Distribution Of Hitting Time Research Articles

Reproduction of initial distributions from the first hitting time distribution for birth-and-death processes

Reproduction of initial distributions from the first hitting time distribution for birth-and-death processes

Identifying the generator matrix of a stationary Markov chain using partially observable data.

Given that most states in real-world systems are inaccessible, it is critical to study the inverse problem of an irreversibly stationary Markov chain regarding how a generator matrix can be identified using minimal observations. The hitting-time distribution of an irreversibly stationary Markov chain is first generalized from a reversible case. The hitting-time distribution is then decoded via the taboo rate, and the results show remarkably that under mild conditions, the generator matrix of a reversible Markov chain or a specific case of irreversibly stationary ones can be identified by utilizing observations from all leaves and two adjacent states in each cycle. Several algorithms are proposed for calculating the generator matrix accurately, and numerical examples are presented to confirm their validity and efficiency. An application to neurophysiology is provided to demonstrate the applicability of such statistics to real-world data. This means that partially observable data can be used to identify the generator matrix of a stationary Markov chain.

First hitting time distribution and cost assessment in a two-unit system with dependent degradation processes subject to imperfect maintenance

This paper proposes a degradation model for a two-unit series system, where the components exhibit dependence and are subject to imperfect maintenance. The interdependence between both components is captured using the trivariate reduction method. The system failure occurs when the degradation level of either component exceeds a predetermined threshold. The main motivation of this work is to develop a preventive maintenance strategy for this system, incorporating periodic imperfect maintenance actions in order to extend its useful lifetime. These actions aim to reduce the accumulated degradation level of each component from its installation in a fixed percentage, following the Arithmetic Reduction of Degradation model of infinite order. Another goal is to derive the distribution of time to the system failure, which provides crucial insights into the system’s reliability and performance, especially in the case of bivariate degradation. Additionally, a cost model for this maintenance strategy is developed and several numerical examples are presented to illustrate the practical implications. The maintenance decision variables are optimized in order to obtain the minimal expected cost rate within a finite time horizon.

Non-homogeneous random walks with stochastic resetting: an application to the Gillis model

We consider the problem of the first passage time to the origin of a spatially non-homogeneous random walk (RW) with a position-dependent drift, known as the Gillis random walk, in the presence of resetting. The walk starts from an initial site x 0 and, with fixed probability r, at each step may be relocated to a given site x r . From a general perspective, we first derive a series of results regarding the first and the second moment of the first hitting time distribution, valid for a wide class of processes, including RWs lacking the property of translational invariance; we then apply these results to the specific model. When resetting is not applied, by tuning the value of a parameter which defines the transition probability of the process, denoted by ε, the recurrence properties of the walk are changed, and we can observe: a transient walk, a null-recurrent walk, or a positive-recurrent walk. When the resetting mechanism is switched on, we study quantitatively in all regimes the improvement of the search efficiency. In particular, in every case resetting allows the system to reach the target with probability one and, on average, in a finite time. If the reset-free system is in the transient or null-recurrent regime, this makes resetting always advantageous and moreover, it assures the existence of an optimal resetting probability which minimizes the mean first hitting time. Instead, when the system is positive-recurrent, the introduction of resetting is not necessarily beneficial. We explain that in this case there exists a threshold for the resetting probability r, above which the resetting mechanism yields a larger mean first hitting time with respect to the reset-free system. We provide a study of , which can be zero for some values of the system parameters, meaning that resetting cannot be beneficial in those situations. All the theoretical findings are corroborated with numerical simulations.

Belief reliability modeling for the two-phase degradation system with a change point under small sample conditions

The sparsity of the detection points and the small data size of samples have brought great difficulties in two-phase degradation modeling reliability analysis. This paper adopts belief reliability theory to cope with insufficient data and proposes a new reliability model for systems that display two-phase degradation. The uncertain process is used to model the degradation, thereby the first hitting time distribution and belief reliability of the two-phase degradation are derived. A uncertain variational inference algorithm based on the uncertain maximum likelihood method is also developed to discover the change-point more precisely and efficiently. The proposed method is verified to be more accurate and stable than the two-phase Wiener stochastic process model through the numerical simulation examples and a case of liquid coupling devices.

Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property

The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of models commonly used for survival data analysis. Other methods, however, usually do not provide a framework for recognizing or dealing with the issue and so the issue is often unintentionally ignored. The theoretical foundations of this work are set out, which presents new and somewhat surprising results for the first hitting time distributions of Lévy processes that have the Esscher property.

