Drift Of Brownian Motion Research Articles

A novel exponential degradation approach for predicting the remaining useful life of roadheader bearings

Based on the optimized exponential-degradation model (OEDM), a novel approach for predicting the remaining useful life(RUL) of roadheader bearings under different working conditions is proposed in this study. Specifically, the exponential process is used to construct the degradation process from a single performance characteristic under variable operating conditions, the generalized expectation maximization is employed to estimate model parameters, and the proposed degradation model is updated after new data is available. In the traditional exponential degradation method, the hyperparameters are only optimized, which leads to low calculation accuracy under severe working conditions. In the proposed method, the Bayesian algorithm and the Drift Brownian motion algorithm were respectively employed to optimize hyperparameters and stochastic parameters to ensure the high accuracy of the prediction results. In addition, degradation characteristics combined with sensory data acquired through condition monitoring were used to continuously update the RUL in the proposed degradation model. Finally, the effectiveness of the proposed model is verified by a simulation case and a case study. The results show that compared with the linear degradation model and the general exponential degradation model, the proposed OEDM performs well in practical applications and has a higher prediction accuracy. This study provides a reference for predictive maintenance of critical parts of tunneling machinery and cost reduction of tunneling.

Reliability assessment for micro inertial measurement unit based on accelerated degradation data and copula theory

With its extensive use in industry, assessing the reliability of the micro inertial measurment unit (MIMU) has become a pressing need. Unfortunately, the MIMU is made up of several components, and the degradation processes of each are intertwined, making it difficult to assess the MIMU’s reliability and remaining useful life. In this research, we offer a reliability assessment approach for the MIMU, which has long-lifetime and multiple performance characteristics (PCs), based on accelerated degradation data and copula theory.Each PC model of MIMU is constructed utilizing drift Brownian motion to depict accelerated degradation process. The copula function is used to model the multivariate dependent accelerated degradation test data and to describe the dependency between multiple MIMU performance parameters. The particle swarm optimization algorithm is used to estimate the unknown parameters in the multi-dependent ADT model. Finally, the storage test and simulation example on MIMU’s accelerated degradation data verify the feasibility and effectiveness of the proposed method.

Optimization Design of Step Stress Accelerated Degradation Test for Motorized Spindle Based on Ds-Optimality

In the accelerated degradation test (ADT) of motorized spindles, it is necessary to apply a variety of stresses to simulate real working conditions. However, the traditional accelerated test scheme optimization method does not consider the weight of various stresses in the test, resulting in the evaluation accuracy of important stress parameters in the model being too low. In order to solve this problem, an optimal design method of the step stress accelerated degradation test (SSADT) scheme for motorized spindles is proposed based on Ds-optimality. Firstly, the fault tree analysis (FTA) method is used to analyze the collected fault data of motorized spindles and screen the main stress. Then, the accelerated degradation model is established by using drift Brownian motion. Based on the Ds-optimality, the optimization variables and constraints in the test are determined, and the optimization model is established with the objective of minimizing the estimated variance of the main stress parameters in the acceleration model; additionally, the optimization steps are given. Finally, an example is given to verify the effectiveness of the method. Sensitivity analysis of the optimization results shows that the method has good robustness.

Inference on the change point under a high dimensional sparse mean shift

We study a plug in least squares estimator for the change point parameter where change is in the mean of a high dimensional random vector under subgaussian or subexponential distributions. We obtain sufficient conditions under which this estimator possesses sufficient adaptivity against plug in estimates of mean parameters in order to yield an optimal rate of convergence $O_{p}(\xi ^{-2})$ in the integer scale. This rate is preserved while allowing high dimensionality as well as a potentially diminishing jump size $\xi $, provided $s\log (p\vee T)=o(\surd (Tl_{T}))$ or $s\log ^{3/2}(p\vee T)=o(\surd (Tl_{T}))$ in the subgaussian and subexponential cases, respectively. Here $s,p,T$ and $l_{T}$ represent a sparsity parameter, model dimension, sampling period and the separation of the change point from its parametric boundary, respectively. Moreover, since the rate of convergence is free of $s,p$ and logarithmic terms of $T$, it allows the existence of limiting distributions under high dimensional asymptotics. These distributions are then derived as the argmax of a two sided negative drift Brownian motion or a two sided negative drift random walk under vanishing and non-vanishing jump size regimes, respectively, thereby allowing inference on the change point parameter. Feasible algorithms for implementation of the proposed methodology are provided. Theoretical results are supported with monte-carlo simulations.

