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Showing papers on "Stochastic process published in 1991"


Journal Article
21 Mar 1991
TL;DR: In this article, the authors introduce the concept of Stationary Random Processes and Spectral Analysis in the Time Domain and Frequency Domain, and present an analysis of Processes with Mixed Spectra.
Abstract: Preface. Preface to Volume 2. Contents of Volume 2. List of Main Notation. Basic Concepts. Elements of Probability Theory. Stationary Random Processes. Spectral Analysis. Estimation in the Time Domain. Estimation in the Frequency Domain. Spectral Analysis in Practice. Analysis of Processes with Mixed Spectra.

5,238 citations



Journal ArticleDOI
TL;DR: This article is concerned with prediction of a function y(t) over a (multidimensional) domain T, given the function values at a set of “sites” in T, and with the design, that is, with the selection of those sites.
Abstract: This article is concerned with prediction of a function y(t) over a (multidimensional) domain T, given the function values at a set of “sites” {t (1), t (2), …, t (n)} in T, and with the design, that is, with the selection of those sites. The motivating application is the design and analysis of computer experiments, where t determines the input to a computer model of a physical or behavioral system, and y(t) is a response that is part of the output or is calculated from it. Following a Bayesian formulation, prior uncertainty about the function y is expressed by means of a random function Y, which is taken here to be a Gaussian stochastic process. The mean of the posterior process can be used as the prediction function ŷ(t), and the variance can be used as a measure of uncertainty. This kind of approach has been used previously in Bayesian interpolation and is strongly related to the kriging methods used in geostatistics. Here emphasis is placed on product linear and product cubic correlation func...

789 citations


Journal ArticleDOI
TL;DR: In this paper, an analytical characterization of polarization dispersion measurements is presented, where the authors report the solution of Poole's stochastic dynamical equation for the evolution of the dispersion vector with fiber length.
Abstract: An analytical characterization of polarization dispersion measurements is presented. The authors report the solution of Poole's stochastic dynamical equation for the evolution of the polarization dispersion vector with fiber length. The authors extend this to a more complete description by considering small, second-order dispersion effects through the frequency derivative of the dispersion vector. The complete analytical solution is seen to accord with what were originally empirically derived features of the joint probability distribution of the polarization dispersion vector and its frequency derivatives. Among the analytically determined properties are the Gaussian probability densities of the three components of the dispersion vector, and the hyperbolic secant (soliton shaped) probability densities of the components of the derivative of the dispersion vector. >

526 citations


Book ChapterDOI
01 Oct 1991
TL;DR: In this paper, the authors discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes, aimed at theoretical probabilists.
Abstract: INTRODUCTION Many different models of random trees have arisen in a variety of applied setting, and there is a large but scattered literature on exact and asymptotic results for particular models. For several years I have been interested in what kinds of “general theory” (as opposed to ad hoc analysis of particular models) might be useful in studying asymptotics of random trees. In this paper, aimed at theoretical probabilists, I discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes. No prior knowledge of this subject is assumed: the paper is intended as an introduction and survey. To give the really big picture in a paragraph, consider a tree on n vertices. View the vertices as points in abstract (rather than d -dimensional) space, but let the edges have length (= 1, as a default) so that there is metric structure: the distance between two vertices is the length of the path between them. Consider the average distance between pairs of vertices. As n → ∞ this average distance could stay bounded or could grow as order n , but almost all natural random trees fall into one of two categories. In the first (and larger) category, the average distance grows as order logn. This category includes supercritical branching processes, and most “Markovian growth” models such as those occurring in the analysis of algorithms. This paper is concerned with the second category, in which the average distance grows as order n ½ .

507 citations


Journal ArticleDOI
TL;DR: In this article, the authors extend Takens' treatment, applying statistical methods to incorporate the effects of observational noise and estimation error, and derive asymptotic scaling laws for distortion and noise amplification.