First hitting times between a run-and-tumble particle and a stochastically gated target.

We study the statistics of the first hitting time between a one-dimensional run-and-tumble particle and a target site that switches intermittently between visible and invisible phases. The two-state dynamics of the target is independent of the motion of the particle, which can be absorbed by the target only in its visible phase. We obtain the mean first hitting time when the motion takes place in a finite domain with reflecting boundaries. Considering the turning rate of the particle as a tuning parameter, we find that ballistic motion represents the best strategy to minimize the mean first hitting time. However, the relative fluctuations of the first hitting time are large and exhibit nonmonotonous behaviors with respect to the turning rate or the target transition rates. Paradoxically, these fluctuations can be the largest for targets that are visible most of the time, and not for those that are mostly invisible or rapidly transiting between the two states. On the infinite line, the classical asymptotic behavior ∝t^{-3/2} of the first hitting time distribution is typically preceded, due to target intermittency, by an intermediate scaling regime varying as t^{-1/2}. The extent of this transient regime becomes very long when the target is most of the time invisible, especially at low turning rates. In both finite and infinite geometries, we draw analogies with partial absorption problems.

Analytical results for the distribution of first hitting times of random walks on random regular graphs

We present analytical results for the distribution of first hitting (FH) times of random walks (RWs) on random regular graphs (RRGs) of degree c ⩾ 3 and a finite size N. Starting from a random initial node at time t = 1, at each time step t ⩾ 2 an RW hops randomly into one of the c neighbors of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. The first time at which the RW enters a node that has already been visited before is called the FH time or the first intersection length. The FH event may take place either by backtracking (BT) to the previous node or by retracing (RET), namely stepping into a node which has been visited two or more time steps earlier. We calculate the tail distribution P(T FH > t) of FH times as well as its mean ⟨T FH⟩ and variance Var(T FH). We also calculate the probabilities P BT and P RET that the FH event will occur via the BT scenario or via the RET scenario, respectively. We show that in dilute networks the dominant FH scenario is BT while in dense networks the dominant scenario is RET and calculate the conditional distributions P(T FH = t|BT) and P(T FH = t|RET), for the two scenarios. The analytical results are in excellent agreement with the results obtained from computer simulations. Considering the FH event as a termination mechanism of the RW trajectories, these results provide useful insight into the general problem of survival analysis and the statistics of mortality rates when two or more termination scenarios coexist.

WHY DOES A HUMAN DIE? A STRUCTURAL APPROACH TO COHORT-WISE MORTALITY PREDICTION UNDER SURVIVAL ENERGY HYPOTHESIS

AbstractWe propose a new approach to mortality prediction under survival energy hypothesis (SEH). We assume that a human is born with initial energy, which changes stochastically in time and the human dies when the energy vanishes. Then, the time of death is represented by the first hitting time of the survival energy (SE) process to zero. This study assumes that SE follows a time-inhomogeneous diffusion process and defines the mortality function, which is the first hitting time distribution function of the SE process. Although SEH is a fictitious construct, we illustrate that this assumption has the potential to yield a good parametric family of cumulative probability of death, and the parametric family yields surprisingly good predictions for future mortality rates.

Tail bounds on hitting times of randomized search heuristics using variable drift analysis

AbstractDrift analysis is one of the state-of-the-art techniques for the runtime analysis of randomized search heuristics (RSHs) such as evolutionary algorithms (EAs), simulated annealing,etc. The vast majority of existing drift theorems yield bounds on the expected value of the hitting time for a target state, for example the set of optimal solutions, without making additional statements on the distribution of this time. We address this lack by providing a general drift theorem that includes bounds on the upper and lower tail of the hitting time distribution. The new tail bounds are applied to prove very precise sharp-concentration results on the running time of a simple EA on standard benchmark problems, including the class of general linear functions. On all these problems, the probability of deviating by anr-factor in lower-order terms of the expected time decreases exponentially withr. The usefulness of the theorem outside the theory of RSHs is demonstrated by deriving tail bounds on the number of cycles in random permutations. All these results handle a position-dependent (variable) drift that was not covered by previous drift theorems with tail bounds. Finally, user-friendly specializations of the general drift theorem are given.

