Stationary Gaussian Process Research Articles

Stochastic modeling of stationary scalar Gaussian processes in continuous time from autocorrelation data

We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given output data in the time domain. We construct the stochastic model in two steps. First, we employ the AAA algorithm to determine a rational function which interpolates the z-transform of the discrete data on the unit circle and use this function to assign the poles of the transfer function of the reduced model. Second, we choose the associated residues as the minimizers of a linear inequality constrained least squares problem which ensures the positivity of the transfer function’s real part for large frequencies. We apply this method to compute extended Markov models for stochastic processes obtained from generalized Langevin dynamics in statistical physics. Numerical examples demonstrate that the algorithm succeeds in determining passive reduced models and that the associated Markov processes provide an excellent match of the given data.

Smallest gaps between zeros of stationary Gaussian processes

In this paper, we study the smallest gaps between successive zeros of nondegenerate smooth stationary centered Gaussian processes on the real line with the assumption that the covariance kernel κ(x) and its derivatives decay to 0 as |x|→∞. We prove that, after rescaling, the smallest gaps converge to a Poisson point process with a specific rate. Moreover, the positions where these smallest gaps occur tend to a uniform distribution. Consequently, we can derive the limiting density for the k-th smallest gap.

Multi-Performance Economical Optimization of Base Isolation System with Tuned Inerter Negative Stiffness Damper Subjected to Non-Stationary Seismic Excitation

Base isolation proves to be an effective vibration control strategy for buildings. However, the base-isolation floor (BIF) may undergo substantial displacements. To improve the seismic performance of base-isolation system, this paper proposes a multi-performance economical optimization procedure (MPEOP) for exploring the optimal parameters of base isolation system with tuned inerter negative stiffness damper (BIS-TINSD) subjected to non-stationary seismic excitation. The simplified analysis model for BIS-TINSD is developed, followed by the formulation of the state-space representation. The non-stationary seismic excitation is represented as a stationary Gaussian process with a time-modulating function based on the Clough–Penzien spectrum, and combining the equations of motion with seismic excitation results in the augmented state-space representation of structure-damper-excitation, followed by the formulation of differential Lyapunov equation. The multi-objective performance economical index is proposed, in which the performance optimization objective is defined as a weighted combination of base isolation floor displacement and superstructure acceleration, and the economic optimization objective is set as the support stiffness. The Pareto optimal fronts (POF) are adopted to deal with optimal objectives. The examination of optimal design parameters is extended under real earthquake records. The results demonstrate the effectiveness of the MPEOP, and indicate the advantages of the BIS-TINSD compared with the inerter amplify damper (IAD) and negative stiffness inerter damper (NSID).

Shrinkage estimators of BLUE for time series regression models

The least squares estimator (LSE) seems a natural estimator of linear regression models. Whereas, if the dimension of the vector of regression coefficients is greater than 1 and the residuals are dependent, the best linear unbiased estimator (BLUE), which includes the information of the covariance matrix Γ of residual process has a better performance than LSE in the sense of mean square error. As we know the unbiased estimators are generally inadmissible, Senda and Taniguchi (2006) introduced a James–Stein type shrinkage estimator for the regression coefficients based on LSE, where the residual process is a Gaussian stationary process, and provides sufficient conditions such that the James–Stein type shrinkage estimator improves LSE. In this paper, we propose a shrinkage estimator based on BLUE. Sufficient conditions for this shrinkage estimator to improve BLUE are also given. Furthermore, since Γ is infeasible, assuming that Γ has a form of Γ=Γ(θ), we introduce a feasible version of that shrinkage estimator with replacing Γ(θ) by Γ(θˆ) which is introduced in Toyooka (1986). Additionally, we give the sufficient conditions where the feasible version improves BLUE. Besides, the results of a numerical studies confirm our approach.

