Showing papers on "Gaussian process published in 1978"

Gaussian Random Processes

Abstract: I Preliminaries.- I.1 Gaussian Probability Distribution in a Euclidean Space.- I.2 Gaussian Random Functions with Prescribed Probability Measure.- I.3 Lemmas on the Convergence of Gaussian Variables.- I.4 Gaussian Variables in a Hilbert Space.- I.5 Conditional Probability Distributions and Conditional Expectations.- I.6 Gaussian Stationary Processes and the Spectral Representation.- II The Structures of the Spaces H(T) and LT(F).- II. 1 Preliminaries.- II.2 The Spaces L+(F) and L-(F).- II.3 The Construction of Spaces LT(F) When T Is a Finite Interval.- II.4 The Projection of L+(F) on L-(F).- II.5 The Structure of the ?-algebra of Events U(T).- III Equivalent Gaussian Distributions and their Densities.- III.1 Preliminaries.- III.2 Some Conditions for Gaussian Measures to be Equivalent.- III.3 General Conditions for Equivalence and Formulas for Density of Equivalent Distributions.- III.4 Further Investigation of Equivalence Conditions.- IV Conditions for Regularity of Stationary Random Processes.- IV.1 Preliminaries.- IV.2 Regularity Conditions and Operators Bt.- IV.3 Conditions for Information Regularity.- IV.4 Conditions for Absolute Regularity and Processes with Discrete Time.- IV.5 Conditions for Absolute Regularity and Processes with Continuous Time.- V Complete Regularity and Processes with Discrete Time.- V.l Definitions and Preliminary Constructions with Examples.- V.2 The First Method of Study: Helson-Sarason's Theorem.- V.3 The Second Method of Study: Local Conditions.- V.4 Local Conditions (continued).- V.5 Corollaries to the Basic Theorems with Examples.- V.6 Intensive Mixing.- VI Complete Regularity and Processes with Continuous Time.- VI.1 Introduction.- VI.2 The Investigation of a Particular Function ?(T ).- VI.3 The Proof of the Basic Theorem on Necessity.- VI.4 The Behavior of the Spectral Density on the Entire Line.- VI.5 Sufficiency.- VI.6 A Special Class of Stationary Processes.- VII Filtering and Estimation of the Mean.- VII.1 Unbiased Estimates.- VII.2 Estimation of the Mean Value and the Method of Least Squares.- VII.3 Consistent Pseudo-Best Estimates.- VII.4 Estimation of Regression Coefficients.- References.

Density Estimation, Stochastic Processes and Prior Information

Abstract: SUMMARY A method is proposed for the non-parametric estimation of a probability density, based upon a finite number of observations and prior information about the smoothness of the density. A logistic density transform and a reproducing inner product from the first-order autoregressive stochastic process are employed to represent prior information that the derivative of the transform is unlikely to change radically within small intervals. The posterior estimate of the density possesses a continuous second derivative; it typically satisfies the frequentist property of asymptotic consistency. A direct analogy is demonstrated with a smoothing method for the time-dependent Poisson process; this is similar in spirit to the normal theory Kalman filter. A procedure for grouped observations in a histogram provides an alternative to the histospline method of Boneva, Kendall and Stefanov. Five practical examples are presented, including two investigations of normality, an analysis of pedestrian arrivals at a Pelican crossing and a histogram smoothing method for mine explosions data.

Study of the Random Vibration of Nonlinear Systems by the Gaussian Closure Technique

Abstract: A technique is developed to study random vibration of nonlinear systems. The method is based on the assumption that the joint probability density function of the response variables and input variables is Gaussian. It is shown that this method is more general than the statistical linearization technique in that it can handle non-Gaussian excitations and amplitude-limited responses. As an example a bilinear hysteretic system under white noise excitation is analyzed. The prediction of various response statistics by this technique is in good agreement with other available results.

