Residence time (statistics) explained

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 is a real, scalar stochastic process with initial value, mean and two critical values, where and . Define the first passage time of from within the interval as

\tau(y _{0)} = inf \{t \ge t _{0} : y(t) \in \{y _{\operatorname{avg} }-y_,\ y_+y_\} \},

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

Because proceeds randomly from its initial value to the boundary, is itself a random variable. The mean of is the residence time,

\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,

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

where the frequency of exceedance is is the variance of the Gaussian distribution,

N _{0} = \sqrt{ (\int ^{inf ty{f} _{0} ^{2} \Phi _{y(f)} df)/() }},and is the power spectral density of the Gaussian distribution over a frequency .

Generalization to multiple dimensions

Suppose that instead of being scalar, has dimension, or . Define a domain that contains and has a smooth boundary . In this case, define the first passage time of from within the domain as

\tau(y _{0)} = inf \{t \ge t _{0} : y(t) \in \partial \Psi \mid y _{0} \in \Psi \}.

In this case, this infimum is the smallest time at which is on the boundary of rather than being equal to one of two discrete values, assuming is within . The mean of this time is the residence time,

\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, the logarithmic residence time of a Gaussian process is defined as

\hat{\mu} = ln \left(N _{0} \bar{\tau} \right) = (min (y _{min} , y _{max} ) ^{2})/(2 \sigma ^{2} _{y}) .

This is closely related to another dimensionless descriptor of this system, the number of standard deviations between the boundary and the mean, .

In general, the normalization factor 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

References

• Aiming Control . Meerkov . S. M. . Runolfsson . T. . 1986 . Proceedings of 25th Conference on Decision and Control . IEEE . 494–498 . Athens.
• Meerkov . S. M. . Runolfsson . T. . Output Aiming Control . 1987 . Proceedings of 26th Conference on Decision and Control . IEEE . 1734–1739 . Los Angeles.
• Richardson . Johnhenri R. . Atkins . Ella M.. Ella Atkins . 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 . 10.2514/1.G000299. 2027.42/140648 . free .