Failure behavior modeling and reliability estimation of product based on vine-copula and accelerated degradation data

Abstract Aiming at the characteristics of failure mechanism coupling of high-reliability and long-lifetime complex products, a method of multivariate failure behavior modeling and reliability assessment is proposed based on vine-copula and accelerated degradation data. The method, considering the coupling modeling under accelerated stress and the relevant acceleration mechanism consistency test, can not only clearly describe the multivariate coupling relationship of the product, but also can carry out the reliability assessment of the product within the affordable time and cost. The failure behavior modeling method consists of two parts which are degradation behavior and correlation relationship among variables. Firstly, degradation behavior description of failure behavior modeling is obtained from, the degradation data of product performance parameters based on the accelerated degradation test, the degradation path of each performance parameter by utilizing drift Brownian motion, then the implementation of accelerated consistency test ensures the true reliability information. Furthermore, the correlation description of failure behavior modeling is constructed by Vine-Copula based multivariate damage coupling modeling method describing the pairwise correlation relationship. Afterward, the exact solution of reliability joint distribution of the product with non-strictly monotonic degradation is figured out by Vine-Copula and the equation of reliability estimation based on conditional probability. Finally, the feasibility of the method is verified by utilizing smart electricity meter as the numerical example.

Multivariate Degradation Modeling of Smart Electricity Meter with Multiple Performance Characteristics via Vine Copulas

For smart electricity meter with multiple performance characteristics (PCs) with coupling relationships because of amounts of components experiencing multiple deteriorating processes, we develop a multivariate degradation modeling method via vine copulas to estimate the reliability of products with multiple PCs reflecting degradation states. In the multivariate model, pair‐copula construction and vine graphical representation are used to describe the mutual relationship among PCs to overcome the lack of multivariate copula in high‐dimensional cases. Each PC model of smart electricity meter is built by using drift Brownian motion to describe degradation processes of each PC on the basis of degradation mechanism analysis, and parameters are estimated by using likelihood estimation method. The Pearson correlation coefficient, Kendall's τ and product information are used to analyze correlation among those PCs. Furthermore, on the basis of conditional probability theory, the vine graph is used to construct a multivariate copula which can be decomposed into pair copulas. Akaike information criterion principle is utilized to choose the forms of pair‐copula functions in the correlation model. Finally, the reliability joint distribution of all PCs of smart electricity meter is obtained with combining all PCs' marginal distributions and copula functions. Copyright © 2016 John Wiley & Sons, Ltd.

Bayesian optimal design of step stress accelerated degradation testing

This study presents a Bayesian methodology for designing step stress accelerated degradation testing (SSADT) and its application to batteries. First, the simulation-based Bayesian design framework for SSADT is presented. Then, by considering historical data, specific optimal objectives oriented Kullback-Leibler (KL) divergence is established. A numerical example is discussed to illustrate the design approach. It is assumed that the degradation model (or process) follows a drift Brownian motion; the acceleration model follows Arrhenius equation; and the corresponding parameters follow normal and Gamma prior distributions. Using the Markov Chain Monte Carlo (MCMC) method and WinBUGS software, the comparison shows that KL divergence is better than quadratic loss for optimal criteria. Further, the effect of simulation outliers on the optimization plan is analyzed and the preferred surface fitting algorithm is chosen. At the end of the paper, a NASA lithium-ion battery dataset is used as historical information and the KL divergence oriented Bayesian design is compared with maximum likelihood theory oriented locally optimal design. The results show that the proposed method can provide a much better testing plan for this engineering application.

Planning of Step-Stress Accelerated Degradation Test with Stress Optimization

For highly-reliable products, Accelerated Degradation Testing (ADT) data can provide useful reliability information. Moreover, Step-Stress Accelerated Degradation Testing (SSADT) usually requires a less sample size, shorter time and less cost than Constant-Stress Accelerated Degradation Testing (CSADT). However, in designing an efficient SSADT, the issue about how to choose accelerated stress level was seldom discussed. In this study, first we use drift Brownian motion to model a typical SSADT performance degradation process. Then, under the situation that total experiment cost is not given while the sample size, total test time and the interval of performance inspection are specified, our objective is to minimize the asymptotic variance of the estimation of the reliability of the pth quantile of product’s lifetime under use condition. Form the derivation of our objective, we find a way to improve the speed of optimal algorithm for SSADT. The optimal plan can give the stress levels and test time at each stress level. Finally a simulation example is used to illustrate the proposed method.

Conditional Bias of Point Estimates Following a Group Sequential Test

Repeated significance testing in a sequential experiment not only increases the overall type I error rate of the false positive conclusion but also causes biases in estimating the unknown parameter. In general, the test statistics in a sequential trial can be properly approximated by a Brownian motion with a drift parameter at interim looks. The unadjusted maximum likelihood estimator can be potentially very biased due to the possible early stopping rule at any interim. In this paper, we investigate the conditional and marginal biases with focus on the conditional one upon the stopping time in estimating the Brownian motion drift parameter. It is found that the conditional bias may be very serious for existing point estimation methods, even if the unconditional bias is satisfactory. New conditional estimators are thus proposed, which can significantly reduce the conditional bias from unconditional estimators. The results of Monte-Carlo studies show that the proposed estimators can provide a much smaller conditional bias and MSE than the naive MLE and a Whitebead's bias reduced estimator.

