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Tool condition monitoring can be employed to ensure safe and full utilization of the cutting tool. Hence, remaining useful life (RUL) prediction of a cutting tool is an important issue for an effective high-speed milling process-monitoring system. However, it is difficult to establish a mechanism model for the life decreasing process owing to the different wear rates in various stages of cutting tool. This study proposes a three-stage Wiener-process-based degradation model for the cutting tool wear estimation and remaining useful life prediction. Tool wear stages classification and RUL prediction are jointly addressed in this work in order to take full advantage of Wiener process, as this three-stage Wiener process definitely constitutes to describe the degradation processes at different wear stages, based on which the overall useful life can be accurately obtained. The numerical results obtained using extensive experiment indicate that the proposed model can effectively predict the cutting tool’s remaining useful life. Empirical comparisons show that the proposed model performs better than existing models in predicting the cutting tool RUL.

Lithium-ion batteries (LiBs) are the most important part of electric vehicle (EV) systems. Because there are two different degradation rates during LiB degradation, there are many two-phase models for LiBs. However, most of these methods do not consider the randomness of the changing point in the two-phase model and cannot update the change time in real time. Therefore, this paper proposes a method based on the combination of the two-phase Wiener model and an extreme learning machine (ELM). The two-phase Wiener model is used to derive the mathematical expression of the remaining useful life (RUL), and the ELM is implemented to adaptively detect the changing point. Based on the Poisson distribution, the distribution of the changing time is derived as a gamma distribution. To evaluate the theoretical results and practicality of the proposed method, we perform both numerical and practical simulations. The results of the simulations show that due to the precise and adaptive detection of changing points, the proposed method produces a more accurate RUL prediction than existing methods. The error of our method for detecting the changing point is about 4% and the mean prediction error of RUL in the second phase is improved from 4.39 cycles to 1.61 cycles.

As crucial rotary components of high-speed trains, wheel treads in realistic operation environment usually suffer severe cyclic shocks, which damage the health status and ultimately cause safety risks. Timely and precise health prognosis based on vibration signals is an effective technology to mitigate such risks. In this work, a new parameter-related Wiener process model is proposed to capture multiple uncertainties existed in on-site prognosis of wheel treads. The proposed model establishes a quantitative relationship between degradation rate and variations, and integrates uncertainties via heterogeneity analysis of both criterions. A maximum-likelihood-based method is presented to initialize the unknown model parameters, followed by a recursive update algorithm with fully utilization of historical lifetime information. An investigation of real-world wheel tread signals demonstrates the superiority of the proposed model in accuracy improvement.

In the spirit of [T. Bjork et al., Finance Stoch., 1 (1997), pp. 141-174], we investigate term structure models driven by Wiener processes and Poisson measures with forward curve dependent volatilities. This includes a full existence and uniqueness proof for the corresponding Heath-Jarrow-Mortontype term structure equation. Furthermore, we characterize positivity preserving models by means of the characteristic coefficients. A key role in our investigation is played by the method of the moving frame, which allows us to transform term structure equations to time-dependent SDEs. [PUBLICATION ABSTRACT]

The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy In this paper, we employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives. We also determine the value of a natural exponent describing in Brownian last passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness tending to zero. To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property. Several results in this article play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.

This article systematically investigates the inverse Gaussian (IG) process as an effective degradation model. The IG process is shown to be a limiting compound Poisson process, which gives it a meaningful physical interpretation for modeling degradation of products deteriorating in random environments. Treated as the first passage process of a Wiener process, the IG process is flexible in incorporating random effects and explanatory variables that account for heterogeneities commonly observed in degradation problems. This flexibility makes the class of IG process models much more attractive compared with the Gamma process, which has been thoroughly investigated in the literature of degradation modeling. The article also discusses statistical inference for three random effects models and model selection. It concludes with a real world example to demonstrate the applicability of the IG process in degradation analysis. Supplementary materials for this article are available online.

