. Weak convergence for random number of stochastic vectors is also studied. Tail probability inequalities for Kolmogorov Smirnov statistics are provided.

Statistical-Physical Models of Man-Made Radio Noise, Part I. First-Order Probability Models of the Instantaneous Amplitude

Abstract: A general statistical-physical model of man-made radio noise processes appearing in the input stages of a typical receiver is described analytically. The first-order statistics of the se random processes are developed in detail for narrow-band reception. These include, principally, the first order probability densities and probability distributions for a) a purely impulsive (poisson) process, and b) an additive mixture of a gauss background noise and impulsive sources. Particular attention is given to the basic waveforms of the emissions. in the course of propagation. including such critical geometric and kinematic factors as the beam patterns of source and receiver, mutual location, Doppler, far-field conditions, and the physical density of the sources, which are assumed independent and poisson distributed in space over a domain A. Apart from specific analytic relations. the most important general result s are that these first-order distributions are analytically tractable and canonical. They are not so complex as to be unusable in communication theory applications; they incorporate in an explicit way the controlling physical parameters and mechanisms which determine the actual radiated and received processes; and finally, they are formally invariant of the particular source location and density, waveform emission, propagation mode, etc., as long as the received disturbance is narrow-band, at least as it is passed by the initial stages of the typical receiver. The desired first-order distributions are represented by an asymptotic development, with additional terms dependent on the fourth and higher moments of the basic interference waveform, which in turn progressively affect the behavior at the larger amplitudes. This first report constitutes an initial step in a program to provide workable analytical models of the general nongaussian channel ubiquitous in practical communications applications. Specifically treated here are the important classes of interference with bandwidths comparable to (or less than) the effective aperture-RF-IF bandwidth of the receiver, the common situation in the case of communication interference.

Diffusive random process approximation in certain nonstationary statistical problems of physics

Abstract: The review considers, on the basis of a unified approach, the problem of Brownian motion in nonlinear dynamic systems, including a linear oscillator acted upon by random forces, parametric resonance in an oscillating system with random parameters, turbulent diffusion of particles in a random-velocity field, and diffusion of rays in a medium with random inhomogeneities of the refractive index. The same method is used to consider also more complicated problems such as equilibrium hydrodynamic fluctuations in an ideal gas, description of hydrodynamic turbulence by the method of random forces, and propagation of light in a medium with random inhomogeneities. The method used to treat these problems consists of constructing equations for the probability density of the system or for its statistical moments, using as the small parameter the ratio of the characteristic time of the random actions to the time constant of the system (in many problems, the role of the time is played by one of the spatial coordinates). The first-order approximation of the method is equivalent to replacement of the real correlation function of the action by a δ function; this yields equations for the characteristics in closed form. The method makes it possible to determine also higher approximations in terms of the aforementioned first-order small parameter.

Stochastic processes for solid tumor kinetics I. surface-regulated growth☆

Abstract: Following an introductory discussion on the implications of stochastic models of carcinogenesis on and of deterministic models for the growth kinetics of solid tumors, and a brief review of the experimental evidence, it is proposed that solid tumor growth kinetics should be represented mathematically by a nonmultiplicative time-homogeneous stochastic process of the birth and death type. Since nutrients have to enter the solid tumor through its surface, it is in a first approximation to a more adequate representation assumed that the birth rate is proportional to the 2 3 power of the tumor cell population size, and the death rate proportional to the population size; both rates are assumed to be age-independent. A solution for the forward Kolmogorov equation for this process could not be obtained, but bounds are given for the probability of extinction, which is shown to tend to unity asymptotically, for the waiting time to extinction, and for the mean and the variance of the population size. Bounds for the asymptotical conditional mean and mode of the tumor size, conditioned on no extinction, are also given, and the results of computer simulations of the process are discussed. It is shown that the (singular) Fokker-Planck equation for the continuous (diffusion) analog of the process has a regular boundary at the origin, which would require the imposition of an arbitrary boundary condition. Since this is at least biologically unfeasible, it is concluded the the surface proportionality assumption for the birth rate should be replaced by a proposition resulting in a better behavior at the origin, and it is indicated that a better formulation has been obtained and will be presented in a subsequent paper.

Modeling of Human Operator Performance Utilizing Time Series Analysis

Abstract: A new method for modeling the human operator from actual input-output data utilizing time series analysis is discussed in this applications oriented paper. The technique first identifies the form of the model, then estimates the parameters of the identified model based on actual data. Finally it checks the fitted model in relation to the data with the aim of revealing model inadequacies, thus providing model improvement. The methodology for applying the time series technique for determining the model of the human element in a feedback system is discussed. In addition, an approach for determining the human model under various levels of stress is discussed. The time series approach is a useful method for modeling any set of discrete observables corrupted with noise, be it human or some other deterministic/stochastic process. Since this is the first time that the human model has ever been obtained from the time series method, it is quite understandable that the results described shed new light on certain aspects of this problem, reveal new insights into the human model, and ask other probing questions.

Effects of multiple scattering on scintillation of transionospheric radio signals

Abstract: Recent development in the optical scintillation theory has been adapted to the ionospheric geometry in order to study the ionospheric scintillation phenomenon in the presence of multiple scattering Under approximations well satisfied in typical ionospheres for a frequency above about 20 MHz, the first through fourth moment equations have been derived and some analytic solutions given The fourth moment equation has also been solved numerically The numerical results show clearly the occurrence of focusing and saturation phenomena The new multiple-scatter effects are emphasized

Distinguishing Between Contagion, Heterogeneity and Randomness in Stochastic Models

Abstract: It is commonly believed that the nature of the generating structure of a stochastic process can be determined by examining the distributional form of the stochastic process in time. For example, it is believed that correlations between time intervals can be used to distinguish between contagion (after-effect) and heterogeneity (stratification). These beliefs are incorrect, and no amount of statistical information internal to the sampled process, is sufficient for such a distinction.

Control of Finite-State, Finite Memory Stochastic Systems

Abstract: A generalized problem of stochastic control is discussed in which multiple controllers with different data bases are present. The vehicle for the investigation is the finite state, finite memory (FSFM) stochastic control problem. Optimality conditions are obtained by deriving an equivalent deterministic optimal control problem. A FSFM minimum principle is obtained via the equivalent deterministic problem. The minimum principle suggests the development of a numerical optimization algorithm, the min-H algorithm. The relationship between the sufficiency of the minimum principle and the informational properties of the problem are investigated. A problem of hypothesis testing with 1-bit memory is investigated to illustrate the application of control theoretic techniques to information processing problems.