Showing papers on "Gaussian process published in 1979"

Polar Quantization of a Complex Gaussian Random Variable

Abstract: We solve numerically the optimum fixed-level non-uniform and uniform quantization of a circularly symmetric complex (or bivariate) Gaussian random variable for the mean absolute squared error criterion. For a given number of total levels, we determine its factorization into the product of numbers of magnitude and phase levels that produces the minimum distortion. We tabulate the results for numbers of "useful" output levels up to 1024, giving their optimal factorizations, minimum distortion, and entropy. For uncoded quantizer outputs, we find that the optimal splitting of rate between magnitude and phase, averaging to 1.52 and 1.47 bits more in the phase angle than magnitude for optimum and uniform quantization, respectively, compares well with the optimal polar coding formula Of 1.376 bits of Pearlman and Gray [3]. We also compare the performance of polar to rectangular quantization by real and imaginary parts for both uncoded and coded output levels. We find that, for coded outputs, both polar quantizers are outperformed by the rectangular ones, whose distortion-rate curves nearly coincide with Pearlman and Gray's polar coding bound. For uncoded outputs, however, we determine that the polar quantizers surpass in performance their rectangular counterparts for all useful rates above 6.0 bits for both optimum and uniform quantization. Below this rate, the respective polar quantizers are either slightly inferior or comparable.

Memoryless discrete-time detection of a constant signal in m-dependent noise

Abstract: The problem of designing memoryless detectors for known signals in stationary m-dependent noise processes is considered. Applying the criterion of asymptotic relative efficiency, the optimal such detector is shown to be characterized by the solution to a Fredholm integral equation whose kernel depends only on the second-order probability distributions of the noise. General expressions are derived for this solution and for the asymptotic efficiency of the optimal detector relative to other memoryless detectors. To illustrate the analysis, specific results are given for the particular case where the noise process is derived by memoryless nonlinear transformation of a Gaussian process. In addition, an extension of the analytical results to the more general case of \phi -mixing noise processes is discussed.

On the dispersion relation of random gravity waves. Part 1. Theoretical framework

Abstract: A theoretical framework is given, upon which to examine the dispersion relation of random gravity waves. First a weakly nonlinear theory is developed to the third-order for a statistically stationary and homogeneous field of random gravity waves. Both the spectrum of forced waves and the nonlinear dispersion relation are expressed in terms of the spectrum of free waves under the assumption of the Gaussian process for the first-order surface displacement. Next a method is proposed by which to separate each of the spectra of free and forced waves from the measured spectrum. This gives practical and powerful means of investigating the statistical structure of wind waves.

Confidence bands for the kaplan-meier survival curve estimate

Abstract: The weak convergence of the Kaplan-Meier estimate for censored survival data to a Gaussian process is used to construct asymptotic confidence bands for the survival curves. The method involves transforming to obtain Brownian motion and using straight line boundaries and hitting probabilities for Brownian motion.

Weak approximations of the empirical process when parameters are estimated

Abstract: Strong approximation results and methodology are used to obtain in-probability representations of the empirical process when the parameters of the underlying distribution function are estimated. These representations are obtained under a null hypothesis and a sequence of alternatives converging to the null hypothesis. The fairly general conditions on the estimators are often satisfied by maximum likelihood estimators. The asymptotic distribution of the estimated empirical process depends, in general, on the true value of the unknown parameters. Some useful methods of overcoming this difficulty are discussed.

Quantization schemes for bivariate Gaussian random variables

Abstract: The problem of quantizing two-dimensional Gaussian random variables is considered. It is shown that, for all but a finite number of cases, a polar representation gives a smaller mean square quantization error than a Cartesian representation. Applications of the results to a transform coding scheme known as spectral phase coding are discussed.

A Markov property for two parameter Gaussian processes.

Abstract: This paper deals with the relationship between two-dimensional parameter Gaussian random fields verifying a particular Markov property and the solutions of stochastic differential equations. In the non Gaussian case some diffusion conditions are introduced, obtaining a backward equation for the evolution of transition probability functions.

Computer generation of correlated Gaussian random variables

Abstract: This paper Presents a means of generating a set of N correlated Gaussian random variables from N or fewer independent Gaussian random variables. In computer generation of pseudorandom variables, this technique sometimes has computational advantages over the more straightforward inverse Gram-Schmidt procedure. As an example, application of the technique in simulation of a pulse frequency modulation (PFM) receiver is discussed.

