Wiener Process Research Articles

Optimized Backstepping Tracking Control Using Reinforcement Learning for a Class of Stochastic Nonlinear Strict-Feedback Systems

In this article, an optimized backstepping (OB) control scheme is proposed for a class of stochastic nonlinear strict-feedback systems with unknown dynamics by using reinforcement learning (RL) strategy of identifier-critic-actor architecture, where the identifier aims to compensate the unknown dynamic, the critic aims to evaluate the control performance and to give the feedback to the actor, and the actor aims to perform the control action. The basic control idea is that all virtual controls and the actual control of backstepping are designed as the optimized solution of corresponding subsystems so that the entire backstepping control is optimized. Different from the deterministic system, stochastic system control needs to consider not only the stochastic disturbance depicted by the Wiener process but also the Hessian term in stability analysis. If the backstepping control is developed on the basis of the published RL optimization methods, it will be difficult to be achieved because, on the one hand, RL of these methods are very complex in the algorithm thanks to their critic and actor updating laws deriving from the negative gradient of the square of approximation of Hamilton-Jacobi-Bellman (HJB) equation; on the other hand, these methods require persistence excitation and known dynamic, where persistence excitation is for training adaptive parameters sufficiently. In this research, both critic and actor updating laws are derived from the negative gradient of a simple positive function, which is yielded on the basis of a partial derivative of the HJB equation. As a result, the RL algorithm can be significantly simplified, meanwhile, two requirements of persistence excitation and known dynamic can be released. Therefore, it can be a natural selection for stochastic optimization control. Finally, from two aspects of theory and simulation, it is demonstrated that the proposed control can arrive at the desired system performance.

A Wiener Process Model With Dynamic Covariate for Degradation Modeling and Remaining Useful Life Prediction

Most industrial products degrade with usage. To capture degradation patterns and schedule condition-based maintenance actions, it is necessary to model the degradation process and predict the system remaining useful life (RUL). In practice, the operating environments and loading conditions are dynamic and stochastic, which introduces extra randomness in the degradation process. To account for the dynamic environmental impacts, this article proposes a nonlinear Wiener process model with a random time-varying covariate. We model the dynamic covariate with an Ornstein–Uhlenbeck process, and link it to the time-varying degradation rate by an exponential form covariate-effect function. Therefore, the proposed Wiener process has a time-varying degradation rate impacted by the dynamic environment. We propose to estimate the model parameters by maximum likelihood estimation based on given observations of the degradation and covariate. We also propose a simulation-based procedure to obtain the RUL for the proposed model. The model is verified by Monte Carlo simulations. We apply the proposed model to a dust-cleaning fan degradation dataset and a lithium-ion battery degradation dataset, and the results show that the proposed model outperforms the existing degradation models in fitting the real data and RUL prediction.

Wiener integrals with respect to the generalized Hermite process (gHp). Applications: SDEs with gHp noise

Abstract In this paper, we introduce Wiener integrals with respect to the generalized Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an application, we investigate the corresponding stochastic differential equations with the generalized Hermite process as a driving noise, we prove the existence and the uniqueness of the solution, and we give a generalization of the Hermite Ornstein–Uhlenbeck process and the Hermite-driving Vasicek process.

A Computational Scheme for Stochastic Non-Newtonian Mixed Convection Nanofluid Flow over Oscillatory Sheet

