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We study a triple of stochastic processes: a Wiener process W t , t ≥ 0 , its running maxima process M t = sup { W s : s ∈ [ 0 , t ] } , and its running minima process m t = inf { W s : s ∈ [ 0 , t ] } . We derive the analytical formula for the corresponding copula and show that it is supported on the hemicube, a convex hexahedron with seven vertices. As an application, we draw out an analytical formula for pricing of a double barrier option. [ABSTRACT FROM AUTHOR]

Several variations of stochastic processes have been studied in the literature to obtain reliability estimations of products and systems from degradation data. As the degradation trajectories may have different degradation rates, it is necessary to consider alternatives to characterize their individual behavior. Some stochastic processes have a constant drift parameter, which defines the mean rate of the degradation process. However, for some cases, the mean rate must not be considered as constant, which means that the rate varies in the different stages of the degradation process. This poses an opportunity to study alternative strategies that allow to model this variation in the drift. For this, we consider the Hjorth rate, which is a failure rate that can define different shapes depending on the values of its parameters. In this paper, the integration of this hazard rate with the Wiener process is studied to individually identify the degradation rate of multiple degradation trajectories. Random effects are considered in the model to estimate a parameter of the Hjorth rate for every degradation trajectory, which allows us to identify the type of rate. The reliability functions of the proposed model is obtained through numerical integration as the function results in a complex form. The proposed model is illustrated in two case studies based on a crack propagation and infrared LED datasets. It is found that the proposed approach has better performance for the reliability estimation of products based on information criteria. [ABSTRACT FROM AUTHOR]

The assessment of rolling bearing reliability is vital for ensuring mechanical operational safety and minimizing maintenance costs. Due to the difficulty in obtaining data on the performance degradation and failure time of rolling bearings, traditional methods for reliability assessment are challenged. This paper introduces a novel hybrid method for the reliability assessment of rolling bearings, combining the convolutional neural network (CNN)‐convolutional block attention module (CBAM)‐ bidirectional long short‐term memory (BiLSTM) network with the Wiener process. The approach comprises three distinct stages: Initially, it involves acquiring two‐dimensional time‐frequency representations of bearings at various operational phases using Continuous Wavelet Transform. Subsequently, the CNN‐CBAM‐BiLSTM network is employed to establish health index (HI) for the bearings and to facilitate the extraction of deep features, serving as input for the Wiener process. The final stage applies the Wiener process to evaluate the bearings’ reliability, characterizing the HI and quantifying uncertainties. The experiment is performed on bearing degradation data and the results indicate the effectiveness and superiority of the proposed hybrid method. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to demonstrate the uniform large deviation principle (UDLP) for the solutions of two-dimensional stochastic Navier–Stokes equations (SNSE) in the vorticity form when perturbed by two distinct types of noises. We first consider an infinite-dimensional additive noise that is white in time and colored in space and then consider a finite-dimensional Wiener process with linear growth coefficient. In order to obtain the ULDP for 2D SNSE in the vorticity form, where the noise is white in time and colored in space, we utilize the existence and uniqueness result from B. Ferrario et. al., Stochastic Process. Appl.,129 (2019), 1568–1604, and the uniform contraction principle. For the finite-dimensional multiplicative Wiener noise, we first prove the existence of a unique local mild solution to the vorticity equation using a truncation and fixed point arguments. We then establish the global existence of the truncated system by deriving a uniform energy estimate for the local mild solution. By applying stopping time arguments and a version of Skorokhod’s representation theorem, we conclude the global existence and uniqueness of a solution to our model. We employ the weak convergence approach to establish the ULDP for the law of the solutions in two distinct topologies. We prove ULDP in the C ([ 0 , T ] ; L p (T 2)) topology, for p > 2 , taking into account the uniformity of the initial conditions contained in bounded subsets of L p (T 2) . Finally, in C ([ 0 , T ] × T 2) topology, the uniformity of initial conditions lying in bounded subsets of C (T 2) is considered. [ABSTRACT FROM AUTHOR]

Proton exchange membrane fuel cell (PEMFC) is considered as one of the most promising green energy devices. Although fruitful results can be available for the remaining useful life (RUL) prediction of PEMFC, stochastic uncertainties have never been considered. To tackle this problem, a hybrid method is proposed in this paper. Wiener process with temporal uncertainty and individual uncertainty is adopted to model the degradation of the state of health (SOH), which is then estimated from monitoring voltage with measurement noise using the unscented Kalman filter (UKF), where unknown filtering and model parameters are jointly identified by expectation-maximization (EM) algorithm and Rauch-Tung-Striebel (RTS) smoother. Finally, gated recurrent unit (GRU) network is employed to realize the RUL prediction with the prediction uncertainty quantified by the Bayesian variational inference technology. The proposed method is verified on the experimental data. Results indicate that smaller mean absolute percentage error (MAPE), root mean square error (RMSE), and mean absolute error (MAE) values can be obtained compared with other methods. When 60% data are used for prediction, the proposed method can achieve a RUL prediction accuracy with 1.63% and 2.17% relative errors under static and dynamic conditions, respectively, which illustrates the feasibility and superiority of the proposed method. [ABSTRACT FROM AUTHOR]