PEM-based random vibration analysis under service-braking conditions on an irregular track

PurposeAccording to relevant research, non-uniform speed has a significant impact on the vehicle-track systems. Up to now, research work on it is still very limited. In this paper, the PEM is adopted to further transform it into a deterministic process to solve the vehicle’s problem of running at a non-uniform speed.Design/methodology/approachThe multi-body vehicle model has 10 degrees of freedom and the track is regarded as a finite long beam supported by lumped sleepers and ballast blocks. They are connected via linear Hertz springs. The vertical track irregularity is a Gaussian stationary process in the space domain. It is transformed into a uniformly modulated nonstationary random process in the time domain with respect to the non-uniform vehicle speed. By solving the equation of motion of the coupled vehicle-track system with the pseudo-excitation method, the pseudo-response and consequently the power spectral density and the standard deviation of the structural response can be obtained.FindingsTwo kinds of vehicle braking programs are taken in the numerical example and some beneficial conclusions are drawn.Originality/valueThe pseudo-excitation method (PEM) was used to perform the random vibration analysis of a coupled non-uniform speed vehicle-track system. Transforming the track irregularity into a uniformly modulated nonstationary random process in time domain with respect to the non-uniform vehicle speed was undertaken. The pseudo-response of the coupled system is solved by applying the Newmark algorithm with constant space integral steps. The random vibration transfer mechanism of the coupled system is fully discussed.

Analysis of the Least Mean Square algorithm with processing delays in the adaptive arm for Gaussian inputs for system identification

This paper analyzes the effect of a processing delay on the Least Mean Squares (LMS) algorithm for a system identification problem when the processing delay is in the adaptive arm of the filter. Thus the sensing of the input signal is delayed. The input is assumed to be a zero mean stationary Gaussian process. The theoretical mean and mean square behavior of the adaptive weight vector is analyzed. The weight vector is shown to be biased and significantly affects the mean square deviation (MSD). Monte Carlo simulations are presented in support of the assumptions used to derive the theoretical model as a function of the delay, bandwidth and step-size for the LMS algorithm. The results suggest bias problems with other more complicated adaptive filtering algorithms such as Normalized LMS and Recursive Least Squares.

Measuring One-Sided Process Capability Index for Autocorrelated Data in the Presence of Random Measurement Errors

Abstract Many quality characteristics in manufacturing industry are of one sided specifications. The well-known process capability indices C P ⁢ U C_{PU} and C P ⁢ L C_{PL} are often used to measure the performance of such type of production process. It is usually assumed that process observations are independent and measurement system is free of errors. But actually in many industry it has been proven that auto-correlation is an inherent nature of the production process, especially for chemical processes. Moreover, even with the use of highly sophisticated advanced measuring instruments some amount of measurement error is always present in the observed data. Hence gauge measurement error also needs to be considered. In this paper we discuss some inferential properties of one-sided process capability indices for a stationary Gaussian process in the presence of measurement errors. As a particular case of a stationary Gaussian process, we discuss the case of a stationary AR ⁡ ( 1 ) \operatorname{AR}(1) process where measurement error follows an independent Gaussian distribution.

Extreme value analysis of high-dimensional Gaussian vector processes

The analysis of the extreme value pertaining to first-passage failure is a key consideration for engineering systems undergoing random vibration. Most of the literature focus on a univariate stochastic process, however for system reliability analysis, the extreme value of multiple dependent processes is required. Due to the problem complexity, analytical methods are scarce, and they are limited to low-dimensional problems (dimension refers to the number of component processes). This paper presents an analytical method for extreme analysis of multivariate stationary Gaussian processes. The method can efficiently solve high-dimensional problems, thus it has diverse applications such as analysing dynamic systems with many degrees-of-freedom, or evaluating space-time extremes of random fields by discretizing the spatial domain. A maximum process is defined, representing the maxima of the components at each time instant. The exact upcrossing rate of the maximum process is derived, and the Poisson approximation is invoked to estimate the extreme value distribution. The method is fast as it involves multiple one-dimensional integrals, which can be computed using Gaussian quadrature. Moreover, an accurate approximation is proposed that obviates the need for numerical integration. The proposed method is implemented on two examples, specifically a dynamic system and spreading ocean waves. Monte Carlo simulations, which are computationally demanding, are performed to verify the accuracy and correctness of the proposed method.

