Wiener Process Research Articles

A rolling bearing state evaluation method based on deep learning combined with Wiener process

A rolling bearing state evaluation method based on deep learning combined with Wiener process

Linear combinations of i.i.d. strictly stable variables with random coefficients and their application to anomalous diffusion processes

We show that linear combinations of independent and identically distributed strictly stable variables with positive random coefficients is equal in distribution to a function of these random coefficients times a random variable from the same stable distribution. Furthermore, this result is used to show that a random linear combination of independent standard Wiener processes has the distribution of a function of these random coefficients times one standard Wiener process. In opposition to the central limit theorem, this result does not require a large number of terms but it holds with two or more terms. This has implications to simplify stochastic differential equations with a finite number of noises with random coefficients that can be used in modeling anomalous diffusion.

The Brownian motion, Wiener process, and Pu migration as examples of random walk

The theory of random walks holds significance across various domains. To this end, this paper discusses the application of random walk theory in the modeling and analysis of complex systems, including the Brownian motion, Wiener process, and Pu migration. The first part mainly explains the concepts of Brownian motion and random walks, as well as their relationship. The core characteristic is the independence and normal distribution of increments, which makes it a special model of random walks. The next part is the Wiener process, which is a continuous time stochastic process. It can describe random phenomena and solve complex problems in finance, encompassing areas such as stock price prediction, option pricing, and risk management. The last part is about Pu migration, with the clear introduction of Pu migration in biology. The authors discuss why the random walk theory can be regarded as the better approach to analysis this problem. Thus, this paper highlights the importance of the random walk in solving many scientific issues.

GRU-AE-wiener: A generative adversarial network assisted hybrid gated recurrent unit with Wiener model for bearing remaining useful life estimation

Bearings play a pivotal role in various mechanical systems, and their health directly impacts the reliability and safety of these systems. Consequently, extensive research has been dedicated to the estimation of Bearing Remaining Useful Life (RUL) through the lens of information fusion theory. The absence of comprehensive life-cycle degradation data for bearings, a common challenge within the information fusion domain, can hinder the accuracy and reliability of RUL prediction models. A novel hybrid data and model approach named GRU-AE-Wiener has been developed to address this limitation. This approach combines the power of Gated Recurrent Unit (GRU) and Wiener process models within the information fusion framework. Firstly, a Loop Generative Adversarial Network (Loop-GAN) is introduced to synthesize pseudo data to enhance the quality of synthetic data. Next, a bidirectional GRU model is structurally integrated with the Wiener process. In this design, the GRU model is configured in an Auto-Encoder-like structure, with the Wiener process serving as the hidden layer within this Auto-Encoder. Importantly, both the GRU and Wiener processes are jointly optimized with the assistance of Loop-GAN, emphasizing the collaborative nature of information fusion in this approach. The effectiveness of the proposed GRU-AE-Wiener is validated using the PHM 2012 dataset and XJTU-SY dataset. Experimental results underscore its superior RUL predictive performance compared to other deep learning models, highlighting the practical application of information fusion principles in bearing health assessment.

Application of Ant Colony Programming Approach for Solving Systems of Stochastic Differential Equations

Stochastic differential equations (SDE) have wide applications in natural phenomena, engineering, finance, and biological models. Obtaining analytic solutions for an SDE is often complex, and the complexity increases for an SDE system. The paper introduces ant colony programming (ACP) as a novel approach for solving SDE system. Ant colony programming was developed in two directions, the first is to add the Wiener process as a variable to the terminals and functions, and the second is to construct the appropriate fitness function . ACP constructs mathematical expressions and evaluates them using the fitness function . The ACP proposed effectiveness has been demonstrated by applying to 2,3 and 4-dimensional SDE systems. The most important finding of this work is that ACP generates symbolic stochastic processes that represent solutions for SDE system. Methods for solving SDE systems are important tools for study phenomena that involve noise or randomness.

