The Wiener Process with a Random Non-Monotone Hazard Rate-Based Drift

The degradation data of highly reliable products are usually analyzed by stochastic process models, such as Wiener process, gamma process and inverse Gaussian process models. If such a specific degradation model is wrongly assumed, then poor analysis results of reliability assessment would be obtained. Therefore, a class of exponential-dispersion processes, named Tweedie exponential-dispersion process (TEDP), is proposed to describe the products’ degradation paths. The TEDP model which comprises the aforementioned stochastic processes as its special cases, is more flexible and applicable for degradation modeling. Considering the nonlinear characteristics of degradation paths and the unit-to-unit variability among the product units, random-effect models are established based on the nonlinear TEDP models with random drift and dispersion parameters. To improve the mathematical tractability of these models, the variational inference, expectation maximization algorithm and differential evolution algorithm are used to estimate the unknown model parameters. Furthermore, two nonlinear TEDP models with accelerated factors and random effects are developed for accelerated degradation analysis. Finally, a simulation study and three real applications are presented to show the effectiveness and superiority of the proposed models and methods.

Owing to operating condition changing, physical mutation, and sudden shocks, degradation trajectories usually exhibit multi-phase features, and the abrupt jump often appears at the changing time, which makes the traditional methods of lifetime estimation unavailable. In this paper, we mainly focus on how to estimate the lifetime of the multi-phase degradation process with abrupt jumps at the change points under the concept of the first passage time (FPT). Firstly, a multi-phase degradation model with jumps based on the Wiener process is formulated to describe the multi-phase degradation pattern. Then, we attain the lifetime’s closed-form expression for the two-phase model with fixed jump relying on the distribution of the degradation state at the change point. Furthermore, we continue to investigate the lifetime estimation of the degradation process with random effect caused by unit-to-unit variability and the multi-phase degradation process. We extend the results of the two-phase case with fixed parameters to these two cases. For better implementation, a model identification method with off-line and on-line parts based on Expectation Maximization (EM) algorithm and Bayesian rule is proposed. Finally, a numerical case study and a practical example of gyro are provided for illustration.

An artificial neural network supported stochastic process for degradation modeling and prediction

Analysis of time-to-failure data for a repairable system subject to degradation

Accelerated degradation testing (ADT) is an efficient tool to conduct material service reliability and safety evaluations by analyzing performance degradation data. Traditional stochastic process models are mainly for linear or linearization degradation paths. However, those methods are not applicable for the situations where the degradation processes cannot be linearized. Hence, in this paper, a general ADT model based on the Wiener process is proposed to solve the problem for accelerated degradation data analysis. The general model can consider the unit-to-unit variation and temporal variation of the degradation process, and is suitable for both linear and nonlinear ADT analyses with single or multiple acceleration variables. The statistical inference is given to estimate the unknown parameters in both constant stress and step stress ADT. The simulation example and two real applications demonstrate that the proposed method can yield reliable lifetime evaluation results compared with the existing linear and time-scale transformation Wiener processes in both linear and nonlinear ADT analyses.

Harmonic reducers exhibit non-stationary and phase-dependent degradation behavior during long-term service, challenging the ability of classical stochastic degradation models to accurately assess reliability. To address phase-dependent differences in degradation behavior, this paper proposes a reliability assessment model based on a two-phase hybrid stochastic degradation process. In the proposed framework, the Wiener process is employed to characterize early-phase gradual degradation dominated by stochastic fluctuations, while the Inverse Gaussian process is used to describe later-phase monotonically accelerated degradation driven by cumulative damage. The framework allows for sample-level variability in transition times to more realistically capture individual degradation behavior. The Schwarz Information Criterion is also adopted to detect change points. Maximum likelihood estimation is performed for model parameter inference, and analytical expressions for the reliability function, cumulative distribution function, and probability density function are derived. Numerical results indicate that a change point exists for each tested product and that the proposed model achieves the best goodness of fit among the considered candidates, demonstrating its superiority in capturing phase-dependent characteristics of harmonic reducer degradation. In terms of reliability assessment bias, the proposed model (0.06%) significantly outperforms the Wiener degradation model (32%) and the IG degradation model (9.9%). These results further confirm that, under an identical failure threshold, the proposed approach yields more accurate and realistic reliability assessment outcomes.