An asymptotic formula for the variance of the number of zeroes of a stationary Gaussian process

We study the variance of the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description under mild mixing conditions. This allows us to characterise minimal and maximal growth. We show that a small (symmetrised) atom in the spectral measure at a special frequency does not affect the asymptotic growth of the variance, while an atom at any other frequency results in maximal growth.

Spatio-temporal DeepKriging for interpolation and probabilistic forecasting

Gaussian processes (GP) and Kriging are widely used in traditional spatio-temporal modelling and prediction. These techniques typically presuppose that the data are observed from a stationary GP with a parametric covariance structure. However, processes in real-world applications often exhibit non-Gaussianity and nonstationarity. Moreover, likelihood-based inference for GPs is computationally expensive and thus prohibitive for large datasets. In this paper, we propose a deep neural network (DNN) based two-stage model for spatio-temporal interpolation and forecasting. Interpolation is performed in the first step, which utilizes a dependent DNN with the embedding layer constructed with spatio-temporal basis functions. For the second stage, we use Long-Short Term Memory (LSTM) and convolutional LSTM to forecast future observations at a given location. We adopt the quantile-based loss function in the DNN to provide probabilistic forecasting. Compared to Kriging, the proposed method does not require specifying covariance functions or making stationarity assumptions and is computationally efficient. Therefore, it is suitable for large-scale prediction of complex spatio-temporal processes. We apply our method to monthly PM2.5 data at more than 200,000 space–time locations from January 1999 to December 2022 for fast imputation of missing values and forecasts with uncertainties.

Rates of convergence for the number of zeros of random trigonometric polynomials

In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein-1 distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous function, we use the Kac-Rice formula to express it as the limit of an integral and, in this way, we approximate it by locally Lipschitz continuous functions.

Winding number for stationary Gaussian processes using real variables

Winding number for stationary Gaussian processes using real variables

Data-Driven Reconstruction of Dynamical Forces and Responses through Bayesian Expectation-Maximization and Modal Reduction Methods

This paper presents a new Bayesian approach for estimating input and state in linear structures using response-only measurements. The proposed approach benefits from a modally reduced state-space model, which circumvents the dimensionality of dynamical responses in complex structures through a low-dimensional subspace. It also substitutes unknown physical forces with a set of equivalent modal forces, which is beneficial when the magnitude and location of input forces are unknown. In this work, these forces are characterized through the conventional random walk model and a class of stationary Gaussian processes. Subsequently, an augmented state-space model is constructed to describe modal states and input loads. Based on this model, a Bayesian expectation-maximization (BEM) methodology is developed to identify the input, state, and noise parameters. This noise identification perspective activates uncertainty quantification in joint input-state estimation problems and enables quantifying the degree of confidence in the estimated quantities. When the proposed method is tested using numerical and experimental examples, accurate estimations and reasonable uncertainty bounds are acquired for the dynamical state and input forces. Although the literature reports the superiority of the Gaussian process latent force model over the random walk model without using a unified noise calibration strategy, this study, to our best knowledge, is the first effort to compare and interpret the results on a consistent basis where the noise and input characteristics are all identified from the data through BEM.

On the Ergodicity of Certain Markov Chains in Random Environments

We study the ergodic behaviour of a discrete-time process X which is a Markov chain in a stationary random environment. The laws of X_t are shown to converge to a limiting law in (weighted) total variation distance as trightarrow infty . Convergence speed is estimated, and an ergodic theorem is established for functionals of X. Our hypotheses on X combine the standard “drift” and “small set” conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain “maximal process” of the random environment. We are able to cover a wide range of models that have heretofore been intractable. In particular, our results are pertinent to difference equations modulated by a stationary (Gaussian) process. Such equations arise in applications such as discretized stochastic volatility models of mathematical finance.

On Nonstationary Gaussian Process Model for Solving Data-Driven Optimization Problems.

