Showing papers on "Stochastic process published in 1985"

Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences

Abstract: The Handbook of Stochastic Methods covers systematically and in simple language the foundations of Markov systems, stochastic differential equations, Fokker-Planck equations, approximation methods, chemical master equations, and quatum-mechanical Markov processes Strong emphasis is placed on systematic approximation methods for solving problems Stochastic adiabatic elimination is newly formulated The book contains the "folklore" of stochastic methods in systematic form and is suitable for use as a reference work

Uncertainty explains many aspects of visual contrast detection and discrimination.

Abstract: More than 20 years ago, Tanner [Ann NY Acad Sci 89, 752 (1961)] noted that observers asked to detect a signal act as though they are uncertain about the physical characteristics of the signal to be detected The popular assumptions of probability summation and decision variable, taken together, imply this uncertainty This paper defines and uncertainty model of visual detection that assumes that the observer is uncertain among many signals and chooses the likeliest With only four parameters, the uncertainty model explains why d' is approximately a power function of contrast ("nonlinear transduction") and accurately predicts effects of summation, facilitation, noise, subjective criterion, and task for near-threshold contrast Thus the uncertainty model offers a synthesis of much of our current understanding of visual contrast detection and discrimination

Path integral approach to birth-death processes on a lattice

Abstract: The Fock space formalism for classical objects first introduced by Doi is cast in a path integral form and applied to general birth-death processes on a lattice. The introduction of suitable auxiliary variables allows one to formulate random walks with memory and irreversible aggregation processes in a Markovian way, which is treatable in this formalism. Existing field theories of such processes are recovered in the continuum limit. Implications of the method for their asymptotic behaviour are briefly discussed.

Mixture autoregressive hidden Markov models for speech signals

Abstract: In this paper a signal modeling technique based upon finite mixture autoregressive probabilistic functions of Markov chains is developed and applied to the problem of speech recognition, particularly speaker-independent recognition of isolated digits. Two types of mixture probability densities are investigated: finite mixtures of Gaussian autoregressive densities (GAM) and nearest-neighbor partitioned finite mixtures of Gaussian autoregressive densities (PGAM). In the former (GAM), the observation density in each Markov state is simply a (stochastically constrained) weighted sum of Gaussian autoregressive densities, while in the latter (PGAM) it involves nearest-neighbor decoding which in effect, defines a set of partitions on the observation space. In this paper we discuss the signal modeling methodology and give experimental results on speaker independent recognition of isolated digits. We also discuss the potential use of the modeling technique for other applications.

External Theory for Stochastic Processes.

Abstract: : The purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes. The theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts. Emphasis is placed on stationary cases but other important classes (e.g. Mark of sequences) are included. Significant ideas and methods are described rather than details, and in particular the nature and role of important underlying point processes (such as exceedances and upcrossings) are emphasized. Applications are given to particular classes of process (e.g. normal, moving average) and connections with related theory (such as convergence of sums) are indicated.

A random fuse model for breaking processes

Abstract: We introduce and study a simple idealized model to describe breaking processes by analysing the current-carrying properties of a random network consisting of insulators and fuses. By increasing the value of the external voltage applied across the network, a sequence of fuses will « burn out » and change irreversible into insultating bonds. This process terminates when a conducting path no longer exists in the network. We monitor several basic quantities during this breaking process, such as the conductivity of the network and the value of the voltage needed to break the hottest fuse. Two new exponents describing the behaviour of these quantities near the percolation threshold are reported.

The foundations of finite sample estimation in stochastic processes

Abstract: Generally, in the literature on stochastic processes, estimation is investigated in terms of asymptotic properties. In this paper we establish some finite sample optimal estimation. Let {Y1, Y2, ...} be a discrete stochastic process. Since we will be discussing results for samples of finite size n, we restrict the process to R'. In the following, the term 'parameter' is used in the same broad sense as by Godambe & Thompson (1984). Let Y be a class of probability distributions F on R' and 0 = 0(F), F E jZ be a real parameter. Let hi be a real function of y 1, yi and 0 be such that

A multivariate Weierstrass-Mandelbrot function

Abstract: The univariate Weierstrass-Mandelbrot function is generalized to many variables to model higher dimensional stochastic processes such as undersea topography. Because this topography is difficult to measure at small length scales over the many large regions that affect long-ranged acoustic propagation in the ocean, one needs a stochastic description that can be extrapolated to both large and small features. Fractal surfaces are a convenient framework for such a description. Computer-generated plots for the two-variable case are presented. It is shown that in the continuum limit the multivariate function is equivalent to the multivariate fractional Brownian motion.

The Wasserstein distance and approximation theorems

Abstract: By an extension of the idea of the multivariate quantile transform we obtain an explicit formula for the Wasserstein distance between multivariate distributions in certain cases For the general case we use a modification of the definition of the Wasserstein distance and determine optimal ‘markov-constructions’ We give some applications to the problem of approximation of stochastic processes by simpler ones, as eg weakly dependent processes by independent sequences and, finally, determine the optimal martingale approximation to a given sequence of random variables; the Doob decomposition gives only the ‘one-step optimal’ approximation

Learning by statistical cooperation of self-interested neuron-like computing elements.

