Nonstationary Gaussian Process Research Articles

On the weak convergence of a class of estimators of the variance-time curve of a weakly stationary point process

The second-moment structure of an estimator V*(t) of the variance-time curve V(t) of a weakly stationary point process is obtained in the case where the process is Poisson. This result is used to establish the weak convergence of a class of estimators of the form Tβ (V*(tTα ) – V(tTα )), 0 < α < 1, to a non-stationary Gaussian process. Similar results are shown to hold when α = 0 and in the case where V(tTα ) is replaced by a suitable estimator.

On the weak convergence of a class of estimators of the variance-time curve of a weakly stationary point process

The second-moment structure of an estimator V*(t) of the variance-time curve V(t) of a weakly stationary point process is obtained in the case where the process is Poisson. This result is used to establish the weak convergence of a class of estimators of the form Tβ(V*(tTα) – V(tTα)), 0 < α < 1, to a non-stationary Gaussian process. Similar results are shown to hold when α = 0 and in the case where V(tTα) is replaced by a suitable estimator.

Tests of randomness in two dimensions

In an earlier paper, the author shows that a suitably normalized estimator of the variance function of a two-dimensional Poisson process converges to a two-dimensional nonstationary Gaussian process . In this paper, distributions of the functionals and are obtained, and computing formulas for sample analogues are given. The first is a normal random variable; the characteristic function, the first two moments and an approximation to the upper tail of the distribution function of the second are given. Finally, tests of “marginal randomness” based upon are considered.

Reliability of Prismatic Bars in Tension

Gaussian processes with absorbing barriers have been the subject of much study because of their numerous applications. The reliability of a brittle prismatic bar under uniform tension and whose strength has a stationary Gaussian distribution leads to one such problems. A particular choice of the autocorrelation function of strength allows translating the problem into a partial differential equation that admits an expedient numerical solution. Therein lies the main attractiveness of the problem. By changing the boundary, a generalization will furnish the reliability of prismatic bars under variable tension and of nonprismatic bars, as well as the solution to other problems of nonstationary Gaussian processes with absorbing barriers. The formulation can also be reinterpreted as pertaining to non-Gaussian probability distribution of strength provided the autocorrelation function is treated appropriately.

A Study of Lifting Rotor Flapping Response Peak Distribution in Atmospheric Turbulence

Studies are made of the peak statistics of rigid-blade flapping responses to atmospheric turbulence at high advance ratios. The rotor model is characterized by a finite dimensional linear system with periodically varying parameters and with feedback controls. System inputs represent a modulated nonstationary Gaussian process. The response description includes: 1) the ratio between the total number of peaks and of zero-level up-crossings per period, 2) the ranges of lower level thresholds within which the response process could deviate from being a narrow-band process, and 3) the accuracy of the approximate formulae of peak distribution conditional on the occurrence of a peak. The conditional probability density of peak magnitude indicates that high-level peaks which cause significant damage to system components are most likely to occur within narrow ranges of the azimuth angle. The Rayleigh density law provides a satisfactory approximation to this conditional prpbability density only within such narrow azimuth ranges. A similar approximation by a widely used narrow-band formula is not satisfactory. Numerical analysis further substantiates an earlier finding that the flapping response is a quasi-coherent narrow-band process with small phase angle variance.

鋼圧縮材の座屈強度の確率論的方法による研究 : ランダムな初期たわみを有する圧縮材

The initial imperfectionls of a steel column are probabilistic in nature, and the backling strength of a column has a scatter. The principal objective of this paper is to find the buckling behavior of a column under the influence of random initial deflections. Considering a pinned-end column, the initial deflection shape is treated as a function of a nonstationary Gaussian process which satisfies the boundary conditions. For a medium column (Slenderness ratio from 50 to 150), which fails in the elasto-plastic range, the variation of initial deflections has a considerable effect on the scatter of buckling strength. As it is difficult to treat the problem theoretically, a computerized simulation is presented, in which the nonlinear analytical computer program by the F.E.M. is utilized. Since conditions for the computation have been chosen herein, the general result is not obtained. However, the some qualitative results of the buckling strength of columns become clear of the investigation on the statistical treatment of the simulations.

On Some Global Measures of the Deviations of Density Function Estimates

We consider density estimates of the usual type generated by a weight function. Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity. Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics. In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters. A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved.

Some Limit Theorems for Maxima of Nonstationary Gaussian Processes

Let $Z_n = \max_{1\leqq j\leqq n} X_j$ where $\{X_n: 1 \leqq n < \infty\}$ is a Gaussian process. Some known limit theorems for $Z_n$ when $\{X_n\}$ is stationary are extended to the nonstationary case.

An iterated logarithm law for maxima of nonstationary Gaussian processes

An iterated logarithm type theorem is obtained for maxima of nonstationary discrete-parameter Gaussian processes.

An iterated logarithm law for maxima of nonstationary Gaussian processes

An iterated logarithm type theorem is obtained for maxima of nonstationary discrete-parameter Gaussian processes.

First-excursion probability in non-stationary random vibration

The first-excursion probability of a non-stationary Gaussian process with zero mean has been studied. Within the framework of the point process approach, a variety of analytical approximations applicable to stationary random processes is extended herein to non-stationary random processes. The extension is possible owing to a recent definition of non-stationary envelope processes proposed by the author. With the aid of numerical examples, merits of each approximation are examined by comparing with the results of simulation. It is found that under non-stationary excitations with short duration, the Markov approximation is the best among all the approximations discussed in this paper. The efficient numerical computation of excursion statistics and the clear advantage of discretization of time domain are discussed.

