Wiener Process Research Articles

Optimizing the maintenance threshold in presence of shocks: A numerical framework for systems with non-monotonic degradation

Shocks have attracted considerable interest in reliability and maintenance engineering because of their impact on vulnerable systems. Most industrial systems suffer from both internal degradation caused by fatigue and wear-out, and external shocks that often occur randomly due to harsh weather conditions, overloading, etc. Developing maintenance optimization models without taking these stochastic shocks into account is often ineffective. This paper develops a model to optimize the maintenance alarm threshold for a single-component continuously monitored system which is exposed to both fatal and non-fatal shocks in the presence of lead time for hard time maintenance. The shocks occur randomly according to a homogeneous Poisson process during the whole degradation process and have a stochastic impact on the degradation level, while the system resistance to shocks decreases as the system approaches failure. We propose a new numerical maintenance optimization model to find the solution without Monte-Carlo simulation and the model is compared to the Wiener process. A numerical example and a real-time experimental case study on roller bearings are used to demonstrate the effectiveness of the model. The results show that the model is capable of improving maintenance decision-making in terms of failure probability and risk perspective.

A hybrid prognosis scheme for rolling bearings based on a novel health indicator and nonlinear Wiener process

This paper proposes a novel hybrid method aiming at the fault prognosis of bearings. A nonlinear health indicator (HI) is first constructed using Complete Ensemble Empirical Mode Decomposition with Adaptive Noise and Kernel Principal Component Analysis to reflect the health state of a bearing accurately and convincingly. Subsequently, multi-domain features are extracted from vibration signals and the Dual-Channel Transformer Network with the Convolutional Block Attention Module is applied for constructing HIs of the rest bearings. Moreover, the 3σ criterion is employed to establish the condition monitoring interval of health state and detect the First Prediction Time, with which degradation modeling and probabilistic Remaining Useful Life (RUL) prediction are conducted with the assistance of nonlinear Wiener process with random effects. The superior performance of the proposed hybrid prognostic method confirms that the method contributes to the accurate RUL prediction and uncertainty quantification.

Pre-electoral coalition agreement from the Black–Scholes point of view

A political party can be considered as a company whose value depends on the voters support i.e. on the percentage of population supporting the party. Dynamics of the support is thus as a stochastic process with a deterministic growth rate perturbed by a white noise modeled through the Wiener process. This is in an analogy with the option modeling where the stock price behaves similarly as the voters’ support. While in the option theory we have the question of fair price of an option, the question that we ask here is what is a reasonable level of support that the coalition of a “major” party (safely above the election threshold) and a “minor” party (under or around the election threshold) should achieve in order for the “minor” party to get one more representative. We shall elaborate some of the conclusions in the case of recent elections in Montenegro (June, 2023) which are particularly interesting due to lots of political subjects entering the race.

Approximate parameter estimation and mis-specification analysis of degradation model with asymmetric random effects

Asymmetric distributions such as the skew-normal (SN) distribution have been used for modeling the unit-to-unit variation of the degradation rate of products, and its superiority for model fitting and lifetime estimation has been demonstrated in literature. However, there still exist two issues that need to be further studied. That is, how to simply estimate the parameters of the degradation model with asymmetric random effects and whether the asymmetry or skewness affects the remaining useful life (RUL) prediction. Driven by these two issues, the parameter estimation and RUL prediction of the degradation model with general random effects are first introduced and a two-step maximum likelihood estimation (MLE) method for parameter estimation is proposed, which could greatly simplify the parameter estimation of the degradation model with asymmetric random effects. Then, the SN distribution is used as an example to analyze the second issue regarding the effect of model mis-specification on RUL prediction based on regression models and Wiener processes with different asymmetric random effects in theory. Finally, the parameter estimation and mis-specification analysis of degradation models with other asymmetric random effects are conducted through numerical examples and two case studies. The results demonstrate that the proposed two-step MLE method is satisfactory and the effect of asymmetry on RUL prediction is negligible. Therefore, there is no need to consider the asymmetry of random effects for RUL prediction.

