Stationary Gaussian Stochastic Process Research Articles

Stochastic Theory of Edge Diffraction

We introduce an original formulation for the electromagnetic field diffraction by a knife edge with random roughness: the formulation, based on the asymptotic physical optics approach, leads to closed form evaluations of the statistics of the diffracted field. The edge roughness is described by a stationary zero-mean Gaussian stochastic process with standard deviation sigma and correlation length L. The physical optics approximation is used to evaluate surface currents; mean and variance of the diffracted field are evaluated by means of asymptotic techniques under the hypotheses lambda/r rarr 0, where lambda is the wavelength and r is the distance from the edge, sigma not large with respect to lambda, and sigma/r Lt 1. The main advantages of the proposed method are its simplicity and the fact that, differently from other approaches, the obtained total field is explicitly written as the sum of incident, reflected and diffracted fields. A very interesting conclusion is that, for moderate edge roughness, the diffracted field propagation can be described in terms of the same ray congruence as in the straight (smooth) edge case, with the only difference that the field associated to each ray is a random variable whose statistics are, in this paper, computed in closed form. The presented approach can be then considered as a first step toward a general stochastic theory of edge diffraction. The proposed theoretical results are amenable of interesting practical applications: for instance, the obtained diffracted field statistics can be used to predict the maximum accuracy that can be expected for ray-tracing algorithms that are based on the straight edge assumption.

Stability of suspension bridge catwalks under a wind load

A nonlinear numerical method was developed to assess the stability of suspension bridge catwalks under a wind load. A section model wind tunnel test was used to obtain a catwalk\'s aerostatic coefficients, from which the displacement-dependent wind loads were subsequently derived. The stability of a suspension bridge catwalk was analyzed on the basis of the geometric nonlinear behavior of the structure. In addition, a full model test was conducted on the catwalk, which spanned 960 m. A comparison of the displacement values between the test and the numerical simulation shows that a numerical method based on a section model test can be used to effectively and accurately evaluate the stability of a catwalk. A case study features the stability of the catwalk of the Runyang Yangtze suspension bridge, the main span of which is 1490 m. Wind can generally attack the structure from any direction. Whenever the wind comes at a yaw angle, there are six wind load components that act on the catwalk. If the yaw angle is equal to zero, the wind is normal to the catwalk (called normal wind) and the six load components are reduced to three components. Three aerostatic coefficients of the catwalk can be obtained through a section model test with traditional test equipment. However, six aerostatic coefficients of the catwalk must be acquired with the aid of special section model test equipment. A nonlinear numerical method was used study the stability of a catwalk under a yaw wind, while taking into account the six components of the displacement-dependent wind load and the geometric nonlinearity of the catwalk. The results show that when wind attacks with a slight yaw angle, the critical velocity that induces static instability of the catwalk may be lower than the critical velocity of normal wind. However, as the yaw angle of the wind becomes larger, the critical velocity increases. In the atmospheric boundary layer, the wind is turbulent and the velocity history is a random time history. The effects of turbulent wind on the stability of a catwalk are also assessed. The wind velocity fields are regarded as stationary Gaussian stochastic processes, which can be simulated by a spectral representation method. A nonlinear finite-element model set forepart and the Newmark integration method was used to calculate the wind-induced buffeting responses. The results confirm that the turbulent character of wind has little influence on the stability of the catwalk.

Estimates for the distribution of the supremum of square-Gaussian stochastic processes defined on noncompact sets

Estimates for the distribution of the supremum of square-Gaussian stochastic processes defined on R+ are found in the paper. Using these results, we find estimates for the deviation in the uniform metric between the correlogram and the correlation function of a real stationary Gaussian stochastic process. A criterion for testing a hypothesis concerning the correlation function is also constructed.

Development and calibration of a stochastic dynamics model for the design of a MEMS inertial switch

The development and calibration to experimental data of a nonlinear stochastic dynamics model for a Micro-Electro-Mechanical System (MEMS) inertial switch is discussed. The MEMS switch is modeled as a classical vibro-impact dynamic system: a single degree-of-freedom oscillator subject to impact with a single rigid barrier. An applied load, modeled as a stationary Gaussian stochastic process with prescribed power spectral density (PSD), excites the device and causes repetitive impacts with the barrier. A subset of the model parameters are described as correlated random variables to represent the significant unit-to-unit variability observed during testing of a collection of the switches. Experimental measurements from linear modal and nonlinear transient tests on multiple nominally-identical units are used to calibrate the probabilistic model. The calibrated model for the MEMS inertial switch is then used for probabilistic design studies, where the metric of performance is the amount of time the switch remains closed when subject to the applied stochastic load.

