In statistics, the residence time is the average amount of time it takes for a random process to reach a certain boundary value, usually a boundary far from the mean.

Definition

Suppose y(t) is a real, scalar stochastic process with initial value y(t₀) = y₀, mean y_avg and two critical values {y_avg − y_min, y_avg + y_max}, where y_min > 0 and y_max > 0. Define the first passage time of y(t) from within the interval (−y_min, y_max) as

${\displaystyle \tau (y_{0})=\inf\{t\geq t_{0}:y(t)\in \{y_{\operatorname {avg} }-y_{\min },\ y_{\operatorname {avg} }+y_{\max }\}\},}$

where "inf" is the infimum. This is the smallest time after the initial time t₀ that y(t) is equal to one of the critical values forming the boundary of the interval, assuming y₀ is within the interval.

Because y(t) proceeds randomly from its initial value to the boundary, τ(y₀) is itself a random variable. The mean of τ(y₀) is the residence time,^[1][2]

${\displaystyle {\bar {\tau ))(y_{0})=E[\tau (y_{0})\mid y_{0}].}$

For a Gaussian process and a boundary far from the mean, the residence time equals the inverse of the frequency of exceedance of the smaller critical value,^[2]

${\displaystyle {\bar {\tau ))=N^{-1}(\min(y_{\min },\ y_{\max })),}$

where the frequency of exceedance N is

${\displaystyle N(y_{\max })=N_{0}e^{-y_{\max }^{2}/2\sigma _{y}^{2)),}$

(1)

σ_y² is the variance of the Gaussian distribution,

${\displaystyle N_{0}={\sqrt {\frac {\int _{0}^{\infty }{f^{2}\Phi _{y}(f)\,df)){\int _{0}^{\infty }{\Phi _{y}(f)\,df)))),}$

and Φ_y(f) is the power spectral density of the Gaussian distribution over a frequency f.

Generalization to multiple dimensions

Suppose that instead of being scalar, y(t) has dimension p, or y(t) ∈ ℝ^p. Define a domain Ψ ⊂ ℝ^p that contains y_avg and has a smooth boundary ∂Ψ. In this case, define the first passage time of y(t) from within the domain Ψ as

${\displaystyle \tau (y_{0})=\inf\{t\geq t_{0}:y(t)\in \partial \Psi \mid y_{0}\in \Psi \}.}$

In this case, this infimum is the smallest time at which y(t) is on the boundary of Ψ rather than being equal to one of two discrete values, assuming y₀ is within Ψ. The mean of this time is the residence time,^[3][4]

${\displaystyle {\bar {\tau ))(y_{0})=\operatorname {E} [\tau (y_{0})\mid y_{0}].}$

Logarithmic residence time

The logarithmic residence time is a dimensionless variation of the residence time. It is proportional to the natural log of a normalized residence time. Noting the exponential in Equation (1), the logarithmic residence time of a Gaussian process is defined as^[5][6]

${\displaystyle {\hat {\mu ))=\ln \left(N_{0}{\bar {\tau ))\right)={\frac {\min(y_{\min },\ y_{\max })^{2)){2\sigma _{y}^{2))}.}$

This is closely related to another dimensionless descriptor of this system, the number of standard deviations between the boundary and the mean, min(y_min, y_max)/σ_y.

In general, the normalization factor N₀ can be difficult or impossible to compute, so the dimensionless quantities can be more useful in applications.