A reliability estimation method based on combination of failure mechanism and ANN supported wiener processes

For some engineering application, accurately estimating reliability only depend on the history data or failure mechanism is difficult to implement, due to the lack of data and imperfect theory of failure mechanism. Namely, both history data and failure mechanism should be utilized to improve the reliability estimation accuracy for engineering applications. Hence, we construct a reliability estimation method by fusing the failure mechanism and artificial neural network (ANN) supported Wiener processes for utilizing both history data and failure mechanism. ANN and failure mechanism are integrated into Wiener process with random effects, respectively. Bayesian model averaging (BMA) method is adapted to combine the failure mechanism with ANN supported Wiener processes, as well as to update the model parameters by fusing data. Based on a typical aviation hydraulic pump's actual dataset, we illustrate the advantages of our approach by comparing to Wiener process supported only by ANN or failure mechanism in engineering practices. The proposed method shows superiorities on reliability estimation considering the estimation accuracies comparing the other two models.

Model-Based Smoothing with Integrated Wiener Processes and Overlapping Splines

In many applications that involve the inference of an unknown smooth function, the inference of its derivatives is also important. To make joint inferences of the function and its derivatives, a class of Gaussian processes called pth order Integrated Wiener’s Process (IWP), is considered. Methods for constructing a finite element (FEM) approximation of an IWP exist but only focus on the case p = 2 and do not allow appropriate inference for derivatives. In this article, we propose an alternative FEM approximation with overlapping splines (O-spline). The O-spline approximation applies for any order p ∈ Z + , and provides consistent and efficient inference for all derivatives up to order p – 1. It is shown both theoretically and empirically that the O-spline approximation converges to the IWP as the number of knots increases. We further provide a unified and interpretable way to define priors for the smoothing parameter based on the notion of predictive standard deviation, which is invariant to the order p and the knot placement. Finally, we demonstrate the practical use of the O-spline approximation through an analysis of COVID death rates where the inference of derivative has an important interpretation in terms of the course of the pandemic. Supplementary materials for this article are available online.

Interactive Hybrid Model for Remaining Useful Life Prediction With Uncertainty Quantification of Bearing in Nuclear Circulating Water Pump

Journal bearings are the key components of the nuclear circulating water pump (NCWP), and accurate remaining useful life (RUL) prediction is of great significance for improving the reliability, safety, and maintenance planning of NCWP. However, it is difficult to quantify the uncertainty of bearing RUL based on the current deep learning (DL) model, resulting in a lack of credibility and effective convincing for RUL predicted by the model. Meanwhile, all existing hybrid models are basically simple combinations, and they cannot solve the uncertainty quantification problem of RUL predicted by DL. Hence, the bearing RUL prediction method based on a dynamic interactive hybrid model is proposed. Firstly, a degradation model based on a nonlinear enhanced generalized Wiener process (EGWP) is proposed, which combines gated neural networks and time-varying drift coefficients to describe the nonlinear degradation process of bearing. Then, a corrective gated recurrent unit (CGRU) network is designed to learn and predict real-time degradation increments, and the parameters of the degradation model are dynamically updated through the history and prediction of degradation increments. Finally, the bearing RUL prediction is given by the CGRU network, and the probability density function (PDF) of RUL is given by the proposed hybrid model. The performance of the proposed method is evaluated using the PHM 2012 bearing dataset and the NCWP journal bearing dataset. The results show that our proposed method can effectively predict bearing RUL and its uncertainty.

Uncertainly Analysis of Prior Distribution in Accelerated Degradation Testing Bayesian Evaluation Method Based on Deviance Information Criterion

The accelerated degradation testing (ADT) Bayesian evaluation method comprehensively utilizes product degradation data under accelerated stress levels collected over a short period of time and multiple sources of prior information, such as historical information, similar product information, simulation information, etc., to conduct life and reliability evaluation. Through the prior distribution, prior information affects the ADT Bayesian evaluation results ultimately. However, different evaluators may obtain different prior distributions based on the same prior information due to varying experiences or rules, which may lead to differences in the ADT Bayesian evaluation results. Therefore, it is necessary to analyze and study the impact of prior distribution uncertainty on the ADT Bayesian evaluation results while also finding criteria to judge the quality of prior distributions. This paper focuses on the ADT Bayesian evaluation method based on the Wiener process and the Arrhenius relation, studying the influence of different prior distributions on the robustness of ADT Bayesian evaluation results. Additionally, based on the deviance information criterion (DIC), a criterion for selecting prior distributions in the ADT Bayesian evaluation method is proposed. Through carrying out uncertainty analysis of prior distribution in the ADT Bayesian evaluation method, a theoretical system and framework for analyzing prior uncertainty in ADT Bayesian evaluation based on DIC are established, providing a better foundation for the practical application of the ADT Bayesian evaluation method in engineering.