In degradation tests, it is common to see that the initial degradation levels of test units are heterogeneous. Moreover, the degradation rate of a path may also be correlated with the initial value of the degradation measure. Motivated by this observation, in this paper, we introduce a time shift to the traditional stochastic process models, which presumes that the product has experienced some degradation at the beginning of the test. Such a modeling technique can also capture the correlation between the initial degradation and the degradation rate, when the degradation rate of each path does vary from unit to unit. We apply this technique to the three popular stochastic process models, i.e., the Wiener process, the gamma process, and the inverse Gaussian process, and develop the corresponding parameter inference procedures. Monte Carlo simulations are implemented to validate the proposed models and the estimation procedures. Applications to the degradation analysis of block error rates data and GaAs laser data reveal good performance of the proposed models.

In this paper, a novel approach is proposed to determine target reliability index based on modeling performance degradation of concrete structures as a stochastic process. First, the Wiener process with drift is used to simulate the degradation process of relative dynamic elastic modulus of concrete, and the drift coefficient and diffusion coefficient of Wiener process are estimated according to the experimental data of degradation of relative dynamic elastic modulus. Then, based on the relations between concrete compressive strength, dynamic elastic modulus, and relative dynamic elastic modulus, the relationship between compressive strength and the amount of degradation of relative dynamic elastic modulus are deduced, and then, the statistical parameters of concrete compressive strength under different service time can be calculated. Finally, based on the established limit state equation of flexural capacity of the weak section and the statistical parameters of compressive strength at different times, the reliability index of a structure under different service life is calculated. On the other hand, according to the drift Wiener process model of relative dynamic elastic modulus degradation of concrete, the time‐varying failure probability of structural durability degradation is calculated based on the first‐passage failure probability formula of drift Wiener process. Taking a concrete plant structure as an example, the time‐varying reliability index calculated based on the limit state equation and the time‐varying failure probability obtained based on the first‐passage failure probability analysis of drift Wiener process are compared, and then the recommended values of target reliability index of structural bearing capacity are comprehensively determined.

Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper—which led to the awarding of the 1997 Nobel prize in Economic Sciences—Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its relative simplicity, the description of Brownian motion in advanced textbooks sometimes lacks an intuitive basis. The present exposition attempts to provide a simplified construction of standard Brownian motion based on a gambling analogy. This is followed by a description and explicit solution of two stochastic differential equations (known as arithmetic and geometric Brownian motion processes) that are driven by the standard Brownian motion process. The paper also illustrates the use of the Maple computer algebra system to simulate the standard and geometric Brownian motion processes.

Background Systems operating in industrial environments are often exposed to two concurrent failure mechanisms: gradual degradation and sudden traumatic events. Maintenance decisions must account for these competing risks while controlling inspection, replacement, and failure costs. This study develops a quantitative framework to determine an economically efficient maintenance strategy under such conditions. Methods A discrete-state model is formulated with three operational conditions: Good, Degraded, and Failed. Transitions between states are driven by the system’s degradation trajectory and the occurrence of traumatic failures. A long-term expected cost model is established, incorporating inspection costs, preventive replacement costs, and failure-related losses. Dynamic programming is used to identify the policy that minimizes the expected cost per unit time. The optimisation evaluates how inspection intervals, degradation rates, and traumatic event probabilities influence replacement decisions. Results The optimisation results indicate that the cost-effective policy depends strongly on the interaction between degradation progression and the frequency of traumatic events. Higher rates of traumatic events lead to earlier preventive replacement, while intermediate degradation rates make the inspection interval the primary driver of cost reduction. The model delineates the parameter regions in which periodic inspection is justified and quantifies the cost effects of different maintenance schedules. Conclusions The proposed dynamic programming approach provides a structured method for selecting inspection and replacement strategies in systems subject to multiple failure mechanisms. The results offer decision-support guidance for maintenance planning, particularly in environments where degradation and traumatic events jointly affect system reliability and operating costs.

Degradation of many products in practical applications is often subject to unit-to-unit heterogeneity. Such heterogeneity can be attributed to the heterogeneous quality of the raw materials and the fluctuation of the manufacturing process, as well as the heterogeneous usage conditions and environments. The heterogeneity leads to the scattering of the degradation rates and diffusion intensities in the population. To model this phenomenon, this study proposes a general random-effects Wiener process model that accounts for the unit-to-unit heterogeneity in the degradation drift and the volatility simultaneously. In particular, the drift of the Wiener process is characterized by a normal distribution and the diffusion parameter is characterized by an independent inverse Gaussian distribution. The proposed model is flexible for characterization of heterogeneous degradation, and permits an analytically tractable model inference. An EM algorithm incorporating the variational Bayesian method is developed to estimate the model parameters, and a parametric bootstrap approach is proposed to construct confidence intervals. The performance of the proposed model and the estimation algorithm is validated by Monte Carlo simulations. The degradation data of an infrared LED device and the wear data of the magnetic head of a hard disk drive are studied based on the proposed model. With comprehensive comparative studies, the good performance of the proposed model in fitting the real degradation data is validated.