Abstract: Since the usual approaches to cooperative computation in networks of neuron-like computating elements do not assume that network components have any "preferences", they do not make substantive contact with game theoretic concepts, despite their use of some of the same terminology. In the approach presented here, however, each network component, or adaptive element, is a self-interested agent that prefers some inputs over others and "works" toward obtaining the most highly preferred inputs. Here we describe an adaptive element that is robust enough to learn to cooperate with other elements like itself in order to further its self-interests. It is argued that some of the longstanding problems concerning adaptation and learning by networks might be solvable by this form of cooperativity, and computer simulation experiments are described that show how networks of self-interested components that are sufficiently robust can solve rather difficult learning problems. We then place the approach in its proper historical and theoretical perspective through comparison with a number of related algorithms. A secondary aim of this article is to suggest that beyond what is explicitly illustrated here, there is a wealth of ideas from game theory and allied disciplines such as mathematical economics that can be of use in thinking about cooperative computation in both nervous systems and man-made systems.

Review of 'Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory' (Kushner, H.J.; 1984)

Abstract: Theorem 3: For kernels of the type defined in (15a) and (15b), we have where m k = c c mk(i>> qEP(k) icf(q) (19)

ARMA Representation of Random Processes

Abstract: Auto-regressive moving-average (ARMA) models of the same order for AR and MA components are used for the characterization and simulation of stationary Gaussian multivariate random processes with zero mean. The coefficient matrices of the ARMA models are determined so that the simulated process will have the prescribed correlation function matrix. To accomplish this, the two-stage least squares method is used. The ARMA representation thus established permits one, in principle, to generate sample functions of infinite length and with such a speed and computational mode that even real time generations of the sample functions can be easily achieved. The numerical example indicates that the sample functions generated by the method presented herein reproduce the prescribed correlation function matrix extremely well despite the fact that these sample functions are all very long. This is seen from the closeness between the analytically prescribed auto- and cross-correlation functions and the corresponding sample correlations computed from the generated sample functions.

S-1, S Policies for Perishable Inventory

Abstract: We consider S-1, S policies for a single item whose lifetime is fixed and known with certainty. Demands are generated by a stationary Poisson process and there is a positive leadtime for replenishment. We believe this study gives the only analysis for perishables with a positive order leadtime. The analysis involves the derivation of the stationary distribution of the S-dimensional stochastic process corresponding to the time elapsed since the last S orders were placed. This distribution is then used to obtain an expression for the expected cost rate of operating the system in steady state as a function of S. A computer program has been developed to compute optimal S values and expected annual costs. We report a computation for a variety of system parameters which show some of the unusual features of the problem. Finally, we show how this model can be used in the context of a problem of optimizing availability of operating equipment subject to scheduled maintenance as well as random failure.

Additive control of stochastic linear systems with finite horizon

Abstract: We consider a dynamic system whose state is governed by a linear stochastic differential equation with time-dependent coefficients. The control acts additively on the state of the system. Our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process. It is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space-time domain. By using some decomposition arguments, the problems of a two-sided control, i.e. optimal corrections, and the case with constraints on the resources, i.e. finite fuel, can be reduced to a simpler case of only one-sided control, i.e. a monotone follower. These results are applied to solving some examples by the so-called method of similarity solutions.

The behavior of intra-daily exchange rates

Abstract: The behavior of exchange rates is examined as they evolve continuously over time. The data consist of Swiss franc/U.S. dollar rates for nine days during the years 1978–1980 as quoted by a major Swiss dealer operating on the interbank market. Since this market is highly organized, the observations are market prices at the same time. The distributions of relative changes in exchange rates measured over one minute are highly leptokurtic. The normal distribution is rather rapidly approached when the measurement interval is lengthened from one up to ten minutes. Time series analysis reveals that the natural logarithms of exchange rates are adequately described by a random walk, the same stochastic process as has been found for daily, weekly, monthly and quarterly observations. For short time intervals, significant autocorrelations sometimes occur at the first few lags, which are, however, not stable enough over time to form a basis for reliable forecasts.

Memory function approaches to stochastic problems in condensed matter

Abstract: Theoretical Foundations Basic Description of the Rules Leading to the Adiabatic Elimination of Fast Variables Continued Fractions in the Theory of Relaxation Memory Function Methods in Solid State Physics Molecular Dynamics: Intense External Fields Non Linear Effects in Molecular Dynamics of the Liquid State Dynamical Properties of Hydrogen-Bonded Liquids Slow Motion EPR Spectra in Terms of a Generalized Langevin Equation The Theory of Chemical Reaction Rates Experimental Investigation on the Effect of Multiplicative Noise by Means of Electric Circuits Interdisciplinary Subjects: The Time Properties of a Model of Random Fluctuating Selection Stochastic Processes in Astrophysics: Stellar Formation and Galactic Evolution Author and Subject Indexes.

