Showing papers on "Stochastic process published in 1987"

Predicting chaotic time series

Abstract: We present a forecasting technique for chaotic data. After embedding a time series in a state space using delay coordinates, we ``learn'' the induced nonlinear mapping using local approximation. This allows us to make short-term predictions of the future behavior of a time series, using information based only on past values. We present an error estimate for this technique, and demonstrate its effectiveness by applying it to several examples, including data from the Mackey-Glass delay differential equation, Rayleigh-Benard convection, and Taylor-Couette flow.

Linear stochastic systems

Abstract: Stochastic Processes Linear Stochastic Systems Estimation Theory Stochastic Realization Theory System Identification: Foundations and Basic Concepts Least Squares Parameter Estimation Maximum Likelihood Estimation of Gaussian Armax and State-Space System Minimum Prediction Error Identification Methods Non-Stationary System Identification Feedback, Causality, and Closed Loop System Identification Linear-Quadratic Stochastic Control Stochastic Adaptive Control Appendix 1: Probability Theory Appendix 2: System Theory Appendix 3: Harmonic Analysis.

Principles of statistical radiophysics

Abstract: This book is a four-volume series that introduces the newcomer to the theory of random functions. It aims at providing the background necessary to understand papers and monographs on the subject and to carry out independent research in fields where fluctuations are of importance, e.g. radiophysics, optics, astronomy, and acoustics. Volume 2, Correlation Theory of Random Processes, presents the correlation theory of nonstationary processes paying particular attention to periodically nonstationary processes. Physical phenomena like interference, coherence and polarisation of random oscillations, thermal noise in discrete dynamical systems, and the spectral representations of random actions on discrete systems are dealt with.

$U$-Processes: Rates of Convergence

Abstract: This paper introduces a new stochastic process, a collection of $U$-statistics indexed by a family of symmetric kernels. Conditions are found for the uniform almost-sure convergence of a sequence of such processes. Rates of convergence are obtained. An application to cross-validation in density estimation is given. The proofs adapt methods from the theory of empirical processes.

Probability, random processes, and ergodic properties

Abstract: Probability, Random Processes, and Ergodic Properties is for mathematically inclined information/communication theorists and people working in signal processing. It will also interest those working with random or stochastic processes, including mathematicians, statisticians, and economists. Highlights Second edition of classic text Complete tour of book and guidelines for use given in Introduction, so readers can see at a glance the topics of interest Structures mathematics for an engineering audience, with emphasis on engineering applications. New in the Second Edition Much of the material has been rearranged and revised for pedagogical reasons. The original first chapter has been split in order to allow a more thorough treatment of basic probability before tackling random processes and dynamical systems. The final chapter has been broken into two pieces to provide separate emphasis on process metrics and the ergodic decomposition of affine functionals. Completion of event spaces and probability measures is treated in more detail. More specific examples of random processes have been introduced. Many classic inequalities are now incorporated into the text, along with proofs; and many citations have been added. From the Authors Preface... This book has a long history. It began over two decades ago as the first half of a book on information and ergodic theory. The intent was and remains to provide a reasonably self-contained advanced (at least for engineers) treatment of measure theory, probability theory, and random processes, with an emphasis on general alphabets and on ergodic and stationary properties of random processes that might be neither ergodic nor stationary. The intended audience was mathematically inclined engineers who had not had formal courses in measure theoretic probability or ergodic theory. Much of the material is familiar stuff for mathematicians, but many of the topics and results had not then previously appeared in books. The original project grew too large and the first part contained much that would likely bore mathematicians and discourage them from the second part. Hence I finally followed a suggestion to separate the material and split the project in two. The resulting manuscript fills a unique hole in the literature. Personal experience indicates that the intended audience rarely has the time to take a complete course in measure and probability theory in a mathematics or statistics department, at least not before they need some of the material in their research. I intended in this book to provide a catalogue of many results that I have found need of in my own research together with proofs that I could follow. I also intended to clarify various connections that I had found confusing or insufficiently treated in my own reading. If the book provides similar service for others, it will have succeeded.

Stochastic processes in the neurosciences

Abstract: Deterministic Theories and Stochastic Phenomena in Neurobiology Synaptic Transmission Early Stochastic Models for Neuronal Activity including Poisson Processes and Random Walks Discontinuous Markov Processes with Exponential Decay One-dimensional Diffusion Processes Stochastic PDEs Statistical Analysis of Stochastic Neural Activity Channel Noise Wiener Kernel Expansions Stochastic Activity of Neuronal Populations.

