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(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.

See also

• Cumulative frequency analysis
• Extreme value theory
• First-hitting-time model
• Frequency of exceedance
• Mean time between failures

Notes

1. Meerkov & Runolfsson 1987, pp. 1734–1735.
2. Richardson et al. 2014, p. 2027.
3. Meerkov & Runolfsson 1986, p. 494.
4. Meerkov & Runolfsson 1987, p. 1734.
5. Richardson et al. 2014, p. 2028.
6. Meerkov & Runolfsson 1986, p. 495, an alternate approach to defining the logarithmic residence time and computing N₀

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. hdl: 2027.42/140648 .