Wiener Process Research Articles

A hybrid reliability assessment method based on health index construction and reliability modeling for rolling bearing

AbstractThe assessment of rolling bearing reliability is vital for ensuring mechanical operational safety and minimizing maintenance costs. Due to the difficulty in obtaining data on the performance degradation and failure time of rolling bearings, traditional methods for reliability assessment are challenged. This paper introduces a novel hybrid method for the reliability assessment of rolling bearings, combining the convolutional neural network (CNN)‐convolutional block attention module (CBAM)‐ bidirectional long short‐term memory (BiLSTM) network with the Wiener process. The approach comprises three distinct stages: Initially, it involves acquiring two‐dimensional time‐frequency representations of bearings at various operational phases using Continuous Wavelet Transform. Subsequently, the CNN‐CBAM‐BiLSTM network is employed to establish health index (HI) for the bearings and to facilitate the extraction of deep features, serving as input for the Wiener process. The final stage applies the Wiener process to evaluate the bearings’ reliability, characterizing the HI and quantifying uncertainties. The experiment is performed on bearing degradation data and the results indicate the effectiveness and superiority of the proposed hybrid method.

Multiple stresses optimization design of constant-stress accelerated degradation test based on Wiener process

Multiple stresses optimization design of constant-stress accelerated degradation test based on Wiener process

Remaining useful life prediction method based on stacked autoencoder and generalized wiener process for degrading bearing

Abstract Remaining Useful Life (RUL) prediction using deep learning networks primarily produces point estimates of RUL, but capturing the inherent uncertainty in RUL prediction is difficult. The use of the stochastic process approach can reflect the uncertainty in RUL predictions. However, the amount of data generated during equipment operation cannot be effectively utilized. This paper aims to propose an adaptive RUL prediction method tailored for extensive datasets and prediction uncertainty, effectively harnessing the strengths of deep learning methods in managing massive data and stochastic process techniques in quantifying uncertainties. RUL prediction method, based on Stacked Autoencoder (SAE) combined with Generalized Wiener Process, employs SAE to extract profound underlying features from the monitoring signals. Principal Component Analysis (PCA) is then used to select highly trending features as inputs. The output of PCA accurately reflects health status. A Generalized Wiener Process is used to construct a model for the evolution of the health indicators. The estimation values for the model parameters are determined using the Maximum Likelihood Estimation method. Furthermore, an adaptive update is performed based on Bayesian theory. Utilizing the sense of the first hitting time concept, the Probability Density Function for RUL prediction is derived accurately. Finally, the effectiveness and superiority of the proposed method is verified using numerical simulations and experimental studies of bearing degradation data. The method improves the life prediction accuracy while reducing the prediction uncertainty.

A novel equipment remaining useful life prediction approach considering dynamic maintenance threshold

AbstractIn conventional remaining useful life (RUL) prediction approaches grounded on maintenance, the maintenance threshold is typically established as a stationary value. However, the actual maintenance threshold may exceed its preset value due to the uncertainty of degradation and other factors. Therefore, it is necessary to consider the dynamic maintenance threshold to improve the precision of remaining useful life prediction. By considering the Wiener process, the maintenance threshold error is introduced to reflect the dynamic nature of the maintenance threshold. The influence of maintenance on degradation amount, degradation rate, and degradation path are comprehensively considered to establish a multi‐stage maintenance‐affected degradation process model. The RUL formula of the equipment is derived using the first hitting time (FHT). The maximum likelihood estimation (MLE) approach and Bayesian theory are employed to estimate the model's parameters. The proposed approach is validated using simulation data and gyroscope degradation data. The outcomes reveal that the proposed approach can significantly enhance the precision of life prediction for the equipment.

Doubly perturbed uncertain differential equations

In this paper, we study a class of doubly perturbed uncertain differential equations driven by canonical process that is the counterpart of Wiener process in the framework of uncertain theory. We give some sufficient conditions for the existence and uniqueness of solutions of the present problem under some weak conditions, where a convex combination of the Nagumo and Osgood conditions is considered. Subsequently, we consider the stability properties with regard to the initial value and coefficients. Finally, an example is provided to illustrate the effectiveness of our main results.