Uncertainty is a significant challenge in tsunami hazard analysis. Tsunami heights are affected by complex factors and change constantly during propagation. The heights of tsunami have random characteristics. This study proposes that the water depths (related to seabed topography) are the most important factors that affect tsunami height. But across the globe, a considerable area of seabed topography has not been measured. So it is necessary to use the method of uncertainty to consider the water depth. The Wiener process is utilized to quantify the random changes of the water depth, which can better describe the situation that water depths change in a non-monotonic way. Considering the uncertainty of water depth, a Weiner process-based probabilistic model was established for predicting the maximum tsunami height, which is different from the maximum tsunami height deterministic or stochastic model previously studied with higher prediction efficiency and good prediction accuracy. The probability distribution of maximum tsunami heights was calculated using the stochastic model. The mean value of the maximum tsunami heights was very similar to the average value of 165 actual observations of maximum tsunami heights collected from 1997 to 2017.

With the popularity of various social media, the propagation of rumors is becoming a social threat. Here, the proposed mathematical model signifies the dynamics of rumor propagation on social media with the influence of counter-rumor spreaders in regulating the transmission process as well as controlling its harmful effect. The total number of users is divided into four categories: (i) newcomer, (ii) spreaders, (iii) counter-rumor spreaders, iv ) stiflers. The spreading threshold ( R 0 ) of rumor transmission regulates the condition of the prevalence of rumor. ( R 0 ) < 1 assures the stability of rumor-free state, while ( R 0 ) > 1 assures that one prevailing state exists uniquely with stable nature. Condition for global stability of prevailing state for deterministic system is derived. Subsequently, the corresponding stochastic model demonstrates the effect of random external factors (Wiener process) on rumor propagation dynamics. The global existence and uniqueness of the solution are established to study the asymptotic behavior of that solution around the steady-states. We have also compared the persistence criterion of rumor propagation for the modified system with the deterministic system and derived the condition for the extinction of rumor. Furthermore, scatter plots indicate the significant impact of parameters and numerical simulations are presented to validate the analytical studies. Numerical results assure that environmental noise plays a significant role in suppressing rumor propagation.

PurposeThis paper models the deterioration process of a multi-component system. Each deterioration process is modelled by the Wiener process. The purposes of this paper are to address these issues and consider the cost process based on the multi-component system.Design/methodology/approachCondition-based Maintenance is a method for reducing the probability of system failures as well as the operating cost. Nowadays, a system is composed of multiple components. If the deteriorating process of each component can be monitored and then modelled by a stochastic process, the deteriorating process of the system is a stochastic process. The cost of repairing failures of the components in the system forms a stochastic process as well and is known as a cost process.FindingsWhen a linear combination of the processes, which can be the deterioration processes and the cost processes, exceeds a pre-specified threshold, a replacement policy will be carried out to preventively maintain the system.Originality/valueUnder this setting, this paper investigates maintenance policies based on the deterioration process and the cost process. Numerical examples are given to illustrate the optimisation process.

Remaining useful life (RUL) prediction is central to the prognostics and health management (PHM) of lithium-ion batteries. This paper proposes a novel RUL prediction method for lithium-ion batteries based on the Wiener process with measurement error (WPME). First, we use the truncated normal distribution (TND) based modeling approach for the estimated degradation state and obtain an exact and closed-form RUL distribution by simultaneously considering the measurement uncertainty and the distribution of the estimated drift parameter. Then, the traditional maximum likelihood estimation (MLE) method for population based parameters estimation is remedied to improve the estimation efficiency. Additionally, we analyze the relationship between the classic MLE method and the combination of the Bayesian updating algorithm and the expectation maximization algorithm for the real time RUL prediction. Interestingly, it is found that the result of the combination algorithm is equal to the classic MLE method. Inspired by this observation, a heuristic algorithm for the real time parameters updating is presented. Finally, numerical examples and a case study of lithium-ion batteries are provided to substantiate the superiority of the proposed RUL prediction method.