This paper proposes a new way of modelling imperfect maintenance in degradation models, by assuming that maintenance affects only a part of the degradation process. More precisely, the global degradation process is the sum of two dependent Wiener processes with drift. Maintenance has an effect of the ARD1$ARD_1$‐type on only one of these processes: it reduces the degradation level of a quantity which is proportional to the amount of degradation of this process accumulated since previous maintenance. Two particular cases of the model are considered: perturbed ARD1$ARD_1$ and partial replacement models. The usual ARD1$ARD_1$ model is also a specific case of this new model. The system is regularly inspected in order to measure the global degradation level. Two observation schemes are considered. In the complete scheme, the degradation levels are measured both between maintenance actions and at maintenance times (just before and just after). In the general scheme, the degradation levels are measured only between maintenance actions. The maximum likelihood estimation of the model parameters is studied for both observation schemes in both particular models. The quality of the estimators is assessed through a simulation study. [ABSTRACT FROM AUTHOR]

The optical soliton and exact solutions of the stochastic concatenation model are obtained by using the modified extended direct algebraic method. This method provides a wide variety of stochastic solutions, including stochastic bright solitons, stochastic dark solitons, stochastic dark combo solitons, stochastic singular combo solitons, stochastic periodic solutions, stochastic rational solutions, stochastic Jacobi elliptic functions solutions and stochastic Weierstrass elliptic functions solutions. For a variety of nonlinear partial differential equations, this method offers a practical and effective method for determining exact solutions. The impact of the noise is illustrated graphically using examples of some of the retrieved solutions with various noise strengths. [ABSTRACT FROM AUTHOR]

This manuscript elaborate a novel characteristic of stochastic multi breather type, periodic, hybrid periodic and different type of waves solutions for the Shrödinger–Hirota (SH) equation. The mentioned equation is studied by a Wiener process and white noise which is used for the sudden and large-scale fluctuation. By applying the wave transformation technique the considered equation is transformed into ordinary differential equations, gives meaningful analysis. From the solution of the SH equation various type of results, including breather like waves, hybrid periodic waves, and singular solitons. Breather waves are applied in optical fiber communications, oceanography, and plasma physics. Singular waves find use in medical imaging, seismic research, and nonlinear optics. Hybrid singular-periodic waves have applications in photonics, acoustics, and material science, enabling advancements in diverse fields. To visually capture these intricate behaviors, we used Mathematica software to generate 2D and 3D graphs of the analytical stochastic solutions. This combination of mathematical rigor and numerical simulations provides a comprehensive understanding of the phenomena under study, making a valuable contribution to the field of nonlinear wave equations and stochastic processes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we apply stochastic differential equations with the Wiener process to investigate the soliton solutions of the Chaffee–Infante (CI) equation. The CI equation, a fundamental model in mathematical physics, explains concepts such as wave propagation and diffusion processes. Exact soliton solutions are obtained through the application of the modified extended tanh (MET) method. The obtained wave figures in 3D, 2D, and contour are highly localized and determine an individual frequency shift under the behavior of sharp peak, periodic wave, and singular soliton. The MET method shows to be a valuable analytical tool for obtaining soliton solutions, essential for understanding the dynamics of nonlinear wave phenomena. Numerical simulations enable us to explore soliton solutions in two and three dimensions, shedding light on their properties over time. Our results have wide applications in various domains, including stochastic processes and nonlinear dynamics, impacting advancements in physics, engineering, finance, biology, and beyond.

