Showing papers on "Stochastic process published in 1986"

The statistics of peaks of Gaussian random fields

Abstract: A set of new mathematical results on the theory of Gaussian random fields is presented, and the application of such calculations in cosmology to treat questions of structure formation from small-amplitude initial density fluctuations is addressed. The point process equation is discussed, giving the general formula for the average number density of peaks. The problem of the proper conditional probability constraints appropriate to maxima are examined using a one-dimensional illustration. The average density of maxima of a general three-dimensional Gaussian field is calculated as a function of heights of the maxima, and the average density of 'upcrossing' points on density contour surfaces is computed. The number density of peaks subject to the constraint that the large-scale density field be fixed is determined and used to discuss the segregation of high peaks from the underlying mass distribution. The machinery to calculate n-point peak-peak correlation functions is determined, as are the shapes of the profiles about maxima.

Comparison methods for queues and other stochastic models

Abstract: Studies stochastic models of queueing, reliability, inventory, and sequencing in which random influences are considered. One stochastic mode--rl is approximated by another that is simpler in structure or about which simpler assumptions can be made. After general results on comparison properties of random variables and stochastic processes are given, the properties are illustrated by application to various queueing models and questions in experimental design, renewal and reliability theory, PERT networks and branching processes.

Structural reliability under incomplete probability information

Abstract: A comprehensive framework is set forth for the analysis of structural reliability under incomplete probability information. Under stipulated requirements of consistency, invariance, operability, and simplicity, a method is developed to incorporate in the reliability analysis incomplete probability information on random variables, including moments, bounds, marginal distributions, and partial joint distributions. The method is consistent with the philosophy of Ditlevsen’s generalized reliability index and complements existing second-moment and full-distribution structural reliability theories.

Stochastic subsurface hydrology from theory to applications

Abstract: Research on stochastic analysis of subsurface flow has developed rapidly in the last decade, but applications of this approach have been very limited. The purpose of this paper is to illustrate how currently available techniques and results can be used to answer important questions about the large-scale behavior of naturally heterogeneous aquifers. Perturbation-based spectral theory, which presumes local statistical homogeneity, provides generic theoretical results for the head variance, effective conductivity tensor, and macrodispersivity tensor in a field. These results emphasize the key role of the variance and spatial correlation scales of the log hydraulic conductivity field. Field information of variances and correlation scales of natural materials is summarized. The validity of some of the generic stochastic results is evaluated through comparisons with Monte Carlo simulations and field observations. A specific field application example is developed to illustrate how the stochastic results are used to estimate large-scale parameters and determine the reliability of three-dimensional numerical simulations. Using typical log conductivity covariance parameters, the effective hydraulic conductivity tensor, and the macrodispersivity tensor are estimated. The calculated head variance, based on the simulated mean hydraulic gradient, is used as a measure of the adequacy of the calculation of the steady state flow model. Discussion emphasizes limitations and extensions of this approach, and ongoing field evaluations of the results.

Maximum Likelihood Estimation of Generalized Ito Processes with Discretely Sampled Data

Abstract: In this paper, we consider the parametric estimation problem for continuous time stochastic processes described by general first-order nonlinear stochastic differential equations of the Ito type We characterize the likelihood function of a discretely-sampled set of observations as the solution to a functional partial differential equation The consistency and asymptotic normality of the maximum likelihood estimators are explored, and several illustrative examples are provided

A statistical method for identifying transition states in high dimensional problems

Abstract: A statistical method based on Monte Carlo sampling of paths associated with Markovian stochastic processes is proposed for the location of transition states in molecularly complex reactions. It is discussed in what respects the proposed method would identify transition state regions, and an algorithm for Metropolis sampling of Metropolis paths (based on the Metropolis random walk process) is devised. Physical aspects of this new algorithm are examined, and the relation of this method to simulated annealing algorithms for physical design of circuits is noted.