Bayesian Fusion of Degradation and Failure Time Data for Reliability Assessment of Industrial Equipment Considering Individual Differences

In the field of industrial equipment reliability assessment, dependency on either degradation or failure time data is common. However, practical applications often reveal that single-type reliability data for certain industrial equipment are insufficient for a comprehensive assessment. This paper introduces a Bayesian-fusion-based methodology to enhance the reliability assessment of industrial equipment. Operating within the hierarchical Bayesian framework, the method innovatively combines the Wiener process with available degradation and failure time data. It further integrates a random effects model to capture individual differences among equipment units. The robustness and applicability of this proposed method are substantiated through an in-depth case study analysis.

Performance reliability degradation analysis and lifetime prediction based on generalized normal grey cloud Bayesian Wiener process

AbstractThis study addresses the problem of performance reliability degradation and lifetime prediction in complex systems with high reliability but insufficient information. It comprehensively considers the effects of various uncertainties, such as measurement, cognitive, and stochastic uncertainties, and proposes a generalized normal grey cloud Bayesian Wiener process (GNGCBWP) stochastic degradation model. First, based on the historical degradation information of the system, the interval grey number sequence of the degradation increment is constructed, and the performance degradation path of the system is analyzed to determine the degradation model type. Then, combining the normal grey cloud and Bayesian estimation theory, the relevant definitions and theorems of generalized normal grey cloud Bayesian estimation (GNGCBE) are proposed. Finally, the parameter estimation results obtained using the GNGCBE method are substituted into the grey Wiener process degradation model to dynamically analyze performance reliability degradation and predict the remaining useful life (RUL) in real‐time. And the validity and scientificity of the proposed model is verified through a case study of laser equipment. Moreover, the comparison results of the models indicate that the proposed model has advantages in dynamic analysis of performance reliability degradation and real‐time RUL prediction for highly reliable complex systems with limited or poor degradation information. It can significantly reduce the impact of multiple uncertainties on the analysis results and improve the accuracy and reliability of the final prediction results.

Remaining useful life prediction of proton exchange membrane fuel cell based on Wiener process and Bayesian GRU network considering multi-source uncertainties

ABSTRACT Proton exchange membrane fuel cell (PEMFC) is considered as one of the most promising green energy devices. Although fruitful results can be available for the remaining useful life (RUL) prediction of PEMFC, stochastic uncertainties have never been considered. To tackle this problem, a hybrid method is proposed in this paper. Wiener process with temporal uncertainty and individual uncertainty is adopted to model the degradation of the state of health (SOH), which is then estimated from monitoring voltage with measurement noise using the unscented Kalman filter (UKF), where unknown filtering and model parameters are jointly identified by expectation-maximization (EM) algorithm and Rauch-Tung-Striebel (RTS) smoother. Finally, gated recurrent unit (GRU) network is employed to realize the RUL prediction with the prediction uncertainty quantified by the Bayesian variational inference technology. The proposed method is verified on the experimental data. Results indicate that smaller mean absolute percentage error (MAPE), root mean square error (RMSE), and mean absolute error (MAE) values can be obtained compared with other methods. When 60% data are used for prediction, the proposed method can achieve a RUL prediction accuracy with 1.63% and 2.17% relative errors under static and dynamic conditions, respectively, which illustrates the feasibility and superiority of the proposed method.

