Generalized Wiener Process Research Articles

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 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.

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.

Data-Driven Discovery of Stochastic Differential Equations

Stochastic differential equations (SDEs) are mathematical models that are widely used to describe complex processes or phenomena perturbed by random noise from different sources. The identification of SDEs governing a system is often a challenge because of the inherent strong stochasticity of data and the complexity of the system’s dynamics. The practical utility of existing parametric approaches for identifying SDEs is usually limited by insufficient data resources. This study presents a novel framework for identifying SDEs by leveraging the sparse Bayesian learning (SBL) technique to search for a parsimonious, yet physically necessary representation from the space of candidate basis functions. More importantly, we use the analytical tractability of SBL to develop an efficient way to formulate the linear regression problem for the discovery of SDEs that requires considerably less time-series data. The effectiveness of the proposed framework is demonstrated using real data on stock and oil prices, bearing variation, and wind speed, as well as simulated data on well-known stochastic dynamical systems, including the generalized Wiener process and Langevin equation. This framework aims to assist specialists in extracting stochastic mathematical models from random phenomena in the natural sciences, economics, and engineering fields for analysis, prediction, and decision making.

A new class of multi-stress acceleration models with interaction effects and its extension to accelerated degradation modelling

Most products operate under multiple stresses. The influences of multi-stress factors on products are commonly not independent and promote a more violent degradation through interactions, referred to as stress interaction effects. However, few studies consider such effects in multi-stress acceleration models, which may reduce the extrapolation precision and further lead to an inaccurate reliability assessment. In this paper, a new class of multi-stress acceleration models with interaction effects is designed to extrapolate more accurate reliability metrics under multi-stress operating conditions. The main stress effect terms are determined by two criteria: physical stress-based ageing law and statistical correlation. The interaction effects are interpreted as the influences of other stress variables on the main stress effects. In particular, the explicit functions of interaction effects are alternative, which can be identified by adaptive optimization using the maximum likelihood criterion. Furthermore, this multi-stress acceleration model is extended to an accelerated degradation model by integrating a generalized Wiener process with nonlinear time scale functions and random effects. The acceleration factor constant principle is utilized to identify the stress-dependent parameters, facilitating a more appropriate model development. Finally, simulation and a real-world case are performed to validate the effectiveness and practical values of the proposed model.

An adaptive and generalized Wiener process model with a recursive filtering algorithm for remaining useful life estimation

In this paper, we propose a generalized Wiener process-based degradation model with an adaptive drift to characterize the degradation behavior exhibiting nonlinearity, temporal uncertainty, item-to-item variability, and time-varying degradation. A recursive Bayesian filtering algorithm is derived to update the drift distribution. The expectation-maximization algorithm is utilized to estimate all other model parameters online whenever a new degradation measurement from the system under consideration is available without requiring population-based degradation data from identical systems in the same batch. This renders both the hidden drift and model parameters adaptive to the newly acquired degradation data. An analytical approximation to the RUL distribution considering the uncertainty of the hidden drift is derived in a closed form which is proved to encompass some existing formulae as its special cases. A numerical example is provided to illustrate the implementation procedure of the proposed RUL estimation method, and a practical milling dataset is adopted to testify to the superior performance of the proposed method against previous similar methods in remaining useful life estimation.

A generalized Wiener process with dependent degradation rate and volatility and time-varying mean-to-variance ratio

In degradation analysis, there exist two natural features for the degradation data. The first is the dependence of degradation rate and degradation volatility, and the other is the time-varying mean-to-variance ratio. Ignoring them may lead to a significant bias in assessing lifetime information of materials. This paper proposes a generalized Wiener degradation model, which puts these two characteristics and unit-to-unit variation into consideration simultaneously. The proposed model includes many existing Wiener degradation models as special cases. Then, a generalized closed-form approximated residual lifetime distribution is given for the proposed model. Statistical inference for the model parameters is conducted based on the expectation maximizing (EM) method, and two simple auxiliary frameworks are developed for the determination of initial values and the forms of the time-scaled functions. The developed methodologies are then illustrated and verified in a simulation study and two real data analysis.

An Improved Random Effects Wiener Process Accelerated Degradation Test Model for Lithium-Ion Battery

For the accelerated degradation test (ADT) of lithium-ion battery, the degradation path models are usually applied with the fixed parameters. To optimize the degradation path models, so many battery cells are required for the ADT models based on the degradation path. However, for some lithium-ion batteries with long cycle life and high reliability, the samples are limited due to experimental cost constraints. Meanwhile, owing to the differences in raw materials and manufacturing processes, it is easy to observe variations among the degradation path of lithium-ion battery cells. Fortunately, Wiener process model considering the individual difference can express the variations and be modeled with limited samples for lithium-ion battery. Motivated by this issue, an improved method with random effects Wiener process (REWP) is proposed in this article. First, to verify the degradation process following the Wiener process, the Shapiro-Wilk test is carried out on the basis of a significance level of 0.05. Second, we analyze the varieties of every lithium-ion battery cell. To select a suitable Wiener process model, we apply the coefficient of variation as a measure of random parameters selection for Wiener process. Meanwhile, the maximum likelihood estimation (MLE) is used to optimize parameters of the proposed model. Next, the three different REWP models and general Wiener process model are compared. Finally, to demonstrate the proposed method, the lithium-ion battery cells used for a spacecraft in low Earth orbit (LEO) are conducted for ADTs. The results suggest that the REWP model has a better performance for assessing reliability of lithium-ion battery with long cycle life.