In data-driven evolutionary optimization, most existing Gaussian processes (GPs)-assisted evolutionary algorithms (EAs) adopt stationary GPs (SGPs) as surrogate models, which might be insufficient for solving most optimization problems. This article finds that GPs in the optimization problems are nonstationary with great probability. We propose to employ a nonstationary GP (NSGP) surrogate model for data-driven evolutionary optimization, where the mean of the NSGP is allowed to vary with the decision variables, while its residue variance follows an SGP. In this article, the nonstationarity of GPs in the tested functions is theoretically analyzed. In addition, this article constructs an NSGP where the SGP is a degenerate case. Performance comparisons of the NSGP with the SGP and the NSGP-assisted EA (NSGP-MAEA) with the SGP-assisted EA (SGP-MAEA) are carried out on a set of benchmark problems and an antenna design problem. These comparison results demonstrate the competitiveness of the NSGP model.

A Scoping Review of Cerebral Doppler Arterial Waveforms in Infants.

Cerebral Doppler ultrasound has been an important tool in pediatric diagnostics and prognostics for decades. Although the Doppler spectrum can provide detailed information on cerebral perfusion, the measured spectrum is often reduced to simple numerical parameters. To help pediatric clinicians recognize the visual characteristics of disease-associated Doppler spectra and identify possible areas for future research, a scoping review of primary studies on cerebral Doppler arterial waveforms in infants was performed. A systematic search in three online bibliographic databases yielded 4898 unique records. Among these, 179 studies included cerebral Doppler spectra for at least five infants below 1 y of age. The studies describe variations in the cerebral waveforms related to physiological changes (43%), pathology (62%) and medical interventions (40%). Characteristics were typically reported as resistance index (64%), peak systolic velocity (43%) or end-diastolic velocity (39%). Most studies focused on the anterior (59%) and middle (42%) cerebral arteries. Our review highlights the need for a more standardized terminology to describe cerebral velocity waveforms and for precise definitions of Doppler parameters. We provide a list of reporting variables that may facilitate unambiguous reports. Future studies may gain from combining multiple Doppler parameters to use more of the information encoded in the Doppler spectrum, investigating the full spectrum itself and using the possibilities for long-term monitoring with Doppler ultrasound.

Efficient Method for Approximating the Joint Extreme Value Distribution of Multivariate Stationary Gaussian Processes

The joint extreme value distribution (JEVD) of multivariate random processes is important for evaluating the system reliability of a structure subjected to random vibrations. There are very limited analytical approaches for predicting the JEVD, and these approaches are only computationally viable for problems up to three dimensions. This paper presents an efficient approximate method, which is not limited to low-dimensional problems, for estimating the JEVD of multivariate stationary Gaussian processes. The proposed method modifies an existing method by using the Gauss-Legendre quadrature to evaluate the extreme value correlation coefficients under the bivariate Poisson assumption. Then, the JEVD is approximated using the Nataf transformation. The effectiveness of the proposed method is demonstrated via two numerical examples. The first example concerns the airgap problem of an offshore structure subjected to random waves, in which the extreme wave elevations are evaluated at six locations, and failure is defined as threshold exceedance for any location. The second example is a random vibration problem comprising a three degrees-of-freedom system. Previous studies focused on series system reliability; here three system reliabilities are examined, including series, parallel, and hybrid systems. The proposed method solves the problems efficiently, and it is found to provide an accurate prediction of the JEVD by comparison with Monte Carlo simulation.

On the finiteness of the moments of the measure of level sets of random fields

General conditions on smooth real valued random fields are given that ensure the finiteness of the moments of the measure of their level sets. As a by product a new generalized Kac-Rice formula (KRF) for the expectation of the measure of these level sets is obtained when the second moment can be uniformly bounded. The conditions involve (i) the differentiability of the trajectories up to a certain order k, (ii) the finiteness of the moments of the k-th partial derivatives of the field up to another order, (iii) the boundedness of the joint density of the field and some of its derivatives. Particular attention is given to the shot noise processes and fields. Other applications include stationary Gaussian processes, Chi-square processes and regularized diffusion processes. AMS2000 Classifications: Primary 60G60. Secondary 60G15.

