Lifetime Estimation for Multi-Phase Deteriorating Process with Random Abrupt Jumps.

Due to the environment changing, working state switch, and physical mutation, the degradation process often exhibits multi-phase feather and the random jump usually exists at the change point, which may make traditional lifetime estimation approach for single-stage degradation process unsuitable. In this paper, we concentrate on how to model the multi-phase degradation process and estimate its lifetime. Under this consideration, we firstly formulate a general degradation framework based on the multi-phase Wiener model to describe such the degradation process. Unlike traditional methods, we provide an analytical form of lifetime estimation for two-phase model with fixed jump, and a single-integral form for two-phase model with random jump via taking full advantage of the uncertainty of the degradation state at the change point. In addition, we extend these results from two-phase model to multi-phase model and obtain a general result of lifetime estimation. Finally, a numerical case is provided for illustration.

Lifetime prognostics for power module bonding wires under non-constant stress conditions: A compound stochastic process degradation modeling approach

Measurement error and random failure threshold widely exist in the real degradation process. Aiming at the current methods about the remaining useful lifetime (RUL) estimation without considering the influence of measurement error and random failure threshold, an approximate analytical RUL distribution in a closed-form of a linear Wiener based degradation process with measurement errors is proposed. The expectation maximization (EM) algorithm is used to estimate the unknown parameters in the model, and the Bayesian method is used to update the model parameters. Then, the probability density function (PDF) of RUL is derived based on the random failure threshold. The simulation results show that considering measurement errors and random failure threshold can significantly improve the accuracy of RUL estimation.

Randomness accelerates the dynamic clearing process of the COVID-19 outbreaks in China

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.

The study of the First Passage Time (FPT) problem (also known as first passage problem, FPP) started more than a century ago, but its diverse applications in science and engineering mostly emerged in the last two to three decades. Assuming that X(t) is a one-dimensional stochastic process, the First Passage Time is defined as the time (T) when X(t) first crosses a threshold. Engineering reliability is obviously a suitable application domain, and indeed applications such as optimal dam design in hydrology and analysis of structural failure in civil and mechanical engineering are typical examples. Although we envision that the FPT problem has great potential in network and software reliability, it should be more useful for network security and survivability because the approaches developed for the FPT problem are mostly analytical. The assumption for this inference is that in reliability analysis, experimental or historical data are often more readily available, which makes statistical approaches such as survival analysis more convenient and likely more realistic. In contrast, data is generally more difficult to obtain in security and survivability analyses, and analytical approaches can be leveraged to play more important roles. Furthermore, security and survivability often have to deal with malicious actions that may be driven by sophisticated cognition and behavioral processes, which are highly variable over time and very difficult to detect with short term data. If the behavior of an intruder can be characterized with some stochastic process such as Brownian motion, then the FPT approach may be applied to find the closed-form solution of the probability density function (PDF) of the first passage time, which can be the time when the system breaks down or when the hacker is successful in compromising a network. In addition, the solutions to FPT depend on boundary and initial conditions of the corresponding partial differential equations, and they also describe the evolution of PDF over time. This may suggest that it is possible to model the behavior changes of an intruder over time and circumstances. Another advantage of FPT analysis is that it may help solve some non-Markov stochastic process problems in reliability analysis and survival analysis. In this article, we first briefly introduce the FPT problem with Brownian motion as an example, and then suggest its potential applications in software reliability and network security.

Reliability analysis for a Wiener degradation process model under changing failure thresholds

A molecular motor is made of either a single macromolecule or a macromolecular complex.Just like their macroscopic counterparts, molecular motors "transduce" input energy into mechanical work.Their diffusion is biased by the energy pumped in by the "burning fuel".In this talk I'll begin by giving an elementary introduction to the most essential and relevant features of molecular motors [1, 2] and then present three sets of examples to establish the ubiquity of first-passage times in intracellular processes.Some motors walk along filamentous proteins carrying molecular cargo.The stochastic pause-and-translocation of such a porter is characterized by the distribution of the times of its dwell at the successive positions on its track.The dwell times are essentially first-passage times and their fluctuation provides a lower bound on the number of kinetic states of the motor.For a motor that can step both forward and backward the kinetics of translocation is characterized by the distributions of four distinct conditional dwell times.DNA polymerase is a motor that can not only step forward and backward mechanically, but also step backward by catalyzing a different chemical reaction; its stepping kinetics is characterized by nine distinct conditional dwell times.As the first example of first passage times, I'll define these conditional dwell times and summarize our analytical results as well as their physical implications [1, 2].In the nucleus DNA is stored in a hierarchical structure called chromatin.The basic unit of chromatin consists of 146 base pairs of a double stranded DNA (dsDNA) wrapped around a spool formed by proteins.The helical path of the dsDNA on a spool is also called its "footprint".We have investigated the mechanism of footprint traversal by a chromatin-remodeling enzyme (CRE), a motor that translocates along the dsDNA thereby unwrapping it from the spool.A CRE-induced biased diffusion of a small DNA loop along the footprint can lead to the sliding of the spool along the DNA.The average time needed for the traversal of the footprint, for the first time, by sliding is essentially the corresponding mean first passage time.As an example of second class of first-passage problems, I'll present our model and fuel-dependence of the mean foot-print traversal time [3].The third type of example I'll present occurs in chromosome segregation before cell division.Each of the sister chromatids, that result from chromosome replication, is bound to a structure, called kinetochore (kt) that, in turn, is coupled to the plus ends of stiff protein filaments called microtubules (MT).The lifetime of a kt-MT attachment is also a first-passage time.I'll present our recent results that explain the experimentally observed, apparently counter-intuitive, dependence of the mean lifetime on the externally applied tension that tends to detach MT from the kinetochore [4,5].

