Wiener Process Research Articles

OSS Sustainability Assessment Based on the Deep Learning Considering Effort Wiener Process Data

This paper focuses on the sustainability based on the effort by using the fault big data of open source software (OSS). The fault detection phenomenon depends on the maintenance effort, because the number of software fault is influenced by the effort expenditure. Actually, the software reliability growth models with testing-effort have been proposed in the past. In this paper, we apply the deep learning approach to the OSS fault big data. Also, we propose the reliability assessment measure of sustainability. Then, we show several sustainability assessment measure based on the deep learning. Moreover, several numerical illustrations based on the proposed deep learning model are shown in this paper.

Reliability modeling based on multiple wiener degradation-shock competing failure process and dynamic failure threshold

Given multiple degradation failure and shock failure processes within the complex system during operation, a reliability model that combines the degradation-shock competing failure process and dynamic failure threshold is modeled. The Wiener process with random effects is considered the degradation process model, which includes random effects to account for the heterogeneity among system units. Then, the extreme shock model with a dynamic failure threshold is used to depict the random shock. The copula function is carried out to illustrate the correlation between multiple degradation processes, the reliability model is constructed further. To verify the application of this model, a numerical case and a micro-electro-mechanical system comprising two micro-mechanical resonators are employed. The parameter sensitivity of the proposed model is analyzed. The study results show that the reliability model, which combines the Wiener process with random effects and the dynamic failure threshold, more accurately reflects the actual operational state of the complex system.

The new wave structures to the perturbed NLSE via Wiener process with its wide-ranging applications

This article extracts stochastic soliton waves for the perturbed nonlinear Schödinger’s equation (PNLSE) forced by multiplicative noise through the Itô sense by utilizing two unified solver methods. The presented solutions involve three types: rational function, trigonometric function, and hyperbolic function solutions. These stochastic solutions are critical for studying numerous complicated phenomena in heat transfer, new physics, and many other fields of applied science. We demonstrate the effect of multiplicative noise on the solution of the stochastic PNLSE, which have never been studied before. The study and acquired solutions clarify that the unified solver technique is sturdy and efficient. Finally, several 2D and 3D graphs for selected solutions are shown.

The impact of the Wiener process on solutions of the potential Yu–Toda–Sasa–Fukuyama equation in a two-layer liquid

The impact of the Wiener process on solutions of the potential Yu–Toda–Sasa–Fukuyama equation in a two-layer liquid

The impact of the Wiener process on solutions of the potential Yu-Toda-Sasa-Fukuyama equation in a two-layer liquid

Изучается ($3+1$)-мерное стохастическое потенциальное уравнение Ю-Тоды-Сасы-Фукуямы, определяющееся в смысле Ито мультипликативным винеровским процессом. Для получения решений этого уравнения использованы метод вспомогательных обыкновенных дифференциальных уравнений Риккати-Бернулли и вариационный метод Хе. Решения выражаются через тригонометрические, гиперболические и рациональные функции. Рассматриваемое уравнение описывает нелинейные волны и солитоны в средах с дисперсией, физике плазмы и гидродинамике и может объяснить многие интересные физические явления. Графически показано, как винеровский процесс влияет на точные решения.

Simultaneous modeling of choice, confidence, and response time in visual perception.

How can choice, confidence, and response times be modeled simultaneously? Here, we propose the new dynamical weighted evidence and visibility (dynWEV) model, an extension of the drift-diffusion model of decision-making, to account for choices, reaction times, and confidence simultaneously. The decision process in a binary perceptual task is described as a Wiener process accumulating sensory evidence about the choice options bounded by two constant thresholds. To account for confidence judgments, we assume a period of postdecisional accumulation of sensory evidence and parallel accumulation of information about the reliability of the present stimulus. We examined model fits in two experiments, a motion discrimination task with random dot kinematograms and a postmasked orientation discrimination task. A comparison between the dynWEV model, two-stage dynamical signal detection theory, and several versions of race models of decision-making showed that only dynWEV produced acceptable fits of choices, confidence, and reaction time. This finding suggests that confidence judgments depend not only on choice evidence but also on a parallel estimate of stimulus discriminability and postdecisional accumulation of evidence. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