Electromagnets 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. [ABSTRACT FROM AUTHOR]

This study addresses the problem of performance reliability degradation and lifetime prediction in complex systems with high reliability but insufficient information. It comprehensively considers the effects of various uncertainties, such as measurement, cognitive, and stochastic uncertainties, and proposes a generalized normal grey cloud Bayesian Wiener process (GNGCBWP) stochastic degradation model. First, based on the historical degradation information of the system, the interval grey number sequence of the degradation increment is constructed, and the performance degradation path of the system is analyzed to determine the degradation model type. Then, combining the normal grey cloud and Bayesian estimation theory, the relevant definitions and theorems of generalized normal grey cloud Bayesian estimation (GNGCBE) are proposed. Finally, the parameter estimation results obtained using the GNGCBE method are substituted into the grey Wiener process degradation model to dynamically analyze performance reliability degradation and predict the remaining useful life (RUL) in real‐time. And the validity and scientificity of the proposed model is verified through a case study of laser equipment. Moreover, the comparison results of the models indicate that the proposed model has advantages in dynamic analysis of performance reliability degradation and real‐time RUL prediction for highly reliable complex systems with limited or poor degradation information. It can significantly reduce the impact of multiple uncertainties on the analysis results and improve the accuracy and reliability of the final prediction results. [ABSTRACT FROM AUTHOR]

The use of Constant‐stress accelerated degradation tests (CSADT) modeling is an effective way to assess the reliability of products. However, when conducting reliability experiments, the data collected may come from various sources such as different manufacturing batches, equipment, and operators. This can result in reduced accuracy of the test evaluations. In this research paper, we propose optimal designs for the Wiener constant‐stress accelerated degradation model that take into account the heterogeneity of manufacturing batches and individual differences. Our goal is to minimize the variability of the mean time to failure (MTTF) while working within a specified budget, test time, and available test units. To achieve this, we utilize particle swarm optimization (PSO) to find the best solution based on a complex objective function. We also compare our model with others using degradation data from LEDs, showing that our model has a better goodness‐of‐fit. Finally, we present examples of optimal CSADT plans under different constraints and conduct sensitivity analysis. [ABSTRACT FROM AUTHOR]

In a finite-dimensional space, we consider a linear stochastic differential equation in Itô form with a singular constant matrix on the left-hand side. Taking into account various economic applications of such equations, they are classified as Leontief type equations, since under some additional assumptions, a deterministic analog of the equation in question describes the famous Leontief input–output balance model taking into account reserves. In the literature, these systems are more often called differential–algebraic or descriptor systems. In general, to study this type of equations, one needs higher-order derivatives of the right-hand side. This means that one must consider derivatives of the Wiener process, which exist in the generalized sense. In the previous papers, these equations were studied using the technique of Nelson mean derivatives of random processes, whose description does not require generalized functions. It is well known that mean derivatives depend on the -algebra used to find them. In the present paper, the study of this equation is carried out using mean derivatives with respect to a new -algebra that was not considered in the previous papers. [ABSTRACT FROM AUTHOR]

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 . 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 } . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q-Wiener processes. [ABSTRACT FROM AUTHOR]

The fault big data sets of many open source software (OSS) are recorded on the bug tracking systems. In the past, we have proposed the effort assessment method under the assumption that the fault detection phenomenon depends on the maintenance effort, because the number of software fault is influenced by the effort expenditure. The past research in terms of the effort assessment method of OSS is based on the effort data sets. On the other hand, we propose the deep learning approach to the OSS fault big data. In the past, the existing method without Wiener process cannot estimate within the range of existing data only. The proposed method assumes that the fault detection process follows the Wiener process such as the imperfect debugging and Markov property. Thereby, the proposed method can estimate the exceeding values by adding the white noise based on the Wiener process. Then, the proposed method make it possible for the OSS managers to assess the values exceeding from the existing data. Then, we show several reliability assessment measures based on the fault modification time based on the deep learning. Moreover, several numerical illustrations based on the proposed deep learning model are shown in this paper. [ABSTRACT FROM AUTHOR]

A differential pursuit game in a stochastic descriptor linear system is analyzed. The dynamic of the system is described by Ito's stochastic differential algebraic equation. Solutions of the equation are presented by the stochastic formula of the variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct pursuer's control bringing the dynamic vector of the system to the terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transients in a radio engineering filter with random disturbances in the form of white noise. [ABSTRACT FROM AUTHOR]

In engineering practice, many products exhibit multiple and dependent degrading performance characteristics (PCs). It is common to observe that these PCs' initial measurements are nonconstant and sometimes correlated with the subsequent degradation rate, which typically varies from one unit to another. To accommodate the unit-wise heterogeneity, PC-wise dependency, and "initiation-growth" correlation, this article proposes a broad class of multi-dimensional degradation models under a framework of hierarchical multivariate Wiener processes. These models incorporate dual multi-normally distributed random effects concerning the initial values and degradation rates. To infer model parameters, expectation-maximization (EM) algorithms and several tools for model validation and selection are developed. Various simulation studies are carried out to assess the performance of the inference method and to compare different models. Two case studies are conducted to demonstrate the applicability of the proposed methodology. The of this article contain derivations of EM estimators, additional numerical results, and R codes. [ABSTRACT FROM AUTHOR]

