Showing papers on "Gaussian process published in 1988"

Statistical inference for spatial processes

Abstract: Introduction 1. Likelihood analysis for spatial Gaussian processes 2. Edge correction for spatial point processes 3. Parameter estimation for Gibbsian point processes 4. Modelling spatial images 5. Summarizing binary images.

Estimation and model identification for continuous spatial processes

Abstract: SUMMARY Formal parameter estimation and model identification procedures for continuous domain spatial processes are introduced. The processes are assumed to be adequately described by a linear model with residuals that follow a second-order stationary Gaussian random field and data are assumed to consist of noisy observations of the process at arbitrary sampling locations. A general class of two-dimensional rational spectral density functions with elliptic contours is used to model the spatial covariance function. An iterative estimation procedure alleviates many of the computational difficulties of conventional maximum likelihood estimation for non-lattice data. The procedure is applied to several generated data sets and to an actual ground-water data set.

A method for two-electron Gaussian integral and integral derivative evaluation using recurrence relations

Abstract: An efficient method is presented for evaluating two‐electron Cartesian Gaussian integrals, and their first derivatives with respect to nuclear coordinates. It is based on the recurrence relation (RR) of Obara and Saika [J. Chem. Phys. 84, 3963 (1986)], and an additional new RR, which are combined together in a general algorithm applicable to any angular momenta. This algorithm exploits the fact that the new RR can be applied outside contraction loops. It is shown, by floating point operation counts and comparative timings, to be generally superior to existing methods, particularly for basis sets containing d functions.

Large Sample Theory of Empirical Distributions in Biased Sampling Models

Abstract: Vardi (1985a) introduced an $s$-sample model for biased sampling, gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator $\mathbb{G}_n$ of the common underlying distribution $G$ and discussed numerical methods for calculating the estimator. Here we examine the large sample behavior of the NPMLE $\mathbb{G}_n$, including results on uniform consistency of $\mathbb{G}_n$, convergence of $\sqrt n (\mathbb{G}_n - G)$ to a Gaussian process and asymptotic efficiency of $\mathbb{G}_n$ as an estimator of $G$. The proofs are based upon recent results for empirical processes indexed by sets and functions and convexity arguments. We also give a careful proof of identifiability of the underlying distribution $G$ under connectedness of a certain graph $\mathbf{G}$. Examples and applications include length-biased sampling, stratified sampling, "enriched" stratified sampling, "choice-based" sampling in econometrics and "case-control" studies in biostatistics. A final section discusses design issues and further problems.

Regularity of solutions of linear stochastic equations in hilbert spaces

Abstract: The paper gives sufficient, and in some cases necessary and sufficient, conditions for the mild solution of a stochastic linear equation to be regular both "in time" and "in space" or to be the strong solution

Statistics of the geomagnetic secular variation for the past 5 m.y.

Abstract: A new statistical model is proposed for the geomagnetic secular variation over the past 5Ma. Unlike previous models, the model makes use of statistical characteristics of the present day geomagnetic field. The spatial power spectrum of the non-dipole field is consistent with a white source near the core-mantle boundary with Gaussian distribution. After a suitable scaling, the spherical harmonic coefficients may be regarded as statistical samples from a single giant Gaussian process; this is the model of the non-dipole field. The model can be combined with an arbitrary statistical description of the dipole and probability density functions and cumulative distribution functions can be computed for declination and inclination that would be observed at any site on Earth's surface. Global paleomagnetic data spanning the past 5Ma are used to constrain the statistics of the dipole part of the field. A simple model is found to be consistent with the available data. An advantage of specifying the model in terms of the spherical harmonic coefficients is that it is a complete statistical description of the geomagnetic field, enabling us to test specific properties for a general description. Both intensity and directional data distributions may be tested to see if they satisfy the expected model distributions.

On bilinear forms in Gaussian random variables and Toeplitz matrices

Abstract: We improve a result of Szegő on the asymptotic behaviour of the trace of products of Toeplitz matrices.

A Bayesian Approach to the Design and Analysis of Computer Experiments

Abstract: We consider the problem of designing and analyzing experiments for prediction of the function y(t), telement ofT, where y is evaluated by means of a computer code (typically by solving complicated equations that model a physical system), and T represents the domain of inputs to the code. We use a Bayesian approach, in which uncertainty about y is represented by a spatial stochastic process (random function); here we restrict attention to stationary Gaussian processes. The posterior mean function can be used as an interpolating function, with uncertainties given by the posterior standard deviations. Instead of completely specifying the prior process, we consider several families of priors, and suggest some cross-validational methods for choosing one that performs relatively well on the function at hand. As a design criterion, we use the expected reduction in the entropy of the random vector y(T*), where T*is contained inT is a given finite set of ''sites'' (input configurations) at which predictions are to be made. We describe an exchange algorithm for constructing designs that are optimal with respect to this criterion. To demonstrate the use of these design and analysis methods, several examples are given, including one experiment on a computer model of a thermalmore » energy storage device and another on an integrated circuit simulator. 23 refs., 14 figs., 1 tab.« less

Boole-Bonferroni inequalities and linear programming

Abstract: We present a method to obtain sharp lower and upper bounds for the probability that at least one out of a number of events in an arbitrary probability space will occur. The input data are some of the binomial moments of the occurrences, such as the sum of the probabilities of the individual events, or the sum of the joint probabilities of all pairs of events. We develop a special, very simple linear programming algorithm to obtain these bounds. The method allows us to compute good bounds in an optimal way, utilizing only the first few terms in the inclusion-exclusion formula. Possible applications include obtaining bounds for the reliability of a stochastic system, solving algorithmically some stochastic programming problems, and approximating multivariate probabilities in statistics. In a numerical example we approximate the probability that a Gaussian process runs below a given level in a number of consecutive epochs.