505 citations


Journal ArticleDOI
TL;DR: This paper presents general properties of periodically driven Brownian motion, such as the long-time behavior of correlation functions and the existence of a ``supersymmetric'' partner system, and discovers a secondary resonance at smaller values of the noise strength in the regime of intermediate driving frequencies.
Abstract: Stochastic resonance is a cooperative effect of noise and periodic driving in bistable systems. It can be used for the detection and amplification of weak signals embedded within a large noise background. In doing so, the noise triggers the transfer of power to the signal. In this paper we first present general properties of periodically driven Brownian motion, such as the long-time behavior of correlation functions and the existence of a ``supersymmetric'' partner system. Within the framework of nonstationary stochastic processes, we present a careful numerical study of the stochastic resonance effect, without restrictions on the modulation amplitude and frequency. In particular, in the regime of intermediate driving frequencies which has not yet been covered by theories, we have discovered a secondary resonance at smaller values of the noise strength.

501 citations


Book
Han-Fu Chen1, Lei Guo1
01 Nov 1991
TL;DR: In this paper, the authors proposed a Kalman filter-based adaptive control algorithm for tracking systems with Random Variables and Stochastic Integral Equations (RIE) to achieve stability and consistency.
Abstract: 1 Probability Theory Preliminaries.- 1.1 Random Variables.- 1.2 Expectation.- 1.3 Conditional Expectation.- 1.4 Independence, Characteristic Functions.- 1.5 Random Processes.- 1.6 Stochastic Integral.- 1.7 Stochastic Differential Equations.- 2 Limit Theorems on Martingales.- 2.1 Martingale Convergence Theorems.- 2.2 Local Convergence Theorems.- 2.3 Estimation for Weighted Sums of a Martingale Difference Sequence.- 2.4 Estimation for Double Array Martingales.- 3 Filtering and Control for Linear Systems.- 3.1 Controllability and Observability.- 3.2 Kalman Filtering for Systems with Random Coefficients.- 3.3 Discrete-Time Riccati Equations.- 3.4 Optimal Control for Quadratic Costs.- 3.5 Optimal Tracking.- 3.6 Model Reference Control.- 3.7 Control for CARIMA Models.- 4 Coefficient Estimation for ARMAX Models.- 4.1 Estimation Algorithms.- 4.2 Convergence of ELS Without the PE Condition.- 4.3 Local Convergence of SG.- 4.4 Convergence of SG Without the PE Condition.- 4.5 Convergence Rate of SG.- 4.6 Removing the SPR Condition By An Overparameterization Technique.- 4.7 Removing the SPR Condition By Using Increasing Lag Least Squares.- 5 Stochastic Adaptive Tracking.- 5.1 SG-Based Adaptive Tracker With d = 1.- 5.2 SG-Based Adaptive Tracker With d ?1.- 5.3 Stability and Optimality of Astrom-Wittenmark Self-Tuning Tracker.- 5.4 Stability and Optimality of ELS-Based Adaptive Trackers.- 5.5 Model Reference Adaptive Control.- 6 Coefficient Estimation in Adaptive Control Systems.- 6.1 Necessity of Excitation for Consistency of Estimates.- 6.2 Reference Signal With Decaying Richness.- 6.3 Diminishingly Excited Control.- 7 Order Estimation.- 7.1 Order Estimation by Use of a Priori Information.- 7.2 Order Estimation by not Using Upper Bounds for Orders.- 7.3 Time-Delay Estimation.- 7.4 Connections of CIC and BIC.- 8 Optimal Adaptive Control with Consistent Parameter Estimate.- 8.1 Simultaneously Gaining Optimality and Consistency in Tracking Systems.- 8.2 Adaptive Control for Quadratic Cost.- 8.3 Connection Between Adaptive Controls for Tracking and Quadratic Cost.- 8.4 Model Reference Adaptive Control With Consistent Estimate.- 8.5 Adaptive Control With Unknown Orders, Time-Delay and Coefficients.- 9 ARX(?) Model Approximation.- 9.1 Statement of Problem.- 9.2 Transfer Function Approximation.- 9.3 Estimation of Noise Process.- 10 Estimation for Time-Varying Parameters.- 10.1 Stability of Random Time-Varying Equations.- 10.2 Conditional Richness Condition.- 10.3 Analysis of Kalman Filter Based Algorithms.- 10.4 Analysis of LMS-Like Algorithms.- 11 Adaptive Control of Time-Varying Stochastic Systems.- 11.1 Preliminary Results.- 11.2 Systems with Random Parameters.- 11.3 Systems with Deterministic Parameters.- 12 Continuous-Time Stochastic Systems.- 12.1 The Model.- 12.2 Parameter Estimation.- 12.3 Adaptive Control.- References.