Characterisation of radar clutter as a spherically invariant random process

Abstract: A statistical characterisation of clutter as a complex random process is needed in the design of optimum detection schemes. The paper considers modelling complex clutter as a spherically invariant random process (SIRP), namely assuming that its PDFs can be expressed as non-negative definite quadratic forms, a generalisation of a Gaussian process. Relevant properties of SIRPs are summarised, and shown to comply with basic requirements such as circular symmetry of the joint PDF of the in-quadrature components or, equivalently, the uniformity of the phase distribution. A constraint of admissibility must be imposed on the envelope distribution, but most commonly used envelope distributions, including Weibull, contaminated Rayleigh and K-distribution are shown to be admissible. Although a general SIRP is not ergodic, a characterisation of the clutter process as an SIRP scanned in the ensemble is finally proposed, which restores ergodicity. The interpretation of this model in the light of already proposed composite scattering models is also discussed.

Introduction to Random Processes, With Applications to Signals and Systems

Abstract: (1987). Introduction to Random Processes, With Applications to Signals and Systems. Technometrics: Vol. 29, No. 2, pp. 245-246.

Improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm

Abstract: Convergence analysis of stochastic gradient adaptive filters using the sign algorithm is presented in this paper. The methods of analysis currently available in literature assume that the input signals to the filter are white. This restriction is removed for Gaussian signals in our analysis. Expressions for the second moment of the coefficient vector and the steady-state error power are also derived. Simulation results are presented, and the theoretical and empirical curves show a very good match.

Likelilood ratio gradient estimation: an overview

Abstract: The likelihood ratio method for gradient estimation is briefly surveyed. Two applications settings are described, namely Monte Carlo optimization and statistical analysis of complex stochastic systems. Steady-state gradient estimation is emphasized, and both regenerative and non-regenerative approaches are given. The paper also indicates how these methods apply to general discrete-event simulations; the idea is to view such systems as general state space Markov chains.

A space‐time stochastic model of rainfall for satellite remote‐sensing studies

Abstract: A model of the spatial and temporal distribution of rainfall is described that produces random spatial rainfall patterns with these characteristics: (1) the model is defined on a grid with each grid point representing the average rain rate over the surrounding grid box, (2) rain occurs at any one grid point, on average, a specified percentage of the time and has a lognormal probability distribution, (3) spatial correlation of the rainfall can be arbitrarily prescribed, and (4) time stepping is carried out so that large-scale features persist longer than small-scale features. Rain is generated in the model from the portion of a correlated Gaussian random field that exceeds a threshold. The portion of the field above the threshold is rescaled to have a lognormal probability distribution. Sample output of the model designed to mimic radar observations of rainfall during the Global Atmospheric Research Program Atlantic Tropical Experiment (GATE), is shown. The model is intended for use in evaluating sampling strategies for satellite remote-sensing of rainfall and for development of algorithms for converting radiant intensity received by an instrument from its field of view into rainfall amount.

Average Run Lengths of an Optimal Method of Detecting a Change in Distribution.

Abstract: : Suppose one accumulates independent observations from a certain process. Initially, the process is at State 0. At some unknown point in time something occurs (e.g., a breakdown ) which puts the process in State 1, and consequently the stochastic behavior of the observations changes. It is of interest to declare that a change took place (to raise an alarm) as soon as possible after its occurrence, subject to a restriction on the rate of false detections. It is assumed that the aforementioned observations are the only information one has about the process, and the problem is to construct a good detection scheme. Practical examples of this problem arise in areas such as health, quality control, ecological monitoring, etc. For instance, consider surveillance for congenital malformations in newborn infants. Under normal circumstances, the percentage of babies born with a certain type of malformation has a known value. Should something occur (such as an environmental change, the introduction of a new drug to the market, etc.) the percentage may increase. One would want to raise an alarm as quickly as possible after a change would have taken place, subject to an acceptable rate of false alarms.

The practice of fast conditional simulations through the LU decomposition of the covariance matrix

Abstract: The LU-matrix approach to conditional simulations allows fast generation of large numbers of realizations for a given stochastic process Simplicity, flexibility, and quality are its main advantages Its implementation for cases where dense grids and/or large numbers of conditioning data cause computational problems is discussed A case study is presented

Monte Carlo simulations for scattering of electromagnetic waves from perfectly conductive random rough surfaces.