Auto‐Regressive Model for Nonstationary Stochastic Processes

Abstract: An autoregressive model for univariate, one‐dimensional, nonstationary, Gaussian random processes with evolutionary power spectra is introduced. At the same time, an efficient technique for numerically generating sample functions of such nonstationary processes is developed. The technique uses a recursive equation which: (1) Reflects the nature of the nonstationarity of the process whose sample functions are to be generated; and (2) involves a normalized univariate, one‐dimensional white noise sequence. The coefficients of the recursive equation are determined using the autocorrelation function of the process, which in turn is calculated from the evolutionary power spectrum at every time instant. Using the recursive equation with those coefficients, sample functions over a specified domain can be generated with substantial computational ease. Univariate, one‐dimensional, nonstationary processes with three different forms of the evolutionary power spectrum are modeled, and their sample functions are genera...

The Logistic Normal Distribution for Bayesian, Nonparametric, Predictive Densities

Abstract: This article models the common density of an exchangeable sequence of observations by a generalization of the process derived from a logistic transform of a Gaussian process. The support of the logistic normal includes all distributions that are absolutely continuous with respect to the dominating measure of the observations. The logistic-normal family is closed in the prior to posterior Bayes analysis, with the observations entering the posterior distribution through the covariance function of the Gaussian process. The covariance of the Gaussian process plays the role of a smoothing kernel. Three features of the model provide a flexible structure for computing the predictive density: (a) The mean of the Gaussian process corresponds to the prior mean of the random density: (b) The prior variance of the Gaussian process controls the influence of the data in the posterior process. As the variance increases, the predictive density has greater fidelity to the data, (c) The prior covariance of the Gau...

On the expectation of the product of four matrix-valued Gaussian random variables

Abstract: The formula for the expectation of the product of four scalar real Gaussian random variables is generalized to matrix-valued (real or complex) Gaussian random variables. As an application of the extended formula, a simple derivation is presented of the covariance matrix of instrumental variable estimates of parameters in multivariable regression models. >

Minimum model error estimation for poorly modeled dynamic systems

Abstract: A novel strategy (which we call "minimum model error'* estimation) for postexperiment optimal state estimation of discretely measured dynamic systems is developed and illustrated for a simple example. The method is especially appropriate for postexperiment estimation of dynamic systems whose presumed state governing equations are known to contain, or are suspected of containing, errors. The hew method accounts for errors in the system dynamic model equations in a rigorous manner. Specifically, the dynamic model error terms in the proposed method do not require the usual Kalman filter-smoother process noise assumptions of zero-mean, symmetrically distributed random disturbances, nor do they require representation by assumed parameterized time series (such as Fourier series); Instead, the dynamic model error terms require no prior assumptions other than piecewise continuity. Estimates of the state histories, as well as the dynamic model errors, are Obtained as part of the solution of a two-point boundary value problem. The state estimates are continuous and optimal in a global sense, yet the algorithm processes the measurements sequentially. The example demonstrates the method and shows it to be quite accurate for state estimation of a poorly modeled dynamic system.

Strong Embedding of the Estimator of the Distribution Function under Random Censorship

Abstract: In this paper the asymptotic behaviour of the product limit estimator $F_n$ of an unknown distribution is investigated. We give an approximation of the difference $F_n(x) - F(x)$ by a Gaussian process and also by the average of i.i.d. processes. We get almost as good an approximation of the stochastic process $F_n(x) - F(x)$ as one can get for the analogous problem when the maximum likelihood estimator is approximated by a Gaussian random variable or by the average of i.i.d. random variables in the parametric case.

Equivalent Wind Spectrum Technique: Theory and Applications

Abstract: The equivalent wind spectrum technique is a mathematical model according o which wind is schematized as a stochastic stationary Gaussian process made up of a mean-speed profile on which an equivalent turbulent fluctuation, perfectly coherent in space, is superimposed. The equivalent criterion is formulated by defining a fictitious velocity fluctuation, random function of time only, giving rise to power spectra of fluctuating modal force that approximate, optimally, the corresponding modal spectra related to the actual turbulence configuration. This paper presents the basic assumptions and the theoretical steps leading to the characterization of the equivalent velocity fluctuation through a power spectrum assigned in closed form. The method proposed herein allows one to estimate the dynamic along-wind response of structures, both in frequency and in time domain, with a high level of precision and simplicity; furthermore it makes it possible to treat wind effects, as well as those of earthquakes, through the well-known response spectrum technique.