Simulation of Multivariate Stationary Gaussian Stochastic Processes: Hybrid Spectral Representation and Proper Orthogonal Decomposition Approach

By observing that the optimal basis for the proper orthogonal decomposition (POD) can be obtained from the cross power spectral density (XPSD) matrix of a multivariate stationary Gaussian stochastic process, the computational efficiency, in both time and memory consumption, of simulations of this process is improved by using a hybrid spectral representation and POD approach with negligible loss of accuracy. This hybrid approach actually simulates another multivariate process with many fewer variables in an optimal subspace obtained by the POD. This approach is straightforward, effective, and does not place any conditions on the XPSD matrices. Furthermore, the error induced by the reduction of variables is predictable and controllable prior to the simulation procedure. The spectral representation method (SRM) is discussed in a heuristic way. In this paper, a specific POD theorem is formally stated, proved, and related to XPSD matrices. A numerical example is given to demonstrate the effectiveness of this hybrid approach. This approach may also have potential applications for simulations of nonstationary non-Gaussian processes.

A test problem for molecular dynamics integrators

We derive a test problem for evaluating the ability of time-stepping methods to preserve the statistical properties of systems in molecular dynamics. We consider a family of deterministic systems consisting of a finite number of particles interacting on a compact interval. The particles are given random initial conditions and interact through instantaneous energy- and momentum-conserving collisions. As the number of particles, the particle density, and the mean particle speed go to infinity, the trajectory of a tracer particle is shown to converge to a stationary Gaussian stochastic process. We approximate this system by one described by a system of ordinary differential equations and provide numerical evidence that it converges to the same stochastic process. We simulate the latter system with a variety of numerical integrators, including the symplectic Euler method, a fourth-order Runge-Kutta method, and an energyconserving step-and-project method. We assess the methods' ability to recapture the system's limiting statistics and observe that symplectic Euler performs significantly better than the others for comparable computational expense.

Maximum likelihood estimation of amplitude-modulated time series

The concern of this paper is to study a class of nonstationary signals of the form x( t) c( t) where x( t) is a stationary Gaussian stochastic process and c( t) is a deterministic signal. The process x( t) is modeled by an autoregressive (AR) process. The deterministic signal c( t) is a known function of a finite-dimensional unknown vector. Closed-form expressions are derived for the finite-sample Cramér–Rao bound. Algorithms for the maximum likelihood estimation of c( t) and the spectral density of x( t) are developed. The proposed methods are applied to the problem of estimating abrupt change in multiplicative noise.

Nonparametric approach to Wiener system identification

A Wiener system, i.e., a system consisting of a linear dynamic subsystem followed by a memoryless nonlinear one is identified. The system is driven by a stationary white Gaussian stochastic process and is disturbed by Gaussian noise. The characteristic of the nonlinear part can be of any form. The dynamic subsystem is asymptotically stable. The a priori information about both the impulse response of the dynamic part of the system and the nonlinear characteristics is nonparametric. Both subsystems are identified from observations taken at the input and output of the whole system. The kernel regression estimate is applied to estimate the invertible part of the nonlinearity. An estimate to recover the impulse response of the dynamic part is also given. Pointwise consistency of the first and consistency of the other estimate is shown. The results hold for any nonlinear characteristic, and any asymptotically dynamic subsystem. Convergence rates are also given.

Midbroken reinforced concrete shear frames due to earthquakes. A hysteretic model to quantify damage at the storey level

A nonlinear hysteretic model for the response and local damage analyses of reinforced concrete shear frames subject to earthquake excitation is proposed, and, the model is applied to analyse midbroken reinforced concrete (RC) structures due to earthquake loads. Each storey of the shear frame is represented by a Clough and Johnston hysteretic oscillator with degrading elastic fraction of the restoring force. The local damage is numerically quantified in the domain [0,1] using the maximum softening damage indicators which are defined in closed form based on the variation of the eigenfrequency of the local oscillators due to the local stiffness and strength deterioration. The proposed method of response and damage analyses is illustrated using a sample 5 storey shear frame with a weak third storey in stiffness and/or strength subject to sinusoidal and simulated earthquake excitations for which the horizontal component of the ground motion is modeled as a stationary Gaussian stochastic process with Kanai-Tajimi spectrum, multiplied by an envelope function.

Designing random vibration tests

The paper provides information about two concepts of digital simulation of samples of the stationary Gaussian stochastic processes possessing multi-modal spectra. They have been applied in order to imitate dynamic loads arising on an airplane undergoing gusty flying conditions. Therefore the particular spectra typical for an airplane gust study were involved, reflecting also elastic properties of the flying vehicle. The presented details are devoted to the problem of solving the system of algebraic non-linear equations describing the desired linear filter. At this stage the given results can be also applied in studying earthquakes, modelling gusty winds for civil engineering and other purposes.

Polynomial estimates of the correlation function in measurements at random times

Estimates are proposed for the correlation function of a stationary Gaussian stochastic process when the measurement times form a Poisson stream of events, in which the measurement times are known only within error limits.

Simulation accuracy of stationary Gaussian stochastic processes inL 2(0,T)

Models of stationary Gaussian stochastic processes with discrete and continuous spectra are constructed. Simulation of stationary Gaussian processes with a continuous spectrum is considered for the following cases: when the covariance function of the stochastic process is expandable in a Fourier series with positive coefficients; when the spectrum of the stationary Gaussian stochastic process is concentrated on the interval [0, Λ]; and in the general case. The stationary Gaussian process is simulated with prescribed reliability and accuracy in L2(0, T).