Stochastic Modelling and Control

Abstract: 1 Probability and linear system theory.- 1.1 Probability and random processes.- 1.2 Linear system theory.- Notes and references.- 2 Stochastic models.- 2.1 A general output process.- 2.2 Stochastic difference equations.- 2.3 ARMA noise models.- 2.4 Stochastic dynamical models.- 2.5 Innovations representations.- 2.6 Predictor models.- Notes and references.- 3 Filtering theory.- 3.1 The geometry of linear estimation.- 3.2 Recursive estimation.- 3.3 The Kalman filter.- 3.4 Innovations representation of state-space models.- Notes and references.- 4 System identification.- 4.1 Point estimation theory.- 4.2 Models.- 4.3 Parameter estimation for static systems.- 4.4 Parameter estimation for dynamical systems.- 4.5 Off-line identification algorithms.- 4.6 Algorithms for on-line parameter estimation.- 4.7 Bias arising from correlated disturbances.- 4.8 Three-stage least squares and order determination for scalar ARMAX models.- Notes and references.- 5 Asymptotic analysis of prediction error identification methods.- 5.1 Preliminary concepts and definitions.- 5.2 Asymptotic properties of the parameter estimates.- 5.3 Consistency.- 5.4 Interpretation of identification in terms of systems approximation.- Notes and references.- 6 Optimal control for state-space models.- 6.1 The deterministic linear regulator.- 6.2 The stochastic linear regulator.- 6.3 Partial observations and the separation principle.- Notes and references.- 7 Minimum variance and self-tuning control.- 7.1 Regulation for systems with known parameters.- 7.2 Pole/zero shifting regulators.- 7.3 Self-tuning regulators.- 7.4 A self-tuning controller with guaranteed convergence.- Notes and references.- Appendix A A uniform convergence theorem and proof of Theorem 5.2.1.- Appendix B The algebraic Riccati equation.- Appendix C Proof of Theorem 7.4.2.- Appendix D Some properties of matrices.- Appendix E Some inequalities of Holder type.- Author index.

A criterion of convergence of measure-valued processes: Application to measure branching processes

Abstract: In this paper martingale properties of a Measure Branching process are investigated. Uniqueness and continuity of this process are proven by a martingale approach. For the existence, we approximate the measure branching process by a sequence of infinite particle branching diffusion processes, and show the convergence in distribution by a new criterion for measure‐valued processes. We also give properties about local structure of the process.

A weighted occupation time for a class of measured-valued branching processes

Abstract: A weighted occupation time is defined for measure-valued processes and a representation for it is obtained for a class of measure-valued branching random motions on Rd. Considered as a process in its own right, the first and second order asymptotics are found as time t→∞. Specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d, and when infinite, a central limit type renormalization is considered, yielding Gaussian or asymmetric stable generalized random fields in the limit. In one Gaussian case the results are contrasted in high versus low dimensions.

Probability: An Introduction

Abstract: This new undergraduate text offers a concise introduction to probability and random processes. Exercises and problems range from simple to difficult, and the overall treatment, though elementary, includes rigorous mathematical arguments. Chapters contain core material for a beginning course in probability, a treatment of joint distributions leading to accounts of moment-generating functions, the law of large numbers and the central limit theorem, and basic random processes.

Recursive estimation and time-series analysis

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The fractal random array

Abstract: A novel method of random array synthesis is presented. This new method relies on the application of the underlying self-similarity inherent in random fractals to the problem of antenna array theory. Several examples demonstrate the synthesis procedure and its usefulness in producing robust, low sidelobe arrays.

Introduction to Random Processes

Abstract: Introduction to random processes , Introduction to random processes , کتابخانه دیجیتال جندی شاپور اهواز

Linear transport theory in a random medium

Abstract: The time‐independent linear transport problem in a purely absorbing (no scattering) random medium is considered. A formally exact equation for the ensemble averaged distribution function 〈Ψ〉 is derived. Under the assumption of a two‐fluid statistical mixture, with the transition from one fluid to the other assumed to be determined by a Markov process, an exact solution to this equation for 〈Ψ〉 is obtained. In the source‐free case, this solution is shown to agree with the result obtained by ensemble averaging simple exponential attenuation. Several approximations to the exact equation for 〈Ψ〉 are considered, and numerical results given to assess the accuracy of these approximations.