A stochastic solvent model for electron exchange reactions

Abstract: Outer sphere electron transfer reactions are investigated in the framework of a semiclassical theory with a density matrix representation The coupling of the electron exchange system to the polar solvent is represented as a stochastic Gaussian process Expressions for the transition probability in the nonadiabatic case are derived in terms of the correlation function for the fluctuating coupling Some models for the correlation functions are discussed together with explicit results for the transition probabilities

Hydrologic Generating Model Selection

Abstract: When modeling seasonal river flows for generating possible flow sequences for use in reservoir design, it is often necessary to first invoke a deterministic transformation to remove seasonality and thus eliminate the need for differencing. To select which stationary stochastic model to fit to the resulting transformed data, a two-stage decision-making process is recommended. The first stage consists of eliminating those models which do not possess a proper statistical fit to the data, while further discrimination can be done at the second stage by judging the remaining models according to the economic criteria of the particular reservoir disign. Simulation procedures are used to obtain designs using various hydrologic generating models for a hydroelectric reservoir complex on the South Saskatchewan River in Canada. Both Box-Jenkins and Fractional Gaussian noise processes are considered in the model selection studies. A new simulation procedure is developed for use with a Fractional Gaussian noise model.

On resolution and exponential discrimination between Gaussian stationary vector processes and dynamic models

Abstract: We derive new necessary and sufficient conditions for exponentially convergent discrimination between two stationary vector Gaussian processes, and relate them to previously studied conditions for parameter identifiability and consistent discrimination.

Estimating the Lag Structure of a Nonlinear Time Series Model

Abstract: Suppose that the time series {Y(t)} is the output of a linear plus quadratic filter observed with additive noise. The relationship is a second-order approximation to a nonlinear distributed lag model. Assume that the input to the filter {X(t)} is a stationary Gaussian process. This article presents a relatively simple procedure for determining the values of the parameters of the quadratic filter by using the sample cross bispectrum between the two series. The asymptotic properties of the parameter estimates are derived when the spectrum of {X(t)} is known. Artificial data results are presented.

A Cramer-Rao bound for nonlinear filtering problems with additive Gaussian measurement noise

Abstract: Versions of the Cramer-Rao bound for nonlinear filtering problems with additive Gaussian measurement noise are presented. This new approach is based upon a modified Cramer-Rao bound for treating nuisance parameters.

How accurate are dispersion predictions

Abstract: Past widespread use of Gaussian dispersion models to predict stack gas plume dispersions has led to the misconception that dispersion models predict plume concentrations within a factor of two or three of real world concentrations. The consistent prediction of actual plume concentrations within a factor of 10 is a more realistic assessment for the short time models. Constraints and assumptions involved in the Gaussian models, and areas of potential inaccuracies are discussed. A hypothetical example is presented to demonstrate that seemingly minor changes in only a few variables can result in an overprediction ratio ranging from a factor of six to a factor of 80. Gaussian dispersion models are precise and useful tools, but their precision must not be confused with accuracy. (1 graph, 5 references, 1 table)

On the strong information singularity of certain stationary processes (Corresp.)

Abstract: In an exploratory paper, T. Berger studied discrete random processes which generate information slower than linearly with time. One of his objectives was to provide a physically meaningful definition of a deterministic process, and to this end he introduced the notion of strong information singularity. His work is supplemented by demonstrating that a large class of convariance stationary processes are strongly information singular with respect to a class of stationary Gaussian processes. One important consequence is that for a large class of covariance stationary processes the information rate equals that of the process associated with the Brownian motion component of the spectral representation. In an exploratory paper, T. Berger studied discrete random processes which generate information slower than linearly with time. One of his objectives was to provide a physically meaningful definition of a deterministic process, and to this end he introduced the notion of strong information singularity. His work is supplemented by demonstrating that a large class of convariance stationary processes are strongly information singular with respect to a class of stationary Gaussian processes. One important consequence is that for a large class of covariance stationary processes the information rate equals that of the process associated with the Brownian motion component of the spectral representation. In an exploratory paper, T. Berger studied discrete random In an exploratory paper, T. Berger studied discrete random processes which generate information slower than linearly with time. One of his objectives was to provide a physically meaningful definition of a deterministic process, and to this end he introduced the notion of strong information singularity. His work is supplemented by demonstrating that a large class of convariance stationary processes are strongly information singular with respect to a class of stationary Gaussian processes. One important consequence is that for a large class of covariance stationary processes the information rate equals that of the process associated with the Brownian motion component of the spectral representation.

