Nonstationary Gaussian Process Research Articles

Adaptive‐optimal control under time‐varying stochastic uncertainty using past learning

SummaryAn adaptive‐optimal control architecture is presented for adaptive control of constrained aerospace systems with matched uncertainties that are subject to dynamic stochastic change. The architecture brings together three key elements, ie, model predictive control–based reference command shaping, Gaussian process (GP)–based Bayesian nonparametric model reference adaptive control (MRAC), and online GP clustering over nonstationary GPs. Model predictive control optimizes reference model and its shaped output is passed into GP–based MRAC, which is used to learn the model in presence of significant time‐varying stochastic uncertainty while maintaining stability. Based on a likelihood ratio test, the changepoints are detected and learned. Lastly, the models are created and clustered by non‐Bayesian clustering algorithm. The key salient feature of our architecture is that not only can it detect changes but also it uses online GP clustering to enable the controller to utilize past learning of similar models to significantly reduce learning transients. Furthermore, persistence of excitation conditions are significantly relaxed due to the use of GP‐MRAC. Stability of the architecture is argued theoretically and performance is validated empirically on different scenarios for wing rock dynamics.

A modified spectral representation method to simulate non-Gaussian random vector process considering wave-passage effect

Wave-passage effect is always of great importance when performing the dynamic analysis of lager scale structures. This effect can be easily considered in the simulation of Gaussian vector process, but there is not an efficient way to consider it in simulating non-Gaussian vector process. In this paper, a method is proposed to generate stationary non-Gaussian vector process considering wave-passage effect. In the simulation of vector process, wave-passage effect leads to a complex-defined cross spectral density matrix (CSDM). To separate the phase part from the CSDM, an improved form of spectral representation method (SRM) is proposed, in which the phase part representing the wave-passage effect can be directly expressed in an explicit form in the simulation formula. Then the characteristic of phase part representing the wave-passage effect during the nonlinear translation (translating underlying Gaussian vector process to the prescribed non-Gaussian vector process) is investigated. Interestingly, it is found that, the phase part is not changed. Based on this good feature, the modified SRM and the iterative scheme can be combined to simulate non-Gaussian vector process considering wave-passage effect. Two examples, involving the simulation of seismic ground motions and wind velocity time histories, are presented to show the characteristics of the simulated vector process and to verify the capabilities of the proposed method.

Time-dependent reliability analysis of RC beams shear and flexural strengthened with CFRP subjected to harsh environmental deteriorations

Time-dependent reliability analysis of existing reinforced concrete (RC) beams strengthened with externally bonded carbon fiber reinforced polymer (CFRP) wraps is conducted. Corrosion effect in terms of remaining cross sectional area of steel reinforcement is modelled as a non-stationary Gaussian process and the probability of failure is estimated as a solution to the first passage probability from the safe domain. Moreover, flexural and shear limit states are modeled in a series system formulation. It was found that the steel stirrup ratio strongly influenced the reliability of beams subject to harsh environments and in some cases beam may experience brittle fracture during its service life. Furthermore, it is found that the upper bound of the system probability of failure is significantly higher than its corresponding individual limit states due to correlation between flexural and shear failure modes.

Multiscale sparse microcanonical models

We study approximations of non-Gaussian stationary processes having long range correlations with microcanonical models. These models are conditioned by the empirical value of an energy vector, evaluated on a single realization. Asymptotic properties of maximum entropy microcanonical and macrocanonical processes and their convergence to Gibbs measures are reviewed. We show that the Jacobian of the energy vector controls the entropy rate of microcanonical processes. Sampling maximum entropy processes through MCMC algorithms require too many operations when the number of constraints is large. We define microcanonical gradient descent processes by transporting a maximum entropy measure with a gradient descent algorithm which enforces the energy conditions. Convergence and symmetries are analyzed. Approximations of non-Gaussian processes with long range interactions are defined with multiscale energy vectors computed with wavelet and scattering transforms. Sparsity properties are captured with \mathbf{l}^1 norms. Approximations of Gaussian, Ising and point processes are studied, as well as image and audio texture synthesis.