Non‐Markovian activated rate processes: Comparison of current theories with numerical simulation data

Abstract: We calculate the barrier crossing rate constants for a Brownian particle in a double well potential experiencing a non‐Markovian friction kernel using a full stochastic simulation. We compare the simulation results with recently proposed interpolation formulas which are based on the Grote–Hynes theory and the energy diffusion mechanism. We find that such formulas can fail by orders of magnitude in a physically interesting regime. Slow activation in an effective dynamic double well potential is probably responsible for the deviations observed.

Testing for spatial association between a point process and another stochastic process

Abstract: Motivated by a problem in geology, this paper proposes some tests of the spatial association between a point process and some other stochastic process of geometric structures, G. All the tests are performed conditionally on the realization of G. Under the null hypothesis that the point process is a stationary Poisson process independent of G, some of these statistics have well‐known distributional properties, even in small samples. The Poisson assumption is relaxed using a conditional Monte Carlo test suggested by Lotwick and Silverman (1982). The tests are applied to a geological data set.

Some doubly stochastic time series models

Abstract: . We consider time series models obtained by replacing the parameters of autoregressive models by stochastic processes. Special attention is given to the problem of finding conditions for stationarity and to the problem of forecasting. For the first problem we are only able to obtain solutions in special cases, and the emphasis is on techniques rather than obtaining the most general results in each case. For the second problem more complete results are obtained by exploiting similarities with discrete time (nonlinear) filtering theory. The methods introduced are illustrated on two standard examples, one of state space type and one where the parameter process is a Markov chain.

An algorithm for detecting a change in a stochastic process

Abstract: The problem of detecting a change from one given stationary and ergodic stochastic process to another such process is considered. It is assumed that both stochastic processes are processes with memory and that they are mutually independent. A sequential test is proposed and analyzed. It is proved that the proposed test is asymptotically optimal in a mathematically precise sense.

On the Convergence in Distribution of Measurable Multifunctions Random Sets Normal Integrands, Stochastic Processes and Stochastic Infima

Abstract: The concept of the distribution function of a closed-valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi-convergence in distribution of normal integrands and stochastic processes; in particular various compactness criteria are exhibited. The connections with the classical convergence theory for stochastic processes are analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of Donsker's Theorem Brownian motion as a limit of normalized random walks. We also suggest the potential application of the theory to the study of the convergence of stochastic infima.

Stationary min-stable stochastic processes

Abstract: We consider the class of stationary stochastic processes whose margins are jointly min-stable. We show how the scalar elements can be generated by a single realization of a standard homogeneous Poisson process on the upper half-strip [0,1] x R+ and a group of L1 — isometries. We include a Dobrushin-like result for the realizations in continuous time.

Parameter drift in LMS adaptive filters

Abstract: We examine general conditions under which the LMS adaptive filter generates unbounded parameter estimates when driven by bounded sequences. This unexpected parameter divergence, or drift, is related to the inadequacy of excitation in the input sequence and is characterized by slow (i.e., nonexponential) escape of the parameter estimate vector to infinity in spite of all other signals (inputs, outputs, prediction errors) remaining bounded or even decaying to zero. The analysis proceeds by showing that, in a general adaptive filtering setting, the sequence of regressors (information vectors) provides a natural decomposition of the parameter estimate space into subspaces, each corresponding to a characteristic class of filter excitation. This subspace decomposition is applied to the LMS adaptive filter, yielding direct links between filter behavior and modes of excitation. In particular, stability and instability of the parameter estimates in the presence of bounded disturbances and prediction errors is obtained for various classes of excitation. This behavior is examined in detail for the first-and second-order cases, and is sufficient to characterize the behavior of higher order adaptive filters. The instability (drift) results are due to modes of "decaying" excitation, and bounds on the rates of drift are derived. This drift mechanism is inherent in the algorithm and is not due to numerical implementation problems or violation of small step-size conditions. Examples are presented where drift may occur in restricted complexity filtering and in the related stochastic gradient algorithm. An analysis of leakage in terms of input excitation reveals a tradeoff in performance between parameter and prediction errors. A modified form of leakage, using the subspace decomposition, is suggested to remove this difficulty.

