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Residence time (statistics)

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Statistical parameter of random process evolution

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 yavg and two critical values {yavg − ymin, yavg + ymax}, where ymin > 0 and ymax > 0. Define the first passage time of y(t) from within the interval (−ymin, ymax) as

τ ( y 0 ) = inf { t ≥ t 0 : y ( t ) ∈ { y avg − y min ,   y avg + y max } } , {\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,

τ ¯ ( y 0 ) = E [ τ ( y 0 ) ∣ y 0 ] . {\displaystyle {\bar {\tau }}(y_{0})=E.}

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,

τ ¯ = N − 1 ( min ( y min ,   y max ) ) , {\displaystyle {\bar {\tau }}=N^{-1}(\min(y_{\min },\ y_{\max })),}

where the frequency of exceedance N is

N ( y max ) = N 0 e − y max 2 / 2 σ y 2 , {\displaystyle N(y_{\max })=N_{0}e^{-y_{\max }^{2}/2\sigma _{y}^{2}},} 1

σy is the variance of the Gaussian distribution,

N 0 = ∫ 0 ∞ f 2 Φ y ( f ) d f ∫ 0 ∞ Φ y ( f ) d f , {\displaystyle N_{0}={\sqrt {\frac {\int _{0}^{\infty }{f^{2}\Phi _{y}(f)\,df}}{\int _{0}^{\infty }{\Phi _{y}(f)\,df}}}},}

and Φy(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. Define a domain Ψ ⊂ ℝ that contains yavg and has a smooth boundary ∂Ψ. In this case, define the first passage time of y(t) from within the domain Ψ as

τ ( y 0 ) = inf { t ≥ t 0 : y ( t ) ∈ ∂ Ψ ∣ y 0 ∈ Ψ } . {\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,

τ ¯ ( y 0 ) = E ⁡ [ τ ( y 0 ) ∣ y 0 ] . {\displaystyle {\bar {\tau }}(y_{0})=\operatorname {E} .}

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

μ ^ = ln ⁡ ( N 0 τ ¯ ) = min ( y min ,   y max ) 2 2 σ y 2 . {\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(ymin, ymax)/σy.

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

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. 37 (6). AIAA: 2026–2030. doi:10.2514/1.G000299. hdl:2027.42/140648.
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