This research paper tackles with the optical soliton solutions of stochastic the third-order nonlinear Schrödinger equation having multiplicative white noise via Ito calculus. Two efficient analytical procedures, which are the unified Riccati equation expansion method and the new Kudryashov method, are performed to structure the stochastic soliton solutions of introduced model. By making this choice, it has been intended to take advantage of the properties of these methods at the same time without using accessional operation and time in problem solving. Under favour of these techniques, bright and dark soliton forms are derived and their behavioral alterations on soliton dynamics under the effect of stochastic and the third-order dispersion terms are figured by 2D, contour and 3D diagrams. Considering the fact that the stochastic form of the equation has not been studied before in the absence of the chromatic dispersion term and the importance of noise effect for nonlinear models it is expected that the results which are obtained in this study will make a contribution to the studies in this field. [ABSTRACT FROM AUTHOR]

A differential pursuit game in a stochastic descriptor linear system is analyzed. The system dynamics is described by Ito’s stochastic differential algebraic equation. Solutions of the equation are presented by the formula of variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct a pursuer control bringing the dynamic vector of the system to a terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transient states in a radio technical filter with random perturbations in the form of white noise. [ABSTRACT FROM AUTHOR]

It is necessary to utilize certain stochastic methods while finding the soliton solutions since several physical systems are by their very nature stochastic. By adding randomness into the modeling process, researchers gain deeper insights into the impact of uncertainties on soliton evolution, stability, and interaction. In the realm of dynamics, deterministic models often encounter limitations, prompting the incorporation of stochastic techniques to provide a more comprehensive framework. Our attention was directed towards the short-wave intermediate dispersive variable (SIdV) equation with the Wiener process. By integrating advanced methodologies such as the modified Kudrayshov technique (KT), the generalized KT, and the sine-cosine method, we delved into the exploration of diverse solitary wave solutions. Through those sophisticated techniques, a spectrum of the traveling wave solutions was unveiled, encompassing both the bounded and singular manifestations. This approach not only expanded our understanding of wave dynamics but also shed light on the intricate interplay between deterministic and stochastic processes in physical systems. Solitons maintained stable periodicity but became vulnerable to increased noise, disrupting predictability. Dark solitons obtained in the results showed sensitivity to noise, amplifying variations in behavior. Furthermore, the localized wave patterns displayed sharp peaks and periodicity, with noise introducing heightened fluctuations, emphasizing stochastic influence on wave solutions. [ABSTRACT FROM AUTHOR]

We are concerned with the well-posedness and stability for a class of non-autonomous stochastic evolution equations of parabolic-type with additive noise. After proving the local existence of mild solutions of non-autonomous stochastic evolution equations with arbitrarily initial time and initial value, we obtain the blowup alternative property, global existence of mild solutions as well as asymptotic stability of mild solutions in $ p $-th moment for the associated initial value problem when the evolution family is noncompact. [ABSTRACT FROM AUTHOR]

Synchronization of stochastic reaction-diffusion Hopfield neural networks with s-delays via sliding mode control is investigated in this article. To begin with, we choose suitable functional space for state variables, then the system is transformed into a functional differential equation in an infinite-dimensional Hilbert space by using appropriate functional analysis technique. Based on above preliminary preparation, sliding mode control (SMC) is constructed to drive the error trajectory into the designed switching surface. Specifically, the switching surface is constructed as linear combination of state variables, which is related to control gains. Then novel SMC law is designed which involving delay, reaction diffusion term, and reaching law. Furthermore, the criterion of mean-square exponential synchronization for stochastic delayed reaction-diffusion Hopfield neural networks with s-delays is given in the form of matrix form. This criterion is less restrictive and easy to check in computer. Meanwhile, a different novel Lyapunov-Krasovskii functional (LKF) mixed with Itô's formula, Young inequality, Hanalay inequality is employed in this proof procedure. At last, a numerical example is presented to validate the availability of theoretical result. The simulation is based on the finite difference method, and numerical result coincides with the theoretical result proposed. [ABSTRACT FROM AUTHOR]

Kernel-type nonparametric estimates of Poisson regression function are considered. We establish the conditions of uniform consistency and the limit theorems for continuous functionals connected with this function on C [ a , 1 - a ] , 0 < a < 1 2 . [ABSTRACT FROM AUTHOR]