Some Developments in the Use of Empirical Orthogonal Functions for Mapping Meteorological Fields

Abstract: Some current uses of empirical orthogonal functions (EOF) are briefly summarized, together with some relations with spectral and principal component analyses. Considered as a mean square estimation technique of unknown values within a random process or field, the optimization of error variance leads to a Fredholm integral equation. Its kernel is the autocorrelation function, which in many practical cases is only known as discrete values of interstation correlation coefficients computed from a sample of independent realizations. The numerical solution in one or two dimensions of this integral equation is approximated in a new and more general framework that requires, in practice, the a priori choice of a set of generating functions. Developments are provided for piecewise constant, facetlike linear, and thin plate type spline functions. The first part of the paper ends with a review of the mapping, archiving and stochastic simulating possibilities of the EOF method. A second part includes a case s...

Stochastic Teams with Nonclassical Information Revisited: When is an Affine Law Optimal?

Abstract: In this paper we consider a parameterized family of two-stage stochastic control problems with nonclassical information patterns, which includes the famous 1968 counterexample of Witsenhausen. We show that the parameter region can be partitioned into two regions, in one of which the optimal solution is linear whereas in the other it is inherently nonlinear. In the latter, the best piecewise-constant solution does not always outperform the best linear solution, whereas a linear plus piecewise-constant policy leads to a uniformly better performance in that region. Extensive numerical computations complement the study.

Levy walks with applications to turbulence and chaos

Abstract: Diffusion on fractal structures has been a popular topic of research in the last few years with much emphasis on the sublinear behavior in time of the mean square displacement of a random walker. Another type of diffusion is encountered in turbulent flows with the mean square displacement being superlinear in time. We introduce a novel stochastic process, called a Levy walk which generalizes fractal Brownian motion, to provide a statistical theory for motion in the fractal media which exists in a turbulent flow. The Levy walk describes random (but still correlated) motion in space and time in a scaling fashion and is able to account for the motion of particles in a hierarchy of coherent structures. We apply our model to the description of fluctuating fluid flow. When Kolmogorov's − 5 3 law for homogeneous turbulence is used to determine the memory of the Levy walk then Richardson's 4 3 law of turbulent diffusion follows in the Mandelbrot absolute curdling limit. If, as suggested by Mandelbrot, that turbulence is isotropic, but fractal, then intermittency corrections to the − 5 3 law follow in a natural fashion. The same process, with a different space-time scaling provides a description of chaos in a Josephson junction.

Stochastic thermodynamics of nonequilibrium steady states in chemical reaction systems

Abstract: We present a stochastic theory of entropy production for steady states in chemical reaction systems. Small scale internal fluctuations around steady states are considered in the Gaussian regime. It is shown that in addition to the usual Gibbsian form of entropy production, there is an entropy production due to fluctuation which is of order O(V0). This comes from the non‐Poisson character of the probability distribution in a nonequilibrium system. Two approaches are considered: in the first, we use an entropy balance equation based on the master equation; in the second, we use a stochastic potential related to the probability distribution and built from the generalized Einstein relation. We show that both approaches give the same result for the entropy production of fluctuation (diS/dt) f . Next we consider a simple one‐component nonequilibrium system under the perturbation of a macroscopically large external fluctuation as a power generator. We interpret (diS/dt) f in terms of net power gain factor under...

A note on shot-noise and reliability modeling

Abstract: We discuss a reliability model that reflects the dynamic dependency between system failure and system stress For example, the mortality rate for individuals who have had a heart attack declines with time In particular, we use a shot-noise process to model "residual system stress," which in turn drives a doubly stochastic Poisson process model for system failures Intuitively, residual stress or susceptibility to failure may vary in a random manner yet be essentially unobservable, while system failures may be readily detectable and observable Shot-noise distributions have a richness and subtlety that suggest untapped potential for applications The doubly stochastic Poisson process provides a reasonable framework for modeling randomly varying rates of occurrence in a broad variety of settings