Data-model linkage prediction of tool remaining useful life based on deep feature fusion and Wiener process

Accurately predicting the tool remaining useful life (RUL) is critical for maximizing tool utilization and saving machining costs. Various physical model-based or data-driven prediction methods have been developed and successfully applied in different machining operations. However, many uncertain factors affect tool RUL during the cutting process, making it challenging to create a precise physical model to characterize the degradation of tool performance. The success of the purely data-driven technique depends on the amount and quality of the training samples, it does not consider the physical law of tool wear, and the interpretability of the prediction results is poor. This paper presents a data-model linkage approach for tool RUL prediction based on deep feature fusion and Wiener process to address the above limitations. A convolutional stacked bidirectional long short-term memory network with time-space attention mechanism (CSBLSTM-TSAM) is developed in the data-driven module to fuse the multi-sensor signals collected during the cutting process and then obtain the mapping relationship between signal features and tool wear values. In the physical modeling module, a three-stage tool RUL prediction model based on the nonlinear Wiener process is established by considering the evolution law of different wear stages and multi-layer uncertainty, and the corresponding probability density function is derived. The real-time estimated tool wear of the data-driven module is used as the observed value of the physical model, and the model parameters are dynamically updated by the weight-optimized particle filter (WOPF) algorithm under a Bayesian framework, thereby realizing the data-model linkage tool RUL prediction. Milling experiments demonstrate that the proposed method not only improves RUL prediction accuracy, but also has good generalization ability and robustness for prediction tasks under different working conditions.

Optimal design of constant‐stress accelerated degradation tests based on the Wiener process with manufacturing batches heterogeneity and individual differences

AbstractThe use of Constant‐stress accelerated degradation tests (CSADT) modeling is an effective way to assess the reliability of products. However, when conducting reliability experiments, the data collected may come from various sources such as different manufacturing batches, equipment, and operators. This can result in reduced accuracy of the test evaluations. In this research paper, we propose optimal designs for the Wiener constant‐stress accelerated degradation model that take into account the heterogeneity of manufacturing batches and individual differences. Our goal is to minimize the variability of the mean time to failure (MTTF) while working within a specified budget, test time, and available test units. To achieve this, we utilize particle swarm optimization (PSO) to find the best solution based on a complex objective function. We also compare our model with others using degradation data from LEDs, showing that our model has a better goodness‐of‐fit. Finally, we present examples of optimal CSADT plans under different constraints and conduct sensitivity analysis.

Lifetime prediction and maintenance assessment of Lithium-ion batteries based on combined information of discharge voltage curves and capacity fade

Current two-stage Lithium-ion battery degradation models commonly treat the change point (CP) in two ways. First, it is a random variable, which increases model complexity and the computational cost for predicting the remaining useful life (RUL). Second, it is a deterministic value, which will simplify the degradation models. The value of CP needs to be well identified for an accurate description of the degradation process. However, the capacity data is generally non-monotonic due to energy regeneration, which makes it hard to use in determining the CP. In addition, our laboratory data on the discharge voltage profile shows a potential for CP detection. Thus, we developed a hybrid method that combines dynamic time warping and nonnegative matrix factorization to detect the CPs of battery cells by using voltage discharge profiles. Then, we proposed a two-stage Wiener process incorporating CP detection method to describe the battery capacity degradation pattern. The proposed model is applied to our lab data and NASA data to predict RUL. Finally, maintenance strategies are analyzed to enhance the management of Lithium-ion batteries by minimizing the long-term cost rate under different replacement policies.

Efficient single-run implementation of generalized Einstein relation to compute transport coefficients: A binary-based time sampling.

A robust and simple implementation of the generalized Einstein formulation using single equilibrium molecular dynamics simulation is introduced to compute diffusion and shear viscosity. The unique features underlying this framework are as follows: (1) The use of a simple binary-based method to sample time-dependent transport coefficients results in a uniform distribution of data on a logarithmic time scale. Although we sample "on-the-fly," the algorithm is readily applicable for post-processing analysis. Overlapping same-length segments are not sampled as they indicate strong correlations. (2) Transport coefficients are estimated using a power law fitting function, a generalization of the standard linear relation, that accurately describes the long-time plateau. (3) The use of a generalized least squares (GLS) fitting estimator to explicitly consider correlations between fitted data points results in a reliable estimate of the statistical uncertainties in a single run. (4) The covariance matrix for the GLS method is estimated analytically using the Wiener process statistics and computed variances. (5) We provide a Python script to perform the fits and automate the procedure to determine the optimal fitting domain. The framework is applied to two fluids, binary hard sphere and a Lennard-Jones near the triple point, and the validity of the single-run estimates is verified against multiple independent runs. The approach should be applicable to other transport coefficients since the diffusive limit is universal to all of them. Given its rigor and simplicity, this methodology can be readily incorporated into standard molecular dynamics packages using on-the-fly or post-processing analysis.

