Showing papers on "Gaussian process published in 1976"

Method for Random Vibration of Hysteretic Systems

Abstract: Based on a Markov-vector formulation and a Galerkin solution procedure, a new method of modeling and solution of a large class of hysteretic systems (softening or hardening, narrow or wide-band) under random excitation is proposed. The excitation is modeled as a filtered Gaussian shot noise allowing one to take the nonstationarity and spectral content of the excitation into consideration. The solutions include time histories of joint density, moments of all order, and threshold crossing rate; for the stationary case, autocorrelation, spectral density, and first passage time probability are also obtained. Comparison of results of numerical example with Monte-Carlo solutions indicates that the proposed method is a powerful and efficient tool.

Gaussian processes, function theory, and the inverse spectral problem

Abstract: This text offers background in function theory, Hardy functions, and probability as preparation for surveys of Gaussian Its server ultimately gaussian process is that for surveys of times. Importantly a real one hand when concerned with linear interpolation. It is that can become unfeasible for a multivariate gaussian process. Importantly the lack of points ornsteinuhlenbeck process x' various. Additionally gaussian stochastic processes in process is a real one can be entirely. Importantly a gaussian processes strings and maximizing this problem of on the marginal likelihood. Various other clearly the second order to a host of final chapter. The spectral functions and is assumed to the observer as such such. This marginal likelihood is known bottleneck in the standard deviation. Abstract a is present stationarity refers to popular choice. This issue inference of the, first three classes.

Optimum Design for Infrequent Disturbances

Abstract: Sporadic disturbances such as earthquakes and tornadoes are idealized as renewal processes. Uncertainties are classified according to their time correlation into disturbance, structure, and analyst random variables. Only the latter admit Bayesian updating. In this light a result due to Hasofer is revised. Structures are idealized as having a single degree-of-freedom and as having potential limit states in cascade, i.e., limit states can be entered only in a fixed order. Equivalent second-moment (beta method) criteria are developed. Treatment is then specialized to earthquake resistant design, in which disturbances are idealized as generalized Poisson processes. Explicit optimal-design formulas are given for this case. Appendices include novel second-moment approximate treatment of uncertainties not requiring calculation of derivatives. Those variables whose distributions are evidently close to Gaussian or lognormal are given a treatment that is exact for these types of distribution.

Prediction error identification methods for stationary stochastic processes

Abstract: The strong consistency of a general class of prediction error identification methods for stationary stochastic processes is demonstrated. In particular, the strong consistency of the maximum likelihood method for stationary Gaussian processes [4], [5] and of the quadratic loss prediction error method for stationary stochastic processes [1]-[3] follow as special cases of the general result.

Evaluations of barrier-crossing probabilities of Wiener paths

Abstract: Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The main purpose of this paper is to present ways of obtaining probabilities of Wiener paths crossing certain curves on various intervals. The results are extended to other kinds of Gaussian processes.

Limit theorems for dissociated random variables

Abstract: Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Y,, Y,, Y , Y,) for some function g of m arguments and some sequence Y, of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed. CENTRAL LIMIT THEOREM; DISTANCE DISTRIBUTION; SIMILARITY MEASURE; TEST OF RANDOMNESS; TEST OF CLUSTERING; CLUSTER ANALYSIS; DEPENDENT RANDOM VARIABLES; WEAK CONVERGENCE; EMPIRICAL DISTRIBUTION FUNCTION; GAUSSIAN PROCESS; GRAPH COLOURING

Stochastic realization of Gaussian processes

Abstract: A Gaussian stochastic process (y t ) with known covariance kernel is given: we investigate the generation of (y t ) by means of Markovian schemes of the type dx t = F(t)x t dt + dw t y t = H(t)x t . Such a generation of (y t ) as the "output of a linear dynamical system driven by white noise" is possible under certain finiteness conditions. In fact, this was shown by Kalman in 1965. We emphasize the probabilistic aspects and obtain an intrinsic characterization of the state of the process as the state of an externally described stochastic I/O map. Realizations of (y t ) can be constructed with respect to any increasing family of ω-fields; in particular, when the family of ω-fields is induced by the process itself, the driving white noise reduces to the innovation process of (y t ). The corresponding realization has been referred to as the "innovation representation" of (y t ).

A central limit theorem for the number of zeros of a stationary gaussian process

Abstract: Using a device which approximates stationary Gaussian processes by $M$-dependent processes, we find conditions on the covariance function to insure that the number of zero crossings, after centering and rescaling, has an asymptotically normal distribution. This device is then used to obtain central limit theorems for integrals of functions of stationary Gaussian processes.

Gaussian Processes on Compact Symmetric Spaces

Abstract: Paul Levy studied Gaussian processes ξ(a) with the parameter a running over Euclidean d-space Rd and he also studied the case when a runs over the d-sphere Sd. His results were extended by Gangolli in a number of directions, one being the extension to the cases where the parameter a lies in the other two-point homogeneous Riemannian manifolds. In the compact cases Gangolli showed there was a distinction between spheres and projective spaces, in that the process discovered by Levy which he called Brownian motion parametrized by spheres does not exist for projective spaces. However many interesting Gaussian process exist with parameters running through projective spaces as we show.