Stochastic simulations enable researchers to incorporate uncertainties beyond numerical discretization errors in computational fluid dynamics (CFD). Here, the authors provide examples of stochastic simulations of incompressible flows and numerical solutions for validating these newly emerging stochastic modeling methods. A numerical scheme is constructed for finding solutions to stochastic parabolic equations. The scheme is second-order accurate in time for the constant coefficient of the Wiener process term. The stability analysis of the scheme is also provided. The scheme is applied to the dimensionless heat and mass transfer model of mixed convective non-Newtonian nanofluid flow over oscillatory sheets. Both the deterministic and stochastic energy equations use temperature-dependent thermal conductivity. The stochastic model is more general than the deterministic model. The results are calculated for both flat and oscillatory plates. Casson parameter, mixed convective parameter, thermophoresis, Brownian motion parameter, Prandtl number, Schmidt number, and reaction rate parameter all impact the velocities, temperatures, and concentrations shown in the graphs. Under the influence of the oscillating plate, the results reveal that the concentration profile decreases with increasing Brownian motion parameters and increases with increasing thermophoresis parameters. The behavior of the velocity profile for the deterministic and stochastic models is provided, and contour plots for the stochastic model are also displayed. This article aims to provide a state-of-the-art overview of recent achievements in the field of stochastic computational fluid dynamics (SCFD) while also pointing out potential future avenues and unresolved challenges for the computational mathematics community to investigate.

Remaining useful life prediction for nonlinear two-phase degradation process with measurement errors and imperfect prior information

Remaining useful life (RUL) prediction is one of the most important issues of prognostic and health management, which can improve the reliability and security of the system. Due to the changeable internal mechanism and external environmental factors, the two-phase degradation process is frequently seen in practice. In addition, measurement errors in degradation signals and the case with imperfect prior degradation information are common, which could decrease the accuracy of RUL prediction. However, the current studies on two-phase degradation usually assume that each phase is linear. Furthermore, the effect of measurement errors and the possibility of incomplete prior degradation data are generally not taken into account simultaneously. Therefore, this paper proposes a novel linear–nonlinear two-phase Wiener process with a measurement errors degradation model, and obtains the probability density function expression of the RUL by fully considering the unknown degradation state at the change point. Meanwhile, in the absence of multiple sets of historical data, a parameter estimation method which only requires a set of prior information is proposed based on an expectation maximization (EM) algorithm and Kalman smoothing. Finally, a numerical example and two practical examples are used to illustrate the effectiveness and superiority of the proposed method.

New reliability model for complex systems based on stochastic processes and survival signature

For systems with complicated structures, reliability analysis based on survival signature has been carried out by modelling time-to-failure data with specific distributions. However, for highly reliable systems, only little or no failure data may be available. To enable reliability analysis without failure data, a new generalised reliability method is proposed for complex systems, based on the survival signature and using stochastic processes to model degradation. The combination of the survival signature and stochastic processes enables the proposed method to be applied to complex systems with different structures and stochastically degrading components. First, system reliability is analysed based on the survival signature and a generalised stochastic process. Then, component reliability analysis based on the generalised stochastic process is introduced using Wiener and Gamma processes. Finally, the approach presented in this paper is illustrated using two numerical examples, and the estimation results are compared with those calculated using failure time distribution functions.

Remaining useful life evaluation for accelerated Wiener degradation process model with mixed random effects and measurement errors

AbstractThis paper evaluates the remaining useful life (RUL) for the accelerated Wiener process with mixed random effects and measurement errors. In the discussed model, the random effects are included in both the drift and diffusion parameters of the Wiener process to model the heterogeneity among different degradation paths of products. The measurement errors are introduced to describe the inaccuracy during the measurement process. The effect of the accelerated stress level is included in the drift parameter. The expectation‐maximization (EM) algorithm is applied to obtain the estimation of the model parameters. The expressions of the probability density functions (PDFs) and cumulative distribution functions (CDFs) of the lifetime and RUL distributions of products under the normal usage stress level are derived theoretically. Finally, an example of stress relaxation data is used to illustrate the effectiveness of the developed model and method.

