Showing papers on "Gaussian process published in 1991"

Bayesian Prediction of Deterministic Functions, with Applications to the Design and Analysis of Computer Experiments

Abstract: This article is concerned with prediction of a function y(t) over a (multidimensional) domain T, given the function values at a set of “sites” {t (1), t (2), …, t (n)} in T, and with the design, that is, with the selection of those sites. The motivating application is the design and analysis of computer experiments, where t determines the input to a computer model of a physical or behavioral system, and y(t) is a response that is part of the output or is calculated from it. Following a Bayesian formulation, prior uncertainty about the function y is expressed by means of a random function Y, which is taken here to be a Gaussian stochastic process. The mean of the posterior process can be used as the prediction function ŷ(t), and the variance can be used as a measure of uncertainty. This kind of approach has been used previously in Bayesian interpolation and is strongly related to the kriging methods used in geostatistics. Here emphasis is placed on product linear and product cubic correlation func...

The conditional dissipation rate of an initially binary scalar in homogeneous turbulence

Abstract: It is shown that a necessary and sufficient condition for the scalar dissipation rate, conditioned on scalar value φ, to be independent of φ is that the one‐point scalar probability distribution function (pdf) is Gaussian. It is then shown that the amplitude mapping closure yields a closed‐form, separable expression for the φ dependence of the conditional dissipation rate in the case of an initial double‐delta scalar pdf. If the initial binary scalar field is located at φ=±1, the solution is exp{ − 2[erf−1(φ)]2}, a result that is strongly supported by earlier direct numerical simulations.

Brownian Dynamics Simulations Without Gaussian Random Numbers

Abstract: We point out that in a Brownian dynamics simulation it is justified to use arbitrary distribution functions of random numbers if the moments exhibit the correct limiting behavior prescribed by the Fokker-Planck equation. Our argument is supported by a simple analytical consideration and some numerical examples: We simulate the Wiener process, the Ornstein-Uhlenbeck process and the diffusion in a Φ4 potential, using both Gaussian and uniform random numbers. In these examples, the rate of convergence of the mean first exit time is found to be nearly identical for both types of random numbers.

Vector process out-crossing as parallel system sensitivity measure

Abstract: The mean rate of vector processes out-crossing safe domains is calculated using methods from time-independent reliability theory. The method is founded on a result for scalar up-crossing derived by Madsen. The out-crossing is formulated as a zero down-crossing of a continuously differentiable scalar process, and the mean crossing rate is obtained as a sensitivity measure of the probability for an associated parallel system domain. The vector process may be Gaussian, non-Gaussian, stationary or nonstationary, and the failure function defining the boundary of the safe domain may be time-dependent. A method for calculation of the expected number of crossings in a time interval through the introduction of an auxiliary uniformly distributed variable is presented. For stochastic failure surfaces the ensemble averaged rate is determined. A closed-form expression for the mean crossing rate of a non-stationary Gaussian vector process crossing into a time-dependent convex polyhydral set is derived. The method is demonstrated to give good results by examples.

Gaussian reference fluid and interpretation of quantum geometrodynamics.

Abstract: The Wheeler-DeWitt equation of vacuum geometrodynamics is turned into a Schr\"odinger equation by imposing the normal Gaussian coordinate conditions with Lagrange multipliers and then restoring the coordinate invariance of the action by parametrization. This procedure corresponds to coupling the gravitational field to a reference fluid. The source appearing in the Einstein law of gravitation has the structure of a heat-conducting dust. When one imposes only the Gaussian time condition but not the Gaussian frame conditions, the heat flow vanishes and the dust becomes incoherent. The canonical description of the fluid uses the Gaussian coordinates and their conjugate momenta as the fluid variables. The energy density and the momentum density of the fluid turn out to be homogeneous linear functions of such momenta. This feature guarantees that the Dirac constraint quantization of the gravitational field coupled to the Gaussian reference fluid leads to a functional Schr\"odinger equation in Gaussian time. Such an equation possesses the standard positive-definite conserved norm.For a heat-conducting fluid, the states depend on the metric induced on a given hypersurface; for an incoherent dust, they depend only on geometry. It seems natural to interpret the integrand of the norm integral as the probability density for the metric (or the geometry) to have a definite value on a hypersurface specified by the Gaussian clock. Such an interpretation fails because the reference fluid is realistic only if its energy-momentum tensor satisfies the familiar energy conditions. In the canonical theory, the energy conditions become additional constraints on the induced metric and its conjugate momentum. For a heat-conducting dust, the total system of constraints is not first class and cannot be implemented in quantum theory. As a result, the Gaussian coordinates are not determined by physical properties of a realistic material system and the probability density for the metric loses thereby its operational significance. For an incoherent dust, the energy conditions and the dynamical constraints are first class and can be carried over into quantum theory. However, because the geometry operator considered as a multiplication operator does not commute with the energy conditions, the integrand of the norm integral still does not yield the probability density. The interpretation of the Schr\"odinger geometrodynamics remains viable, but it requires a rather complicated procedure for identifying the fundamental observables. All our considerations admit generalization to other coordinate conditions and other covariant field theories.