RUL prediction for two-phase degrading systems considering physical damage observations

This paper focuses on a specific type of two-phase degrading system commonly encountered in industrial practice. The first phase is moderate with a low degradation rate while the second is rapid with a high rate. Current studies usually rely solely on sensor measurements to divide phases and predict the remaining useful life (RUL), ignoring the utilization of actual physical damage observations, such as wear depth and crack length. These observations, available during system shutdown periods, directly reflect system states and phase changes. To this end, we propose a novel RUL prediction framework consisting of offline training and online prediction processes. In the offline training process, the physical damage observations and sensor measurements are utilized to estimate the parameters of a two-phase Wiener process and a bijective function matrix. In the online prediction process, real-time sensor measurements are transformed into virtual damage observations for RUL prediction. To enhance the accuracy of phase change point detection, a change point detection algorithm is proposed for both processes. The effectiveness is demonstrated using a simulation and a real case study.

Statistical inference for a competing failure model based on the Wiener process and Weibull distribution.

Competing failure models with degradation phenomena and sudden failures are becoming more and more common and important in practice. In this study, the generalized pivotal quantity method was proposed to investigate the modeling of competing failure problems involving both degradation and sudden failures. In the competing failure model, the degradation failure was modeled through a Wiener process and the sudden failure was described as a Weibull distribution. For point estimation, the maximum likelihood estimations of parameters $ \mu $ and $ \sigma^2 $ were provided and the inverse estimation of parameters $ \eta $ and $ \beta $ were derived. The exact confidence intervals for parameters $ \mu $, $ \sigma^2 $, and $ \beta $ were obtained. Furthermore, the generalized confidence interval of parameter $ \eta $ was obtained through constructing the generalized pivotal quantity. Using the substitution principle, the generalized confidence intervals for the reliability function, the $ p $th percentile of lifetime, and the mean time to failure were also obtained. Simulation technique was carried out to evaluate the performance of the proposed generalized confidence intervals. The simulation results showed that the proposed generalized confidence interval was effective in terms of coverage percentage. Finally, an example was presented to illustrate the application of the proposed method.

Simultaneous Model Change Detection in Multivariate Linear Regression With Application to Indonesian Economic Growth Data

In this paper, we study asymptotic model change detection in multivariate linear regression by using the Kolmogorov–Smirnov function of the partial sum process of recursive residuals. We approximate the rejection region and also the power function of the test by establishing a functional central limit theorem for the sequence of the partial sum processes of the recursive residuals of the observations. When the assumed model is true, the limit process is given by the standard multivariate Brownian motion which does not depend on the regression functions. However, when the assumed model is not true (some models change), the limit process is represented by a vector of deterministic trend plus the standard multivariate Brownian motion. The finite sample size rejection region and the power of the test are investigated by means of Monte Carlo simulation. The simulation study shows evidence that the proposed test is consistent in the sense that it attains the power larger than the size of the test when the hypothesis is not true. We also demonstrate the application of the proposed test method to Indonesian economic growth data in which we test the adequacy of three‐variate low‐order polynomial model. The test result shows that the growth of the Indonesian economy is neither simultaneously constant nor linear. The test has successfully detect the appearance of a change in the model which is mainly caused by the COVID‐19 pandemic in 2020.

The Trapezoidal Numerical Method for the CIR Model Driven by Standard Brownian Motion

The Trapezoidal Numerical Method for the CIR Model Driven by Standard Brownian Motion

Pricing forward-start style exotic options under uncertain stock models with periodic dividends

<p>In this study, we derived pricing formulas for various forward-start style exotic options based on an uncertain stock models with periodic dividends. Specifically, we present valuations for forward-start, Cliquet/Ratchet, and spread options. In addition, we conducted numerical simulations of these formulas and compared them to pricing formulas for the same options based on a dividend-paying stock model driven by standard Brownian motion.</p>