Asymptotic design of symmetric triangular tests for the drift of brownian motion

For testing the drift of Brownian motion with equal error probabilities. We provided an asymptotic solution, within the class of symmetric triangular stopping boundaries, to the Kiefer-Weiss problem of minimizing the maximum expected stopping time. The resulting asymptotic minimax triangular test (AMTT) is a slight improvement over Lordenś2-SPRT.

Minimum variance unbiased estimation of the drift of brownian motion with linear stopping boundaries

We consider a Brownian motion X(t) with drift θ and various linear stopping boundaries, including boundaries popular for sequential testing. We show that the stopping time T and the observed value X(T) are jointly complete for the drift. We then derive the UMVUE of the drift and of some functions of the drift, including the variance of the drift estimate. The estimates are also truncation-adaptable, with uniformly minimum variance among such estimates.

On the maximum likelihood estimate for the drift of brownian motion following a symmetric sequential probability ratio test

Bias and variance are evaluated explicitly for the maximum likelihood estimate (MLE) of the drift of a Brownian motion following a symmetric sequential probability ratio test (SPRT). The MLE is shown to be asymptotically efficient when the boundary of the SPRT tends to infinity.

Dynamic Sampling Plan in Shiryayev-Roberts Procedure for Detecting a Change in the Drift of Brownian Motion

In this paper, a dynamic sampling plan in the Shiryayev-Roberts procedure is considered. It is shown that a two-rate dynamic sampling plan is optimal in the sense that it minimizes the stationary average delay time (SADT). Analytical results as well as numerical comparisons show that it is significantly superior to the fixed sampling plan. The comparison also shows that it is as powerful as the dynamic sampling procedure of Assaf and Ritov. The generalizations to the fast initial response and to the CUSUM procedure are also briefly discussed.

Adaptive sampling for detecting a change point in the past

Consider the problem of estimating a change point in the drift of Brownian motion which is known to have occurred at some time during the time interval [0,1], Rather than observing the process, we use adaptive (dynamic) sampling which allows us to continuously select the time points at which increments of the motion may be observed. Our main results are that the steepest descent method will continuosly select the current Bayes estimator of the change point as the next time point to observe. The method results in a very convergence rate. In addition, a discrete formulation is given. In this case, a slightly different method based on the mean and the standard deviation is suggested, and is proved to result in an exponential rate of convergence.

Dynamic Sampling Procedures for Detecting a Change in the Drift of Brownian Motion: A Non-Bayesian Model

We consider dynamic procedures for sampling from a process (a Brownian motion) and stopping it after a change is detected. The basic idea is to conduct a sequence of similar SPRT's, each one of them done in negligible time, while not sampling at all between them. The procedures detect the change point much faster than the standard procedures with the same sampling rate and time to false alarm, but hold the sampling rate constant.

The effect of truncation on a sequential test for the drift of brownian motion

Lerche (1986) investigated a sequential testing problem concerned with deciding the sign of a normal mean. He demonstrated that a parabolic boundary which corresponds to

A Dynamic Sampling Approach for Detecting a Change in Distribution

The problem of detecting a change in drift of Brownian motion is considered in the Bayesian framework with the time of change having a (prior) exponential distribution. To the well known problem of finding an optimal stopping rule for "declaring a change," we add the option of continuously controlling the sampling rates--resulting in controlling the variance coefficient of the process. The combined problem of finding an optimal rate function (dynamic sampling) together with an optimal stopping rule is solved and explicit expressions for the quantities of interest are derived. The dynamic sampling procedure is shown to be significantly superior to constant rate sampling. The comparison is most favorable when the expected time until change tends to infinity, where the relative efficiency between the two procedures tends to infinity.

A Diffusion Process and Its Applications to Detecting a Change in the Drift of Brownian Motion

A Diffusion Process and Its Applications to Detecting a Change in the Drift of Brownian Motion

A diffusion process and its applications to detecting a change in the drift of Brownian motion

Abstract : The purpose of the present paper is to make a quantitative comparison of the Shiryayev-Roberts and Page procedures. We do this in the context of continuous time in order to use the machinery of diffusion processes to perform explicitly certain calculations, which seem impossible in discrete time. Although the continuous time results are not especially good approximations to the corresponding quantities in discrete time, they provide very useful comparative information on which to base selection of a stopping rule. This paper is organized as follows. The Shiryayev-Roberts process is defined and shown to be a novel diffusion process with some surprising properties. We also specify more precisely the basis for our comparison of the two procedures and give the results of some elementary calculations. These developments contain an asymptotic evaluation. We define a modification of our basic procedure and give an asymptotic evaluation of its average run length. Numerical comparisons and a discussion of their significance are contained next. Our conclusions are roughly these. In simple situations where the two procedures can be directly compared, neither seems dramatically superior to the other. However, the Shiryayev-Roberts procedure is more easily adapted to complex circumstances and consequently warrants additional study.