Degradation modeling using condition monitoring (CM) data is fundamental for prognostics and health management (PHM). However, due to the variations in manufacturing materials and operating environments, degradation heterogeneity makes life prediction difficult. Motivated by this problem, a flexible nonlinear Wiener process with random drift diffusion is proposed for degradation modeling. Different from traditional methods, this approach regards both drift and diffusion coefficients as random parameters to describe the heterogeneity of the degradation rate and volatility simultaneously. In addition, the model supports the selection of appropriate distribution types for the random parameters according to the statistical characteristics of the actual data to improve fitting performance. To effectively overcome the parameter estimation difficulties caused by model assumptions, we propose a two-stage maximum likelihood estimation (MLE) algorithm embedded with a distribution selection strategy to estimate the model parameters. Specifically, the method is first used to estimate the drift and diffusion coefficients of each unit. Then, the estimated coefficients are used to select the distribution types and perform MLE for parameters in the selected distributions. The effectiveness of the proposed parameter estimation algorithm is demonstrated with both simulated datasets and real turbofan engine datasets. A comparison of results show that compared to simplified versions and traditional methods, the proposed degradation model improves the fitting performance and life prediction accuracy. [ABSTRACT FROM AUTHOR]

Remaining 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. [ABSTRACT FROM AUTHOR]

Axle box vibration serves as the main source of excitations for rail vehicles. Due to the wear of wheel/rail contact and the re-profiling procedure, the axle box vibration usually degrades periodically with the increased mileage in the service. This could significantly impact the estimation of vibration fatigue when the component is subjected to the axle box vibration. This paper develop a method to describe the periodic evolution of axle box vibration spectrum to better characterise the vibration spectrum of axle box. In this study, a Wiener process incorporating with four random parameters was employed to model the non-linearity of the degradation process. The maximum likelihood estimation (MLE) algorithm is used to estimate the initial values of the random parameters, and a Bayesian approach is employed to update the parameters based on newly obtained data. Finally, the proposed methodology is tested using long-term field test data from a high-speed train, and the results demonstrate that it accurately estimates the evolution of the axle box acceleration spectral density (ASD) spectrum. This could aid in predicting the residual service life of structures subjected to axle box vibration and further contribute to the development of maintenance strategies and top-down design of the structure. [ABSTRACT FROM AUTHOR]

Maintenance is an important technical measure to maintain and restore the performance status of equipment and ensure the safety of the production process in industrial production, and is an indispensable part of prediction and health management. However, most of the existing remaining useful life (RUL) prediction methods assume that there is no maintenance or only perfect maintenance during the whole life cycle; thus, the predicted RUL value of the system is obviously lower than its actual operating value. The complex environment of the system further increases the difficulty of maintenance, and its maintenance nodes and maintenance degree are limited by the construction period and working conditions, which increases the difficulty of RUL prediction. An RUL prediction method for a multi-omponent system based on the Wiener process considering maintenance is proposed. The performance degradation model of components is established by a dynamic Bayesian network as the initial model, which solves the uncertainty of insufficient data problems. Based on the experience of experts, the degree of degradation is divided according to Poisson process simulation random failure, and different maintenance strategies are used to estimate a variety of condition maintenance factors. An example of a subsea tree system is given to verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Engineered Cementitious Composite (ECC), is a fiber-reinforced cementitious composite with excellent ductility, toughness, impact resistance and multi-crack development characteristics. In addition to the excellent mechanical properties, the durability of ECC has also received equal attention. In order to investigate the frost resistance of polyethylene fiber-reinforced cementitious composites (PE-ECC), PE-ECC and normal concrete (NC) specimens were subjected to different numbers of freeze-thaw cycles, and the changes in compressive strength, mass loss rate and relative dynamic modulus of elasticity of the specimens before and after freezing and thawing were tested, so as to compare the freeze-resistant properties of PE-ECC and NC. A freeze-thaw damage model was established using the two-parameter Weibull distribution function, and the freeze-thaw reliability of NC and PE-ECC was analyzed based on the Wiener process analysis, based on which a life prediction model was established to predict the life of NC and PE-ECC under freeze-thaw environment. The results show that after 200 freeze-thaw cycles, the compressive strength loss rate, mass loss rate and the relative dynamic elastic modulus loss rate of NC and PE-ECC are 32.97%, 2.35%, 50.28% and 17.28%, 0.71%, 6.57%, respectively, and the damage indexes of PE-ECC are lower than that of NC, and the NC after 200 freeze-thaw cycles has reached the standard of freeze-thaw damage, while PE-ECC is still at a lower level of freeze-thaw damage; it shows that the frost resistance of PE-ECC is better than that of NC, and the addition of PE fibers can effectively improve the frost resistance of concrete; the results of the damage model and the reliability analysis coincide well with the experimental results, and the results of life prediction are given in some cold regions, which can provide a basis for the application of PE-ECC in practice.