A Hybrid Approach Combining the Lie Method and Long Short-Term Memory (LSTM) Network for Predicting the Bitcoin Return

This paper introduces hybrid models designed to analyze daily and weekly bitcoin return spanning the periods from 18 July 2010 to 28 December 2023 for daily data, and from 18 July 2010 to 24 December 2023 for weekly data. Firstly, the fractal and chaotic structure of the selected variables was explored. Asymmetric Cantor set, Boundary of the Dragon curve, Julia set z2 −1, Boundary of the Lévy C curve, von Koch curve, and Brownian function (Wiener process) tests were applied. The R/S and Mandelbrot–Wallis tests confirmed long-term dependence and fractionality. The largest Lyapunov test, the Rosenstein, Collins and DeLuca, and Kantz methods of Lyapunov exponents, and the HCT and Shannon entropy tests tracked by the Kolmogorov–Sinai (KS) complexity test determined the evidence of chaos, entropy, and complexity. The BDS test of independence test approved nonlinearity, and the TeraesvirtaNW and WhiteNW tests, the Tsay test for nonlinearity, the LR test for threshold nonlinearity, and White’s test and Engle test confirmed nonlinearity and heteroskedasticity, in addition to fractionality and chaos. In the second stage, the standard ARFIMA method was applied, and its results were compared to the LieNLS and LieOLS methods. The results showed that, under conditions of chaos, entropy, and complexity, the ARFIMA method did not yield successful results. Both baseline models, LieNLS and LieOLS, are enhanced by integrating them with deep learning methods. The models, LieLSTMOLS and LieLSTMNLS, leverage manifold-based approaches, opting for matrix representations over traditional differential operator representations of Lie algebras were employed. The parameters and coefficients obtained from LieNLS and LieOLS, and the LieLSTMOLS and LieLSTMNLS methods were compared. And the forecasting capabilities of these hybrid models, particularly LieLSTMOLS and LieLSTMNLS, were compared with those of the main models. The in-sample and out-of-sample analyses demonstrated that the LieLSTMOLS and LieLSTMNLS methods outperform the others in terms of MAE and RMSE, thereby offering a more reliable means of assessing the selected data. Our study underscores the importance of employing the LieLSTM method for analyzing the dynamics of bitcoin. Our findings have significant implications for investors, traders, and policymakers.

Multi-UAVs Tracking Non-Cooperative Target Using Constrained Iterative Linear Quadratic Gaussian

This study considers the problem of controlling multi-unmanned aerial vehicles (UAVs) to consistently track a non-cooperative ground target with uncertain motion in a hostile environment with obstacles. An active information acquisition (AIA) problem is formulated to minimize the uncertainty of the target tracking task. The uncertain motion of the target is represented as a Wiener process. First, we optimize the configuration of the UAV swarm considering the collision avoidance, horizontal field of view (HFOV), and communication radius to calculate the reference trajectories of the UAVs. Next, a novel algorithm called Constrained Iterative Linear Quadratic Gaussian (CILQG) is introduced to track the reference trajectory. The target’s state with uncertainty and the UAV state are described as beliefs. The CILQG algorithm utilizes the Unscented Transform to propagate the belief regarding the UAVs’ motions, while also accounting for the impact of navigation errors on the target tracking process. The estimation error of the target position of the proposed method is under 4 m, and the error of tracking the reference trajectories is under 3 m. The estimation error remains unchanged even in the presence of obstacles. Therefore, this approach effectively deals with the uncertainties involved and ensures accurate tracking of the target.

Numerical analysis of a time discretized method for nonlinear filtering problem with Lévy process observations

In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.

A model-data-fusion method for real-time continuous remaining useful life prediction of lithium batteries

Ensuring the reliable operation of electric vehicles relies on accurate predictions of the remaining useful life (RUL) for lithium batteries. Existing model-based prediction methods face limitations due to insufficient full-life experimental data, while online monitoring data-driven methods lack empirical guidance and interpretability of the training data. To address these challenges, a model-data-fusion prediction method is proposed. Firstly, the degradation process is characterized using a generalized Wiener process model. Secondly, A Whale Optimization Particle Search Filter (WOS-PF) is introduced for real-time parameter updating of the degradation model. This approach ensures that even with a small sample size, parameters can be reliably estimated. Finally, leveraging the continuously updated parameters, a probability density function is derived to predict the RUL of lithium batteries in real-time. Experimental results indicate that this method enhances prediction accuracy compared to other existing methods, leading to better real-time monitoring of lithium battery health.

Two-stage Remaining Useful Life Prediction Based on the Wiener Process With Multi-feature Fusion and Stage Division

Remaining life prediction (RUL) is a critical link of maintenance decision-making, the accurate RUL prediction is an important means to monitor the operating status and achieve the safe operation of equipment. However, existing studies rarely considered the multi-stage characteristics of indicator fusion in the degradation process, and directly used the Wiener process to establish degradation model, which results in significant errors in RUL prediction results. Therefore, to solve above issues, a two-stage RUL prediction method of bearing based on the Wiener process model with data fusion and stage division is proposed in the paper. Firstly, the concept of multi-feature fusion is introduced to construct a comprehensive health indicator (CHI) that considers indicator performance. After that, a two-stage RUL prediction model based on the CHI is developed, and a method for detecting change points and dividing stages is proposed. Finally, the effectiveness and predictability of the proposed method and CHI are demonstrated based on the bearing test datasets.