Quality estimation for models of a generalized Wiener process

Quality estimation for models of a generalized Wiener process

Analysis of multivariate dependent accelerated degradation data using a random-effect general Wiener process and D-vine Copula

A modern product usually shows multiple performance characteristics that degrade simultaneously. It is quite common that these degradation processes are dependent due to some common factors such as internal structure, operating conditions, and working history. Technically, the essence of modeling such dependent degradation processes is to find appropriate marginal degradation models and capture the dependence structure among the multiple performance characteristics. This paper develops a multivariate dependent accelerated degradation test model based on a random-effect general Wiener process and D-vine copula. The proposed model and statistical inference method analyze nonlinear accelerated degradation data considering three sources of uncertainty and construct the marginal distributions. To overcome the lack of flexibility and parameter restrictions of standard multivariate copulas, the pair-copula constructions and their vine graphical representations are applied to reveal and fully understand the complex and hidden dependence patterns in the multivariate degradation data. A simulation example and a real application on the accelerated degradation data of a tuner are provided to illustrate the performance and benefits of the proposed model and statistical estimation method.

Degradation data analysis based on a generalized Wiener process subject to measurement error

Wiener processes have received considerable attention in degradation modeling over the last two decades. In this paper, we propose a generalized Wiener process degradation model that takes unit-to-unit variation, time-correlated structure and measurement error into considerations simultaneously. The constructed methodology subsumes a series of models studied in the literature as limiting cases. A simple method is given to determine the transformed time scale forms of the Wiener process degradation model. Then model parameters can be estimated based on a maximum likelihood estimation (MLE) method. The cumulative distribution function (CDF) and the probability distribution function (PDF) of the Wiener process with measurement errors are given based on the concept of the first hitting time (FHT). The percentiles of performance degradation (PD) and failure time distribution (FTD) are also obtained. Finally, a comprehensive simulation study is accomplished to demonstrate the necessity of incorporating measurement errors in the degradation model and the efficiency of the proposed model. Two illustrative real applications involving the degradation of carbon-film resistors and the wear of sliding metal are given. The comparative results show that the constructed approach can derive a reasonable result and an enhanced inference precision.

Stochastic Modeling and Analysis of Multiple Nonlinear Accelerated Degradation Processes through Information Fusion.

Accelerated degradation testing (ADT) is an efficient technique for evaluating the lifetime of a highly reliable product whose underlying failure process may be traced by the degradation of the product’s performance parameters with time. However, most research on ADT mainly focuses on a single performance parameter. In reality, the performance of a modern product is usually characterized by multiple parameters, and the degradation paths are usually nonlinear. To address such problems, this paper develops a new s-dependent nonlinear ADT model for products with multiple performance parameters using a general Wiener process and copulas. The general Wiener process models the nonlinear ADT data, and the dependency among different degradation measures is analyzed using the copula method. An engineering case study on a tuner’s ADT data is conducted to demonstrate the effectiveness of the proposed method. The results illustrate that the proposed method is quite effective in estimating the lifetime of a product with s-dependent performance parameters.

A Generalized Wiener Process Degradation Model with Two Transformed Time Scales

Degradation analysis is very useful in reliability assessment for complex systems and highly reliable products, because few or even no failures are expected in a reasonable life test span for them. In order to further our study on degradation analysis, a novel Wiener process degradation model subject to measurement errors is proposed. Two transformed time scales are involved to depict the statistical property evolution over time. A situation where one transformed time scale illustrates a linear form for the degradation trend and the other transformed time scale shows a generalized quadratic form for the degradation variance is discussed particularly. A one‐stage maximum likelihood estimation of parameters is constructed. The statistical inferences of this model are further discussed. The proposed method is illustrated and verified in a comprehensive simulation study and two real applications for indium tin oxide (ITO) conductive film and light emitting diode (LED). The Wiener process model with mixed effects is considered as a reference. Comparisons show that the proposed method is more general and flexible, and can provide reasonable results, even in considerably small sample size circumstance. Copyright © 2016 John Wiley & Sons, Ltd.

Residual Life Estimation Based on a Generalized Wiener Process with Skew-normal Random Effects

For reliability-critical and expensive products, it is necessary to estimate their residual lives based on available information, such as the degradation data, so that proper maintenance actions can be arranged to reduce or even avoid the occurrence of failures. In this work, by assuming that the product-to-product variability of the degradation is characterized by a skew-normal distribution, a generalized Wiener process-based degradation model is developed. Following that, the issue of residual life (RL) estimation of the target product is addressed in detail. The proposed degradation model provides greater flexibility to capture a variety of degradation processes, since several commonly used Wiener process-based degradation models can be seen as special cases. Through the EM algorism, the population-based degradation information is used to estimate the parameters of the model. Whenever new degradation measurement information of the target product becomes available, the degradation model is first updated based on the Bayesian method. In this way, the RL of the target product can be estimated in an adaptive manner. Finally, the developed methodology is demonstrated by a simulation study.

