ⓘ First-hitting-time model. Events are often triggered when a stochastic or random process first encounters a threshold. The threshold can be a barrier, boundary ..


ⓘ First-hitting-time model

Events are often triggered when a stochastic or random process first encounters a threshold. The threshold can be a barrier, boundary or specified state of a system. The amount of time required for a stochastic process, starting from some initial state, to encounter a threshold for the first time is referred to variously as a first hitting time. In statistics, first-hitting-time models are a sub-class of survival models. The first hitting time, also called first passage time, of the barrier set B {\displaystyle B} with respect to an instance of a stochastic process is the time until the stochastic process first enters B {\displaystyle B}.

More colloquially, a first passage time in a stochastic system, is the time taken for a state variable to reach a certain value. Understanding this metric allows one to further understand the physical system under observation, and as such has been the topic of research in very diverse fields, from economics to ecology.

The idea that a first hitting time of a stochastic process might describe the time to occurrence of an event has a long history, starting with an interest in the first passage time of Wiener diffusion processes in economics and then in physics in the early 1900s. Modeling the probability of financial ruin as a first passage time was an early application in the field of insurance. An interest in the mathematical properties of first-hitting-times and statistical models and methods for analysis of survival data appeared steadily between the middle and end of the 20th century.


1. Examples

A common example of a first-hitting-time model is a ruin problem, such as Gamblers ruin. In this example, an entity often described as a gambler or an insurance company has an amount of money which varies randomly with time, possibly with some drift. The model considers the event that the amount of money reaches 0, representing bankruptcy. The model can answer questions such as the probability that this occurs within finite time, or the mean time until which it occurs.

First-hitting-time models can be applied to expected lifetimes, of patients or mechanical devices. When the process reaches an adverse threshold state for the first time, the patient dies, or the device breaks down.


2. First passage time of a 1D Brownian Particle

One of the simplest and omnipresent stochastic systems is that of the Brownian particle in one dimension. This system describes the motion of a particle which moves stochastically in one dimensional space, with equal probability of moving to the left or to the right. Given that Brownian motion is used often as a tool to understand more complex phenomena, it is important to understand the probability of a first passage time of the Brownian particle of reaching some position distant from its start location. This is done through the following means.

The probability density function PDF for a particle in one dimension is found by solving the one-dimensional diffusion equation. This equation states that the position probability density diffuses outward over time. It is analogous to say, cream in a cup of coffee if the cream was all contained within some small location initially. After a long time the cream has diffused throughout the entire drink evenly. Namely,

∂ p x, t ∣ x 0 ∂ t = D ∂ 2 p x, t ∣ x 0 ∂ x 2, {\displaystyle {\frac {\partial px,t\mid x_{0}}{\partial t}}=D{\frac {\partial ^{2}px,t\mid x_{0}}{\partial x^{2}}},}

given the initial condition p x, t = 0 ∣ x 0 = δ x − x 0 {\displaystyle px,t={0}\mid x_{0}=\delta x-x_{0}} ; where x t {\displaystyle xt} is the position of the particle at some given time, x 0 {\displaystyle x_{0}} is the tagged particles initial position, and D {\displaystyle D} is the diffusion constant with the S.I. units m 2 s − 1 {\displaystyle m^{2}s^{-1}} an indirect measure of the particles speed. The bar in the argument of the instantaneous probability refers to the conditional probability. The diffusion equation states that the rate of change over time in the probability of finding the particle at x t {\displaystyle xt} position depends on the deceleration over distance of such probability at that position.

It can be shown that the one-dimensional PDF is

p x, t ; x 0 = 1 4 π D t exp ⁡ − x − x 0 2 4 D t). {\displaystyle px,t;x_{0}={\frac {1}{\sqrt {4\pi Dt}}}\exp \left-{\frac {x-x_{0}^{2}}{4Dt}}\right).}

This states that the probability of finding the particle at x t {\displaystyle xt} is Gaussian, and the width of the Gaussian is time dependent. More specifically the Full Width at Half Maximum FWHM – technically, this is actually the Full Duration at Half Maximum as the independent variable is time – scales like

F W H M ∼ t. {\displaystyle {\rm \right)\right),}

for x < x c {\displaystyle x 0 {\displaystyle X0=x_{0}> 0\,}.


3. Operational or analytical time scale

The time scale of the stochastic process may be calendar or clock time or some more operational measure of time progression, such as mileage of a car, accumulated wear and tear on a machine component or accumulated exposure to toxic fumes. In many applications, the stochastic process describing the system state is latent or unobservable and its properties must be inferred indirectly from censored time-to-event data and/or readings taken over time on correlated processes, such as marker processes. The word regression’ in threshold regression refers to first-hitting-time models in which one or more regression structures are inserted into the model in order to connect model parameters to explanatory variables or covariates. The parameters given regression structures may be parameters of the stochastic process, the threshold state and/or the time scale itself.

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