On the probabilistic representations of solutions of pantograph equations and triangle coefficients

The probabilistic representations of solutions of pantograph equations are presented. In some cases these can be functionals of standard Brownian motion, while in others they are functionals of an appropriately constructed Poisson point process. The study of the probabilistic representations lead to surprising phenomena of the so called triangle coefficients, which are generated by a recurrence formula and form some interesting structures.

The “logarithmic scale” Minkowski dimension of the most visited sites of two-dimensional Brownian motion

This paper studies the sets of visit times to points x on the plane, where a standard two-dimensional Brownian motion makes a substantial number of visits. For this purpose, we introduce the concept of the “logarithmic scale” Minkowski dimension as a tool for measuring the sets of visit times. We prove that, almost surely, there is a point x such that the “logarithmic scale” Minkowski dimension of the set of visit times to x is 1.

Failure time prediction for vehicle dynamics under performance degradation of dampers and track evolution

During long-term service, railway vehicles will experience damper performance degradation and track evolution, both have significant effects on the vehicle dynamics performance. This study aims to develop a mathematical model for assessing the failure time of the vehicle system dynamics when considering two random factors. As for the methodology, Wiener processes are used to propose a damper degradation model. Moreover, a continuous-time Markov chain is adopted to describe the state transaction of the track irregularities. In addition, the Monte Carlo method is employed to sample the dynamic responses of the vehicle using stochastic parameters. Finally, the Bayesian theory is utilized to establish the reliability model of the Weibull mixtures under the Dirichlet process prior. The results are obtained through a numerical example: the dampers’ individual differences directly affect effect on the rate of degradation of the vehicle dynamics performance. Furthermore, different indexes positively correlate with each other, mixture models are composed of a small number of components, and Weibull mixtures closely approximate the distribution of the failure time. Compared to other dampers, the anti-yaw dampers have a more significant impact on the vehicle system dynamics. Other results show that each type of damper can adopt a uniform replacement period whereas anti-yaw dampers should be more effectively monitored.

Systematic investigations on ion dynamics with noises in Paul trap

Ions confined in a Paul trap serve as crucial platforms in various research fields, including quantum computing and precision spectroscopy. However, the ion dynamics is inevitably influenced by different types of noise, which require accurate computations and general analytical analysis to facilitate diverse applications based on trapped ions with white or colored noise. In the present work, we investigate the motion of ions in a Paul trap via the Langevin equation using both analytical and numerical methods, systematically studying three different types of noise: the white noise, the colored noise via the Ornstein–Uhlenbeck process and the Wiener process. For the white noise of the case, we provide a recursion method to calculate ion motion for a wide range of parameters. Furthermore, we present an analytical solution to the more realistic stochastic process associated with the colored noise, verified by the Monte Carlo simulation. By comparing the results of the colored noise with those of the white noise, and additionally considering another limit of noise parameters corresponding to the Wiener process, we summarize the effects of different noise types on the ion dynamics.

Remaining useful life prediction for degradation processes based on the Wiener process considering parameter dependence

AbstractRemaining useful life prediction (RUL) is a critical procedure in the application of prognostics and health management for devices or systems. It is difficult to predict the RUL in a time‐varying external environment. Specifically, many mechanical systems typically experience various operating conditions, which have impacts on the degradation process and degradation rate. In particular, the linear degradation modeling of the Wiener process‐based RUL prediction method has attracted considerable attention recently. However, the dependency of degradation rate and operating conditions is generally ignored in the current degradation modeling, which leads to inaccurate issues in the RUL prediction. Therefore, to solve the above issues, a novel RUL prediction method based on the Wiener process considering parameter dependence is proposed in this paper. At first, a linear Wiener process degradation model considering parameter dependence is constructed to describe the dependency of the drift coefficient and operating conditions. Secondly, the probability density function of RUL is derived under the concept of first hit time. After that, the collaboration between the Bayesian update and expectation maximization algorithm is introduced to update and estimate the model parameters. Finally, the validity and applicability of the proposed method are verified by a numerical simulation and three case studies of bearings.