Abstract: Numerical calculations of mean scattered intensities by simulation of one-dimensional perfectly conductive random rough surfaces are presented Results relative to backscattering enhancement and more accurate criteria for the validity of the Kirchhoff approximation are obtained This method can also be used for assessing perturbative theories and for further experiments

Generalized stochastic subdivision

Abstract: Stochastic techniques have assumed a prominent role in computer graphics because of their success in modeling a variety of complex and natural phenomena. This paper describes the basis for techniques such as stochastic subdivision in the theory of random processes and estimation theory. The popular stochastic subdivision construction is then generalized to provide control of the autocorrelation and spectral properties of the synthesized random functions. The generalized construction is suitable for generating a variety of perceptually distinct high-quality random functions, including those with non-fractal spectra and directional or oscillatory characteristics. It is argued that a spectral modeling approach provides a more powerful and somewhat more intuitive perceptual characterization of random processes than does the fractal model. Synthetic textures and terrains are presented as a means of visually evaluating the generalized subdivision technique.

Forecasting Aggregated Vector ARMA Processes

Abstract: 1. Prologue.- 1.1 Objective of the Study.- 1.2 Survey of the Study.- 2. Vector Stochastic Processes.- 2.1 Discrete-Time, Stationary Vector Stochastic Processes.- 2.1.1 General Assumptions.- 2.1.2 The Wold or Moving Average Representation.- 2.1.3 Autoregressive Representation.- 2.1.4 Spectral Representation.- 2.2 Nonstationary Processes.- 2.3 Vector Autoregressive Moving Average Processes.- 2.3.1 Stationary Processes.- 2.3.2 Nonstationary Processes.- 2.4 Estimation.- 2.4.1 Maximum Likelihood Estimation of Stationary Gaussian Vector ARMA Processes of Known Order.- 2.4.2 Estimation of Vector Autoregressive Processes of Known Order.- 2.4.3 Multivariate Least Squares Estimation of AR Processes with Unknown Order.- 2.4.4 Nonstationary Processes.- 2.5 Model Specification.- 2.5.1 AR Order Determination.- 2.5.2 Subset Autoregressions.- 2.5.3 The Box-Jenkins Approach.- 2.6 Summary.- 3. Forecasting Vector Stochastic Processes.- 3.1 Forecasting Known Processes.- 3.1.1 Predictors Based on the Moving Average Representation.- 3.1.2 Predictors Based on the Autoregressive Representation.- 3.1.3 Forecasting Known Vector ARMA Processes.- 3.2 Forecasting Vector ARMA Processes with Estimated Coefficients.- 3.2.1 The General Case.- 3.2.2 Finite Order AR Processes.- 3.3 Forecasting Autoregressive Processes of Unknown Order.- 3.3.1 The Asymptotic MSE Matrix.- 3.3.2 Proof of Proposition 3.2.- 3.4 Forecasting Nonstationary Processes.- 3.4.1 Known Processes.- 3.4.2 Estimated Coefficients.- 3.4.3 Unknown Order.- 3.5 Comparing Forecasts.- 3.6 Summary.- 4. Forecasting Contemporaneously Aggregated Known Processes.- 4.1 Linear Transformations of Vector Stochastic Processes.- 4.2 Forecasting Linearly Transformed Stationary Vector Stochastic Processes.- 4.2.1 The Predictors.- 4.2.2 Comparison of the Predictors.- 4.2.3 Equality of the Predictors.- 4.2.4 Granger-Causality.- 4.3 Forecasting Linearly Transformed Nonstationary Processes.- 4.4 Linearly Transformed Vector ARMA Processes.