Neural stochastic differential equations network as uncertainty quantification method for EEG source localization

EEG source localization remains a challenging problem given the uncertain conductivity values of the volume conductor models (VCMs). As uncertain conductivities vary across people, they may considerably impact the forward and inverse solutions of the EEG, leading to an increase in localization mistakes and misdiagnoses of brain disorders. Calibration of conductivity values using uncertainty quantification (UQ) techniques is a promising approach to reduce localization errors. The widely-known UQ methods involve Bayesian approaches, which utilize prior conductivity values to derive their posterior inference and estimate their optimal calibration. However, these approaches have two significant drawbacks: solving for posterior inference is intractable, and choosing inappropriate priors may lead to increased localization mistakes. This study used the Neural Stochastic Differential equations Network (SDE-Net), a combination of dynamical systems and deep learning techniques that utilizes the Wiener process to minimize conductivity uncertainties in the VCM and improve the inverse problem. Results revealed that SDE-Net generated a lower localization error rate in the inverse problem compared to Bayesian techniques. Future studies may employ new stochastic dynamical systems-based techniques as a UQ technique to address further uncertainties in the EEG Source Localization problem. Our code can be found here: https://github.com/rrwabina/SDENet-UQ-ESL.

Threshold estimation for jump-diffusions under small noise asymptotics

We consider parameter estimation of stochastic differential equations driven by a Wiener process and a compound Poisson process as small noises. The goal is to give a threshold-type quasi-likelihood estimator and show its consistency and asymptotic normality under new asymptotics. One of the novelties of the paper is that we give a new localization argument, which enables us to avoid truncation in the contrast function that has been used in earlier works and to deal with a wider class of jumps in threshold estimation than ever before.

A reliability analysis method based on the mixed correlated competition model considering multi-performance degradation and sudden failures

Complex products usually fail due to competing multi-performance degradation and sudden failure modes. The correlations between competing failure modes, including among multi-performance degradation failures and between degradation and sudden failures, must be considered in reliability analysis. This paper proposes a general reliability analysis method that considers such cases of complex mixed correlated competition. Firstly, a nonlinear Wiener process model based on a time-scale transformation function is applied to model performance degradation failure, and a multi-performance-degradation correlated-competition model based on a Copula function is constructed, which considers the correlated competition between multi-performance degradation failures. Secondly, a three-parameter Weibull distribution is used to model sudden failures, and an evaluation method based on nonlinear planning-maximum likelihood estimation is proposed to estimate its parameters. Thirdly, a mixed correlated-competition model of multi-performance degradation and sudden failures is proposed for product reliability evaluation. Finally, a servo system experiment is used for demonstration. The experimental results demonstrate that the proposed method achieves higher evaluation accuracy and has wider applicability than existing methods, which ignore mixed correlated competition.

A new Wiener process with bathtub‐shaped degradation rate in the presence of random effects

AbstractThis paper proposes a new time scaled Wiener process with random effects that has been specially designed to allow the description of non‐monotonic degradation phenomena with bathtub shaped degradation rate, here intended as derivative of the mean function. Two different parameterizations of the proposed Wiener process are suggested, and the main features of the process are illustrated and discussed. The maximum likelihood estimation of its parameters is addressed. A failure threshold model is adopted to formulate and estimate the probability distribution function of the remaining useful life, which constitutes the core prognostic tool in condition‐based maintenance. As a motivating example, the proposed model is applied to a set of real degradation data of metal‐oxide‐semiconductor field‐effect transistors, where, as a result of preliminary analyses, some of the process parameters are assumed to be random in order to describe the considerable heterogeneity observed between the degradation paths of different units. Obtained results demonstrate the affordability and utility of the proposed model, especially for the estimation of unit‐specific features performed on the basis of degradation data collected in the early phase of the life of a unit belonging to the considered heterogeneous population.