Addendum to “Hitting time distributions for efficient simulations of drift‐diffusion processes”

Addendum to “Hitting time distributions for efficient simulations of drift‐diffusion processes”

Taboo rate and hitting time distribution of continuous-time reversible Markov chains

The taboo rate is first defined, which satisfies with the Chapman–Kolmogorov equation. Then the differentials of hitting time distribution are expressed by many different taboo rates, which deeply reveal the intrinsic relationship between the transition rate matrix and the hitting time distribution in continuous-time reversible Markov chains. As an example, the explicit expressions of the differentials of the hitting time distribution at a single state are provided for the birth and death chain, hence the transition rate matrix can be identified. Such differentials improve the theory of statistical identification of continuous-time reversible Markov chains.

Inverse functions for Monte Carlo simulations with applications to hitting time distributions

AbstractRandom sampling is a ubiquitous tool in simulations and modeling in a variety of applications. There are efficient algorithms for several known distributions, but in general, one must resort to computing or approximating the inverse of the distributions to generate random samples. In certain physical and biomedical applications with which we have been particularly concerned, it has proven to be more efficient to provide random times for a walk of a fixed length, rather than the conventional random step lengths in a given time step for the walker. For these, the hitting‐time distributions must be sampled, but the problem is that the distributions contain complicated expressions for which no efficient method exists to compute an inverse. In this article, we explore a well‐known probability (the F‐ratio distribution)—whose inverse is efficiently computable—as an alternative to generating look‐up tables and interpolations to obtain the required stochastic time samples. We transform the two‐parameter F‐distribution into a four‐parameter distribution and we match the first four moments of the distribution to the moments of the exact hitting‐time distribution. We find that this simple procedure approximates the hitting‐time distribution well and produces very small errors. Future Monte Carlo simulations in a number of fields of applications may benefit from our method.

Expected Discounted Valuation With Gamma and Brownian Hitting-Time Distributions — Extends the Life Span and Reduces the Premium

The gamma and hitting-time distributions extend the geometric and exponential distributions allowing for greater flexibility in modelling termination events and their premia. In an example, the termination premia were calculated using prior beliefs and mean-variance approximation in addition to using the gamma and hitting-time distributions; they gave risk premia that were within 4 basis points of each other because skewness, kurtosis and higher order cumulants did not contribute significantly. Since the mean and variance are the significant contributors, the gamma and hitting-time distributions can be used interchangeably if the mean and variance of one distribution correspond with those of the other distribution. If spreadsheets are used, it is not necessary to add a termination premium to the discount rate; instead, the spreadsheet cells can be truncated at some point before the expected life span of the income stream and the income discounted without the termination premium added to the discount rate. Two applications demonstrate the use of the hitting-time and gamma distributions. The hitting-time distribution was used for a problem that involved an expenditure rate, variance of expenditure and time when the funds are fully expensed. The gamma distribution was used in another application where the survivability value contributed to the life span and consequently a lower termination premium. It demonstrated that a fund with a higher rate of negative monthly performance can have a lower termination premium than another with a lower rate of negative performance due to the survivability value. The examples used in the article assumed that the termination premium was not included in the CAPM-based discount rate; if it was, the discount rate would have to be adjusted to remove the termination premium. A regression of returns on the risk-free rate and excess market returns is a possible test to determine whether the termination premium was included.

On the Transition Density and First Hitting Time Distributions of the Doubly Skewed CIR Process

In this paper, we study doubly skewed CIR processes, which are extensions of skew Brownian motion. We use modified spectral expansion to obtain some properties, including the transition densities and first hitting time distributions, of doubly skewed CIR processes.

Hitting time distributions for efficient simulations of drift‐diffusion processes

Numerical solutions to partial differential equations in anisotropic, heterogeneous media obtained by their probabilistic representations are useful for a number of purposes, including our own interests in biomedical simulations. These solutions are obtained by a walk with both random and deterministic components hitting a boundary. Hitting time distributions are required to efficiently simulate such processes. The distributions for hitting timeTand placeXTon a surface for a particle undergoing both diffusion and drift are presented here, generalizing the previous work. Importantly, the distributions directly obtained are, on the whole, not useful for numerical work, being expressed as very poorly and nonuniformly convergent series. Resummed series with rapid convergence for probabilities over a useful range of the hitting time to the boundary are here obtained in different ways. A numerical inverse is constructed to implement the random walk. We have used these hitting time distributions, combined with efficient inverse function construction, to obtain speed‐up of simulations of fluid and particle transport by permitting the steps allowed for a given accuracy to be as large as possible.