Fitting a Filtered Poisson Process

A technique has been developed for generating an approximate characterization of a class of non-Gaussian stochastic processes. The class of processes considered are basically those non-Gaussian stationary processes for which the fourth moments of the univariate distributions are greater than the fourth moments of corresponding Gaussian approximations to the processes. In addition, the processes are assumed to be one-dimensional with continuous sample functions and the corresponding autocovariance functions are assumed to go to zero within finite time intervals. The characterization is accomplished by fitting a filtered Poisson process to the non-Gaussian process by matching the autocovariance function and an arbitrary number of the lower order univariate moments of the two processes. In matching the autocovariance function, a problem mathematically equivalent to the “factorization” of the power spectrum of a stationary process is solved numerically using an iterative procedure. The main advantage of using the filtered Poisson process is that linear operations on these processes yield filtered Poisson processes. This then simplifies the determination of probabilistic information of the output of a stochastic process passed through linear filters. Such considerations might be required, for example, in reliability problems involving random vibrations of linear mechanical systems.

Optimum estimation of nonstationary Gaussian signals in noise

This paper treats the problem of estimating a signal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(t)</tex> in the presence of noise <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(t)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(t)</tex> are independent nonstationary Gaussian processes. Specifically, we present the maximum likelihood and the minimum mean-square estimates of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(t)</tex> for each <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t, T_{1} \leq t \leq T_{2}</tex> , by observing the entire signal-plus-noise waveform <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(\cdot)</tex> during the interval <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[T_{1}, T_{2}]</tex> . Under the condition that the signal cannot be detected perfectly in the presence of noise, we explicilyty prove that both estimates, denoted by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ŝ(t)</tex> , are given by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E\{s(t)|x(\cdot)\}</tex> , the conditional expectation of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(t)</tex> given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(\cdot)</tex> . With the use of simultaneously orthogonal expansions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(t)</tex> , we further obtain explicit expressions in the form of infinite series for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ŝ(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon(t) \equiv E|s(t) - ŝ(t)|^{2}</tex> , the minimum mean-square error. Moreover, if the integral equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sum_{j=0}^{p} \int H̃_{j}(s, u) \frac{\delta^{j}}{\delta u^{j}}X(u, t) du = S(s, t)</tex> admits a formal solution <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{H̃_{j}(s,t)\}</tex> , then <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ŝ(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon(t)</tex> have the closed-form expressions <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ŝ(t) = \sum_{j=0}^{p} \int H̃_{j} (t,u)x^{j}(u)du,</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon(t) = \sum_{j=0}^{p} \int H̃_{j}(t,u)\frac{\delta^{j}}{\delta u^{j}}N(u,t)du,</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S(s,t), N(s,t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X(s,t)</tex> are the covariances of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(t), n(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x(t)</tex> , respectively, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> is the largest integer for which the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2p</tex> th partial derivatives of the covariances are continuous. The last result is a generalization of the classical Wiener filtering theory for stationary processes, and it is valid without the condition of imperfect detection if existence of the formal solution is assumed instead. Finally, we exhibit a general solution of the integral equation in the case where both <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s(t)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n(t)</tex> have rational power spectra.

Simulation of Nonstationary Random Process

A method is presented of simulating a class of nonstationary Gaussian random processes by passing a Gaussian white noise through a system introducing desirable nonstationarity at some phase of simulation. The method might have wide applications in stochastic mechanics such as in analysis of aircraft structures subjected to severe, random aerodynamic loading and in earthquake engineering. The relationship between the proposed nonstationary Gaussian process and the filtered Poisson process is examined in detail. Examples show that, if the convolution integrals, involved in the output of the system and also of the structure, can be avoided by particular choice of the impulse response functions, then a set of numerical simulations of the nonstationary process and the structural response can be performed rapidly on a computer.

On a simplified estimate of correlogram for a stationary non-Gaussian process

On a simplified estimate of correlogram for a stationary non-Gaussian process

On crossings of arbitrary curves by certain Gaussian processes

under conditions which are very close to the necessary ones. Here N(T) is the number of crossings of the level u in (0, T) by the stationary Gaussian process x(t), with covariance function r(-r). The symbol E denotes expectation. The treatment of this problem given by Bulinskaya is essentially a rigorization of the method used by Grenander and Rosenblatt [4], which in turn extends an argument due to Kac [7]. For some applications, however, it is important to consider crossings of a curve, rather than a fixed level, by such a process, and to consider also the same problem for certain Gaussian, but nonstationary processes. In ?2 we shall obtain the formula corresponding to (1) for the case where { x(t) } is a stationary Gaussian process and u = u(t) is an arbitrary (differentiable) curve, instead of a fixed level. In ?4, the same problem will be considered for a nonstationary process z(t) = f,x(s)ds where { x(t) } is, as before, a stationary Gaussian process. The discussion of this z(t)-process has application to the study of the probabilistic behaviour of certain physical systems. It would be possible to state a result corresponding to (1) for curve crossings by a member of a wide class of (nonstationary) Gaussian processes. However, this could be stated in very general terms only, and moreover it is obvious from the derivation for the z(t)-case how such a general result would be formulated.