Five ruminally cannulated steers (body weight = 390 ± 7.86 kg) were used in two experiments to evaluate the effects of flake density and starch retrogradation on in situ ruminal degradation of steam-flaked corn. In experiment 1, sifted flakes with flake densities of 257, 296, 335, 373, and 412 g/L (enzymatic starch availabilities: 87%, 76%, 66%, 43%, and 49%, respectively) were evaluated in a randomized complete block design experiment. In experiment 2, the experimental design was a randomized complete block design with a 3 × 2 factorial arrangement of treatments. Three steam-flaked corn fractions corresponding to different particle sizes were used: flakes + fines (not sifted; >4 and <4 mm), sifted flakes (>4 mm), and sifted fines (<4 mm). Particle size fractions were stored for 3 d at either 23 °C or 55 °C (starch availabilities averaged across particle sizes: 53.3% and 25.5%, respectively) in heat-sealed foil bags. Samples were ruminally incubated for 0, 3, 6, 12, 24, 48, 72, or 96 h. Degradation data were modeled to obtain the rate and extent of degradation and passage rate was set to 6% per hour. In experiment 1, the rate of degradation decreased linearly (P < 0.01) and in situ ruminal dry matter (DM) degradability decreased linearly (P < 0.01) from 78.9% to 57.3% as flake density increased from 257 to 412 g/L. In experiment 2, storage of steam-flaked corn samples at 55 °C for 3 d decreased (P < 0.01) the rate of degradation by 37.6% across all particle sizes. Storing samples at 55 °C for 3 d decreased (P < 0.01) in situ ruminal DM degradability of flakes + fines, sifted flakes, and sifted fines by 20.9%, 22.6%, and 14.7%, respectively. Using data from experiment 1 and 2, enzymatic starch availability of sifted flakes was positively correlated (R2 = 0.97; P < 0.01) with in situ ruminal DM degradability. The results demonstrate that decreased starch availability resulting from either starch retrogradation or increased flake density is associated with decreased ruminal digestibility. Decreases in starch availability and in situ ruminal degradability may indicate that increasing flake density or starch retrogradation could potentially alter the site of digestion in cattle. Using prediction equations, decreases in ruminal starch digestibility of steam-flaked corn caused by increasing flake density or increasing starch retrogradation could increase energetic efficiency, depending on the rate of passage and if small intestinal starch digestibility is maintained.

The dynamical evolution of dislocations in plastically deformed metals is controlled by both deterministic factors arising out of applied loads and stochastic effects appearing due to fluctuations of internal stress. Such types of stochastic dislocation processes and the associated spatially inhomogeneous modes lead to randomness in the observed deformation structure. Previous studies have analyzed the role of randomness in such textural evolution, but none of these models have considered the impact of a finite decay time (all previous models assumed instantaneous relaxation which is "unphysical") of the stochastic perturbations in the overall dynamics of the system. The present article bridges this knowledge gap by introducing a colored noise in the form of an Ornstein-Uhlenbeck noise in the analysis of a class of linear and nonlinear Wiener and Ornstein-Uhlenbeck processes that these structural dislocation dynamics could be mapped on to. Based on an analysis of the relevant Fokker-Planck model, our results show that linear Wiener processes remain unaffected by the second time scale in the problem, but all nonlinear processes, both the Wiener type and Ornstein-Uhlenbeck type, scale as a function of the noise decay time τ. The results are expected to ramify existing experimental observations and inspire new numerical and laboratory tests to gain further insight into the competition between deterministic and random effects in modeling plastically deformed samples.

&lt;p&gt;In this paper, we have discussed a class of second-order neutral stochastic differential evolution systems, based on the Wiener process, with random impulses and state-dependent delay. The system is an extension of impulsive stochastic differential equations, since its random effect is not only from stochastic disturbances but also from the random sequence of the impulse occurrence time. By using the cosine operator semigroup theory, stochastic analysis theorem, and the measure of noncompactness, the existence of solutions was obtained. Then, giving appropriate assumptions, the approximate controllability of the considered system was inferred. Finally, two examples were given to illustrate the effectiveness of our work.&lt;/p&gt;

Structural fatigue reliability evaluation based on probability analysis of the number of zero-crossings of stochastic response process