It is necessary to utilize certain stochastic methods while finding the soliton solutions since several physical systems are by their very nature stochastic. By adding randomness into the modeling process, researchers gain deeper insights into the impact of uncertainties on soliton evolution, stability, and interaction. In the realm of dynamics, deterministic models often encounter limitations, prompting the incorporation of stochastic techniques to provide a more comprehensive framework. Our attention was directed towards the short-wave intermediate dispersive variable (SIdV) equation with the Wiener process. By integrating advanced methodologies such as the modified Kudrayshov technique (KT), the generalized KT, and the sine-cosine method, we delved into the exploration of diverse solitary wave solutions. Through those sophisticated techniques, a spectrum of the traveling wave solutions was unveiled, encompassing both the bounded and singular manifestations. This approach not only expanded our understanding of wave dynamics but also shed light on the intricate interplay between deterministic and stochastic processes in physical systems. Solitons maintained stable periodicity but became vulnerable to increased noise, disrupting predictability. Dark solitons obtained in the results showed sensitivity to noise, amplifying variations in behavior. Furthermore, the localized wave patterns displayed sharp peaks and periodicity, with noise introducing heightened fluctuations, emphasizing stochastic influence on wave solutions.

This paper deals with the phase noise affecting communication systems, where local oscillators are employed to obtain reference signals for carrier and timing synchronizations. The most common discrete-time phase noise channel model is analyzed, with the aim to fill the gap between measurements and analytical models. In particular, the power loss and the intersymbol interference due to the presence of phase noise is evaluated with reference to the measurements parameters and to the system bandwidth. Moreover, the impact on the communication systems’ performance of the phase noise originating from the oscillator non-idealities is considered, in case of free-running and phase-locked oscillators. The proposed analysis allows to extrapolate useful information about the performance of practical systems by investigating the power spectral density of the oscillator phase noise. An expression for the variance of the residual phase error after tracking, which depends on the main parameters of practical oscillators, is derived, and used to study the dependence of the performance on the symbol rate.

Owing to operating condition switching and internal degradation mechanisms, the degradation processes of some lithium-ion batteries (LIBs) exhibit non-monotone and two-phase patterns, which are composed of a linear first phase and a nonlinear second phase. The existing Gamma process and Inverse Gaussian process methods are limited to modeling the monotone degradation data. Besides, traditional single-phase nonlinear models and two-phase linear models are insufficient to describe such a degradation process effectively. Therefore, degradation modeling and remaining useful life (RUL) prediction of the hybrid deteriorating LIBs is still a compelling practical issue. In this paper, a two-phase hybrid degradation model with a linear first phase and a nonlinear second phase is formulated based on the widely used Wiener process-based model. Taking into account the random effects caused by the unit heterogeneity and the uncertainty of the degradation state at the changing point, we obtain the analytical solutions of the lifetime estimation and RUL prediction under the concept of the first passage time (FPT). In addition, to conduct model parameter identification, the expectation maximization (EM) algorithm in conjunction with a profile log-likelihood function method are utilized for offline parameter estimation. Subsequently, the Bayesian rule is adopted to conduct the online parameter updating. Finally, the numerical and practical experiments are provided for verification and show that the proposed method could achieve high estimation accuracy for the RUL prediction of the two-phase hybrid deteriorating LIBs.

Competing failure models with degradation phenomena and sudden failures are becoming more and more common and important in practice. In this study, the generalized pivotal quantity method was proposed to investigate the modeling of competing failure problems involving both degradation and sudden failures. In the competing failure model, the degradation failure was modeled through a Wiener process and the sudden failure was described as a Weibull distribution. For point estimation, the maximum likelihood estimations of parameters $ \mu $ and $ \sigma^2 $ were provided and the inverse estimation of parameters $ \eta $ and $ \beta $ were derived. The exact confidence intervals for parameters $ \mu $, $ \sigma^2 $, and $ \beta $ were obtained. Furthermore, the generalized confidence interval of parameter $ \eta $ was obtained through constructing the generalized pivotal quantity. Using the substitution principle, the generalized confidence intervals for the reliability function, the $ p $th percentile of lifetime, and the mean time to failure were also obtained. Simulation technique was carried out to evaluate the performance of the proposed generalized confidence intervals. The simulation results showed that the proposed generalized confidence interval was effective in terms of coverage percentage. Finally, an example was presented to illustrate the application of the proposed method.