473 citations


Journal ArticleDOI
TL;DR: In this paper, a generalized Brownian motion is given by creation and annihilation operators on a "twisted" Fock space of L2(ℝ), and the distribution of these operators with respect to the vacuum expectation is a generalized Gaussian distribution, in the sense that all moments can be calculated from second moments with the help of a combinatorial formula.
Abstract: We present an example of a generalized Brownian motion. It is given by creation and annihilation operators on a “twisted” Fock space ofL2(ℝ). These operators fulfill (for a fixed −1≦μ≦1) the relationsc(f)c*(g)−μc*(g)c(f)=〈f,g〉1 (f, g ∈L2(ℝ)). We show that the distribution of these operators with respect to the vacuum expectation is a generalized Gaussian distribution, in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula. We also indicate that our Brownian motion is one component of ann-dimensional Brownian motion which is invariant under the quantum groupS ν U(n) of Woronowicz (withμ =v2).

390 citations


Journal ArticleDOI
TL;DR: In this paper, a second-order autoregressive equation is used to model the acceleration of fluid particles in turbulence in order to study the effect of Reynolds number on Lagrangian turbulence statistics.
Abstract: A second‐order autoregressive equation is used to model the acceleration of fluid particles in turbulence in order to study the effect of Reynolds number on Lagrangian turbulence statistics. It is shown that this approach provides a good representation of dissipation subrange structure of Lagrangian velocity and acceleration statistics. The parameters of the model, two time scales representing the energy‐containing and dissipation scales, are determined by matching the model velocity autocorrelation function to Kolmogorov similarity forms in the inertial subrange and the dissipation subrange. The model is tested against the Lagrangian statistics obtained by Yeung and Pope [J. Fluid Mech. 207, 531 (1989)] from direct numerical simulations of turbulence. Agreement between the model predictions and simulation data for second‐order Lagrangian statistics such as the velocity structure function, the acceleration correlation function, and the dispersion of fluid particles is excellent, indicating that the main departures from Kolmogorov’s theory of local isotropy shown by the simulation data are due to low Reynolds number. For Reynolds numbers typical of laboratory experiments and direct numerical simulations of turbulence the root‐mean‐square dispersion of marked particles is changed from the Langevin equation (i.e., infinite Reynolds number) prediction by up to about 50% at large times. Most of this change can be accounted for by the change in the Lagrangian integral time scale. It is also shown that Reynolds number effects in laboratory dispersion or Lagrangian turbulence measurements can cause significant errors (typically of order 50%) when the value of the Kolmogorov Lagrangian structure function constant C0 is estimated by fitting the predictions of the Langevin equation to these data. A value C0 = 7 is obtained by fitting the new model to the direct simulation data.

347 citations


Book
01 Jan 1991
TL;DR: Probability Concepts.
Abstract: Probability Concepts. Distributions. Random Processes. Discrete-Time Random Processes. Statistical Decision Theory. Parameter Estimation. Filtering. Representation of Signals. The General Gaussian Problem. Detection and Parameter Estimation. Adaptive Thresholding CFAR Detection. Distributed CFAR Detection.

Proceedings ArticleDOI
03 Jun 1991
TL;DR: A highly parallel incremental stochastic minimization algorithm is presented which has a number of advantages over previous approaches and the incremental nature of the scheme makes it dynamic and permits the detection of occlusion and disocclusion boundaries.
Abstract: A novel approach to incrementally estimating visual motion over a sequence of images is presented. The authors start by formulating constraints on image motion to account for the possibility of multiple motions. This is achieved by exploiting the notions of weak continuity and robust statistics in the formulation of a minimization problem. The resulting objective function is non-convex. Traditional stochastic relaxation techniques for minimizing such functions prove inappropriate for the task. A highly parallel incremental stochastic minimization algorithm is presented which has a number of advantages over previous approaches. The incremental nature of the scheme makes it dynamic and permits the detection of occlusion and disocclusion boundaries. >