The development of models and methods for cure rate estimation has recently burgeoned into an important subfield of survival analysis. Much of the literature focuses on the standard mixture model. Recently, process-based models have been suggested. We focus on several models based on first passage times for Wiener processes. Whitmore and others have studied these models in a variety of contexts. Lee and Whitmore (Stat Sci 21(4):501-513, 2006) give a comprehensive review of a variety of first hitting time models and briefly discuss their potential as cure rate models. In this paper, we study the Wiener process with negative drift as a possible cure rate model but the resulting defective inverse Gaussian model is found to provide a poor fit in some cases. Several possible modifications are then suggested, which improve the defective inverse Gaussian. These modifications include: the inverse Gaussian cure rate mixture model; a mixture of two inverse Gaussian models; incorporation of heterogeneity in the drift parameter; and the addition of a second absorbing barrier to the Wiener process, representing an immunity threshold. This class of process-based models is a useful alternative to the standard model and provides an improved fit compared to the standard model when applied to many of the datasets that we have studied. Implementation of this class of models is facilitated using expectation-maximization (EM) algorithms and variants thereof, including the gradient EM algorithm. Parameter estimates for each of these EM algorithms are given and the proposed models are applied to both real and simulated data, where they perform well.

Existing strategies for forecasting the remaining lifespan of deteriorating equipment are not suitable for situations where the measurement intervals for the deteriorating equipment are unevenly distributed, and the monitoring data's measurement frequency does not align with historical data. Moreover, they have not taken into account the fluctuation of adaptive drift parameters during the period of decline. In light of this, a nonlinear prediction model for the remaining lifespan of deteriorating equipment that incorporates historical observational data, based on adaptive drift parameters, is proposed. Initially, the degradation process is described using a real-time state-space model with drift parameters adjusted using historical data to establish a nonlinear degradation model following the Wiener process. Subsequently, an analytical expression for the remaining life in terms of first-passage time is derived, and the parameters of the model are determined through the expectation maximization algorithm. Finally, the proposed method's efficacy and applicability are confirmed using degradation data from a reset spring under impact.

BackgroundPrevious studies indicate that rates of cognitive decline in Alzheimer’s disease (AD) and other dementias may accelerate significantly as brain pathology increases. The objectives of this study included to: 1) characterize changepoints and related slope parameters across four cognitive composites in a longitudinal cohort study; and 2) examine how these parameters varied across predictors of interest.MethodsDementia‐free participants at baseline from the Wisconsin Registry for Alzheimer’s Prevention (WRAP) with > = 3 cognitive assessments were eligible (n≈1071; mean(sd) baseline age = 58.4(6.4)). We used newly‐developed Bayesian random change point mixed models (BRCPMM) to estimate fixed (group‐level) and random (person‐level) change points (CPs), slopes pre‐ and post‐CP, and intercepts at CP for four cognitive composites: three‐test preclinical Alzheimer’s cognitive composite (PACC3; primary outcome), immediate learning, delayed learning, and executive function. We used non‐parametric statistics to test whether PACC3 estimates differed by: last cognitive status observed (cognitively unimpaired‐stable or ‐declining (CU‐S, CU‐D), MCI, and dementia); sex/gender, and APOE‐e4 carrier status.ResultsPosterior median fixed and random effect estimates (corresponding 95% credible intervals from 20000 BRCPMM runs) are summarized in Table 1 by outcome. The PACC3 group‐level CP was earliest at ∼75.56 [72.98,78.44] years old with 95% of individuals hitting CP between 59.69 and 91.86. Timing of CP was significantly associated with last observed cognitive status (Table 2); those 37 (3.46%) progressing to MCI/dementia had estimated CP’s before or during study participation while those remaining unimpaired had estimated CP’s after their last visit. All random effects (person‐level) PACC3 estimates differed by worst cognitive status (Figure_1) and sex/gender (Figure_2). For example, while the CU‐D and MCI groups had similar slopes before CP, CP occurred earlier, estimated PACC3 at CP was lower, and post‐CP slope was worse in the MCI group compared to CU‐D (Figure_1). APOE‐e4 random‐effects estimates differed by CP and post‐CP slopes (Figure 3).ConclusionsThe timing and rates of change in different cognitive composites differ by cognitive status, gender, and APOE‐e4 status. Predicting individual cognition trajectories and change points may provide an opportunity for early intervention by diagnosing high‐risk subjects at the right time for treatment and/or enrollment in a clinical trial.

