Residence Time (statistics)
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In statistics, the residence time is the average amount of time it takes for a
random process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Stoc ...
to reach a certain boundary value, usually a boundary far from the mean.


Definition

Suppose is a real, scalar
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
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\, where "inf" is the
infimum In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique ...
. 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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
. The mean of is the residence time, : \bar(y_0) = E tau(y_0)\mid y_0 For a
Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The di ...
and a boundary far from the mean, the residence time equals the inverse of the frequency of exceedance of the smaller critical value, : \bar = N^(\min(y_,\ y_)), where the frequency of exceedance is is the variance of the Gaussian distribution, : N_0 = \sqrt, 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\. 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(y_0) = \operatorname tau(y_0)\mid y_0


Logarithmic residence time

The logarithmic residence time is a
dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
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 = \ln \left(N_0 \bar \right) = \frac. 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 Extreme value theory or extreme value analysis (EVA) is the study of extremes in statistical distributions. It is widely used in many disciplines, such as structural engineering, finance, economics, earth sciences, traffic prediction, and Engin ...
* First-hitting-time model * Frequency of exceedance * Mean time between failures


Notes


References

* * * {{cite journal , last1=Richardson , first1=Johnhenri R. , last2=Atkins , first2=Ella M., author2-link=Ella Atkins , last3=Kabamba , first3=Pierre T. , last4=Girard , first4=Anouck R. , year=2014 , title=Safety Margins for Flight Through Stochastic Gusts , journal=Journal of Guidance, Control, and Dynamics , publisher=AIAA , volume=37 , issue=6 , pages=2026–2030 , doi=10.2514/1.G000299, hdl=2027.42/140648 , hdl-access=free Extreme value data Survival analysis Reliability analysis