Journal ArticleDOI
TL;DR: In this article, a spectral expansion of the nodal random variables is introduced involving a basis in the space of random variables, which consists of the polynomial chaoses that are polynomials orthogonal with respect to the Gaussian probability measure.
Abstract: An approach for the solution of problems of structural mechanics involving material variability is proposed. The material property is modeled as a stochastic process. The Karhunen-Loeve expansion is used to represent this process in a computationally expedient manner by means of a set of random variables. Further, the well-established deterministic finite-element method is used to discretize the differential equations governing the structural response. A spectral expansion of the nodal random variables is introduced involving a basis in the space of random variables. The basis consists of the polynomial chaoses that are polynomials orthogonal with respect to the Gaussian probability measure. The new formulation allows the computation of the probability distribution functions of the response variables in an expeditious manner. Two problems from structural mechanics are investigated using the proposed approach. The derived results are found in good agreement with data obtained by a Monte Carlo simulation solution of these problems.

Proceedings ArticleDOI
Farid N. Najm1
01 Jun 1991
TL;DR: A new measure of activity, called the transition density, is proposed, which may be defined as the “average switching rate” at a circuit node, based on a stochastic model of logic signals and an algorithm to propagate it from the primary inputs to internal and output nodes is presented.
Abstract: Reliability assessment is an important part of the design process of digital integrated circuits. We observe that a common thread that runs through most causes of run-time failure is the extent of circuit activity, i.e., the rate at which its nodes are switching. We propose a new measure of activity, called the transition density, which may be defined as the “average switching rate” at a circuit node. Based on a stochastic model of logic signals, we rigorously define the transition density and present an algorithm to propagate it from the primary inputs to internal and output nodes. This algorithm may be thought of as a simulation of the circuit, and has been implemented in a prototype density simulator. We present some results of this implementation to verify the theoretical results and assess the feasibility of the approach. In order to obtain the same density information by traditional means, the circuit would need to be simulated for thousands of input transitions. Thus this approach is very efficient and makes possible the analysis of VLSI circuits, which are traditionally too big to simulate for long input sequences.

Journal ArticleDOI
TL;DR: An approach based on a modular state-transition representation of a parallel system called the stochastic automata network (SAN) is developed, which is automatically derived using tensor algebra operators, under a format which involves a very limited storage cost.
Abstract: A methodology for modeling a system composed of parallel activities with synchronization points is proposed. Specifically, an approach based on a modular state-transition representation of a parallel system called the stochastic automata network (SAN) is developed. The state-space explosion is handled by a decomposition technique. The dynamic behavior of the algorithm is analyzed under Markovian assumptions. The transition matrix of the chain is automatically derived using tensor algebra operators, under a format which involves a very limited storage cost. >

Book
01 Oct 1991
TL;DR: General Schemes for Constructing Scalar and Vector Monte Carlo Algorithms for Solving Boundary Value Problems: Random Walks on Boundary and Inside the Domain Algorithm and Numerical Experiments.
Abstract: General Schemes for Constructing Scalar and Vector Monte Carlo Algorithms for Solving Boundary Value Problems: Random Walks on Boundary and Inside the Domain Algorithms. Random Walks and Approximations of Random Processes. Monte Carlo Algorithms for Solving Integral Equations: Algorithms Based on Numerical Analytical Continuation. Asymptotically Unbiased Estimates Based on Singular Approximation of the Kernel. The Eigen-Value Problems for Integral Operators. Alternative Constructions of the Resolvent: Modifications and Numerical Experiments. Monte Carlo Algorithms for Solving Boundary Value Problems of the Potential Theory: The Walk on Boundary Algorithms for Solving Interior and Exterior Boundary Value Problems. Walk Inside the Domain Algorithms. Numerical Solution of Test and Applied Problems of Potential Theory in Deterministic. Monte Carlo Algorithms for Solving High-order Equations and Problems in Elasticity: Biharmonic Problem. Metaharmonic Equations. Spatial Problems of Elasticity Theory. Applications to Stochastic Elasticity Problems. Diffusion Problems: Walk on Boundary Algorithms for the Heat Equation. The Walk Inside the Domain Algorithms. Particle Diffusion in Random Velocity Fields. Applications to Diffusion Problems.

