Residence time (statistics)

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 $y (t 0) = y 0$ , 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

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

where "inf" in the infimum. This is the smallest time after the initial time $t 0$ that $y (t)$ is equal to one of the critical values forming the boundary of the interval, assuming $y 0$ is within the interval.

Because $y (t)$ proceeds randomly from its initial value to the boundary, $τ(y 0)$ is itself a random variable. The mean of $τ(y 0)$ is the residence time,^[1]^[2]

{\bar {\tau }}(y_{0})=E[\tau (y_{0})|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]

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

where the frequency of exceedance $N$ is

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

(1)

$σ y 2$ is the variance of the Gaussian distribution,

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 $y (t) \in ℝ p$ . Define a domain $Ψ \subset ℝ p$ that contains $y avg$ and has a smooth boundary $\partialΨ$ . In this case, define the first passage time of $y (t)$ from within the domain $Ψ$ as

\tau (y_{0})=\inf\{t\geq t_{0}:y(t)\in \partial \Psi |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 $y 0$ is within $Ψ$ . The mean of this time is the residence time,^[3]^[4]

{\bar {\tau }}(y_{0})=E[\tau (y_{0})|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]

{\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 $N 0$ can be difficult or impossible to compute, so the dimensionless quantities can be more useful in applications.

Notes

↑ Meerkov 1987, pp. 1734-1735.
1 2 Richardson 2014, p. 2027.
↑ Meerkov 1986, p. 494.
↑ Meerkov 1987, p. 1734.
↑ Richardson 2014, p. 2028.
↑ Meerkov 1986, p. 495, an alternate approach to defining the logarithmic residence time and computing $N 0$

References

Meerkov, S. M.; Runolfsson, T. (1986). Aiming Control. Proceedings of 25th Conference on Decision and Control. Athens: IEEE. pp. 494–498.
Meerkov, S. M.; Runolfsson, T. (1987). Output Aiming Control. Proceedings of 26th Conference on Decision and Control. Los Angeles: IEEE. pp. 1734–1739.
Richardson, Johnhenri R.; Atkins, Ella M.; Kabamba, Pierre T.; Girard, Anouck R. (2014). "Safety Margins for Flight Through Stochastic Gusts". Journal of Guidance, Control, and Dynamics. AIAA. 37 (6): 2026–2030. doi:10.2514/1.G000299.

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