This paper introduces two techniques for computing the distribution of the absorption or first passage time of the drifted Wiener diffusion subject to Poisson resetting times to an upper hard wall barrier and to a lower absorbing barrier. The first method, which we call the Padé-partial fraction approximation, starts with the Padé approximation to the Laplace transform of the first passage time distribution, which is then exactly inverted by means of the partial fraction decomposition. The second method, which we call the multiresolution algorithm, is a Monte Carlo technique that exploits the properties of the Wiener process to generate Brownian bridges at increasing levels of resolution. Our numerical study reveals that the multiresolution algorithm has higher efficiency than standard Monte Carlo, whereas the faster Padé-partial fraction method is accurate in various circumstances and provides an analytical formula. Also, a closed-form exact expression for the expected first passage time is derived.

To investigate effects of preoperative walking on bowel function recovery for patients after gynecological malignancy laparoscopy. The 156 patients with gynecoligical cancers after laparoscopy in Jiangsu from June 2020 to September 2021 were selected as research subjects, who were randomized into an experimental group (n = 78) and a control group (n = 78). Both of the groups received routine nursing care during the study. In addition, the experimental group underwent low‐moderate intensity walking exercise 1 week before surgery. The bowel function (including the time of first defecation, the time of first passage of flatus/”gas‐out time” and the recovery time of bowel sound), adverse events (nausea, vomiting abdominal distension and abdominal pain), as well as postoperative complications (ileus symptoms, deep venous thrombosis, infections and etc.), were measured daily. The time of first defecation, the time of first passage of flatus and the recovery time of bowel sound in experimental group were less than the control group after treatment (p < .05). Repeated measures analysis of variance showed that the adverse reactions (nausea, vomiting, abdominal distension, abdominal pain, and ileus symptoms) of the experimental group were weaker than those of the control group at different time points after the intervention (p < .05). Walking before surgery can effectively promote the recovery of bowel function and reduce the adverse reactions, as well as the risk of ileus related to gynecological malignancy laparoscopy.

The spread of influence in a network is a fundamental issue in many complex systems. For instance, the word of-mouth effect in a social network is crucial to the emerging viral market. The process was previously analyzed with a voter model where an individual weighs the opinions of his acquaintances equally. However, one would expect that an individual's opinion is often more swayed by her/his better connected peers. As such, the influence of a node x on another node should be proportional to x's degree. This paper studies the spread of influence under the latter model - called Preferential Voter Model (PVM) for easier reference. We first present the exact form of the random walk stationary distribution for PVM, based on which, we show that the First Passage Time (FPT) of PVM follows an exponential decay. Furthermore, compared against the conventional (nonpreferential) voter model, we show that nodes with larger degrees exhibit faster decay rate and lower mean First Passage Time in PVM. These new results have applications to network-based complex systems. For instance, in the context of viral marketing in a social network, picking nodes of larger degrees as the starting nodes of a sales campaign not only maximizes the expected number of influenced nodes in the network, but also reduces the expected time to spread the influence. Thus, our result confirms analytically that well-connected individuals indeed exert faster and more effective influences in social networks under the preferential model.

The classical theory of chemical reactions can be understood in terms of\ndiffusive barrier crossing, where the rate of a reaction is determined by the\ninverse of the mean first passage time (FPT) to cross a free energy barrier.\nWhenever a few reaction events suffice to trigger a response or the energy\nbarriers are not high, the mean first passage time alone does not suffice to\ncharacterize the kinetics, i.e., the kinetics do not occur on a single\ntime-scale. Instead, the full statistics of the FPT are required. We present a\nspectral representation of the FPT statistics that allows us to understand and\naccurately determine FPT distributions over several orders of magnitudes in\ntime. A canonical narrowing of the first passage density is shown to emerge\nwhenever several molecules are searching for the same target, which was termed\nthe 'few-encounter limit'. The few-encounter limit is essential in all\nsituations, in which already the first encounter triggers a response, such as\nmisfolding-triggered aggregation of proteins or protein transcription\nregulation.\n