Directionality and Reversibility in Time Series

Abstract: Summary This paper gives a review and development of what is currently known about the directionality (irreversibility) of time series models, together with briefer coverage of the still limited statistical methodology. Reversibility is shown to imply stationarity; Weiss's result concerning the reversibility of linear Gaussian processes is stressed, and contrasted to the directional nature of much time series data. Reversed ARMA models are explored, and non-linear examples given; the stationarity and invertibility conditions of ARMA models are shown to be implicitly directional, and a consequence of the future-independent nature of such models. Invertibility is extended to the two-sided futuredependent generalised linear model, and applied to reversible moving average models. The directional and reversible implications of autoregressive roots are covered. Work applying directional-sensitive methods of statistical analysis to reversed data series is mentioned; possible dangers in transforming directional series to Gaussian marginal distributions are noted. The directional nature of most non-linear models is invoked to emphasise the current importance of the area.

On convergence rates of suprema

Abstract: It is shown that the convergence rate of suprema of stationary Gaussian and related processes, such as processes defined by the empirical distribution function, is logarithmically slow, even if the rates are to be uniform over as few as three points. It is proved that the bootstrap approximation provides a substantial improvement.

Simulation of Multivariate Nonstationary Random Processes by FFT

Abstract: The fast Fourier transform (FFT) technique is utilized to simulate a multivariate nonstationary Gaussian random process with prescribed evolutionary spectral description. A stochastic decomposition technique facilitates utilization of the FFT algorithm. The decomposed spectral matrix is expanded into a weighted summation of basic functions and time‐dependent weights that are simulated by the FFT algorithm. The desired evolutionary spectral characteristics of the multivariate unidimensional process may be prescribed in a closed form or a set of numerical values at discrete frequencies. The effectiveness of the proposed technique is demonstrated by means of three examples with different evolutionary spectral characteristics derived from past earthquake events. The closeness between the target and the corresponding estimated correlation structure suggests that the simulated time series reflect the prescribed probabilistic characteristics extremely well. The simulation approach is computationally efficient, p...

Effective circle fitting for particle trajectories

Abstract: We present a fast method for circular trajectory fitting. Our method is based on an explicit solution of an nonlinear least-squares problem to fit the circle curvature, direction and position parameters. The advantage with respect to previously published methods is that these parameters are Gaussian behaved which implies more reliable error estimation of the fitted parameters. We present formulae for error estimation as well as for propagation of parameters and error matrix to another point of reference.

Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order

Abstract: We show that under suitable conditions the covariance of a Gaussian reciprocal process satisfies a self-adjoint linear differential equation of second order. We also give a revised definition of a linear stochastic differential equation of second order and necessary and sufficient conditions for the existence of solutions of such equations with Dirichlet boundary conditions. We close with a series of examples of the theory applied to the scalar stationary Gaussian Reciprocal processes which have been completely classified.