Nuclear estimators of the correlation function and power spectrum of a stochastic process with measurements at random times

Nuclear-type estimators of the power spectrum and the correlation function of a stationary Gaussian stochastic process are proposed and investigated for the case in which the measurement times form a recurrent time series.

On the Upper and Lower Classes for a Stationary Gaussian Stochastic Process

We give a complete and rather explicit characterization of the upper and lower classes for a family of stationary Gaussian stochastic processes.

An approximation for performance evaluation of stationary single server queues

This paper provides a method for approximating the probability distributions of stationary statistics in FIFO single server queues. The method is based on the Wiener-Hopf factorization technique, and is applied to semi-Markov queues where the underlying state space is of unlimited size. It has been established that the tail of the distribution of the waiting time or unfinished work in such queues is negative exponential, and in this paper we estimate the parameters of that exponential term. A particularly important case, which is treated here, is a model for a statistical multiplexer where the net input process forms a stationary ergodic Gaussian discrete-time stochastic process. In this case, it is possible to derive analytically a simple formula in a closed form for the approximation. The formula is in terms of three parameters of the net input process: the mean, the variance, and the autocovariance sum. This provides a solid theoretical basis for traffic characterization by these parameters. Comparison with simulation results show that the method is accurate. Also presented is a result for a special case where the arrival process is autoregressive.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Approximation du temps local des surfaces gaussiennes

LetX be a centered stationary Gaussian stochastic process with ad-dimensional parameter (d≧2),F its spectral measure,\(\int\limits_{R^d } {||x||^2 F(dx) = + \infty } \) (‖x‖ denotes the Euclidean norm ofx). We consider regularizations of the trajectories ofX by means of convolutions of the formXe(t)=(Ψe*X)(t) where Ψe stands for an approximation of unity (as e tends to zero) satisfying certain regularity conditions.

Commutator of two projections in prediction theory

Let w be a nonnegative weight function in L1 = L1 (dθ/2π). Let Q and P denote the orthogonal projections to the closed linear spans in L2(wdθ/2π) of {einθ: n ≤ 0} and {einθ: n &gt; 0}, respectively. The commutator of Q and P is studied. This has applications for prediction problems when such a weight arises as the spectral density of a discrete weakly stationary Gaussian stochastic process.

Intensity cross-correlation functions in stimulated Raman scattering of colored chaotic light

The cross-correlation functions of a Stokes wave and the laser field of an arbitrary bandwidth in the process of stimulated Raman scattering were studied. The laser light was described as a stationary Gaussian stochastic process. A normalized second-order intensity cross-correlation function has been obtained both in transient and in steady-state limits. It exhibits an asymmetry with respect to zero, which is due to the buildup of Raman gain. We propose the functional method to be used in calculating the higher-order correlations.

Relativistic charged-particle interactions in a chaotic laser field

Interactions of charged particles in the presence of a chaotic laser field are considered for the case where the quadratic ${A}^{2}$ term in the Volkov solution must be retained. This is necessary whenever the kinematics are relativistic or the masses of the particles involved change during the interaction, such as in a decay, even though the kinematics might be nonrelativistic. Spin terms of the Volkov solution are also considered. The problem reduces to the solution of a stationary Gaussian stochastic process. This allows for an explicit evaluation if the two-time correlation function of the laser field is exponential with an arbitrary correlation time, i.e., for an Ornstein-Uhlenbeck process. The evaluation is performed by two alternative methods, one relying on path integrals and the other one on functional methods. Various limiting cases are discussed, notably that of an infinite correlation time. In the latter case, a cross section in a chaotic field is obtained from that in a coherent field by integrating the latter over the intensity with an exponential weight function. This prescription was already known to hold in the nonrelativistic case. As an application, we discuss high-intensity Compton scattering. A specialization of the present results yields an ensemble average, which is needed in the evaluation of a recent $g\ensuremath{-}2$ experiment for the anomalous magnetic moment of the electron.

Toward a Fundamental Theory of Optimal Feature Selection: Part I

Several authors have studied the problem of dimensionality reduction or feature selection using statistical distance measures, e.g., the Chernoff coefficient, Bhattacharyya distance, I-divergence, and J-divergence because they generally felt that direct use of the probability of classification error expression was either computationally or mathematically intractable. We show that for the difficult problem of testing one weakly stationary Gaussian stochastic process against another when the mean vectors are similar and the covariance matrices (patterns) differ, the probability of error expression may be dealt with directly using a combination of classical methods and distribution function theory. The results offer a new and accurate finite dimensionality information-theoretic strategy to feature selection, and are shown, by use of examples, to be superior to the well-known Kadota-Shepp approach which employs distance measures and asymptotics in its formulation. The present Part I deals with the theory; Part II deals with the implementation of a computer-based real-time pattern classifier which takes into account a realistic quasi-stationarity of the patterns.