Probabilistic remaining useful life prediction without lifetime labels: A Bayesian deep learning and stochastic process fusion method

Trustworthy remaining useful life (RUL) predictions are critical for the long-term safe and reliable operation of degradation systems. The existing deep learning-based methods for RUL prediction are attracting increasing attention but typically face three main challenges. One is the absence of complete run-to-failure data, implying a lack of lifetime labels. Second, it is difficult to directly measure the health indicators (HIs) for field degradation systems. Third, the prediction models output point estimates without uncertainty. To this end, this paper proposes a Bayesian deep learning and stochastic process fusion method for probabilistic RUL prediction without lifetime labels. First, a model-free Bayesian neural network (BNN) is constructed to integrate the quantification of epistemic and aleatoric uncertainties in deep learning. Based on the constructed BNN, it is possible to predict the probability features of HIs. Then, degeneracy modeling is conducted using a nonlinear Wiener process to derive the probability density function of the RUL. Furthermore, model evolution can be achieved through parameter updating during online operations. Finally, the effectiveness and superiority of the proposed prediction method are verified on CALCE battery degradation data.

A novel interactive prognosis framework with nonlinear Wiener process and multi-sensor fusion for remaining useful life prediction

Accurate remaining useful life (RUL) prediction plays a vital role in increasing the system operation safety and reducing maintenance costs. In industrial applications, there is usually a large amount of multi-sensor data generated. Therefore, how to construct an appropriate health index (HI) based on multi-sensor signals is very important for the RUL prediction. However, existing works treat sensor selection, HI construction, and degradation modeling independently as unrelated parts, which may result in the combination of sensors selected not constituting an optimal HI or the constructed HI not matching the degradation model. In addition, most existing works treat prior units as a whole to obtain a unique set of sensor combinations and fusion coefficients, which cannot reflect unit-to-unit heterogeneity, thus affecting the accuracy of RUL prediction. Therefore, a novel interactive feedback framework is established to construct HI, where the sensor selection, fusion coefficient calculation, and nonlinear Wiener process degradation modeling are incorporated into the feedback. Furthermore, an adaptive weight selection method based on particle swarm optimization and leave-one-out cross-validation (PSO-LV) is proposed to adjust the fusion coefficients in real-time. Then, the RUL is estimated by updating model parameters online, detecting degradation trends, and deriving the probability density function (PDF) of the RUL. Finally, two examples of engine datasets are provided to verify the effectiveness of the proposed method.

Optimal designs of accelerated degradation tests with random shock failures and measurement errors

AbstractAccelerated degradation tests (ADTs) are widely used for assessing the reliability of long‐life products. During an ADT, accelerated stresses not only expedite the degradation of test products but also increase the likelihood of encountering traumatic shocks. Moreover, it is important to acknowledge that measurement errors can be inevitable during the observation process of an ADT. Unfortunately, these errors are often overlooked in the optimal design of the ADT, especially when multiple competing failure modes are present. In this article, we propose a new approach to design ADTs when measurement errors exist and test products suffer from degradation failures and random shock failures. We utilize the Wiener process to model the degradation path, incorporating normally distributed measurement errors, and an exponential distribution to fit the time between random shock failures. Given the number of test products and the termination time, we optimize the ADT plans under three common design criteria. The equivalence theorem is used to verify the optimality of the optimal ADT plans. A real‐life example and sensitivity analysis are provided to illustrate our proposed method. The results demonstrate that when competing failure modes are present, the optimal ADT plans, which account for measurement errors, differ significantly from those that do not consider such errors.

A random-effect Wiener process degradation model with transmuted normal distribution and ABC-Gibbs algorithm for parameter estimation

Degradation modeling is a widely used technique for evaluating the reliability of high-quality products. However, unit-to-unit variability, stemming from material fluctuations during manufacturing and environmental factors, can significantly impact the accuracy of this assessment. To tackle this issue, we propose a novel time-transformed Wiener process with a transmuted normal distribution to represent unit-to-unit variability. This distribution extends the normal distribution, offering greater flexibility by capturing a wide range of non-normal, asymmetric behaviors in unit-to-unit variability. We derive closed-form expressions for the probability density function and reliability function for the model in two scenarios: (i) assuming the degradation process observations remain unaffected by measurement errors, and (ii) assuming the degradation process observations influenced by measurement errors. Due to the complexity of the likelihood functions, we employ the Gibbs version of approximate Bayesian computation (ABC-Gibbs) for parameter estimation. The effectiveness and application of the proposed method were demonstrated through numerical examples and practical application with light-emitting diode degradation data. Moreover, we use approximate Bayesian computation model choice (ABC-MC) for model comparison studies. The results indicated that our model offers improved approximations to observed degradation results compared to existing models, showcasing its superiority in generating data under the smallest tolerance threshold.