VI Wave Propagation in Random Media: A System Approach

Abstract: Publisher Summary This chapter discusses a system approach of wave propagation in random media. The chapter presents the review of the properties of electromagnetic wave propagation in a random medium in the limit when the random spatial inhomogeneities in the medium are large in comparison with the wavelength of the radiation and the magnitude of the index of refraction fluctuations (produced by these random inhomogeneities) is small in comparison with unity. It is also being assumed that the electromagnetic field is sufficiently weak, so, non-linear effects can be ignored, and that the propagation path is not so long that there is a saturation of the scintillations. The treatment begins with the vector form of the Maxwell wave equation, which is used to derive a generalized version of the Huygens-Fresnel Principle. This serves as the basis of all of the important results to be obtained.

A Frequency Domain Approach to Water Quality Modeling in Groundwater: Theory

Abstract: A method for the analysis of complex temporal variations of environmental tracers or pollution time series in groundwater is examined using spectral analysis and linear filter theory for stationary stochastic processes. The interpretation of solute fluctuations subject to a time varying source is accomplished via frequency domain solutions to stochastic differential equations for three widely applied transport models: (1) a lumped parameter or linear reservoir model; (2) convective (advective) transport in a curvilinear flow field; and (3) convective-dispersive transport in a uniform flow field. Frequency domain solutions are presented in terms of the theoretical transfer function and phase spectra which describe the amplitude attenuation and phase lag between frequencies in the input and output. A comparison of the frequency response of the three models indicates that the unique filtering characteristics of each may provide a diagnostic tool for matching the appropriate theory to a sampled water quality “signal.” A procedure is suggested for parameter estimation which involves comparison of the theoretical and field estimated transfer function and phase spectra.

Identification of dynamic systems, theory and formulation

Abstract: The problem of estimating parameters of dynamic systems is addressed in order to present the theoretical basis of system identification and parameter estimation in a manner that is complete and rigorous, yet understandable with minimal prerequisites. Maximum likelihood and related estimators are highlighted. The approach used requires familiarity with calculus, linear algebra, and probability, but does not require knowledge of stochastic processes or functional analysis. The treatment emphasizes unification of the various areas in estimation in dynamic systems is treated as a direct outgrowth of the static system theory. Topics covered include basic concepts and definitions; numerical optimization methods; probability; statistical estimators; estimation in static systems; stochastic processes; state estimation in dynamic systems; output error, filter error, and equation error methods of parameter estimation in dynamic systems, and the accuracy of the estimates.

Global optimization via the Langevin equation

Abstract: We provide a simple proof of the convergence of the cooling algorithms, i.e., the annealing algorithm and the Langevin equation. The convergence is established for temperature schedules which are very near to optimal ones. Our methods are based on Differential Equations techniques.

Mean stochastic comparison of diffusions

Abstract: Stochastic bounds are derived for one dimensional diffusions (and somewhat more general random processes) by dominating one process pathwise by a convex combination of other processes. The method permits comparison of diffusions with different diffusion coefficients. One interpretation of the bounds is that an optimal control is identified for certain diffusions with controlled drift and diffusion coefficients, when the reward function is convex. An example is given to show how the bounds and the Liapunov function technique can be applied to yield bounds for multidimensional diffusions.

Frequency-domain implementations of periodically time-varying filters

Abstract: Cyclostationary random processes have statistics that vary periodically in time. Optimum filtering of cyclostationary signals requires a filter whose impulse response also varies periodically in time. This paper describes computationally efficient methods for implementing periodically time-varying filters in the frequency domain. Both fixed and adaptive filtering is discussed. Two frequency-domain approaches are considered-overlap-save and transmultiplexer. A transmultiplexer is an efficient way of implementing a bank of bandpass filters. The overlap-save technique is found to be slightly more efficient than the transmultiplexer approach for fixed filters. For adaptive filtering, however, the transmultiplexer approach has an advantage over the overlap-save technique because the transmultiplexer allows the filter weights associated with each frequency to be adapted independently of weights for other frequencies.

Effect of exchange reactions on transport processes: A comparison of thermodynamic treatment with random walk on lattice points

Abstract: The effect of an AX+A ⇄ A+AX exchange reaction on the transport of X is discussed for a general transport process driven by the gradient in the chemical potential of AX, as well as its special cases of isothermal diffusion and electric conductivity. This treatment based on macrodifferentials is compared with a stochastic treatment of random walk on regular lattice points in one‐, two‐ , and three‐dimensional cases. The basic equivalence is proved for the two methods which ensures the application of the latter, more simple, procedure with general validity. It is also shown that the stochastic treatment reveals some aspects that remain hidden in the thermodynamic approach.