First-hitting-time model

In statistics, first-hitting-time models are a sub-class of survival models. The first hitting time, also called first passage time, of a set $A$ with respect to an instance of a stochastic process is the time until the stochastic process first enters $A$.

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.

Examples

A common example of a first-hitting-time model is a ruin problem, such as Gambler's 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.

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 out over time. It is analogous to say, cream in a cup of coffee if the cream was all contained within some small location initially. In the long time limit the cream has diffused throughout the entire drink evenly.) Namely,

${\frac {\partial p(x,t\mid x_{0})}{\partial t}}=D{\frac {\partial ^{2}p(x,t\mid x_{0})}{\partial x^{2}}},$

given the initial condition $p(x_{0},t={0}\mid x_{0})=\delta (x-x_{0})$; where $x(t)$ is the position of the particle at some given time, $x_{0}$ is the tagged particle's initial position, and $D$ is the diffusion constant with the S.I. units $m^{2}s^{-1}$ (an indirect measure of the particle's speed). The bar in the argument of the instantaneous probability refers to the conditional probability. The diffusion equation states that the speed at which the probability for finding the particle at $x(t)$ is position dependent.

It can be shown that the one-dimensional PDF is

$p(x,t;x_{0})={\frac {1}{\sqrt {4\pi Dt}}}\exp \left(-{\frac {(x-x_{0})^{2}}{4Dt}}\right).$

This states that the probability of finding the particle at $x(t)$ 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

${\rm {{FWHM}\sim {\sqrt {t}}.}}$

Using the PDF one is able to derive the average of a given function, $L$, at time $t$:

$\langle L(t)\rangle \equiv \int _{-\infty }^{\infty }L(x,t)p(x,t)dx,$

where the average is taken over all space (or any applicable variable).

The First Passage Time Density (FPTD) is the probability that a particle has first reached a point $x_{c}$ at time $t$. This probability density is calculable from the Survival probability (a more common probability measure in statistics). Consider the absorbing boundary condition $p(x_{c},0)=0$ (The subscript c for the absorption point $x_{c}$ is an abbreviation for cliff used in many texts as an analogy to an absorption point). The PDF satisfying this boundary condition is given by

$p(x,t;x_{0},x_{c})={\frac {1}{\sqrt {4\pi Dt}}}\left(\exp \left(-{\frac {(x-x_{0})^{2}}{4Dt}}\right)-\exp \left(-{\frac {(x-(2x_{c}-x_{0}))^{2}}{4Dt}}\right)\right).$

The Survival probability is calculated from the PDF, namely

$S(t)\equiv \int _{-\infty }^{x_{c}}p(x,t;x_{0},x_{c})dx,$

which is the probability that a particle is still at a position $x<x_{c}$ at time $t$. The relation between the Survival probability and the FPTD is as follows(the probability that a particle has reached the absorption point between times $t$ and $t+dt$ is $f(t)dt=S(t)-S(t+dt)$. If one uses the first-order Taylor approximation, the definition of the FPTD follows):

$f(t)=-{\frac {\partial S(t)}{\partial t}}.$

By using the diffusion equation and integrating by parts, the explicit FPTD is

$f(t)\equiv {\frac {|x_{c}-x_{0}|}{\sqrt {4\pi Dt^{3}}}}\exp \left(-{\frac {(x_{c}-x_{0})^{2}}{4Dt}}\right).$

For $t\gg {\frac {(x_{c}-x_{0})^{2}}{4D}}$, it follows from above that

$f(t)={\frac {\Delta x}{\sqrt {4\pi Dt^{3}}}}\sim t^{-3/2},$

where $\Delta x\equiv |x_{c}-x_{0}|$. This equation states that the probability for a Brownian particle achieving a first passage at some long time (defined in the paragraph above) becomes increasingly small, but always finite.

The first moment of the FPTD diverges (as it is a so-called heavy-tailed distribution), therefore one cannot calculate the average FPT, so instead, one can calculate the typical time, the time when the FPTD is at a maximum ($\partial f/\partial t=0$), i.e.,

$\tau _{\rm {ty}}={\frac {\Delta x^{2}}{6D}}.$

Latent vs observable

In many real world applications, the process is latent, or unobservable. When first hitting time models are equipped with regression structures, accommodating covariate data, we call such regression structure Threshold regression. The threshold state, parameters of the process, and even time scale may depend on corresponding covariates.

A first-hitting-time (FHT) model has two underlying components: (1) a parent stochastic process $\{X(t)\}\,\,$, and (2) a threshold. The first hitting time is defined as the time when the stochastic process first reaches the threshold. It is very important to distinguish whether the sample path of the parent process is latent (i.e., unobservable) or observable, and such distinction is a characteristic of the FHT model. By far, latent processes are most common. To give an example, we can use a Wiener process $\{X(t),t\geq 0\,\}\,$ as the parent stochastic process. Such Wiener process can be defined with the mean parameter ${\mu }\,\,$, the variance parameter ${\sigma ^{2}}\,\,$, and the initial value $X(0)=x_{0}>0\,$.

See also

• Survival analysis
• Proportional hazards models

References

• Whitmore, G. A. (1986). "First passage time models for duration data regression structures and competing risks". The Statistician. 35: 207–219. JSTOR 2987525.
• Whitmore, G. A. (1995). "Estimating degradation by a Wiener diffusion process subject to measurement error". Lifetime Data Analysis. 1 (3): 307–319. doi:10.1007/BF00985762.
• Whitmore, G. A.; Crowder, M. J.; Lawless, J. F. (1998). "Failure inference from a marker process based on a bivariate Wiener model". Lifetime Data Analysis. 4 (3): 229–251. doi:10.1023/A:1009617814586.
• Redner, S. (2001). A Guide to First-Passage Processes. Cambridge University Press. ISBN 0-521-65248-0.