Book
01 Jan 1991
TL;DR: This paper presents a meta-analyses of Homogeneous Processes in R, a model for multiplicative processes based on Abelian groups, with a focus on processes with Independent Increments.
Abstract: 0. Preliminary Informationh.- 0.1 Probability Space.- 0.2 Random Functions and Processes.- 0.3 Conditional Probabilities.- 0.4 Independence.- 1. Sums of Independent Random Variables.- 1.1 Main Inequalities.- 1.2 Renewal Scheme.- 1.3 Random Walks. Recurrence.- 1.4 Distribution of Ladder Functions.- 2. General Processes with Independent Increments (Random Measures).- 2.1 Nonnegative Random Measures with Independent Values (r.m.i.v.).- 2.2 Random Measures with Alternating Signs.- 2.3 Stochastic Integrals and Countably Additive r.m.i.v.- 2.4 Random Linear Functional and Generalized Functions.- 3. Processes with Independent Increments. General Properties.- 3.1 Decomposition of a Process. Properties of Sample Functions.- 3.2 Stochastically Continuous Processes.- 3.3 Properties of Sample Functions.- 3.4 Locally Homogeneous Processes with Independent Increments.- 4. Homogeneous Processes.- 4.1 General Properties.- 4.2 Additive Functionals.- 4.3 Composed Poisson Process.- 4.4 Homogeneous Processes in R.- 5. Multiplicative Processes.- 5.1 Definition and General Properties.- 5.2 Multiplicative Processes in Abelian Groups.- 5.3 Stochastic Semigroups of Linear Operators in Rd.- Notes.- References.

Journal ArticleDOI
TL;DR: Reduced base model construction methods for stochastic activity networks are discussed and examples which illustrate the method and demonstrate its effectiveness in reducing the size of a state space are presented.
Abstract: Reduced base model construction methods for stochastic activity networks are discussed. The basic definitions concerning stochastic networks are reviewed and the types of variables used in the construction process are defined. These variables can be used to estimate both transient and steady-state system characteristics. The construction operations used and theorems stating the validity of the method are presented. A procedure for generating the reduced base model stochastic process for a given stochastic activity network and performance variable is presented. Some examples which illustrate the method and demonstrate its effectiveness in reducing the size of a state space are presented. >

Journal ArticleDOI
TL;DR: In this article, a simple numerical procedure for estimating the stochastic robustness of a linear time-invariant system is described, and confidence intervals for the scalar probability of instability address computational issues inherent in Monte Carlo simulation.
Abstract: A simple numerical procedure for estimating the stochastic robustness of a linear time-invariant system is described. Monte Carlo evaluation of the system's eigenvalues allows the probability of instability and the related stochastic root locus to be estimated. This analysis approach treats not only Gaussian parameter uncertainties but also nonGaussian cases, including uncertain but bound variations. Confidence intervals for the scalar probability of instability address computational issues inherent in Monte Carlo simulation. Trivial extensions of the procedure admit consideration of alternate discriminants; thus, the probabilities that stipulated degrees of instability will be exceeded or that closed-loop roots will leave desirable regions can also be estimated. Results are particularly amenable to graphical presentation. >

Proceedings ArticleDOI
11 Dec 1991
TL;DR: In this article, the authors derive a novel algorithm to consistently identify stochastic state space models from given output data without forming the covariance matrix and using only semi-infinite block Hankel matrices.
Abstract: The authors derive a novel algorithm to consistently identify stochastic state space models from given output data without forming the covariance matrix and using only semi-infinite block Hankel matrices. The algorithm is based on the concept of principle angles and directions. The authors describe how these can be calculated with only QR and QSVD decompositions. They also provide an interpretation of the principle directions as states of a non-steady-state Kalman filter. With a couple of examples, it is shown that the proposed algorithm is superior to the classical canonical correlation algorithms. >