Sensor degradation in nuclear reactor pressure vessels: the overlooked factor in remaining useful life prediction

Sensor degradation poses a critical yet ‘often overlooked’ challenge in accurately predicting the remaining useful life (RUL) of nuclear reactor pressure vessels (RPVs), hindering safe and efficient plant operation. This paper introduces an approach to RUL estimation that explicitly addresses sensor degradation, a significant departure from conventional methods. We model neutron embrittlement, a dominant degradation process in RPV steel, as a Wiener process and leverage real-world surveillance capsule data for insightful parameterization. Maximum likelihood estimation is utilized to characterize the degradation dynamics in the model. A Kalman filter then seamlessly integrates the degradation model with sensor measurements, effectively compensating for degradation-induced errors and providing refined state estimates. These estimates power a robust RUL prediction framework. Our results expose the profound impact of sensor degradation on conventional RUL predictions. By directly confronting sensor degradation, our method yields substantially more accurate and reliable RUL estimates. This work marks a significant advancement in the field of materials degradation, offering a powerful tool to optimize nuclear power plant safety and longevity.

Modelling and inference for a degradation process with partial maintenance effects

AbstractThis paper proposes a new way of modelling imperfect maintenance in degradation models, by assuming that maintenance affects only a part of the degradation process. More precisely, the global degradation process is the sum of two dependent Wiener processes with drift. Maintenance has an effect of the ‐type on only one of these processes: it reduces the degradation level of a quantity which is proportional to the amount of degradation of this process accumulated since previous maintenance. Two particular cases of the model are considered: perturbed and partial replacement models. The usual model is also a specific case of this new model. The system is regularly inspected in order to measure the global degradation level. Two observation schemes are considered. In the complete scheme, the degradation levels are measured both between maintenance actions and at maintenance times (just before and just after). In the general scheme, the degradation levels are measured only between maintenance actions. The maximum likelihood estimation of the model parameters is studied for both observation schemes in both particular models. The quality of the estimators is assessed through a simulation study.

Multi-failure mode reliability of monorail vehicle gear transmission system based on multi-index staged degradation

The operational condition of monorail vehicle is diverse and influenced by numerous factors, leading to multiple failure modes and complex performance degradation mechanisms in gear transmission system. Addressing the problem that current reliability evaluation method based on single-stage single-index degradation models fail to accurately evaluate the multi-failure mode reliability of monorail vehicle gear transmission system, this study comprehensively considers the stage and nonlinear characteristics of degradation processes, along with the interdependence of indexes at different stages, a multi-index staged degradation model is established based on nonlinear Wiener process and Vine-Copula function. By analyzing the positions of multi-index stage change points, a multi-failure mode reliability evaluation method of monorail vehicle gear transmission system considering segmented dependence is proposed. And using the Bayes-Gibbs stepwise parameter estimation method proposed in this study, model parameters for Wiener process and Copula function are estimated separately. Finally, the applicability and effectiveness of the proposed method are verified, and the results shows that, compared to traditional single-stage single-index models, the proposed method more accurately describes the dependence and stage patterns of the degradation process, yielding more rational and effective reliability evaluation results.

Particles with constant speed and random velocity in 3+1 space-time: separation of the space variables

We consider a particle in 3+1 space-time dimensions which constantly moves at speed of light c, randomly changing its velocity which can be represented by a point on the surface of a sphere of radius c. The velocity performs an isotropic Wiener process on this surface so that the velocity trajectories are almost everywhere continuous although not differentiable, while the position trajectories are continuous and differentiable. Together with the process that describes the particle in the ‘rest frame’, where the diffusion of velocity on the surface of the sphere is isotropic, the entire family of anisotropic processes which result from Lorentz boosts is also described. The focus of this article is on stochastic evolution in space. We identify a reduced set of variables whose stochastic evolution is autonomous from the remaining variables, but, nevertheless, carry all the relevant information concerning the spatial properties of the process. The associated stochastic equations as well the Forward Kolmogorov equation are considerably simplified compared to those of the complete set of position and velocity variables.