Mutual information, strong equivalence, and signal sample path properties for Gaussian processes

Abstract: Let (St) and (Nt), t in [0, T] be stochastic processes with almost all paths in ℒ2[0, T], jointly Gaussian. Relations are obtained between the average mutual information of (St) and (St + Nt), strong equivalence of the measures induced by (St + Nt) and (Nt), and the almost sure sample path properties of (St).

Level dependence of the variance of the number of level-crossing points of a stationary Gaussian process (Corresp.)

Abstract: The variance of the number of level-crossing points of stationary Gaussian processes is calculated numerically for several spectral shapes. It is shown that as the crossing level is changed the variance reaches a maximum at some nonzero level for most spectral shapes of the underlying Gaussian process. An attempt is made to interpret the result using some simple assumptions.

A new model for estimating concentrations of substances emitted from a line source

Abstract: Extensive theoretical and experimental investigations of atmospheric diffusion have shown that the concentration of a substance at a receptor downwind from a source can generally be represented by a gaussian function. Prior implementations of gaussian models have usually been based on the incorrect implicit theory that the vertical disperson constant σz is nearly independent of the time required for the substance to go from source to receptor. In the present study we have explored the behavior of a model based on the theory that σx is some type of function of time. Application of the new model to data from the GM sulfate experiment requires σx to be directly proportional to time. We find that the new model correctly reproduces the trends of the experimental data for the entire set of available downwind values, and it also reproduces the values obtained on the median strip if a minor empirical correction is applied to allow for an upwind thrust of SF6 due to traffic wake. The new model is applicable, there...

Non-gaussianity and non-linearity in electroencephalographic time series

Abstract: In this interdisciplinary contribution we discuss a number of problems related to stochastic models and statistical methods used in electroencephalography The main part of the paper is devoted to the assumption of a Gaussian process We present a variety of methods to check empirically such an assumption, together with examples The deviations from a Gaussian process which occur in EEG analysis are interpreted in terms of non-linear dynamics; the input-output-map is assumed to be well represented by a Volterra series

A correlated random numbers generator and its use to estimate false alarm rates of airplane sensor failure detection algorithms

Abstract: A routine procedure is presented to generate random number series with specified power spectral density and composite Gaussian probability distribution functions. This can be used to simulate airplane sensor outputs in the synthesis and evaluation of failure detection schemes for redundant sensor sets. An example is given comparing some statistics of simulated sensor outputs to their observed counterparts.

Detection and MMSE Estimation of Nonlinear Memoryless Functionals of Random Processes Using the Volterra Functional Expansion

Abstract: : This report considers the general problem of detection and MMSE estimation of nonlinear memoryless functionals of random processes. In all cases considered, the observation process is assumed to be contaminated by additive Gaussian white noise. A Volterra functional expansion is derived for the likelihood ratio used in the detection of a nonlinear memoryless functional of a random process. This expansion is reduced to well known results for the special case of detection of a Gaussian process. For the case of detection of a nonlinear memoryless functional of a stationary Gaussian random process, it is shown that the likelihood ratio has an asymptotic form for which performance can be obtained provided the nonlinearities and processes satisfy Sun's theorem. A Volterra functional expansion for MMSE estimation of a nonlinear memoryless functional of a random process using nonlinear observations is also derived. It is shown that, using linear observations, the Volterra expansion reduces to well known results for the case of MMSE estimation of a zero mean Gaussian process. A stochastic differential equation for the logarithm of the likelihood ratio is also derived to demonstrate agreement with known results. (Author)

A Volterra functional expansion for the logarithm of the likelihood ratio for detection of a memoryless nonlinear functional of a random process in Gaussian white noise

Abstract: A Volterra functional expansion for the logarithm of the likelihood ratio, for detection of a memoryless nonlinear functional of a random process in Gaussian white noise, is presented. For the special case of detection of a zero‐mean Gaussian process in Gaussian white noise, it is shown that the Volterra expansion reduces to well‐known results. An approximation to the Volterra expansion is suggested for some problems with periodic nonlinearities.

Estimator performance for a class of nonlinear estimation problems

Abstract: The state estimation problem for a certain class of nonlinear stochastic systems with white Gaussian plant and observation noise is considered. The optimal (minimum variance) estimators for these systems are recursive and finite dimensional. A particular nonlinear system which contains a polynomial nonlinearity is presented. Both optimal and suboptimal estimators and an estimation lower bound for such a system are derived. The performance of the optimal and suboptimal estimators and the lower bound are compared both analytically and by computer simulation.