PROBABLE CALCULATION OF BUILDING SEISMIC RESISTANCE ON KINEMATIC SUPPORT

Objectives.The article reflects the results of the numerical analysis of the earthquake-resistant building on kinematic supports. To this end, the problem is reduced to solving the nonlinear stochastic Cauchy problem. The solution is constructed by the method of successive approximations. The probabilistic characteristics of the oscillation of the building are determined without the use of linearization techniques. An algorithm for solving this problem, which allows to perform numerical experiments on a computer to study the operation of a earthquake-resistant building on kinematic sup-ports, is given.Method.The acceleration of the earth's surface during an earthquake is represented as a non-stationary random Gaussian process. This approach is now generally accepted and beyond doubt. The study of vibrations of the building on kinematic supports under the influence of strong earthquakes is reduced to the solution of the stochastic nonlinear Cauchy equation. This equation is solved by iteration. The acceleration of the earth's surface is a function of three random variables. The required probability is represented as a triple integral, which is calculated using a computer.Result.The basic information about the considered kinematic supports is given. The Cauchy problem is formulated for the case of oscillations of a earthquake-resistant building on kinematic supports under the influence of strong earthquakes. The algorithm allowing to solve this equation is described in detail. The probability of finding the movements of the building within certain limits is represented as a triple integral. The results of numerical experiments carried out on a computer are given. The corresponding graphs are constructed using real accelerograms of strong earthquakes that occurred in the cities of Taft (USA) and Gazli (Uzbekistan).Conclusion.This article describes the method of calculation of earthquake-resistant buildings on kinematic supports, using the data of real strong earthquakes. Based on the results of numerical experiments conducted on a computer, graphs of the reliability of seismic stability of the building in earthquakes. The constructed algorithm and the developed technique can be used in the calculation and design of earthquake-resistant buildings both on conventional supports and on kinematic supports.

Robust Ocean Acoustic Localization With Sparse Bayesian Learning

Matched field processing (MFP) compares the measures to the modeled pressure fields received at an array of sensors to localize a source in an ocean waveguide. Typically, there are only a few sources when compared to the number of candidate source locations or range-depth cells. We use sparse Bayesian learning (SBL) to learn a common sparsity profile corresponding to the location of present sources. SBL performance is compared to traditional processing in simulations and using experimental ocean acoustic data. Specifically, we localize a quiet source in the presence of a surface interferer in a shallow water environment. This multi-frequency scenario requires adaptive processing and includes modest environmental and sensor position mismatch in the MFP model. The noise process changes likely with time and is modeled as a non-stationary Gaussian process, meaning that the noise variance changes across snapshots. The adaptive SBL algorithm models the complex source amplitudes as random quantities, providing robustness to amplitude and phase errors in the model. This is demonstrated with experimental data, where SBL exhibits improved source localization performance when compared to the white noise gain constraint (–3 dB) and Bartlett processors.

ДІАГНОСТУВАННЯ СУДНОВОГО УСТАТКУВАННЯ ПРИ ВИКОРИСТАННІ РІЗНИХ МОДЕЛЕЙ ВІБРАЦІЇ

Розглянуті моделі діагностики суднового устаткування та доцільність їх використання. Найбільш ефективними методами діагностування суднових машин та механізмів є віброакустичні методи, які використовують ту або іншу модель сигналу вібрації. Проаналізовано конкретні моделі вібрації в якості вібродіагностичних моделей. Проведено аналіз властивостей розглянутих моделей випадкових процесів, що характеризують різні процеси в теорії і при проведенні експериментальних досліджень. На підставі експериментальних досліджень знайшли підтвердження різні моделі, які дозволяють розробити і отримати практичні методи діагностування.

Simultaneous quantile inference for non-stationary long-memory time series

We consider the simultaneous or functional inference of time-varying quantile curves for a class of non-stationary long-memory time series. New uniform Bahadur representations and Gaussian approximation schemes are established for a broad class of non-stationary long-memory linear processes. Furthermore, an asymptotic distribution theory is developed for the maxima of a class of non-stationary long-memory Gaussian processes. Using the latter theoretical results, simultaneous confidence bands for the aforementioned quantile curves with asymptotically correct coverage probabilities are constructed.