Residual life prediction of wet clutch based on binary Wiener process

PurposeIn armored vehicles integrated transmissions, residual life prediction based on oil spectrum data is crucial for condition monitoring and reliability assessment. This paper aims to use the advantages of real-time and accurate prediction of binary Wiener process, the residual life prediction of clutch is studied.Design/methodology/approachFirst, combined with the wet clutch life test, the indicator elements Cu and Pb and the failure threshold of the residual life prediction of the clutch are extracted through the oil replacement correction of the spectral data of the whole life cycle; second, the correlation characteristics of indicating elements are analyzed by MATLAB Copula function, then the correlation function of residual life will be derived; third, according to the inverse Gaussian principle, the performance degradation mathematical models of the unary and binary Wiener processes of the above two indicator elements are established; finally, the maximum likelihood estimation method is used to estimate the parameters, and the monadic and binary performance degradation mathematical models are used to predict the residual life of the tested clutch.FindingsBy comparing the prediction results with the test results, with the passage of time, 81.25% of the predicted value error of the residual life prediction method based on the binary Wiener process is controlled within 20%, while 56.25% of the predicted value error of the residual life prediction method based on the unitary Wiener process is controlled within 20%. At the same time, the prediction accuracy of the binary prediction model is 2%–16.7% higher than that of the unitary prediction model.Originality/valueThis paper studies the residual life prediction theory of wet clutch, which can develop the theory and method of comprehensive transmission health monitoring, and provide theoretical and technical support for the construction of a reliable health management system for high-speed tracked vehicles.

A remaining useful life prediction method of aluminum electrolytic capacitor based on wiener process and similarity measurement

Complex characteristics such as non-linearity and multi-stage are usually presented during the degradation process of aluminum electrolytic capacitors (AECs). Therefore, it is difficult to accurately predict the capacitors’ remaining useful life (RUL), which is significant to the system’s reliability. Based on the model of Wiener process with linear drift, a simple two-stage RUL predicting method for AECs is proposed in this paper. In the offline stage, the Box–Cox transformation (BCT) is first utilized to linearize the degraded data. Then the model parameters of BCT are identified, and the Shapiro–Wilk test is employed to verify whether the transformed data meet the linearity. In the online stage, based on similarity measurement, the drift coefficient of the Wiener process is updated in real-time. By combining BCT and similarity measurement, the nonlinear and multi-stage problems are effectively solved. Finally, the proposed method is verified on NASA’s accelerated degradation data set. Experimental results show that the proposed method can automatically learn the capacitor’s complex degradation characteristics and accurately predict its RUL.

Two-Sample Testing for Tail Copulas with an Application to Equity Indices

A novel, general two-sample hypothesis testing procedure is established for testing the equality of tail copulas associated with bivariate data. More precisely, using a martingale transformation of a natural two-sample tail copula process, a test process is constructed, which is shown to converge in distribution to a standard Wiener process. Hence, from this test process a myriad of asymptotically distribution-free two-sample tests can be obtained. The good finite-sample behavior of our procedure is demonstrated through Monte Carlo simulations. Using the new testing procedure, no evidence of a difference in the respective tail copulas is found for pairs of negative daily log-returns of equity indices during and after the global financial crisis.

Stochastic integrodifferential models of fractional orders and Leffler nonsingular kernels: well-posedness theoretical results and Legendre Gauss spectral collocation approximations

Stochastic fractional integrodifferential models are widely employed to model several natural phenomena these days. This current work focuses on the well-posedness results and numerical solutions of a specific form of these models considering the Leffler nonsingular kernels operator wherein the stochastic term is driven by the standard Brownian motion. Accordingly, a combination of sufficient conditions, topological theorems, and Banach space theory are utilized to construct the well-posedness proof. For treating the numerical issue, a familiar spectral collocation technique relying upon shifted Legendre series expansion theory is proposed. The basic properties of Brownian motion and a linear spline interpolation method are used to simulate the standard Brownian motion at a fixed time value. In addition, the idea of the Gauss-Legendre numerical integration rule is implemented to approximate the finite integral. We also devote our attention to the concept of convergence of the proposed method and demonstrate its analysis. Ultimately, the obtained theoretical results and the presented method are examined with five numerous applications. The obtained results indicate the high accuracy and efficiency of applying this method in solving stochastic models of the above-mentioned form.