Simulation of non‐Gaussian space random functions for modeling transport in groundwater

Abstract: The recent developments and successful applications of covariance-related stochastic methods to groundwater fields experiments suggest that further efforts in developing spatial characterization methods are warranted. One area that deserves further attention is that of characterizing multimodal distributions, such as fractured rock masses, dolomite rocks with dissolution channels, and sand-shale formations. Traditionally used covariance functions may not be enough to fully characterize spatial continuity applications where the pattern of spatial continuity depends on the specific magnitude level of the attribute, for example, hydraulic conductivity. Alternative models based on multiple indicator covariances or mixture of populations offer greater flexibility. Applying such models to the case of perfect stratification, a stochastic simulations is proposed which allows generating different realizations, all sharing the same vertical conductivity covariance yet differing by other spatial statistics and leading to widely different transport characteristics. A mere covariance-based approach would have failed to distinguish between these alternatives. Investigation of tracer transport in bimodal stratified formations suggests a dependence of the mean advection velocity on the dispersion coefficients as a result of a mechanism that enhance longitudinal dispersion directly proportional to lateral pore scale dispersion and entails a diversion of a substantial portion of the plume into the low-conductivity layers.more » This mechanism may not be evident when the class-specific spatial structures of the conductivities are ignored.« less

M Estimators of Location for Gaussian and Related Processes With Slowly Decaying Serial Correlations

Abstract: We investigate the behavior of M estimators of the location parameter for stochastic processes with long-range dependence. The processes considered are Gaussian or one-dimensional transformations of Gaussian processes. It turns out that, up to a constant, all M estimators are asymptotically equivalent to the arithmetic mean. For Gaussian processes this constant is always equal to one, independently of the ψ function. In view of the case of iid observations, the results are surprising. They are related to earlier work by Gastwirth and Rubin. Some simulations illustrate the results.

Modelling Spatially Correlated K-Distributed Clutter

Abstract: K-distributed clutter is represented as the product of a Rayleigh speckle component and the square root of a gamma distributed modulation process. A model describing the modulation process of arbitrary spatial autocorrelation function is presented. The key advantages of this model over previous models are that the spatial correlation is introduced to Gaussian processes, thus enabling well established linear filtering techniques to be used, the resulting clutter process is strictly K-distributed, and it leads to usable expressions for the multivariate description of the modulation and clutter processes.

Higher correlation functions of chaotic dynamical systems-a graph theoretical approach

Abstract: For dynamical systems conjugated to the Bernoulli shift and their higher-dimensional extensions of Kaplan-Yorke type the author calculates higher-order correlation functions by means of a graph theoretical method. The graphs relevant to this problem are forests of incomplete double binary trees. His method has similarities with the Feynman graph approach in quantum field theory: the 'free field' corresponds to a Gaussian random dynamics, the 'interacting field' to a chaotic process with non-trivial higher-order correlations. The 'coupling constant' tau 1/2 is a time scale parameter that measures how much the chaotic process differs from a Gaussian process. He develops a perturbation theory around the Gaussian limit case for sums of iterates of the fully developed logistic map with arbitrary coefficients.

On the mixture of the inverse Gaussian distribution with its complementary reciprocal

Abstract: By generalizing the inverse Gaussian distribution function, we obtain a new three- parameter family of distributions which includes as special cases the inverse Gaussian and the reciprocal inverse Gaussian distributions, while preserving some of the interesting properties of the inverse Gaussian distribution. We derive two representations of the new distribution, one as the mixture of an inverse Gaussian distribution with its complementary reciprocal, and the second as a sum of an inverse Gaussian variable and an independent compound Bernoulli variable. The family is a two-parameter exponential model for known value of the third parameter, and is intimately related with exponential dispersion model theory. We also consider estimation and inference properties for the family, and show that it may have applications for positive right-skewed unimodal data and, in particular, duration or failure-time data.

Impact interactions between two vibration systems under random excitation

Abstract: The exact stationary solutions of the random response of two special vibration systems with impact interactions are formulated in this paper. Between the two systems, the Hertz contact law is used to model the contact process during vibration. A clearance is also introduced. The excitation is assumed to be a stationary white Gaussian process with zero mean and acting on the two systems independently. By assuming the ratio of the excitation and damping parameter of each system to be the same, the exact solutions can be found through solving the time-independent Fokker-Planck equation. The effects of contact stiffness, clearance and the system properties on the response are discussed probabilistically. From this study, it is found that, under some cases, the contact phenomena still play an important role on the response even when the clearance is larger than three times the root mean square response of the corresponding linear systems.