Book
01 Jan 1991
TL;DR: In this paper, the main notions of lemmas on the approximation of distribution of an arbitrary random variable by the normal distribution are defined and the main inequalities of large deviations in terms of Lyapunov's fractions.
Abstract: 1. The main notions.- 2. The main lemmas.- 2.1. General lemmas on the approximation of distribution of an arbitrary random variable by the normal distribution.- 2.2. Proof of lemmas 2.1-2.4.- 3. Theorems on large deviations for the distributions of sums of independent random variables.- 3.1. Theorems on large deviations under Bernstein's condition.- a) Sums of non-identically distributed random variables.- b) Sums of weighted random variables.- 3.2. A theorem of large deviations in terms of Lyapunov's fractions.- 4. Theorems of large deviations for sums of dependent random variables.- 4.1. Estimates of the kth order centered moments of random processes with mixing.- 4.2. Estimates of mixed cumulants of random processes with mixing.- 4.3. Estimates of cumulants of sums of dependent random variables.- 4.4. Theorems and inequalities of large deviations for sums of dependent random variables.- 5. Theorems of large deviations for polynomial forms, multiple stochastic integrals and statistical estimates.- 5.1. Estimates of cumulants and theorems of large deviations for polynomial forms, polynomial Pitman estimates and U-statistics.- 5.2. Cumulants of multiple stochastic integrals and theorems of large deviations.- 5.3. Large deviations for estimates of the spectrum of a stationary sequence.- 6. Asymptotic expansions in the zones of large deviations.- 6.1. Asymptotic expansion for distribution density of an arbitrary random variable.- 6.2. Estimates for characteristic functions.- 6.3. Asymptotic expansion in the Cramer zone for distribution density of sums of independent random variables.- 6.4. Asymptotic expansions in integral theorems with large deviations.- 7. Probabilities of large deviations for random vectors.- 7.1. General lemmas on large deviations for a random vector with regular behaviour of cumulants.- 7.2. Theorems on large deviations for sums of random vectors and quadratic forms.- a) Sums of non-identically distributed random vectors.- b) Sums of weighted random vectors.- c) Sums of random number of random vectors.- d) Quadratic forms.- Appendices.- Appendix 1. Proof of inequalities for moments and Lyapunov's fractions.- Appendix 2. Proof of the lemma on the representation of cumulants.- Appendix 3. Leonov - Shiryaev's formula.- References.

01 Dec 1991
TL;DR: In this article, a simple numerical procedure for estimating the stochastic robustness of a linear time-invariant system is described, and confidence intervals for the scalar probability of instability address computational issues inherent in Monte Carlo simulation.
Abstract: A simple numerical procedure for estimating the stochastic robustness of a linear time-invariant system is described. Monte Carlo evaluations of the system's eigenvalues allows the probability of instability and the related stochastic root locus to be estimated. This analysis approach treats not only Gaussian parameter uncertainties but non-Gaussian cases, including uncertain-but-bounded variation. Confidence intervals for the scalar probability of instability address computational issues inherent in Monte Carlo simulation. Trivial extensions of the procedure admit consideration of alternate discriminants; thus, the probabilities that stipulated degrees of instability will be exceeded or that closed-loop roots will leave desirable regions can also be estimated. Results are particularly amenable to graphical presentation.

Journal ArticleDOI
TL;DR: In this paper, the problem of modeling change in a vector time series is studied using a dynamic linear model with measurement matrices that switch according to a time-varying independent random process.
Abstract: The problem of modeling change in a vector time series is studied using a dynamic linear model with measurement matrices that switch according to a time-varying independent random process. We derive filtered estimators for the usual state vectors and also for the state occupancy probabilities of the underlying nonstationary measurement process. A maximum likelihood estimation procedure is given that uses a pseudo-expectation-maximization algorithm in the initial stages and nonlinear optimization. We relate the models to those considered previously in the literature and give an application involving the tracking of multiple targets.

Journal ArticleDOI
TL;DR: In this article, it was shown that in a Brownian dynamics simulation, it is justified to use arbitrary distribution functions of random numbers if the moments exhibit the correct limiting behavior prescribed by the Fokker-Planck equation.
Abstract: We point out that in a Brownian dynamics simulation it is justified to use arbitrary distribution functions of random numbers if the moments exhibit the correct limiting behavior prescribed by the Fokker-Planck equation. Our argument is supported by a simple analytical consideration and some numerical examples: We simulate the Wiener process, the Ornstein-Uhlenbeck process and the diffusion in a Φ4 potential, using both Gaussian and uniform random numbers. In these examples, the rate of convergence of the mean first exit time is found to be nearly identical for both types of random numbers.