Adaptive Lattice Filters for Systems of Space-Time Processing of Non-Stationary Gaussian Processes

Adaptive systems protecting pulse radars from non-stationary in time (range) clutter echoes are usually tuned using training vectors composed of complex amplitudes of input signals and comprising a finite-length “sliding window” of data. From any current range gate to a subsequent one, a training sample is partially updated (or modified) by means of excluding the “old” training vectors (correspond to the current range gate) and including the “new” ones (correspond to the next range gate). As a consequence, respective estimates of adaptive system parameters are corrected according to a modified sample correlation matrix (CM), which is typically a sum of an initialCMand a modifying matrix of rank K ≥ 1. In this case it is possible to avoid re-computing these parameters based on a new training sample of full size and, instead of this, we correct them in an “economical” way employing K-rank modification of a matrix inverse to the CM estimate. This paper is devoted to comparative analysis of various (K ≥ 1)-rank modification algorithms that correct the parameters of adaptive lattice filters (ALF). Main attention is paid to synthesis as well as theoretical and experimental study of algorithms of direct (K > 1)-rank modification of the ALF parameters. These algorithms attain the said objective omitting the K-fold application of known rank-one (K = 1) modification algorithms. We also synthesize a combined algorithm (CA) of (K ≥ 1)-rank modification of the ALF parameters that is more computationally simple and more numerically robust compared to known algorithms. The ALF employing the CA can serve as an effective tool for solving various tasks of space-time adaptive signal processing in pulse radars of different purpose.

Statistical calibration and uncertainty quantification of complex machining computer models

Abstract Finite element based machining process models are used in research and industry for process design and optimization. These models require a constitutive description of the material behavior to accurately model and predict process responses such as cutting forces, temperatures, and residual stress. Calibration of these models to low-strain uniaxial dynamic compression experiments can be troublesome since the machining process generally imposes much larger strains than uniaxial compression. Calibration of finite element models directly to machining data is generally difficult since the models are computationally expensive and nonlinear optimization methods for estimating the unknown calibration parameters yield non-unique solutions and require many iterations. In this work we utilize a nonstationary Gaussian Process surrogate model to emulate the finite element response and calibrate to experimental orthogonal cutting tests using a Bayesian inference framework. We assume that the material yield behavior can be described by the Johnson-Cook material flow model. We find that the nonstationary Gaussian Process model is an good surrogate for the complex finite element model. Cutting forces measured from orthogonal tube turning experiments were used for calibration. Validation is performed using a separate response variable - the cut chip thickness. Calibration results illustrate a preference for material models with low hardening rates, which alleviates issues such as over-prediction of strain hardening behavior when using the Johnson-Cook material flow model. The Bayesian formulation also captures the uncertainty in the Johnson-Cook parameters, which can be used to quantify the uncertainty in the machining process responses. The methods presented here are general and can be used for more complex constitutive and tribological models for machining and other complex manufacturing processes.

Bayesian nonstationary Gaussian process models via treed process convolutions

The Gaussian process is a common model in a wide variety of applications, such as environmental modeling, computer experiments, and geology. Two major challenges often arise: First, assuming that the process of interest is stationary over the entire domain often proves to be untenable. Second, the traditional Gaussian process model formulation is computationally inefficient for large datasets. In this paper, we propose a new Gaussian process model to tackle these problems based on the convolution of a smoothing kernel with a partitioned latent process. Nonstationarity can be modeled by allowing a separate latent process for each partition, which approximates a regional clustering structure. Partitioning follows a binary tree generating process similar to that of Classification and Regression Trees. A Bayesian approach is used to estimate the partitioning structure and model parameters simultaneously. Our motivating dataset consists of 11918 precipitation anomalies. Results show that our model has promising prediction performance and is computationally efficient for large datasets.