Sampling for Remote Estimation of the Wiener Process over an Unreliable Channel

In this paper, we study a sampling problem where a source takes samples from a Wiener process and transmits them through a wireless channel to a remote estimator. Due to channel fading, interference, and potential collisions, the packet transmissions are unreliable and could take random time durations. Our objective is to devise an optimal causal sampling policy that minimizes the long-term average mean square estimation error. This optimal sampling problem is a recursive optimal stopping problem, which is generally quite difficult to solve. However, we prove that the optimal sampling strategy is, in fact, a simple threshold policy where a new sample is taken whenever the instantaneous estimation error exceeds a threshold. This threshold remains a constant value that does not vary over time. By exploring the structure properties of the recursive optimal stopping problem, a low-complexity iterative algorithm is developed to compute the optimal threshold. This work generalizes previous research by incorporating both transmission errors and random transmission times into remote estimation. Numerical simulations are provided to compare our optimal policy with the zero-wait and age-optimal policies.

A reliability analysis method for electromagnet performance degradation based on FMEA and fuzzy inference system

AbstractElectromagnets are often used in indirect control for industrial applications. The ability of an electromagnet to control objects should decrease with performance degradation. And electromagnets product poses a danger to people and objects in the working environment. So, it is very difficult to analyze the reliability of electromagnetic performance degradation because of the complicated working condition. Failure Mode and Effect Analysis (FMEA) is the most commonly used tool for product reliability analysis. The new version of FMEA uses integer as evaluation value, which cannot represent the hesitation psychology of the evaluator. The Action Priority (AP) table of the FMEA describes the relationship between the evaluation of influencing factors and the risk level of the failure mode, which provides rules for determining the risk level of the failure mode. However, the AP table may result in multiple failure modes having the same ranking, which does not align with the intention of FMEA to prevent failures. Therefore, this paper proposes a reliability analysis method for electromagnetic performance degradation based on FMEA and FIS. Firstly, the Double Hierarchy Hesitant Fuzzy Linguistic Term Set (DHHFLTS) is used as the evaluation language to describe the hesitation psychology of evaluators. Secondly, the AP table of FMEA is used as FIS fuzzy inference rule. In this way, the idea of FMEA to determine the risk level of failure mode is retained and the problem of FIS fuzzy rule making is overcome. Then, FIS defuzzification AP table inference results to determine the risk ranking of failure modes. This avoids situations where the order of failure modes is equal. Finally, a performance degradation model of the electromagnet is constructed based on the Wiener process, and the calculation results of the new method are verified.

Optimal maintenance policy design considering learning effects under successive performance-based contracts

Successive short-term performance-based contracts (PBCs) would not only drive the original equipment manufacturer (OEM) to improve maintenance service quality, but also decrease the financial burdens of the equipment owner. This paper presents the optimal maintenance policy incorporating both preventive maintenance (PM) and upgrade actions for continuously monitored degrading equipment under successive short-term PBCs. The improvement factor model is adopted to characterize the impact of the PM and upgrade actions. We also consider that the OEM’s learning effects will affect the cost and duration of the PM, as well as the upgrade cost. The learning effect increases as the period of the contract accumulates. To characterize the varied equipment production and degradation rates across different PBC periods, a Wiener process with covariates is developed. Meanwhile, we assume that the reward and penalty parameters of the performance-based revenue function varied each contract period, thus motivating the OEM to enhance the equipment availability. Based on that, the optimal maintenance plan for each contract period is determined to maximize the OEM’s expected net profit. Numerical examples show the superiority and examine the sensitivity of the proposed successive short-term PBCs. Managerial insights are also given to guide the implementation of the proposed maintenance policy.