Book
23 Jan 1991
TL;DR: Introduction and Terminology of Fourier Analysis, Random Signal Modeling and Modern Spectral Estimation, and Theory and Application of Cross Correlation and Coherence.
Abstract: Introduction and Terminology. Empirical Modeling and Approximation. Fourier Analysis. Probability Concepts and Signal Characteristics. Introduction to Random Processes and Signal Correlation. Random Signals, Linear Systems, and Power Spectra. Spectral Analysis for Random Signals: Classical Estimation. Random Signal Modeling and Modern Spectral Estimation. Theory and Application of Cross Correlation and Coherence.

Journal ArticleDOI
TL;DR: It is shown that the one-way multigrid algorithm improves upon the complexity of its single-grid variant and is, in a certain sense, optimal.
Abstract: The numerical solution of discrete-time stationary infinite-horizon discounted stochastic control problems is considered for the case where the state space is continuous and the problem is to be solved approximately, within a desired accuracy. After a discussion of problem discretization, the authors introduce a multigrid version of the successive approximation algorithm that proceeds 'one way' from coarse to fine grids, and analyze its computational requirements as a function of the desired accuracy and of the discount factor. They also study the effects of a certain mixing (ergodicity) condition on the algorithm's performance. It is shown that the one-way multigrid algorithm improves upon the complexity of its single-grid variant and is, in a certain sense, optimal. >

Journal ArticleDOI
TL;DR: The mean rate of vector processes out-crossing safe domains is calculated using methods from time-independent reliability theory as a sensitivity measure of the probability for an associated parallel system domain.
Abstract: The mean rate of vector processes out-crossing safe domains is calculated using methods from time-independent reliability theory. The method is founded on a result for scalar up-crossing derived by Madsen. The out-crossing is formulated as a zero down-crossing of a continuously differentiable scalar process, and the mean crossing rate is obtained as a sensitivity measure of the probability for an associated parallel system domain. The vector process may be Gaussian, non-Gaussian, stationary or nonstationary, and the failure function defining the boundary of the safe domain may be time-dependent. A method for calculation of the expected number of crossings in a time interval through the introduction of an auxiliary uniformly distributed variable is presented. For stochastic failure surfaces the ensemble averaged rate is determined. A closed-form expression for the mean crossing rate of a non-stationary Gaussian vector process crossing into a time-dependent convex polyhydral set is derived. The method is demonstrated to give good results by examples.

Journal ArticleDOI
TL;DR: In this paper, a general framework for finite element reliability analysis based on the first-and second-order reliability methods, namely, FORM and SORM, is presented, where new expressions for the required gradients of the response of geometrically nonlinear structures are derived and implemented in an existing finite element code.
Abstract: A general framework for finite element reliability analysis based on the first‐ and second‐order reliability methods, FORM and SORM, is presented. New expressions for the required gradients of the response of geometrically nonlinear structures are derived and implemented in an existing finite element code, which is then merged with a FORM/SORM reliability code. The gradient computation does not require repeated solutions of the nonlinear response and is free of the errors inherent in the perturbation method. The proposed reliability method offers significant advantages over the conventional Monte Carlo simulation approach. The method is illustrated for a plate problem with random field properties, random geometry, and subjected to random static loads. The example represents the first application of the finite element method in conjunction with SORM, for a system reliability problem, and involving non‐Gaussian random fields. Extensive analyses of reliability sensitivities with respect to parameters definin...

Journal ArticleDOI
TL;DR: In this article, a family of multivariate models for the occurrence/nonoccurrence of precipitation at N sites is constructed by assuming a different joint probability of events at the sites for each of a number of unobservable climate states.
Abstract: A family of multivariate models for the occurrence/nonoccurrence of precipitation at N sites is constructed by assuming a different joint probability of events at the sites for each of a number of unobservable climate states. The climate process is assumed to follow a Markov chain. Simple formulae for first- and second-order parameter functions are derived, and used to find starting values for a numerical maximization of the likelihood. The method is illustrated by applying it to data for one site in Washington and to data for a network in the Great Plains.