Estimation of a biometric function

Abstract: In the analysis of life tables one biometric function of interest is the life expectancy at age $x, e_x = E\lbrack X - x\mid X > x\rbrack$. Estimation of $e_x$ is considered, the standard estimator used in life tables is shown to be asymptotically unbiased, uniformly strong consistent, and converges in distribution to a Gaussian process. The connections of the estimator studied in this article and that used in reliability theory are illustrated.

Some properties of the crossings process generated by a stationary χ2 process

Abstract: The process generated by the crossings of a fixed level, u, by the process P,(t) is considered, where P.(t) = X2(t)+ + + X2(t) and the X1(t) are identical, independent, separable, stationary, zero mean, Gaussian processes. A simple formula is obtained for the expected number of upcrossings in a given time interval, sufficient conditions are given for the upcrossings process to tend to a Poisson process as u--0oo, and it is shown that under suitable scaling the distribution of the length of an excursion of P,(t)

Linear prediction in cascade form

Abstract: The autocorrelation and covariance methods of linear prediction are formulated in terms of an inverse digital filter in cascade form, rather than the traditional direct form, to allow pole locations in the system model to be readily estimated and constrained. Iterative solution of the corresponding nonlinear normal equations is described. Applications to speech analysis and the compensation of biomedical signals are briefly discussed.

Equivalence and Singularity of Gaussian Measures and Applications

Abstract: Keywords: reproducing kernel Hilbert spaces;;; equivalence and singularity;;; Gaussian measures;;; expository paper;;; Feldman-Hajek dichotomy for Gaussian measures;;; stationary Gaussian processes;;; absolute continuity and singularity of probability measures Reference GPRO-CHAPTER-1978-003 Record created on 2010-05-25, modified on 2016-08-08

Fast time-invariant implementations of Gaussian signal detectors

Abstract: A new implementation is presented for the optimum likelihood ratio detector for stationary Gaussian signals in white Gaussian noise that uses only two causal time-invariant filters. This solution also has the advantage that fast algorithms based on the Levinson and Chandrasekhar equations can he used for the determination of these time-invariant filters. By using a notion of "closeness to stationarity,' there is a natural extension of the above results for nonstationary signal processes.

Recursive causal linear filtering for two-dimensional random fields

Abstract: The causal estimation of a two-parameter Gaussian random field in the presence of an additive, independent, white Gaussian noise is studied. The dynamics of this random field are modeled by partial differential equations from which the recursive filtering equations and the generalized Riccati equation are derived. A specific example is solved in detail.

A Normalizing Logarithmic Transformation for Inverse Gaussian Random Variables

Abstract: A logarithmic transformation for inverse Gaussian variates which produces approximate normality for large values of the concentration parameter is presented.

Local Nondeterminism and the Zeros of Gaussian Processes

Abstract: The concept of local nondeterminism introduced by Berman is generalized and applied to divided difference sequences generated by a Gaussian process. The resulting estimates are then used to find simple sufficient conditions for the finiteness of the moments of the number of crossings of level zero. In particular it is shown that under mild regularity conditions very little more is required to make all moments finite when the variance is finite. The results are extended to curves $\xi \in \mathscr{L}_2\lbrack 0, T\rbrack$. Finally examples are given in which the variance is finite but the third moment is infinite.

Covariance Matrix Computation of the State Variable of a Stationary Gaussian Process

Abstract: A recursive procedure for the computation of one-step ahead predictions for a finite span of time series data by a Gaussian autoregressive moving average model can be realized by using the Markovian representation of the model. The covariance matrix of the stationary state variable of the Markovian representation is required to implement a computational procedure of the predictions. A simple computational procedure of the covariance matrix which does not need an iterative method is obtained by using a canonical representation of the autoregressive moving average process. The recursive computation of the predictions realized by using this procedure provides a computationary efficient method of exact likelihood evaluation of a Gaussian autoregressive moving average model.

A sufficient condition for consistent discrimination between stationary Gaussian models

Abstract: The uniqueness of the prediction error covariance matrix is shown to be sufficient for a consistent selection among a finite set of stationary Gaussian models, employing the maximum-likelihood criterion. The new consistency condition is considerably easier to verify than previously suggested conditions.