First-Passage Times and Optimal Control of Integrated Jump-Diffusion Processes

Let Y(t) be a one-dimensional jump-diffusion process and X(t) be defined by dX(t)=ρ[X(t),Y(t)]dt, where ρ(·,·) is either a strictly positive or negative function. First-passage-time problems for the degenerate two-dimensional process (X(t),Y(t)) are considered in the case when the process leaves the continuation region at the latest at the moment of the first jump, and the problem of optimally controlling the process is treated as well. A particular problem, in which ρ[X(t),Y(t)]=Y(t)−X(t) and Y(t) is a standard Brownian motion with jumps, is solved explicitly.

"Reliable organisms from unreliable components" revisited: the linear drift, linear infinitesimal variance model of decision making.

Diffusion models of decision making, in which successive samples of noisy evidence are accumulated to decision criteria, provide a theoretical solution to von Neumann's (1956) problem of how to increase the reliability of neural computation in the presence of noise. I introduce and evaluate a new neurally-inspired dual diffusion model, the linear drift, linear infinitesimal variance (LDLIV) model, which embodies three features often thought to characterize neural mechanisms of decision making. The accumulating evidence is intrinsically positively-valued, saturates at high intensities, and is accumulated for each alternative separately. I present explicit integral-equation predictions for the response time distribution and choice probabilities for the LDLIV model and compare its performance on two benchmark sets of data to three other models: the standard diffusion model and two dual diffusion model composed of racing Wiener processes, one between absorbing and reflecting boundaries and one with absorbing boundaries only. The LDLIV model and the standard diffusion model performed similarly to one another, although the standard diffusion model is more parsimonious, and both performed appreciably better than the other two dual diffusion models. I argue that accumulation of noisy evidence by a diffusion process and drift rate variability are both expressions of how the cognitive system solves von Neumann's problem, by aggregating noisy representations over time and over elements of a neural population. I also argue that models that do not solve von Neumann's problem do not address the main theoretical question that historically motivated research in this area.

A reliability estimation method based on signal feature extraction and artificial neural network supported Wiener process with random effects

Accurately predicting degradation and reliability is the key to effective condition-based machinery maintenance. This paper presents a reliability estimation method based on an artificial neural network (ANN)-supported Wiener process model with random effects. A number of time-domain features, frequency-domain features, and intrinsic energy features are considered to reflect the health condition of products. These features can be selected by the PROMETHEE II method considering multi-sample and multi-measure conditions. An ANN-supported health index function is defined to describe the relationship between the health condition and the selected signal features. Based on the health index function, an ANN-supported Wiener process is used to model machine degradation. The corresponding health index function training and process parameter inference approaches are presented. An actual testing dataset of a machine bearing is used to demonstrate the proposed method. The proposed method provides accurate service life and reliability predictions.

Numerical analysis of the Linearly implicit Euler method with truncated Wiener process for the stochastic SIR model

The paper deals with the numerical positivity, convergence and dynamical behaviors (including extinction and persistence) for stochastic SIR model. For the real significance of the numerical analysis on stochastic SIR model, a linearly implicit Euler method with truncated Wiener process is introduced. The numerical positivity is obtained by the truncated Wiener process, which is the basis for the investigation of convergence and dynamical behavior. The numerical dynamical behavior is obtained by an exponential presentation for the nonlinear stochastic stability function and the large number theorem for martingale, which reproduces the existing theoretical results of exact solution. Finally, numerical examples are given to validate our numerical results for stochastic SIR model.

Moderate Deviation Principle for Multiscale Systems Driven by Fractional Brownian Motion

In this paper, we study the moderate deviations principle (MDP) for slow–fast stochastic dynamical systems where the slow motion is governed by small fractional Brownian motion (fBm) with Hurst parameter \(H\in (1/2,1)\). We derive conditions on the moderate deviations scaling and on the Hurst parameter H under which the MDP holds. In addition, we show that in typical situations the resulting action functional is discontinuous in H at \(H=1/2\), suggesting that the tail behavior of stochastic dynamical systems perturbed by fBm can have different characteristics than the tail behavior of such systems that are perturbed by standard Brownian motion.