Parameter-Efficient Fine-Tuning is widely used to transfer models between different domains. However, for some high-reliability-equipment, the degradation is at a slow rate and continually fluctuates, making it difficult to extract features effectively. Moreover, collecting an integrated source domain for high-reliability-equipment is tough due to the small sample of related datasets. Aiming at the transfer problem of the time-series prediction model, this research proposed an LSTM-fine-tune model, where the parameters of the model are explicitly trained and partly frozen, such that a small number of gradient steps with a small amount of training data from a new task will produce good generalization performance on that task. The algorithm is then benchmarked on sinusoidal functions, where data were randomly generated with different phases and amplitudes. The results show that the LSTM-fine-tune model can learn knowledge from different sinusoidal data and fit a new one quickly with high accuracy. This paper also considers solving two actual problems. One is transferring oxygen concentrator data from the experimental condition to the actual service condition, the results show that the accuracy is largely improved. Moreover, this paper tried to extract more general degradation knowledge from the Wiener process and then transferred it to the degradation data of the oxygen concentrator. The results show that the model quickly achieved higher prediction accuracy with the knowledge. The code and data from the test bed are accessible at https://github.com/panjinxin123/Adaptive-finetuning-in-degradation-time-series-forecasting-via-generating-source-domain.

Relying on the pathwise uniqueness property, we prove existence of the strong solution of a one-dimensional reflected stochastic delay equation driven by a mixture of independent Brownian and fractional Brownian motions. The difficulty is that on the one hand we cannot use the fixed-point and contraction mapping methods because of the stochastic and pathwise integrals, and on the other hand the non-continuity of the Skorokhod map with respect to the norms considered. [ABSTRACT FROM AUTHOR]

A low-voltage circuit breaker thermal trip plays a role in power systems, including opening and closing, control, protection, and more. Their reliability directly affects the security and stability of the power distribution system. The Wiener process model is established according to the accelerated degradation data of thermal trips, and remaining life prediction is carried out. In this paper, firstly, the constant stress accelerated degradation test is carried out on a thermal trip with temperature as the accelerated stress and specific thermal deflection as the degradation eigenquantity to characterise the degradation trajectory according to the degradation data and analyse the degradation law; then, it is verified by the test of normal distribution that the degradation data of the thermal trip conform to the Wiener process. The Wiener process, using the great likelihood estimation method to estimate the parameters of the remaining life, calculates the remaining life probability density function and reliability function under different temperature stresses and obtains the remaining life of the thermal trip under each accelerated stress conditions; finally, the remaining life of the thermal trip at the initial moment is taken as the pseudo-failure life, and the Arrhenius accelerated model is utilised to launch the external thermal trip under normal stress. Finally, the Arrhenius acceleration model is used to extrapolate the lifespan of the thermal trip under normal stress. [ABSTRACT FROM AUTHOR]

We investigate existence, uniqueness and approximation of solutions to stochastic delay differential equations (SDDEs) under Carathéodory-type drift coefficients. Moreover, we also assume that both drift f = f (t , x , z) and diffusion g = g (t , x , z) coefficient are Lipschitz continuous with respect to the space variable x , but only Hölder continuous with respect to the delay variable z. We provide a construction of randomized Euler scheme for approximation of solutions of Carathéodory SDDEs, and investigate its upper error bound. Finally, we report results of numerical experiments that confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to obtain the analytic solutions for stochastic fourth-order (2+1)-dimensional nonlinear Schrödinger equation (NLSE) with higher order odd and even terms. Our Study mainly depends on applying the improved modified extended tanh function method (IMETFM) to get the soliton wave solutions and other exact wave solutions of our model of interest. These solutions such as (bright, dark, singular) solitons, hyperbolic solutions, periodic solutions, singular periodic solutions, rational solutions, Jacobi elliptic functions solutions and Weierstrass elliptic functions solutions. Graphical representations of some of the extracted solutions using different noise intensities are shown to demonstrate the influence of the noise. [ABSTRACT FROM AUTHOR]

For products whose performance characteristic (PC) gradually degrades with time, one usually observes its degradation levels repeatedly to predict its remaining useful life (RUL). Due to the limited storage space of the server and the low resolution of a measurement instrument, we seldom record the low-magnitude degradation values at the early degradation stage in applications. Such observation setting introduces left-truncated degradation data, in which the data collection starts later than the unit's installation. This brings sampling biases and complicates the degradation data analysis. Moreover, due to the uncontrollable factors in applications, the degradation drift and the degradation diffusion may differ among various units. Motivated by an application of high-speed train bearings, we propose a Wiener process model for the left-truncated degradation data and jointly consider the drift-diffusion random effects. Closed-form formulas are available in the expectation-maximization (EM) algorithm for estimating the model parameters. We derive the RUL distribution in closed form. We also extend the proposed model to the multivariate degradation process. The parameters are estimated with the help of the Monte Carlo EM (MCEM) algorithm. An additional laser application illustrates the performance of the proposed model in RUL prediction, which may help to design a predictive maintenance strategy [ABSTRACT FROM AUTHOR]