Modeling and emulation of nonstationary Gaussian fields

Geophysical and other natural processes often exhibit nonstationary covariances and this feature is important for statistical models that attempt to emulate the physical process. A convolution-based model is used to represent nonstationary Gaussian processes that allows for variation in the correlation range and the variance of the process across space. We apply this model in two steps: windowed estimates of the covariance function under the assumption of local stationarity and encoded local estimates into a single spatial process model that allows for efficient simulation. We show that nonstationary covariance functions based on the Matérn family can be reproduced by the LatticeKrig (LK) model, a flexible, multi-resolution representation of Gaussian processes. Stationary models based on the Matérn covariance are fit in local windows and these estimates are assembled into a single, global LK model. The LK model is efficient for simulating nonstationary fields even at 105 locations. This work is motivated by the interest in emulating spatial fields derived from numerical model simulations such as Earth system models. We successfully apply these ideas to emulate fields that describe the uncertainty in the pattern scaling of mean summer (JJA) surface temperature from a series of climate model experiments. The spatial covariance structure developed in this paper is not limited to emulation, and could also be used for spatial prediction and conditional simulation for observational data and leverages embarrassingly parallel strategies for computational efficiency.

First passage probability assessment of stationary non-Gaussian process using the third-order polynomial transformation

In this article, an analytical moment-based procedure is developed for estimating the first passage probability of stationary non-Gaussian structural responses for practical applications. In the procedure, an improved explicit third-order polynomial transformation (fourth-moment Gaussian transformation) is proposed, and the coefficients of the third-order polynomial transformation are first determined by the first four moments (i.e. mean, standard deviation, skewness, and kurtosis) of the structural response. The inverse transformation (the equivalent Gaussian fractile) of the third-order polynomial transformation is then used to map the marginal distributions of a non-Gaussian response into the standard Gaussian distributions. Finally, the first passage probabilities can be calculated with the consideration of the effects of clumping crossings and initial conditions. The accuracy and efficiency of the proposed transformation are demonstrated through several numerical examples for both the “softening” responses (with wider tails than Gaussian distribution; for example, kurtosis > 3) and “hardening” responses (with narrower tails; for example, kurtosis < 3). It is found that the proposed method has better accuracy for estimating the first passage probabilities than the existing methods, which provides an efficient and rational tool for the first passage probability assessment of stationary non-Gaussian process.

The harmonic analysis of kernel functions

Kernel-based methods have been recently introduced for linear system identification as an alternative to parametric prediction error methods. Adopting the Bayesian perspective, the impulse response is modeled as a non-stationary Gaussian process with zero mean and with a certain kernel (i.e. covariance) function. Choosing the kernel is one of the most challenging and important issues. In the present paper we introduce the harmonic analysis of this non-stationary process, and argue that this is an important tool which helps in designing such kernel. Furthermore, this analysis suggests also an effective way to approximate the kernel, which allows to reduce the computational burden of the identification procedure.

STATISTICAL ANALYSIS OF ACCELEROGRAMS OF REAL STRONG EARTHQUAKES

Abstract. Objectives The statistical analysis of real accelerograms is considered. For this purpose, graphs of real accelerograms are enlarged in abscissa and ordinate, allowing the appropriate measurements to be made and the accelerograph records to be presented in the form of numerical tables. To process these tables, a mathematical model is constructed that allows statistical studies of actual accelerograms to be carried out. Methods The acceleration of the Earth's surface during an earthquake is represented as a non-stationary random Gaussian process. The non-stationary process describing the acceleration of the Earth's surface is modelled as a function of three random parameters. In this case, the real accelerogram, which is presented in a single copy, is modelled by a random ergodic function. Results Accelerograms of strong real earthquakes and corresponding fragments of tables are given. An algorithm that allows all parameters of correlation functions and spectral densities to be determined corresponding to real earthquakes is described in detail. A constructed algorithm that allows the statistical parameters of an earthquake to be calculated, including the dominant frequency, the standard deviation, the correlation coefficient and a coefficient that takes into account the non-stationary nature of the earthquake process, is discussed. The parameters of the correlation functions of accelerograms of strong earthquakes that occurred in the cities of Taft (USA) and Gazli (Uzbekistan) are calculated. The results of the study are presented in the form of graphs and tables. Conclusion The algorithm constructed by the authors allows the statistical study of strong earthquakes, whose accelerograms are presented in a single copy, to be carried out and the corresponding correlation function parameters to be calculated. The proposed algorithm can be used for the statistical analysis of accelerograms of strong earthquakes.The parameters of correlation functions can find application in the investigation of seismic resistance of buildings, both with passive and active seismic protection.