Correlation analysis of degrading systems based on bivariate Wiener processes under imperfect maintenance

AbstractThis article focuses on the correlation between the degradation levels of the two components that form a system. The degradation evolution of each component is modeled using Wiener processes. Both components are dependent and this dependence is described using the trivariate reduction method. To reduce the degradation and extend the system lifetime, preventive maintenance actions are periodically performed. These preventive maintenance actions are imperfect and they are modeled by using an arithmetic reduction of degradation of infinite order model with a determined maintenance efficiency parameter. The evolution of the maintained system is analysed by assessing the expectation and variance of both degradation processes at successive maintenance times. The novelty of this work is the analysis of the Pearson correlation coefficient between the degradation levels of the two components. Different properties of the monotonicity of the Pearson correlation coefficient between the two degradation paths are obtained by considering equal maintenance efficiency and equal general time scales functions for the two Wiener degradation processes associated to each degrading component.

Change point detection of health stage and remaining useful lifetime prediction for electro-mechanical brake unit on trains

Electro-mechanical brake units (EMBUs) have complex degradation process. There are challenges such as component fault coupling, unclear degradation characteristics, random disturbance and so on. A remaining useful life (RUL) framework for EMBUs is developed in this paper. To eliminate the issue of time lag, a data pre-processing method based on zero-phase filtering is proposed for health indicator (HI) construction. Then, the 3 health stages (HSs) characteristics of degradation process is proposed, and is modeled using the Wiener process. To overcome perturbation of large diffusion coefficient and random fluctuation, the change point (CP) detection methods based on the variance ratio hypothesis test and maximal overlap discrete wavelet transform — multi-resolution analysis (MODWT-MRA) are developed. Besides, a consecutive anomaly window detection strategy using sliding windows is employed. Finally, the MODWT-MRA-based RUL prediction methods is proposed. The proposed methods are verified by the durability test data. The proposed data pre-processing method can avoid 10% time error compared to the common difference method. The proposed MODWT-MRA-based CP detection and RUL prediction method can overcome the interference of large random fluctuations in the non-healthy state of EMBU.

A rolling bearing state evaluation method based on deep learning combined with Wiener process

A rolling bearing state evaluation method based on deep learning combined with Wiener process

Linear combinations of i.i.d. strictly stable variables with random coefficients and their application to anomalous diffusion processes

We show that linear combinations of independent and identically distributed strictly stable variables with positive random coefficients is equal in distribution to a function of these random coefficients times a random variable from the same stable distribution. Furthermore, this result is used to show that a random linear combination of independent standard Wiener processes has the distribution of a function of these random coefficients times one standard Wiener process. In opposition to the central limit theorem, this result does not require a large number of terms but it holds with two or more terms. This has implications to simplify stochastic differential equations with a finite number of noises with random coefficients that can be used in modeling anomalous diffusion.

The Brownian motion, Wiener process, and Pu migration as examples of random walk

The theory of random walks holds significance across various domains. To this end, this paper discusses the application of random walk theory in the modeling and analysis of complex systems, including the Brownian motion, Wiener process, and Pu migration. The first part mainly explains the concepts of Brownian motion and random walks, as well as their relationship. The core characteristic is the independence and normal distribution of increments, which makes it a special model of random walks. The next part is the Wiener process, which is a continuous time stochastic process. It can describe random phenomena and solve complex problems in finance, encompassing areas such as stock price prediction, option pricing, and risk management. The last part is about Pu migration, with the clear introduction of Pu migration in biology. The authors discuss why the random walk theory can be regarded as the better approach to analysis this problem. Thus, this paper highlights the importance of the random walk in solving many scientific issues.

GRU-AE-wiener: A generative adversarial network assisted hybrid gated recurrent unit with Wiener model for bearing remaining useful life estimation

Bearings play a pivotal role in various mechanical systems, and their health directly impacts the reliability and safety of these systems. Consequently, extensive research has been dedicated to the estimation of Bearing Remaining Useful Life (RUL) through the lens of information fusion theory. The absence of comprehensive life-cycle degradation data for bearings, a common challenge within the information fusion domain, can hinder the accuracy and reliability of RUL prediction models. A novel hybrid data and model approach named GRU-AE-Wiener has been developed to address this limitation. This approach combines the power of Gated Recurrent Unit (GRU) and Wiener process models within the information fusion framework. Firstly, a Loop Generative Adversarial Network (Loop-GAN) is introduced to synthesize pseudo data to enhance the quality of synthetic data. Next, a bidirectional GRU model is structurally integrated with the Wiener process. In this design, the GRU model is configured in an Auto-Encoder-like structure, with the Wiener process serving as the hidden layer within this Auto-Encoder. Importantly, both the GRU and Wiener processes are jointly optimized with the assistance of Loop-GAN, emphasizing the collaborative nature of information fusion in this approach. The effectiveness of the proposed GRU-AE-Wiener is validated using the PHM 2012 dataset and XJTU-SY dataset. Experimental results underscore its superior RUL predictive performance compared to other deep learning models, highlighting the practical application of information fusion principles in bearing health assessment.