Showing papers on "Gaussian process published in 1986"

Detection of Intensity Changes with Subpixel Accuracy Using Laplacian-Gaussian Masks

Abstract: We present a system that takes a gray level image as input, locates edges with subpixel accuracy, and links them into lines. Edges are detected by finding zero-crossings in the convolution of the image with Laplacian-of-Gaussian (LoG) masks. The implementation differs markedly from M.I.T.'s as we decompose our masks exactly into a sum of two separable filters instead of the usual approximation by a difference of two Gaussians (DOG). Subpixel accuracy is obtained through the use of the facet model [1]. We also note that the zero-crossings obtained from the full resolution image using a space constant ? for the Gaussian, and those obtained from the 1/n resolution image with 1/n pixel accuracy and a space constant of ?/n for the Gaussian, are very similar, but the processing times are very different. Finally, these edges are grouped into lines using the technique described in [2].

Parameter Uncertainty in Estimation of Spatial Functions: Bayesian Analysis

Abstract: Linear estimation has found many applications in the inference of spatial functions in surface and subsurface hydrology. The effect of parameter uncertainty is examined in a Bayesian framework with emphasis on the derivation of the Bayesian distribution (and its first two moments) of unknown quantities given some measurements. This distribution accounts not only for natural variability but also for parameter uncertainty. For known covariance parameters the Bayesian distribution is Gaussian (for Gaussian processes) with the mean being a given linear function of the data. This linear estimator is equivalent to the conventional Gaussian conditional mean estimator for a priori known drift coefficients and is the same with kriging for diffuse prior distribution of the drift coefficients; however, the developed procedure is more general. When both drift and covariance function parameters are uncertain, the Bayesian distribution is generally not Gaussian, and the Bayesian conditional mean is a nonlinear estimator. The case of diffuse priors is examined in some detail; it is shown that the posterior distribution of the covariance function parameters is given by the restricted likelihood function, i.e., the likelihood function of generalized increments. The results provide insight into the applicability of maximum likelihood versus restricted maximum likelihood parameter estimation, and conventional linear versus kriging estimation. A more general procedure which includes these methods as special cases is presented.

Asymptotic Properties of the Product Limit Estimate Under Random Truncation

Abstract: Many authors have considered the problem of estimating a distribution function when the observed data is subject to random truncation. A prominent role is played by the product limit estimator, which is the analogue of the Kaplan-Meier estimator of a distribution function under random censoring. Wang and Jewell (1985) and Woodroofe (1985) independently proved consistency results for this product limit estimator and showed weak convergence to a Gaussian process. Both papers left open the exact form of the covariance structure of the limiting process. Here we provide a precise description of the asymptotic behavior of the product limit estimator, including a simple explicit form of the asymptotic covariance structure, which also turns out to be the analogue of the covariance structure of the Kaplan-Meier estimator. Some applications are briefly discussed.

Computation of the exact information matrix of Gaussian time series with stationary random components

Abstract: The paper presents an algorithm for efficient recursive computation of the Fisher information matrix of Gaussian time series whose random components are stationary, and whose means and covariances are functions of a parameter vector. The algorithm is first developed in a general framework and then specialized to the case of autoregressive moving-average processes, with possible additive white noise. The asymptotic behavior of the algorithm is explored and a termination criterion is derived. Finally, the algorithm is used to demonstrate the behavior of the exact Cramer-Rao bound (for unbiased estimates) for some ARMA processes, as a function of the number of data points. It is shown that for processes with zeros near the unit circle and short data records, the exact Cramer-Rao bound differs dramatically from its common approximation based on asymptotic theory.

On the Relationship between Two Fatigue-Life Models

Abstract: The relationship between two fatigue life distributions, namely the Birnbaum-Saunders and the Inverse Gaussian, is further investigated An intimate connection exists between the two models, viz, the Birnbaum-Saunders is a mixture of two probability distributions: 1) an Inverse Gaussian random variable, and 2) the reciprocal of an Inverse Gaussian random variable Advantages and disadvantages of the two distributions are discussed The arguments favour the Birnbaum-Saunders distribution from a stochastic modeling point of view, whereas the Inverse Gaussian distribution seems to be the more attractive of the two with respect to statistical analysis and analysis of censored data