Efficient Solution for Calculation of Upcrossing Rate of Nonstationary Gaussian Process

AbstractAlmost all engineering systems are not only uncertain, but also time-variant. As such, it is most appropriate to use a time-dependent reliability method, e.g., first passage probability, in...

A Cameron-Storvick theorem for the analytic Feynman integral associated with Gaussian paths on a Wiener space and applications

The purpose of this paper is to establish a Cameron-Storvick theorem for the analytic Feynman integral of functionals in non-stationary Gaussian processes on Wiener space. As interesting applications, we apply this theorem to evaluate the generalized analytic Feynman integral of certain polynomials in terms of Paley-Wiener-Zygmund stochastic integrals.

Non-stationary sources separation based on maximum likelihood criterion using source temporal–spatial model

Abstract In this paper, we propose a new non-stationary sources separation algorithm, which is referred to as autoregressive hidden Markov Gaussian process (AR-HMGP), in which the sources are non-stationary and temporally correlated. For the proposed algorithm, a generative model is employed to track the non-stationarity of the source where the temporal dependencies of sources are represented by autoregressive model (AR) and the distribution of the associated innovation process is described using non-stationary Gaussian process with hidden Markov model (HMM). We further explore the maximum likelihood (ML) method to estimate the parameters of the source model by using the expectation maximum (EM) algorithm. Our important findings reveal that (a) AR-HMGP algorithm outperforms the other three BSS algorithms for non-stationary sources separation, the instantaneous mixture system is also well corroborated with the effectiveness of our algorithm; (b) both independent and dependent non-stationary sources have been successfully separated; (c) the proposed algorithm is robust with respect to noise, while the other three algorithms are not.

RESEARCH OF EARLY STAGES OF MODELING

In represented article the questions of estimate of accuracy of an average integral characteristics of random process in the course of imitation modeling is considered. For the purposes of analytical treatment of initial stage of modeling a conditionally nonstationary Gaussian process is analyzed as stationary Gaussian process with boundary prehistory. A model of approximant autocorrelation function is recommended. Analytical expression for variance and mathematical expectation of average integral estimation are obtained. Statistical estimation efficiency criterion, the probability of belonging to correct parameter interval is introduced. Dependences of closeness in estimation statistics clearing interval at transient behavior are researched for various types of processes.

A combined first- and second-order theory for the deckwetness prediction of sandglass-type floating body in irregular head waves

Deckwetness is one of the most important problems for seakeeping performance. With regard to the conventional ship-type Floating Production, Storage, and Offloading Units (FPSOs), the deckwetness depends on the relative vertical motion which is dominated usually by the first-order wave frequency loads. However, it can not apply to the deckwetness prediction of the sandglass-type FPSO since the influence of second-order slowly varying loads is remarkable. In this paper, a combined first- and second-order theory accounting for the first-order wave loads and nonlinear second-order slowly varying loads, is established to predict the deckwetness occurrence. By using the second-order Volterra series model, the relative vertical motion between the bow and incident wave surface can be described as a stationary non-Gaussian process. Then general formulas are derived for the prediction of the deckwetness occurrence for the sandglass-type floating body, based on the mathematical characteristics of the relative vertical motion process. As an application, the theory presented is discussed with reference to the deckwetness prediction of the sandglass-type floating model. Therein, a time domain analysis is employed to get the relative vertical motion response whereby the above formulas can be efficiently calculated. The influence factors including the length of the time simulation and the randomness of the wave phase are examined. Next, the convergence of the results is investigated with the increase of the order of non-Gaussian terms included in the formulas. By comparing with the experimental data, the reasonability and validity of the proposed method are proved.