On the reconstruction of the covariance of stationary Gaussian processes observed through zero-memory nonlinearities--Part II (Corresp.)

Abstract: The problem of reconstructing the normalized covariance function R(t) of a zero-mean stationary Gaussian process observed through a zero-memory nonlinearity f(x) is considered, when the nonlinearity and the correlation function or the second-order distribution of the output process are known. Three kinds of results are established. (i) Arbitrary covariances can be reconstructed for certain nonlinearities, including monotonic f , appropriate interval windows, and certain quite general f . (ii) Certain covariances can be reconstructed for arbitrary nonlinearities: included here are positive covariances (\geq 0) , covariances with rational spectral densities, and bandlimited covariances. (iii) Certain covariances, satisfying rather weak conditions, that can easily be checked in terms of the output correlation function, can be reconstructed for certain nonlinearities that include symmetric as well as nonsymmetric f .

Differential Pulse Code Modulation of Stationary Gaussian Inputs

Abstract: This paper is concerned with an exact analysis of DPCM systems with stationary Gaussian inputs. We consider a class of digital communication systems in which DPCM is included as a member. An integral equation for the joint probability of the input and the state of the system is derived first. Solution of the equation is sought in the form of a power series in the elements of the covariance matrix of the input that involves generalized Hermite functions. Then the integral equation is reduced to a set of algebraic equations for the coefficients in the series that are solved recursively. The steady-state distribution of the input and the state is thus found. We are interested particularly in the mean-squared error of the system as a function of step size, the sampling interval and the number of quantization levels. Numerical results are shown for the GaussMarkov input; the MSE is calculated in terms of system parameters and performance of DPCM and PCM is compared with reference to the theoretical bound.

Degradation of an image due to Gaussian motion.

Abstract: Degradation of an image due to its random motion is considered Equations describing the effects of Gaussian random motion on the point-spread function, Strehl ratio, and encircled energy are given in terms of the degraded optical transfer function These equations are used to obtain numerical results for several typical cases of motion of diffraction-limited images formed by imaging systems with obscured circular pupils

Experimental results on the level-crossing intervals of Gaussian processes (Corresp.)

Abstract: The statistical properties of the level-crossing intervals of Gausslan random process are experimentally studied. The probability density functions and variances of the level-crossing intervals and the correlation coefficients between successive level-crossing intervals are measured for Gaussian processes having a seventh-order Butterworth power spectral density.

The probability distribution of a wave at a very large depth within an extended random medium

Abstract: A plane wave is incident normally onto the boundary of a semi-infinite stationary random medium. The statistical moments of the field variable, both at one and at several points, are calculated when the refractive index of the medium at different points has a joint Gaussian probability distribution and Gaussian power spectrum, and the observer is at a very great depth within the medium. From these moments the probability distributions are calculated. The method uses a perturbation expansion. The limit of infinite distance and the limit of infinite depth are considered.

The weak convergence of a class of estimators of the variance function of a two-dimensional Poisson process

Abstract: Results of a previous paper (Liebetrau (1977a)) are extended to higher dimensions. An estimator V*(t1, t2) of the variance function V(t1, t2) of a two-dimensional process is defined, and its firstand second-moment structure is given assuming the process to be Poisson. Members of a class of estimators of the form TT2[V*(t2,t1)V(t2,t)], where t,= t,Tcs and f, = 2ii-3a ) for 0 < a, < 1, are shown to converge weakly to a non-stationary Gaussian process. Similar results hold when the t' are taken to be constants, when V is replaced by a suitable estimator and when the dimensionality of the underlying Poisson process is greater than two. WEAK CONVERGENCE; VARIANCE FUNCTION; TWO-DIMENSIONAL POISSON PROCESS; MULTIDIMENSIONAL POISSON PROCESS; ESTIMATOR

A direct proof of innovations/observations equivalence for Gaussian procedures (Corresp.)

Abstract: A direct argument is given to show that the linear subspaces spanned by an observed process and its linear innovations are the same, and hence that in the Gaussian case these generate the same families of \sigma -fields.