Optimum test condition setting and sample allocation for accelerated degradation tests under the last-exit-time failure mode

ABSTRACT Accelerated degradation tests (ADTs) have been widely adopted for products whose failure is defined as the first-passage-time (FPT) when the degradation path crosses a given threshold. However, for products whose failure is defined as the last-exit-time (LET), few studies related to ADT have been investigated. In this paper, we focus on designing an optimum test condition setting and sample allocation scheme for ADTs under the LET failure mode. In particular, the widely adopted Wiener process model and the generalized exponential acceleration model are used to describe the fluctuating degradation behaviors of products. By minimizing the asymptotic variance of mean-time-to-failure under the normal use condition, we design a two-level optimum test plan and a three-level compromise plan under the constraints of total sample size and stress region. The optimum plans are given in analytical forms, and we prove that the optimum plans under the FPT failure mode and LET failure mode are equivalent under some situations. Finally, a renewed selection method for the upper limit of the stress level in ADTs is proposed based on the prior ADT information from the perspective of accelerated mechanism equivalence. The proposed method is illustrated using a real-world application to rubber rings.

A sliding sequence importance resample filtering method for rolling bearings remaining useful life prediction based on two Wiener-process models

The remaining useful life (RUL) prediction of rolling bearings is an important part of prognostic and health management of mechanical systems. The model based on Wiener process can describe the time variability in the degradation process of bearings. However, in practical engineering, the degradation trends of bearings are often inconsistent, and it is difficult to fit the actual degradation trends of bearings with a single Wiener process model-based filtering method. Therefore, to improve the generalization ability, this paper uses linear model and exponential model based on Wiener process to predict bearing RUL. A sliding sequence importance resample filtering algorithm is proposed to track the degradation state of the bearing and reduce the prediction error by combining the two degradation models. Last, the superiority and effectiveness of the proposed method are illustrated by comparing with other commonly used RUL prediction methods on the basis of PRONOSTIA dataset.

Numerical Analysis of Split-Step Backward Euler Method with Truncated Wiener Process for a Stochastic Susceptible-Infected-Susceptible Model.

This article deals with the numerical positivity, boundedness, convergence, and dynamical behaviors for stochastic susceptible-infected-susceptible (SIS) model. To guarantee the biological significance of the split-step backward Euler method applied to the stochastic SIS model, the numerical positivity and boundedness are investigated by the truncated Wiener process. Motivated by the almost sure boundedness of exact and numerical solutions, the convergence is discussed by the fundamental convergence theorem with a local Lipschitz condition. Moreover, the numerical extinction and persistence are initially obtained by an exponential presentation of the stochastic stability function and strong law of the large number for martingales, which reproduces the existing theoretical results. Finally, numerical examples are given to validate our numerical results for the stochastic SIS model.

Correctness and regularization of stochastic problems

Abstract The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces: (0.1) d ⁢ u ⁢ ( t ) = A ⁢ u ⁢ ( t ) ⁢ d ⁢ t + B ⁢ d ⁢ W ⁢ ( t ) , t > 0 , u ⁢ ( 0 ) = ξ . du(t)=Au(t)dt+BdW(t),\quad t>0,\qquad u(0)=\xi. The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process { W ⁢ ( t ) : t ≥ 0 } {\{W(t):t\geq 0\}} . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q-Wiener processes.