Novel stochastic multi breather type, a-periodic, hybrid periodic and other type of waves of the Shrödinger–Hirota model with Wiener process

Novel stochastic multi breather type, a-periodic, hybrid periodic and other type of waves of the Shrödinger–Hirota model with Wiener process

Degradation Modeling and RUL Prediction in Dynamic Environments Using a Wiener Process With an Autoregressive Rate

Degradation Modeling and RUL Prediction in Dynamic Environments Using a Wiener Process With an Autoregressive Rate

Decoherence time control by interconnection for finite-level quantum memory systems

This paper is concerned with open quantum systems whose dynamic variables have an algebraic structure, similar to that of the Pauli matrices for finite-level systems. The Hamiltonian and the operators of coupling of the system to the external bosonic fields depend linearly on the system variables. The fields are represented by quantum Wiener processes which drive the system dynamics according to a quasilinear Hudson-Parthasarathy quantum stochastic differential equation whose drift vector and dispersion matrix are affine and linear functions of the system variables. This setting includes the zero-Hamiltonian isolated system dynamics as a particular case, where the system variables are constant in time, which makes them potentially applicable as a quantum memory. In a more realistic case of nonvanishing system-field coupling, we define a memory decoherence time when a mean-square deviation of the system variables from their initial values becomes relatively significant as specified by a weighting matrix and a fidelity parameter. We consider the decoherence time maximization over the energy parameters of the system and obtain a condition under which the zero Hamiltonian provides a suboptimal solution. This optimization problem is also discussed for a direct energy coupling interconnection of such systems.

Liouville models of particle-laden flow

Langevin (stochastic differential) equations are routinely used to describe particle-laden flows. They predict Gaussian probability density functions (PDFs) of a particle's trajectory and velocity, even though experimentally observed dynamics might be highly non-Gaussian. Our Liouville approach overcomes this dichotomy by replacing the Wiener process in the Langevin models with a (small) set of random variables, whose distributions are tuned to match the observed statistics. This strategy gives rise to an exact (deterministic, first-order, hyperbolic) Liouville equation that describes the evolution of a joint PDF in the augmented phase-space spanned by the random variables and the particle position and velocity. Analytical PDF solutions for canonical models of particle-laden flows serve to establish a relationship between the Langevin and Liouville approaches. Finally, our framework is used to derive a new analytical PDF model for fluidized homogeneous heating systems.

Sampling of the Wiener Process for Remote Estimation Over a Channel With Unknown Delay Statistics

Sampling of the Wiener Process for Remote Estimation Over a Channel With Unknown Delay Statistics

Averaging principle for Hifer–Katugampola fractional stochastic differential equations

In this paper, we mainly study the averaging principle for a class of Hifer–Katugampola fractional stochastic differential equations driven by standard Brownian motion. Firstly, we establish the existence and uniqueness of mild solution for the considered system using Banach contraction principle. Then, under suitable assumptions, we demonstrate that the solution to the original differential equations converges to that of the averaged differential equations in the sense of mean square and probability as the time scale goes to zero. Finally, an illustrative example is provided to verify our theoretical results.

A numerical investigation on the vibration of a two-deck Euler–Bernoulli beam flooded by a potential flow

Abstract This work concerns the structural vibration of a bladeless wind turbine, modelled by a two-deck Euler–Bernoulli beam, due to a surrounding potential flow. The deflection is governed by the Euler–Bernoulli equation which is studied first by a linear theory and then computed numerically by a finite difference method in space with a collocation method over the arc length, and an implicit Euler method in time. The fluid motion in the presence of gravity is governed by the full Euler equations and solved by the time-dependent conformal mapping technique together with a pseudo-spectral method. Numerical experiments of excitation by a moving disturbance on the fluid surface with/without a stochastic noise are carried out. The random process involved in generating the noise on the water surface is driven by a Wiener Process. A Monte Carlo method is used for stochastic computations. The generated surface waves impinge on the beam causing structural vibration which is presented and discussed in detail. By elementary statistical analysis, the structural response subject to the stochastic hydrodynamic disturbance caused by white noise is found to be Gaussian.

Constant speed random particles spontaneously confined on the surface of an expanding sphere

The particles that we describe here can only move at the speed of light c in three-dimensional space. The velocity, which randomly but continuously changes direction, can be represented as a point on the surface of a sphere of constant radius c, and its trajectories may only connect points of this variety. The Wiener process that we use to describe the velocity dynamics on the surface of the sphere is anisotropic since the infinitesimal variation of the velocity is not only always orthogonal to the velocity itself (which guarantees a constant speed), but also to the position. This choice for the infinitesimal variation of the velocity is the one that best slows down the diffusion of particles in space by random motion at the speed of light. As a result of these dynamics, the position of the particles spontaneously remain confined on the surface of an expanding sphere whose radius increases, for large times, as the square root of time.