Stochastic Optimization and Sensitivity Analysis of the Combined Negative Stiffness Damped Outrigger and Conventional Damped Outrigger Systems Subjected to Nonstationary Seismic Excitation

In recent years, various kinds of damped outrigger systems, particularly with the negative stiffness devices, have been shown to effectively suppress excessive vibration induced by earthquake and wind. To gain insight into such systems, this paper proposes a stochastic optimization method to investigate optimal configurations of combined negative stiffness damped outriggers (NSDOs) and conventional damped outriggers (CDOs) subjected to nonstationary stochastic seismic excitation. The simplified analysis model of various combinations of damped outriggers is developed, and the state-space representation of outrigger systems is then formulated. The nonstationary seismic excitation is modeled as a uniformly modulated stationary Gaussian process with a time-modulating function following Clough–Penzien spectrum, and combining equations of motions of outrigger systems and seismic excitation gives rise to the augmented state-space representation of structure-damper-excitation and subsequently the differential Lyapunov equation. The optimal designs of damped outriggers are defined via the solution of the differential Lyapunov equation. The multiobjective optimization with Pareto optimal fronts is adopted to deal with conflicting objectives between harmful interstory drift and floor absolute acceleration. The sensitivity of negative stiffness devices on optimal objectives is also investigated. The optimal designs are further examined under real typical and near-fault earthquake records. The results demonstrate the efficacy of the proposed method and provide insight into the systems subjected to nonstationary seismic excitation. While the purely NSDO systems could have better performance with adequate negative stiffness devices, the proposed combined NSDO and CDO systems prove to be more effective with limited negative stiffness devices and thus provide more design flexibilities tailored to different application situations.

АНАЛІЗ ЗАСТОСОВНОСТІ МЕТОДУ ДВОЧАСТОТНОЇ ІНТЕРФЕРОМЕТРІЇ ДЛЯ ВИМІРЮВАННЯ КУТА МІСЦЯ ЦІЛЕЙ У ДВОКООРДИНАТНИХ РЛС

Subject and Purpose. The study deals with the dual-frequency radio interferometry technique, which is based on the employment of two fairly close frequencies with the aim to remove ambiguity of the radar target elevation estimation using 2D-radar and eliminate 2 pm-uncertainty of the signal phase difference measurement. Analysis of random noise action on the accuracy of the elevation angle estimation by the dual-frequency radio interferometry and assessment of practical applicability of the method make up the purpose of the paper. Methods and Methodology. The noise action on the elevation angle measurement accuracy is examined through a series of an- alytical calculations with the use of statistical analysis methods. The noise in each receiving channel is modeled in terms of additive, statistically independent stationary Gaussian processes with zero mean values and equal variances. The calculation results are checked via computer simulations with statistics estimations for 106 random noise realizations. Results. A correct condition has been developed for the sector width where the target elevation angle is unambiguously estimated depending on the space separation of the antennas (baselines) and the frequency ratio. Expressions for elevation angle estimation errors have been obtained, showing that the error is mainly contributed by the faults in the determination of the ambiguity interval number. A probability of the correct determination of the ambiguity interval number has been derived depending on the signal- to-noise ratio and the frequency difference, indicating that almost one hundred per cent probability of the correct determination of the ambiguity interval number is only achieved when the signal-to-noise ratio exceeds 30 dB. A comparative analysis has been performed between the methods of dual-frequency interferometry and conventional phase-difference direction finding in the case of close X-band frequencies and the same sectors of survey. Conclusions. The dual-frequency radio interferometry technique with close frequencies has been shown to outperform the stand- ard phase-difference direction-finding method only when the signal-to-noise ratio is sufficiently high (over 30 dB). In principle, the accuracy of the technique seems possible to improve by taking significantly different frequencies selected with regard to the scale negotiation condition. However, it should be mentioned that the implementation of the relevant algorithm in practice is much more complicated than the conventional scheme with a single frequency and several antenna baselines.