Nonlinear and non-Gaussian ocean noise

Abstract: Bispectral analysis is a statistical tool for detecting and identifying a nonlinear stochastic signal generating mechanism from data containing its output. Bispectral analysis can also be employed to investigate whether the observed data record is consistent with the hypothesis that the underlying stochastic process has Gaussian distribution. From estimates of bispectra of several records of ambient acoustic ocean noise, a newly developed statistical method for testing whether the noise had a Gaussian distribution, and whether it contains evidence of nonlinearity in the underlying mechanisms generating the observed noise is applied. Seven acoustic records from three environments are examined: the Atlantic south of Bermuda, the northeast Pacific, and the Indian Ocean. The collection of time series represents both ambient acoustic noise (no local shipping) and noise dominated by local shipping. The three ambient records appeared to be both linear and Gaussian processes when examined over a period on the ord...

Origins of linear and nonlinear recursive restoration algorithms

Abstract: Linear and nonlinear image restoration methods have been studied in depth but have always been treated separately. In this paper several well-known linear and nonlinear restoration methods are written as recursive algorithms, and some new recursive algorithms are developed. The nonlinear restoration algorithms are based on the assumption that the noise is either a Poisson or a Gaussian process. The linear algorithms are shown to be related to the nonlinear methods through the partial derivative, with respect to the object, of a Poisson or a Gaussian likelihood function. A table of results is given, along with applications to real imagery.

The nature of the non‐gaussianity of primary reflection coefficients and its significance for deconvolution*

Abstract: One of the important properties of a series of primary reflection coefficients is its amplitude distribution. This paper examines the amplitude distribution of primary reflection coefficients generated from a number of block-averaged well logs with block thicknesses corresponding to 1 ms (two-way time). The distribution is always essentially symmetric, but has a sharper central peak and larger tails than a Gaussian distribution. Thus any attempt to estimate phase using the bi-spectrum (third-order spectrum) is unlikely to be successful, since the third-order moment is almost identically zero. Complicated tri-spectrum (fourth-order spectrum) calculations are thus required. Minimum Entropy Deconvolution (MED) schemes should be able to exploit this form of non-Gaussianity. However, both these methods assume a white reflectivity sequence; they would therefore mix up the contributions to the trace's spectral shape that are due to the wavelet and those that are due to non-white reflectivity unless corrections are introduced. A mixture of two Laplace distributions provides a good fit to the empirical amplitude distributions. Such a mixture distribution fits nicely with sedimentological observations, namely that clear distinctions can be made between sedimentary beds and lithological units that comprise one or more such beds with the same basic lithology, and that lithological units can be expected to display larger reflection coefficients at their boundaries than sedimentary beds. The geological processes that engender major lithological changes are not the same as those for truncation of bedding. Analyses of sub-sequences of the reflection series are seen to support this idea. The variation of the mixing proportion parameter allows for scale and shape changes in different segments of the series, and hence provides a more flexible description of the series than the generalized Gaussian distribution which is shown to also provide a good fit to the series. Both the mixture of two Laplace distributions and the generalized Gaussian distribution can be expressed as scale mixtures of the ordinary Gaussian distribution. This result provides a link with the ordinary Gaussian distribution which might have been expected to be the distribution of a natural series such as reflection coefficients. It is also important in the consideration of the solution of MED-type methods. It is shown that real (coloured) primary reflection series do not seem to be obtainable as the deconvolution result from MED-type deconvolution schemes.

Nonparametric Bayesian Regression

Abstract: It is desired to estimate a real valued function F on the unit square having observed F with error at N points in the square. F is assumed to be drawn from a particular Gaussian process and measured with independent Gaussian errors. The proposed estimate is the Bayes estimate of F given the data. The roughness penalty corresponding to the prior is derived and it is shown how the Bayesian technique can be regarded as a generalisation of variance components analysis. The proposed estimate is shown to be consistent in the sense that the expected squared error averaged over the data points converges to zero as $N\rightarrow\infty$. Upper bounds on the order of magnitude of magnitude of the expected average squared error are calculated. The proposed technique is compared with existing spline techniques in a simulation study. Generalisations to higher dimensions are discussed.

Langevin equations for L′-Valued Gaussian processes and fluctuation limits of infinite particle systems

Abstract: L′ Gaussian processes of a certain class are shown to satisfy generalized Langevin equations. Examples are fluctuation limits of several infinite particle systems, in particular infinite particle branching Brownian motions with immigration under various scalings and the voter model with hydrodynamic scaling.