Practical design of partially accelerated degradation test with two stress variables under step-stress loading scheme

The number of samples and test duration during the development stage are usually insufficient, since it is time consuming and costly to collect reliability information for highly reliable products. Therefore, a more efficient test design is required to collect as much information as possible. In addition, considering two or more stress variables simultaneously can catalyze the degradation mechanism. Thus, a planning strategy for partially step-stress accelerated degradation tests with two stress variables is developed under the assumption that the degradation characteristic follows a Wiener process. Based on the compromise concept, the mid stress levels of each stress variable is determined, such that asymptotic variance of the maximum likelihood estimator of the qth quantile of the distribution under use conditions is minimized. A scenario analysis for a numerical example considering the experimental situation is provided to demonstrate the practicality of the proposed strategy.

A novel hybrid simulated annealing algorithm for colored bottleneck traveling salesman problem

In the fields, such as intelligent transport and multiple tasks cooperation, many real-world problems can be modeled by colored bottleneck traveling salesman problem (CBTSP). Since CBTSP has been proved belong to the NP-hard problems, the population-based algorithms, such as genetic algorithm (GA), hill climbing GA (HCGA) and simulated annealing GA (SAGA), have been used for solving it. However, in term of convergence speed and solution quality, their performances are limited. Moreover, some intelligent algorithms still have the problems such as “black box problem” or lacking theoretical support. Aiming at these problems, this paper proposes a new hybrid simulated annealing algorithm with generating neighborhood solution and crossover operator (GCSA) for CBTSP. In the iterative process of GCSA, the gene coding (dual-chromosome coding) is used to code feasible solutions, and the improved generating neighborhood solution and crossover operator by ITÖ lemma (Wiener process) are used to explore and exploit new unknown region. During the process, generating neighborhood solution and crossover operator are both controlled by the activity intensity based on Wiener process. The experiments show that GCSA can demonstrate an improvement over the state-of-the-art algorithms for this problem.

Storage reliability prediction of electromechanical components based on virtual manufacturing and testing

Electromechanical components (EMCs) such as relays and contactors have been used extensively in industrial and military areas. The storage reliability of these EMCs has a direct impact on the reliability of the system that contains them. However, during the design phase, it is difficult to predict the storage reliability of EMCs because of few failure rate data of parts, as well as limited testing time and budgets. To address these problems, a virtual Manufacturing and testing method is proposed in this paper, so as to simulate the storage degradation process of batch EMCs. By considering the influence of the quality screening process in the manufacturing process, as well as the unit-to-unit variability of EMCs on the storage degradation paths and the overall life distribution of batch products, the storage failure distribution function is obtained, based on Wiener process. At the same time, the distribution of the diffusion coefficient in the degradation model and the failure distribution model is quantified by introducing testing data of related products as priori information, so as to reflect the uncertainty of the storage degradation process of EMCs. A case study of an electromagnetic relay is carried out to illustrate the effectiveness of the proposed approach.

Pricing formula for a Barrier call option based on stochastic delay differential equation

We derive new explicit pricing formulae for a type of Barrier call option, down and in call option when underlying asset price processes are represented by a stochastic delay differential equation (hereafter “SDDE”). We note the conditional normality of a stochastic integral with respect to a Wiener process to find the joint distribution of the stochastic integral and their minimum. On the basis of this result, we obtain pricing formulae for the Barrier call option which extends ones in the classical Black-Scholes models without delay. Finally, through Monte-Carlo simulations, we demonstrate that our theoretical prices for a Barrier option are correct.

Present value of firm in case of correlated defaults

In this article, the valuation of firm’s present value in case of correlated defaults is studied. We showed that the valuation of portfolio credit risk can be interpreted as a valuation of the multiple contingent option. In this article, some results for valuation of multiple contingent options are generalized in case of stochastic barriers and the financial interpretation is given. Also, the modelling of other financial risks, notably, the market risk (the change of debt value) and the operational risk (the adequacy of selected credibility criteria) are included in these theorems. Using the principle of conservatism in credit risk assessment, the debtor credibility, the debt value and the firm value are assumed to follow geometric Wiener process.