Optimal Constant-Stress Accelerated Degradation Test Plans Using Nonlinear Generalized Wiener Process

Accelerated degradation test (ADT) has been widely used to assess highly reliable products’ lifetime. To conduct an ADT, an appropriate degradation model and test plan should be determined in advance. Although many historical studies have proposed quite a few models, there is still room for improvement. Hence we propose a Nonlinear Generalized Wiener Process (NGWP) model with consideration of the effects of stress level, product-to-product variability, and measurement errors for a higher estimation accuracy and a wider range of use. Then under the constraints of sample size, test duration, and test cost, the plans of constant-stress ADT (CSADT) with multiple stress levels based on the NGWP are designed by minimizing the asymptotic variance of the reliability estimation of the products under normal operation conditions. An optimization algorithm is developed to determine the optimal stress levels, the number of units allocated to each level, inspection frequency, and measurement times simultaneously. In addition, a comparison based on degradation data of LEDs is made to show better goodness-of-fit of the NGWP than that of other models. Finally, optimal two-level and three-level CSADT plans under various constraints and a detailed sensitivity analysis are demonstrated through examples in this paper.

Aerial Infrared Thermography: A Scalable Procedure for Photovoltaics Inspections Based on Efficiency and Flexibility

The objective of this contribution is to explore innovative ways to increase the efficiency of large scale photovoltaic systems (LSPVS) in the medium term through effective management of operating, monitoring and maintenance (OMM) costs based on aerial surveillance with small multi-rotor flight robot (s-MRFR). An effective solution of OMM cost assessment must capture the dynamic interaction between energy market developments, technological progress and investor tolerance to the risk posed by climate changes. The valuation method for OMM cost-risk analysis is based on generalized Wiener process with stochastic jumps. The interest is to increase the value of the investment by using innovative but efficient procedures of PV systems inspection based on aerial infrared thermography. The results are encouraging because the total costs of s-MRFR monitoring are very low compared to the benefits.

Approximation algorithms for optimal purchase/inventory policy when purchase price and demand are stochastic

We consider a purchase/inventory control problem in which the purchase price and demand are stochastic, a common situation encountered by firms that replenish in a foreign currency or from commodity markets. More specifically, we assume that the demand follows a Poisson arrival process and that the log-price evolves according to a general Wiener process. Under these circumstances, the optimal policy is a state dependent base-stock policy that can be described as a series of threshold prices. An iterative procedure for determining the optimal thresholds has been derived earlier but, even for the simplest price process, the solution quickly becomes numerically intractable. To deal with this, we propose an approximation that allows us to derive simple heuristics for finding thresholds that are close to optimal. For certain price processes the heuristics are just a series of closed-form expressions. The computational complexity is reduced significantly, and the numerical study shows that the new heuristics perform considerably better than earlier suggested heuristics.

A Continuous-Time Model for Valuing Foreign Exchange Options

This paper makes use of stochastic calculus to develop a continuous-time model for valuing European options on foreign exchange (FX) when both domestic and foreign spot rates follow a generalized Wiener process. Using the dollar/euro exchange rate as input for parameter estimation and employing our FX option model as a yardstick, we find that the traditional Garman-Kohlhagen FX option model, which assumes constant spot rates, values incorrectly calls and puts for different values of the ratio of exchange rate to exercise price. Specifically, it undervalues calls when the ratio is between 0.70 and 1.08, and it overvalues calls when the ratio is between 1.18 and 1.30, whereas it overvalues puts when the ratio is between 0.70 and 0.82, and it undervalues puts when the ratio is between 0.86 and 1.30.

Comments on “The Generalized Wiener Process for Colored Noise”

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Galleani and Cohen have developed a new approach to the study of random differential equations. Recently, they applied their method to the interesting case that they called the generalized Wiener process. We show that while the phase space equation they derived is correct, their solution to the equation is not the full solution but holds only under certain conditions. We obtain the general solution and discuss under what circumstances their solution is exact, and under what circumstances their solution is a good approximation to the exact solution. In addition, we pinpoint where in their method of solution they neglected a term, and give the correction. In many cases, their approach may be accurate enough, and hence preferable, as it is a simpler calculation than the exact solution. However, for processes with long correlation times (i.e., autocorrelation functions that do not decay rapidly to zero), the complete method of solution presented here may be required. </para>

Reply to 'Comments on 'The generalized Wiener process for colored noise'"

This paper presents the reply of the authors to the Comments on dasiathe generalized Wiener process for colored noise by P. Loughlin (see ibid., vol.14, no.11, 2007). The original paper was by the authors (L. Galleani and L. Cohen, see ibid., vol.13, no.10, p.608-11, 2006).