A Survey of Recent Progress on Level-Crossing Problems for Random Processes

Abstract: Since Blake and Lindsey’s [19] comprehensive survey of results and techniques for level-crossing problems for random processes appeared in 1973, a number of interesting new results addressing both classical and new problem areas have been developed. As there are many diverse areas of application of level crossing results, the theoretical literature is fairly widely dispersed. The present survey is intended to be an update of Blake and Lindsey’s and is intended to be similar to theirs with an emphasis on explicit analytical results for continuous parameter processes. As they discussed many of the techniques and methods available in an accessible tutorial fashion, the present survey will be confined to an overview of the results that are available.

Chi-Square Goodness-if-Fit Tests for Randomly Censored Data

Abstract: Two Pearson-type goodness-of-fit test statistics for parametric families are considered for randomly right-censored data. Asymptotic distribution theory for the test statistics is based on the result that the product-limit process with MLE for nuisance parameters converges weakly to a Gaussian process. The Chernoff-Lehmann (1954) result extends to a generalized Pearson statistic. A modified Pearson statistic is shown to have a limiting chi-square null distribution.

Adaptive identification of a time-varying ARMA speech model

Abstract: We propose an adaptive algorithm to estimate time-varying ARMA parameters for speech signals. It estimates both input excitations and underlying system parameters. The proposed algorithm is an extended form of the Kalman filter algorithm. We assume the input is either a white Gaussian process or a pseudoperiodical pulse-train as commonly adopted in LPC processing. The time variation of parameters is monitored by a likelihood function. In order to estimate optimal parameters in a small amount of data, AR and MA orders of an estimator are set to be higher than those of a true system. Parsimonious ARMA parameters are calculated from parameters obtained by the high-order ARMA model. Examples of synthetic and real speech sounds are given to demonstrate the tracking ability of this algorithm.

Distribution of the filtered output of a quadratic rectifier computed by numerical contour integration

Abstract: The cumulative distribution of the filtered output of a quadratic rectifier whose input is either narrow-band Gaussian noise or Gaussian noise with a low-pass spectral density is to be computed by numerical quadrature of a Laplace inversion integral along a contour in the complex plane chosen to economize the number of steps. The integrand contains the moment-generating function (mgf) of the output. It is expressed in terms of the Fredholm determinant and the resolvent kernel associated with an integral equation involving the autocovariance function of the input and the impulse response of the output filter. A special case is the power of a mean-zero Gaussian process averaged over a finite interval, and when this process has a rational spectral density, the mgf can be expressed as the ratio of certain finite determinants. By this method distributions are calculated for low-pass noise with RLC and second- and fourth-order Chebyshev spectral densities. For rational input spectral densities but arbitrary positive output filtering and an arbitrary additive input signal, the mgf can be calculated by integrating differential equations of the Kalman-Bucy type.

Numerical fixed-point solution for near-optimum regulators of linear quadratic gaussian control problems for singularly perturbed systems

Abstract: The fixed-point method is developed for obtaining an efficient numerical solution of the linear quadratic gaussian problem for singularly perturbed systems. It is shown that each iteration step improves the accuracy by an order of magnitude, that is, the accuracy 0(ek) can be obtained by performing only k — 1 iterations. In addition, only low-order systems are involved in algebraic manipulations and no analyticity requirements are imposed on the system coefficients.

Response of linear systems to quadratic gaussian excitations

Abstract: Probability density functions, mean crossing rates, and other descriptors are developed for the response of linear systems to squares of Gaussian excitations. The analysis is based on discrete approximations of the spectrum of the Gaussian excitation. Accordingly, the response can be expressed as a finite quadratic form in Gaussian variables, whose characteristic function has a closed form. The characteristic function can be inverted by Fast Fourier Transform algorithms to find the first order probability of the response. Several approximations are applied to determine crossing and peak characteristics of the response. The proposed methodology is applied to estimate structural response to wind loads and the mean failure rate of systems subjected to bivariate Gaussian stress processes.

A Bayesian approach to density estimation

Abstract: SUMMARY Assume a priori that the log density is a sample function from a Gaussian process subject to the condition that the density integrates to one. The posterior distribution given a number of observations is then still a Gaussian process with the same condition and the same covariance function. The mean value function is changed according to a simple formula. This prior may thus be regarded as a conjugate prior for an unknown density. The mode of the posterior distribution is given implicitly by a simple formula, which can be solved numerically. The mode is a close approximation to the optimal estimate with squared error loss in the discrete case. Some examples with data are given.