- 4.4.1 Finite Order MA Processes.- 4.4.2 ARMA Processes.- 4.5 Summary and Comments.- 5. Forecasting Contemporaneously Aggregated Estimated Processes.- 5.1 Summary of Assumptions and Predictors.- 5.2 Estimated Coefficients.- 5.2.1 Comparison of $${\rm \hat Y}_{\rm t}^{\rm o} ({\rm h})$$ and $${\rm \hat Y}_{\rm t}^{} ({\rm h})$$.- 5.2.2 Comparison of $${\rm \hat Y}_{\rm t}^{\rm o} ({\rm h})$$ and $${\rm \hat Y}_{\rm t}^{\rm u} ({\rm h})$$.- 5.3 Unknown Orders and Estimated Coefficients.- 5.4 Nonstationary Processes.- 5.5 Small Sample Results.- 5.5.1 Design of the Monte Carlo Experiment.- 5.5.2 Simulation Results for AR Process I.- 5.5.3 Simulation Results for AR Process II.- 5.5.4 Simulation Results for MA Process I.- 5.5.5 Simulation Results for MA Process II.- 5.5.6 Simulation Results for MA Process III.- 5.6 An Empirical Example.- 5.7 Conclusions.- 6. Forecasting Temporally and Contemporaneously Aggregated Known Processes.- 6.1 Macro Processes.- 6.2 Six Predictors.- 6.3 Comparison of Predictors.- 6.4 Nonstationary Processes.- 6.4.1 Differencing to Obtain Stationarity.- 6.4.2 Forecasting Aggregated Nonstationary Processes.- 6.5 Temporally and Contemporaneously Aggregated Vector ARMA Processes.- 6.6 Conclusions and Comments.- 7. Temporal Aggregation of Stock Variables - Systematically Missing Observations.- 7.1 Forecasting Known Processes with Systematically Missing Observations.- 7.2 Processes With Estimated Coefficients.- 7.3 Processes With Unknown Orders and Estimated Coefficients.- 7.4 Nonstationary Time Series with Systematically Missing Observations.- 7.5 Monte Carlo Results.- 7.5.1 Univariate AR Processes.- 7.5.2 Bivariate AR Process.- 7.5.3 MA (m) Processes.- 7.5.4 Univariate MA(1) Process.- 7.5.5 Summary of Small Sample Results.- 7.6 Empirical Examples.- 7.6.1 Consumption Expenditures.- 7.6.2 Investment.- 7.7 Concluding Remarks.- 7.A Appendix: Proof of Relation (7.2.18).- 8. Temporal Aggregation of Flow Variables.- 8.1 Forecasting with Known Processes.- 8.2 Forecasts Based on Processes with Estimated Coefficients.- 8.3 Forecasting with Autoregressive Processes of Unknown Order.- 8.4 Temporally Aggregated Nonstationary Processes.- 8.5 Small Sample Comparison.- 8.5.1 A Univariate AR Process.- 8.5.2 A Univariate MA(2) Process.- 8.5.3 A Univariate MA(3) Process.- 8.5.4 A Bivariate MA Process.- 8.5.5 A System with a Stock and a Flow Variable.- 8.6 Examples.- 8.6.1 Consumption.- 8.6.2 Investment.- 8.7 Summary and Conclusions.- 8.A Appendix: Proof of Relation (8.2.23).- 9. Joint tTemporal and Contemporaneous Aggregation.- 9.1 Summary of Processes and Predictors.- 9.2 Prediction Based on Processes with Estimated Coefficients.- 9.2.1 General Results.- 9.2.2 An Example.- 9.2.3 Conclusions for Processes with Estimated Coefficients.- 9.3 Prediction Based on Estimated Processes with Unknown Orders.- 9.3.1 General Comments.- 9.3.2 Comparison of MSEs.- 9.3.3 Summary and Discussion of Results for Processes with Unknown Orders.- 9.4 Monte Carlo Comparison of Predictors.- 9.4.1 Simulation Results for AR Process.- 9.4.2 Simulation Results for MA Process.- 9.4.3 Discussion of Small Sample Results.- 9.5 Forecasts of U.S. Gross Private Domestic Investment.- 9.5.1 First Differences of Investment Data.- 9.5.2 Aggregation of Original Investment Data.- 9.6 Summary and Conclusions.- 10. Epilogue.- 10.1 Summary and Conclusions.- 10.2 Some Remaining Problems.- Appendix. Data Used for Examples.