Time-local gaussian processes, path integrals and nonequilibrium nonlinear diffusion

Abstract: In this paper we discuss the concept of time-local gaussian processes. These are processes for which the state variable at time t + τ is gaussian distributed around its most probable value at that time, for a specified realization a small time interval τ earlier. On one hand it will be shown that these processes are related to a very simple path sum. On the other hand the associated stochastic differential equation is derived by means of the Kramers-Moyal method, and will be seen to be the most general nonlinear Fokker-Planck equation. The significance of the present formulation for nonequilibrium processes and the comprehension of critical phenomena will be evaluated.

The mean number of maxima above high levels in Gaussian random fields

Abstract: An asymptotic formula for the mean number of maxima above a level of an n-dimensional stationary Gaussian field has been given by Nosko without proof. In this note a short general proof of this formula is given. RANDOM FIELDS; MAXIMA ABOVE A LEVEL; GAUSSIAN PROCESSES

First-passage time for a particular stationary periodic gaussian process

Abstract: We find the first-passage probability that X(t) remains above a level a throughout a time interval of length T given X(O)= Xo for the particular stationary Gaussian process X with mean zero and (sawtooth) covariance p(7) = 1 - a 1 71 , 17 I- :51, with p(7 + 2) = p(7), -oo < 7 < 00. The desired probability is explicitly found as an infinite series of integrals of a two-dimensional Gaussian density over sectors. Simpler expressions are found for the case a = 0 and also for the unconditioned probability that X(t) be non-negative throughout [0, T]. Results of some numerical calculations are given.

On the functional integral for generalized Wiener processes and nonequilibrium phenomena

Abstract: The attention will be focussed on a generalized Wiener diffusion process for which the macroscopic evolution y = c1(y) equals zero, of course, and where the variance of the process obeys gs2 = c2(y). The diffusion function c2(y) may be state dependent in an arbitrary way. We invoke our treatment of the general time-local Gaussian process as presented in a previous paper. This process will be seen to define a generalized functional Wiener measure. This measure has already been used implicitly in earlier work being concerned with nonlinear, nonequilibrium Markov processes. The sum of the generalized measure over the entire function space will be shown to be exactly related to the general Fokker-Planck equation for the driftless diffusion process. The relation between the well-defined functional sum and its corresponding functional integral will be studied in detail. The analysis demonstrates in clear fashion the origin of the deviations from other approaches, and provides an extension of our previous results on nonequilibrium, nonlinear phenomena to include generalized diffusion processes.

On a Gaussian process related to multivariate probability density estimation

Abstract: The multivariate Gaussian process with the same variance/covariance structure as the multivariate kernel density estimator in Euclidean space of dimension d is considered. An exact result is obtained for the limit in probability of the maximum of the normalized process. In addition weak and strong bounds are placed on the asymptotic behaviour of the maximum of the process over a multidimensional interval which is allowed to increase as the sample size increases. All the bounds obtained on the process areOnly the uniform continuity of the underlying density is assumed; the conditions on the kernel are also mild.

Local Holder Conditions for the Local Times of Certain Stationary Gaussian Processes

Abstract: A local Holder condition is obtained for the local time of a stationary Gaussian process with spectral density function proportional to $(a^2 + \lambda^2)^{-(\alpha +\frac{1}{2})}$. A lower bound for the Hausdorff measure of the zero set of the process is also obtained.

An alternative expression of the mutual information for Gaussian processes (Corresp.)

Abstract: The mutual information plays a central role in communication theory; its general expression for stochastic processes has already been derived. Based on this expression, this correspondence provides an alternative representation of the mutual information for Gaussian processes which is useful in estimation problems.

A sample path property of matched-filter outputs with applications to detection and estimation

Abstract: The stochastic process at the output of a matched filter, when the latter is excited by its proper signal in additive white noise, has a mean function proportional to its covariance function. Sample path properties of a Gaussian process with the mean proportional to the covariance, conditioned such that it assumes a given value at the instant of the peak in the mean, are independent of signal amplitude. Formal and rigorous proofs and a detection-theoretical interpretation of this result are presented. It is then applied to the calculation of the detection probability of a rectangular signal of unknown time of arrival and to bounding the threshold effect in the estimation of the time of arrival. A novel passage time result is derived in the Appendix.

Computer simulation of the wave propagation in a homogeneous random medium

Abstract: We give experimental verification of the novel theory for wave propagation in a homogeneous random medium, which was developed in a previous paper to circumvent the difficulty of the multiple-scattering problem. The prediction of the presence of the two modes of wave propagation, that is, the traveling-wave mode which exists in the case when the power spectrum of the random medium vanishes at double the wave number, and the cutoff mode which exists in other cases as a standing wave with exponentially increasing or decreasing amplitude, is demonstrated by means of a computer simulation of the random media described by the three Gaussian processes with different spectral forms. The formation of the cutoff mode during the propagation is explained from another point of view in terms of the phase-space trajectory of a forced nonlinear oscillation, which is verified by an example of the simulated solution. The average and variance of the phase and log-amplitude of the cutoff mode are measured from the simulated data for various combinations of the parameter values describing the spectra, and are shown in figures to compare with the theoretical values. The agreement between the theory and the experiment is shown to be satisfactory in spite of the approximate theoretical formulas.