Discrimination with respect to a Gaussian process

Abstract: Let (N t) and (Y t), t in [0,1], be stochastic processes on (Ω, ℬ, P). Suppose that (N t) is Gaussian, m.s. continuous, zero mean, and vanishes a.s. at t=0. Let v Y and v N be the induced measures on ℝ[0,1]. Conditions are obtained for v Y to be absolutely continuous w.r.t. v N. Expressions for the Radon-Nikodym derivative are derived. Further results on these problems are obtained for measures induced on L 2[0, 1] and on C[0, 1].

A prediction bound for Gaussian ARMA processes

Abstract: A lower bound for the mean accumulated squared errors for the class of the gaussian ARMA processes has been described under the assumption that the parameters of the data generating process are not known.

A nonlinear system under combined periodic and random excitation

Abstract: The anharmonic oscillator under combined sinusoidal and white noise excitation is studied using the Gaussian closure approximation. The mean response and the steady-state variance of the system is obtained by the WKBJ approximation and also by the Fokker-Planck equation. The multiple steadystate solutions are obtained and their stability analysis is presented. Numerical results are obtained for a particular set of system parameters. The theoretical results are compared with a digital simulation study to bring out the usefulness of the present approximate theory.

A second-order Monte Carlo method for the solution of the Ito stochastic differential equation

Abstract: A difference approximation that is second-order accurate in the time step his derived for the general Ito stochastic differential equation. The difference equation has the form of a second-order random walk in which the random terms are non-linear combinations of Gaussian random variables. For a wide class of problems, the transition pdf is joint-normal to second order in h; the technique then reduces to a Gaussian random walk, but its application is not limited to problems having a Gaussian solution. A large number of independent sample paths are generated in a Monte Carlo solution algorithm; any statistical function of the solution (e.g., moments or pdf's) can be estimated by ensemble averaging over these paths

Wave Group Statistics

Abstract: Because of the dispersive property of surface waves in deep water, whitecaps are intermittent, and their lifetime is controlled partly by the lengths of wave groups, or of runs of successive high waves. This paper discusses and unifies two different approaches to the problem of wave grouping: (a) treating the sea state as a gaussian process, with group properties given by the wave envelope function, and (b) treating the sequence of wave-heights as a one-step Markov process. The latter is here related, with examples, to the spectral density function. E(σ), and it is shown that the spectral width parameter V plays an important role in both (a) and (b). It is pointed out, however, that any group analysis implicitly requires a prefiltering of the data, and an appropriate band-pass filter is determined. The Markov predictions that the distribution of high runs is a negative exponential, and that the total group length is the difference between two exponentials, are both confirmed by comparison with numerical data.

A Test of the Gaussian-ness of a Data Set Using Clustering

Abstract: The properties of the ``magnifying glass'' method of clustering are discussed. These properties, which include unbiased and consistent estimation of the mean for Gaussian distributions and biased and inconsistent estimation of the mean for non-Gaussian distributions, lead to the development of a technique for testing data to determine whether or not it is Gaussian. An example of a non-Gaussian distribution is given to show the sensitivity of the proposed Gaussian test.

Elasto‐Plastic Oscillator with Gaussian Excitation

Abstract: A common design criterion for plastic frame or truss structures is that the probability of formation of a mechanism should be below some specified value. However, a mechanism formation need not be catastrophic because the masses of the structure must be accelerated in order to move the mechanism. If the load on the structure varies randomly in time, the load may change such that the plastic movement stops shortly after it has started. The relevant design parameter may therefore be related to the accumulated plastic deformation of the structure rather than to the mere formation of a mechanism. The problem of calculating this plastic movement process is studied for a single degree of freedom linear elastic‐ideal plastic oscillator subject to stationary Gaussian process excitation. It is assumed that the events of plastic movements are rare and of short duration such that the movement process may be modeled as a compound Poisson process. The study concentrates on the calculation of the distribution of the si...

Gaussian beam excitation of TE0 nonlinear guided waves

Abstract: Selective, efficient excitation of TE0 nonlinear guided waves is demonstrated numerically. For a thin film bounded by two self‐focusing media, three different field distributions corresponding to the same flux level can be excited independently by suitable Gaussian input beams.