The topology of large-scale structure. II - Nonlinear evolution of Gaussian models

Abstract: The evolution of non-Gaussian behavior in the large-scale universe from Gaussian initial conditions is studied. Topology measures developed in previous papers are applied to the smoothed initial, final, and biased matter distributions of cold dark matter, white noise, and massive neutrino simulations. When the smoothing length is approximately twice the mass correlation length or larger, the evolved models look like the initial conditions, suggesting that random phase hypotheses in cosmology can be tested with adequate data sets. When a smaller smoothing length is used, nonlinear effects are recovered, so nonlinear effects on topology can be detected in redshift surveys after smoothing at the mean intergalaxy separation. Hot dark matter models develop manifestly non-Gaussian behavior attributable to phase correlations, with a topology reminiscent of bubble or sheet distributions. Cold dark matter models remain Gaussian, and biasing does not disguise this.

Response of linear systems to polynomials of Gaussian processes

Abstract: Moments and mean crossing rates are determined for the response of linear systems subject to polynomial forms in Gauss-Markov processes. Differential equations describing the evolution in time of response moments are developed by the use of Ito’s calculus. Estimators are obtained for mean crossing rates based on higher order moments and approximations of the distribution of the response. The method is applied to the analysis of a simple oscillator subject to the square of an Ornstein-Uhlenbeck process.

A conditionally almost linear filtering problem with non-gaussian initial condition ∗

Abstract: We derive a formula for computing explicitly the optimal non-linear filter for a class of problems which admit finite dimensional filters. The result includes all known results of this kind as spec...

Underwater noises: Statistical modeling, detection, and normalization

Abstract: Knowledge of the noise probability density function (PDF) is central in signal detection problems, not only for optimum receiver structures, but also for processing procedures such as power normalization. Unfortunately, the statistical knowledge must be acquired since the classical assumption of a Gaussian noise PDF is often not valid in underwater acoustics. In this article, statistical modeling is studied using a Gaussian–Gaussian mixture (GGM) for three different underwater noise data sets. It is shown that one of them can be adequately described by a Gaussian–Gaussian mixture, one is very close to a Gaussian model and is described by a mixture with a very small perturbating term, whereas the third one seems closer to a nonstationary version of the Middleton class‐A model. The first noise sample is also studied with emphasis on the normalization needed in the receiver in order to achieve a constant false alarm probability and on the optimal receiver structure for the detection of a deterministic signal...

Gaussian Processes and Almost Spherical Sections of Convex Bodies

Abstract: On presente une preuve simple avec des estimations pointues du theoreme de Dvoretzky sur l'existence des sections presque spheriques ayant une grande dimension dans des corps convexes arbitraires dans R N

Finite-temperature effects on the Gaussian effective potential

Abstract: We derive the finite-temperature Gaussian effective potential (FTGEP) from first principles, resolving some confusions arising in earlier treatments. The advantages of the FTGEP approach over the conventional loop expansion, even for high temperatures, are illustrated in a quantum-mechanical example. Results are presented for phi/sup 6/ theory in two and three dimensions and for the ''autonomous'' version of lambdaphi/sup 4/ theory in four dimensions.

On time-reversibility and the uniqueness of moving average representations for non-Gaussian stationary time series

Abstract: A recent result by Findley (1986) on the uniqueness of moving average representations for non-Gaussian time series is shown to establish a conjecture by Weiss (1975) on the time-reversibility of general linear processes. © 1988 Biometrika Trust.

Testing that a Gaussian Process is Stationary

Abstract: A class of procedures is proposed for testing the stationarity of a Gaussian process or the homogeneity of independent processes. Requiring very limited prior knowledge of model structure, the methods can detect changes or differences in mean, in variance, in covariances and even in law. Although the theory of the stationarity test is worked out only for processes whose realizations are stationary over "epochs" separated by known change points, Monte Carlo evidence indicates that it can be useful also in detecting more general forms of nonstationarity. The test statistic is a quadratic form in differences among epoch means of certain "sensing" functions, the choice of which governs sensitivity to specific forms of nonstationarity or inhomogeneity. The applicability of the general asymptotic theory of the test is verified for two specific forms of sensing function, and small-sample properties of tests of each form are studied by means of simulation.

On time-reversibility and the uniqueness of moving average representations for non-Gaussian stationary time series

Abstract: A recent result by Findley (1986) on the uniqueness of moving average representations for non-Gaussian time series is shown to establish a conjecture by Weiss (1975) on the time-reversibility of general linear processes. © 1988 Biometrika Trust.

From two-state jump to Gaussian stochastic processes

Abstract: A stochastic process, which bridges the two-state jump Markoff process and the Gaussian process, is introduced and fundamental properties of the process are fully analyzed. The resulting expressions are combined with the “partial cumulant” expansion formula to give an exact power spectrum in the form of a continued fraction. A rigorous expression is also obtained for a relaxation function in an analytic form.