Remaining useful lifetime prediction for equipment based on nonlinear implicit degradation modeling

Nonlinearity and implicitness are common degradation features of the stochastic degradation equipment for prognostics. These features have an uncertain effect on the remaining useful life (RUL) prediction of the equipment. The current data-driven RUL prediction method has not systematically studied the nonlinear hidden degradation modeling and the RUL distribution function. This paper uses the nonlinear Wiener process to build a dual nonlinear implicit degradation model. Based on the historical measured data of similar equipment, the maximum likelihood estimation algorithm is used to estimate the fixed coefficients and the prior distribution of a random coefficient. Using the on-site measured data of the target equipment, the posterior distribution of a random coefficient and actual degradation state are step-by-step updated based on Bayesian inference and the extended Kalman filtering algorithm. The analytical form of the RUL distribution function is derived based on the first hitting time distribution. Combined with the two case studies, the proposed method is verified to have certain advantages over the existing methods in the accuracy of prediction.

Nonlinear-Drifted Fractional Brownian Motion With Multiple Hidden State Variables for Remaining Useful Life Prediction of Lithium-Ion Batteries

Lithium-ion rechargeable batteries are widely used in various electronic products and equipment due to their immense benefits in power supplying. The exact remaining useful life (RUL) prediction of lithium-ion batteries has shown excellent achievements in preventing severe economic and security consequences incurred in failing to provide necessary power levels. Recently, the nonlinear-drifted fractional Brownian motion made quite a splash in RUL prediction, since its first hitting time distribution can be approximated by weak convergence theorem and time-space transformation. However, the previous RUL prediction methods based on fractional Brownian motion only considered current state measurement. In this paper, a prediction framework based on nonlinear-drifted fractional Brownian motion with multiple hidden state variables is put forward to estimate RUL. Specifically, all the parameters of nonlinear function are defined as specific hidden state variables of lithium-ion battery degradation model, and all the state measurements are used to posteriorly estimate the distribution of the multiple hidden state variables by unscented particle filter algorithm. Four sets of lithium-ion battery degradation data provided by NASA Ames Research Center are used to validate the proposed prediction framework. According to comparison study with other methods, the proposed prediction framework demonstrates greater precision in the RUL prediction.

Return- and hitting-time distributions of small sets in infinite measure preserving systems

We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $\unicode[STIX]{x1D707}(E_{k})\rightarrow 0$ in recurrent ergodic dynamical systems preserving an infinite measure $\unicode[STIX]{x1D707}$. Some properties which are easy in finite measure situations break down in this null-recurrent set-up. However, in the presence of a uniform set $Y$ with wandering rate regularly varying of index $1-\unicode[STIX]{x1D6FC}$ with $\unicode[STIX]{x1D6FC}\in (0,1]$, there is a scaling function suitable for all subsets of $Y$. In this case, we show that return distributions for the $E_{k}$ converge if and only if the corresponding hitting-time distributions do, and we derive an explicit relation between the two limit laws. Some consequences of this result are discussed. In particular, this leads to improved sufficient conditions for convergence to ${\mathcal{E}}^{1/\unicode[STIX]{x1D6FC}}{\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$, where ${\mathcal{E}}$ and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ are independent random variables, with ${\mathcal{E}}$ exponentially distributed and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ following the one-sided stable law of order $\unicode[STIX]{x1D6FC}$ (and ${\mathcal{G}}_{1}:=1$). The same principle also reveals the limit laws (different from the above) which occur at hyperbolic periodic points of prototypical null-recurrent interval maps. We also derive similar results for the barely recurrent $\unicode[STIX]{x1D6FC}=0$ case.

Random Search with Resetting: A Unified Renewal Approach.

We provide a unified renewal approach to the problem of random search for several targets under resetting. This framework does not rely on specific properties of the search process and resetting procedure, allows for simpler derivation of known results, and leads to new ones. Concentrating on minimizing the mean hitting time, we show that resetting at a constant pace is the best possible option if resetting helps at all, and derive the equation for the optimal resetting pace. No resetting may be a better strategy if without resetting the probability of not finding a target decays with time to zero exponentially or faster. We also calculate splitting probabilities between the targets, and define the limits in which these can be manipulated by changing the resetting procedure. We moreover show that the number of moments of the hitting time distribution under resetting is not less than the sum of the numbers of moments of the resetting time distribution and the hitting time distribution without resetting.