On growth in Gaussian elimination with complete pivoting

Abstract: It has been conjectured that when Gaussian elimination with complete pivoting is applied to a real n-by-n matrix, the maximum possible growth is n. In this note, a 13-by-13 matrix is given, for which the growth is 13.0205. The matrix was constructed by solving a large nonlinear programming problem. Growth larger than n has also been observed for matrices of orders 14, 15, and 16.

Integral-geometric construction of self-similar stable processes

Abstract: Recently, fractional Brownian motions are widely used to describe complex phenomena in several fields of natural science. In the terminology of probability theory the fractional Brownian motion is a Gaussian process {X(t) : t e R} with stationary increments which has a self-similar property, that is, there exists a constant Η (for the Brownian motion Η = 1/2, in general 0 0, two processes {X(ct) :teR} and {cX(t) :teR} are subject to the same law (see [10]). Let us consider the following generalization of fractional Brownian motions.

Goodness-of-fit tests for the second moment funciton of a stationary multidimensional poisson process

Abstract: This paper suggests several tests for checking the nullhypothesis that a point pattern observed in a (large) rectangular sampling window D⊂Rd belongs to a realization of a stationary Poisson Process based on test statistics measuring the distance between the empirical and the true second moment function volume of the unit sphere im R d. We consider the case of known intensity and the case of estimated intensity separately. On the basis of weak convergence of the corresponding diviation processes on [0,R] (as D becomes large) to Gaussian Processes various analogues to the classical goodness-of-fit tests X 2 Kolmogorov-SMIRNOV, CRAMER-VON MISES test) are developed and discussed. Tabulated values of the test statistics based on simulated data compared with the quantiles of their limit distributions illustrate the applicability of the proposed tests

Zero-crossing rates of functions of Gaussian processes

Abstract: Formulas for the expected zero-crossing rates of random processes that are monotone transformations of Gaussian processes can be obtained by using two different techniques. The first technique involves derivation of the expected zero-crossing rate for discrete-time processes and extends the result of the continuous-time case by using an appropriate limiting argument. The second is a direct method that makes successive use of R. Price's (1958) theorem, the chain rule for derivatives, and S.O. Rice's (1954) formula for the expected zero-crossing rate of a Gaussian process. A constant, which depends on the variance of the transformed process and a second-moment of its derivative, is derived. Multiplying Rice's original expression by this constant yields the zero-crossing formula for the transformed process. The two methods can be used for the general level-crossing problem of random processes that are monotone functions of a Gaussian process. >

Rates of clustering for some Gaussian self-similar processes

Abstract: The analogue of Strassen's functional law of the iterated logarithm in known for many Gaussian processes which have suitable scaling properties, and here we establish rates at which this convergence takes place. We provide a new proof of the best upper bound for the convergence toK by suitably normalized Brownian motion, and then continue with this method to get similar bounds for the Brownian sheet and other self-similar Gaussian processes. The previous method, which produced these results for Brownian motion in ℝ1, was highly dependent on many special properties unavailable when dealing with other Gaussian processes.

Multiple Wiener-itô integral expansions for level-crossing-count functionals

Abstract: This paper applies the stochastic calculus of multiple Wiener-Ito integral expansions to express the number of crossings of the mean level by a stationary (discrete- or continuous-time) Gaussian process within a fixed time interval [0,T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-Ito integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.

Time delay estimation with unknown spatially correlated Gaussian noise

Abstract: The problem of estimating the difference in arrival times of a nonGaussian signal at two spatially separated sensors is considered. The signal is assumed to be corrupted by spatially correlated Gaussian noises of unknown cross correlation. The author proposes and analyzes two class methods for time delay estimation based on higher order statistics. The proposed methods are conceptually very similar to the traditional cross correlation based techniques in that the proposed criteria peak at the lag value equals the true delay. At the same time, the proposed methods are based on higher-order cumulant statistics of the data and are therefore unaffected by the Gaussian noises. >