Radically Elementary Probability Theory.

Abstract: Using only the very elementary framework of finite probability spaces, this book treats a number of topics in the modern theory of stochastic processes. This is made possible by using a small amount of Abraham Robinson's nonstandard analysis and not attempting to convert the results into conventional form.

Statistics of the Stokes parameters

Abstract: The probability-density functions and lower-order moments of the four Stokes parameters are obtained as a function of the degree of polarization and the mean intensities of the two field components.

Asymptotic properties of distributed and communication stochastic approximation algorithms

Abstract: The asymptotic properties of extensions of the type of distributed or decentralized stochastic approximation proposed in [1] are developed. Such algorithms have numerous potential applications in decentralized estimation, detection and adaptive control, or in decentralized Monte Carlo simulation for system optimization (where they can exploit the possibilities of parallel processing). The structure involves several isolated processors (recursive algorithms) that communicate to each other asynchronously and at random intervals. The asymptotic (small gain) properties are derived. The communication intervals need not be strictly bounded, and they and the system noise can depend on the (communicating) system state. State space constraints are also handled. In many applications, the dynamical terms are merely indicator functions, or have other types of discontinuities. The “typical” such case is also treated, as is the case where there is noise in the communication. The linear stochastic differential equation ...

Random stress and earthquake statistics: spatial dependence

Abstract: SUMMARY We calculate the 3-D stochastic rotations of focal mechanisms (disorientations) caused by stresses arising from the presence of many small, random, point defects that may surround the tip of an earthquake fault. These random stresses can be shown to be distributed according to a Cauchy distribution; as a consequence the next episode of fracture of a fault cannot be planar. The disorientation of focal mechanisms of these new rupture episodes is closely approximated by a rotational Cauchy distribution. As observed previously, the geometry of fault systems for natural earthquakes is consistent with these observations, since it too can be modelled by the Cauchy distribution. These observations indicate how the 3-D rupture process in rocks and other materials can be modelled. We also calculate distributions of disorientations caused by a uniformly random 3-D rotation of sources. These distributions are governed by symmetry properties of earthquake focal mechanisms. To take into account the source symmetry, we compare the Cauchy distribution which is due to influence of random stresses to the distributions caused by random rotations.

Regression Models for Nonstationary Categorical Time Series: Asymptotic Estimation Theory

Abstract: For the analysis of nonstationary categorical time series, a parsimonious and flexible class of models is proposed. These models are generalizations of regression models for stochastically independent categorical observations. Consistency, asymptotic normality and efficiency of the maximum likelihood estimator are shown under weak and easily verifiable requirements. Some models for binary time series are discussed in detail. To demonstrate asymptotic properties, a theorem is given addressing maximum likelihood estimation for general stochastic processes. Then it is shown that the assumptions of this theorem are consequences of the requirements for categorical time series. For this proof some lemmas are used which may be of interest in similar cases.

First‐order reliability approach to stochastic analysis of subsurface flow and contaminant transport

Abstract: The first-order reliability method is an attractive approach to stochastic analysis of subsurface flow and contaminant transport. The method can be used with either analytical or numerical solutions, allowing a uniform but flexible approach to solving a variety of problems, and it can fully utilize any level of probabilistic information from the minimum knowledge of second moments to complete knowledge of the full joint distribution. Therefore the first-order reliability method is particularly useful when statistical information is incomplete, as is common for problems in the subsurface environment. Additionally, correlation and nonnormal marginal distributions may be incorporated into the solution. Results from a first-order reliability analysis include an estimate of the probability of exceeding a specified performance criteria and measures of sensitivity of the stochastic solution to changes in random variables and their statistical moments. Three subsurface flow and contaminant transport example problems are used to illustrate the capabilities of the method; results from these examples compare well with previously published Monte Carlo simulation results.

Stochastic and deterministic absorption in neutron-interference experiments

Abstract: Experiments with absorbing foils, beam choppers, or an absorbing lattice in one path of a neutron interferometer to expose the difference between stochastic and deterministic absorption in quantum mechanics are reported. The different amplitudes of the interference patterns in stochastic and deterministic absorption when the absorption probability is the same were observed in agreement with prediction. Also the possibility of a gradual transition from deterministic to stochastic absorption was experimentally investigated.

On estimating the diffusion coefficient

Abstract: Random processes of the diffusion type have the property that microscopic fluctuations of the trajectory make possible the identification of certain statistical parameters from one continuous observation. The paper deals with the construction of parameter estimates when observations are made at discrete but very dense time points.

Digital simulation of multivariate two- and three-dimensional stochastic processes with a spectral turning bands method

Abstract: The space domain version of the turning bands method can simulate multidimensional stochastic processes (random fields) having particular forms of covariance functions. To alleviate this limitation a spectral representation of the turning bands method in the two-dimensional case has shown that the spectral approach allows simulation of isotropic two-dimensional processes having any covariance or spectral density function. The present paper extends the spectral turning bands method (STBM) even further for simulation of much more general classes of multidimensional stochastic processes. Particular extensions include: (i) simulation of three-dimensional processes using STBM, (ii) simulation of anisotropic two- or three-dimensional stochastic processes, (iii) simulation of multivariate stochastic processes, and (iv) simulation of spatial averaged (integrated) processes. The turning bands method transforms the multidimensional simulation problem into a sum of a series of one-dimensional simulations. Explicit and simple expressions relating the cross-spectral density functions of the one-dimensional processes to the cross-spectral density function of the multidimensional process are derived. Using such expressions the one-dimensional processes can be simulated using a simple one-dimensional spectral method. Examples illustrating that the spectral turning bands method preserves the theoretical statistics are presented. The spectral turning bands method is inexpensive in terms of computer time compared to other multidimensional simulation methods. In fact, the cost of the turning bands method grows as the square root or the cubic root of the number of points simulated in the discretized random field, in the two- or three-dimensional case, respectively, whereas the cost of other multidimensional methods grows linearly with the number of simulated points. The spectral turning bands method currently is being used in hydrologic applications. This method is also applicable to other fields where multidimensional simulations are needed, e.g., mining, oil reservoir modeling, geophysics, remote sensing, etc.