In the field of industrial equipment reliability assessment, dependency on either degradation or failure time data is common. However, practical applications often reveal that single-type reliability data for certain industrial equipment are insufficient for a comprehensive assessment. This paper introduces a Bayesian-fusion-based methodology to enhance the reliability assessment of industrial equipment. Operating within the hierarchical Bayesian framework, the method innovatively combines the Wiener process with available degradation and failure time data. It further integrates a random effects model to capture individual differences among equipment units. The robustness and applicability of this proposed method are substantiated through an in-depth case study analysis. [ABSTRACT FROM AUTHOR]

Degradation tests are used when there is a quality characteristic related to the life of a product. In this paper, we investigate the model mis-specification effect on the estimation precision of product's mean time to failure (MTTF) and consider a degradation test design problem. The Wiener and inverse Gaussian (IG) processes are two possible models considered. We derive expressions for the mean and variance of the estimated product's MTTF when the true model is an IG process, but is wrongly fitted by a Wiener process. We further discuss the experimental design problem and derive the explicit functional form of the estimation variances. Using the derived functions, optimal degradation test plans assuming a Wiener process model is correctly or wrongly specified are both proposed. The derived plans are applied to a laser data example. We evaluate the test efficiency of the plans derived from a Wiener process assumption when the model is mis-specified. For many optimization criteria, we observe that the obtained plans are robust even when the fitted model is mis-specified. For some criteria that may result in very different test plans under different models, we use a weighted ratio criterion to find practically useful degradation plans under model uncertainty. [ABSTRACT FROM AUTHOR]

We consider an energy saving approximation of a Wiener process under unilateral constraints. We show that, almost surely, on large time intervals the minimal energy necessary for the approximation logarithmically depends on the interval length. We also construct an adaptive approximation strategy optimal in a class of diffusion strategies and providing the logarithmic order of energy consumption. [ABSTRACT FROM AUTHOR]

This work recovers optical soliton solutions to the concatenation model with differential group delay in the presence of white noise along both the components. Two integration algorithms have successfully recovered a full spectrum of solitons solutions to the model. It has been proved that the effect of white noise is visible only in the phase portion of the solitons along both components. [ABSTRACT FROM AUTHOR]

In this study, for the first time, a study on optical soliton solutions of the stochastic perturbed Schrödinger-Hirota equation which models the propagation of solitons through optical fibers having the Kerr law is carried out. For this purpose, the nonlinear partial differential equation of the examined model is first reduced to the nonlinear ordinary differential equation form using the complex wave transformation. Then, using the new Kudryashov and the Kudryashov methods, bright, dark and singular soliton solutions are derived as a basic step. The next part is devoted to the examination of the noise effect, and first of all, soliton solutions are obtained in the absence of noise strength. Afterwards, the same soliton solutions are obtained, taking into account the noise effect, and comparison and comments are made. In addition to the fact that the form of the examined equation has not been presented and studied before. Considering that, in today's world, taking into account the effect of noise is an inevitably important factor for engineering models to be designed correctly and to operate efficiently, it is obvious that the results obtained will contribute to the literature. [ABSTRACT FROM AUTHOR]

Driven by the need for remaining useful life prediction of degraded motor systems with feedback controllers, a real-time updated Wiener stochastic process is adopted to model the performance degradation of motor systems. First, a closed-loop performance index of the motor system is derived incorporating the multiple slow time-varying characteristic of motor parameters. On this basis, the drift coefficient and diffusion coefficient of the Wiener degradation model are updated to obtain the prior maximum likelihood function with available historical data. It is followed by the iterative optimization of nonlinear feature parameter in the Wiener degradation model with taking the prior maximum likelihood function as the cost equation. The effectiveness of proposed remaining useful life prediction architecture for closed-loop motor systems is demonstrated by motor systems. [ABSTRACT FROM AUTHOR]

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A novel approach for predicting remaining useful life (RUL) is proposed for situations where maintenance threshold and failure threshold exhibit dynamic behavior due to uncertainties in degradation and the influence of detection strategies during maintenance processes. The approach introduces maintenance threshold error to establish a multi-stage maintenance-impact degradation model with dynamic maintenance threshold based on the Wiener process. This model considers the impact of maintenance on degradation rate, amount, and path. Moreover, by using the first hitting time (FHT) and introducing failure threshold error to reflect the dynamic behavior of the failure threshold, the formula for predicting equipment RUL is derived. The model parameters are estimated using both the maximum likelihood estimation (MLE) approach and Bayesian formula. The proposed approach was validated with simulation data and gyroscope degradation data, and the results demonstrate its ability to effectively enhance the precision of equipment RUL prediction. [ABSTRACT FROM AUTHOR]